Magnus Academy (Maths Specialist): April 2023

Wednesday, 19 April 2023

THE PLANE

DEFINITION

A plane is a surface such that if any two points are taken on it, the line segment joining them lies completely on the surface. OR

Every point on the line segment joining any two points on a plane lies on the plane.

Type-1

EQUATION OF A PLANE PASSING THROUGH A GIVEN POINT

1) The general equation of a plane passing through a point (x₁ , y₁ ,z₁) is

a(x - x₁)+ b(y - y₁)+ c(z - z₁)= 0, where a, b, and c are constants.

STEP-1.

Write the equation of a plane passing through (x₁ , y₁ ,z₁) as:

a(x - x₁)+ b(y - y₁)+ c(z - z₁)= 0. (i)

STEP -2

If the plane (i) passes through (x₂ , y₂ , z₂) and (x₃ , y₃ , z₃), then

a(x₂ - x₁)+ b(y₂ - y₁)+ c(z₂ - z₁)= 0. (ii) And

a(x₃ - x₁)+ b(y₃ - y₁) + c(z₃ - z₁)= 0. (iii)

STEP-3

Solve equation (ii) and (iii), obtained in STEP 2, by cross multiplication.

STEP -4

Substituting the value of a, b, c , obtained in STEP- 3, in equation (i) in STEP -1 to get the required plane.

REMARK: On eliminating a, b, c from equation (i),(ii),(iii), we get

x - x₁ y- y₁ z - z₁

x₂ - x₁ y₂- y₁ z₂ - z₁ = 0

x₃- x₁ y₃ - y₁ z₃ - z₁ in determinant form

As the equation of the plane passing through three given points (x₁ , y₁, z₁), (x₂, y₂, z₂), (x₃, y₃, z₃)

EXERCISE -A

1) Find the equation of the plane through the points

A) A(2,2, -1), B(3,4,2) and C(7,0,6). 5x + 2y- 3z= 17

B) M(1,1, 0), N(-2,2,-1) and P(1,2,2). 2x + 3y- 3z= 5

C) (2,1, 0), (3,-2, -2) and C(3,2,7). 7x + 3y- z= 17

D) A(1,1, 1), B(1,-1,2) and C(-2,-2,2). x - 3y- 6z+ 8= 0

E) A(0,-1, 0), B(3,3,0) and C(1,1,1). 4x - 3y + 2z= 3

2) Show that the following points are coplanar:

A) (0,-1,-1), (-4,4,4),(4,5,1).

B) (0,4,3),(-1,-5,-3),(-2,-2,1) and (1,1,-1).

C) (0,-1,0), (2,1,-1),(1,1,1) and (3,3,0)

3) Show that the four points (0,-1,-1),(4,5,1),(3,9,4) and (-4,4,4) are coplanar and find the equation of the common plane. 5x - 7y + 11z +4= 0

INTERCEPT FORM OF A PLANE

The equation of a plane intercepting lengths a, b, and c with x-axis, y-axis and z-axis respectively is

x/a + y/b + z/c = 1.

NOTE:

1) The above equation is known as the intercept form of the plane, because the plane intercepts lengths a, b and c with x-axis, y-axis and z-axis.

STEP

To determine the intercepts made by a plane with the coordinate axes

1) For x-intercept:

Put y= 0 , z= 0 in the equation of the plane and obtain the value of x. The value of x is the intercept on x-axis.

For y-intercept:

Put x= 0 , z= 0 in the equation of the plane and obtain the value of y. The value of y is the intercept on y-axis.

For z-intercept:

Put x= 0 , y= 0 in the equation of the plane and obtain the value of y. The value of y is the intercept on x-axis.

EXERCISE -B

1) Write the equation of the plane whose intercept on the coordinates axes are

A) -4, 2 and 3. -3x+6y+4z= 12

B) 2, -3 and 4. 6x-4y+3z= 12

2) Reduce the equation of the following planes in intercept form and find its intercept on the coordinate axes:

A) 2x+ 3y - 4z= 12. x/6 + y/4 + z/-3 = 1. 6, 4 , -3

B) 4x+ 3y - 6z= 12. x/3 + y/4 + z/-2 = 1. 3, 4 , -2

C) 2x+ 3y - z= 6. x/3 + y/2 + z/-6 = 1. 3, 2 , -6

D) 2x - y +z= 5. x/5/2 + y/-5 + z/5 = 1. 5/2, -5, 5

3) Find the equation of the plane passing through the point (2,4,6) and making equal intercept on the coordinate axes. x+y+ z= 12

4) A plane meets the coordinates axes are A, B and C respectively such that the centroid of triangle ABC is (1,-2,3). Find the equation of the plane. 6x- 3y +2z = 18

5) A plane meets the coordinates axes in A, B and C respectively such that the centroid of triangle ABC is (p,q,r). Show that the equation of the plane is x/p +y/q +z/r = 3.

Vector Equation Of A Plane Passing Through A Given Point And Normal To A Given Vector

* The vector equation of a plane passing through a point having position vector a and normal to vector n is (r - a) . n = 0

r. n = a. n

Note 1: It is to note here that vector equation of a plane means a relation involving the position vector r of an arbitrary point on the plane.

Note 2: The above equation can also be written as r . n = d, where d= a. n. This is known as the scalar product form of a plane.

REDUCTION TO CARTESIAN FORM

If r= xi + yj + zk, a= a₁i+ a₂j + a₃k and n= n₁i+ n₂j + n₃k

Then, r - a= (x - a₁)i +(y - a₂)j +(z - a₃)k.

Substituting the value of (r - a) and n in equation (r - a). n = 0, we get

[(x -a₁)i +(y- a₂)j+(z- a₃)]. (n₁i+ n₂j + n₃k)= 0.

=> (x -a₁)n₁ +(y- a₂)n₂+(z- a₃)n₃= 0.

This is the Cartesian equation of a plane passing through (a₁ , a₂, a₃)

NOTE that the coefficient of x,y and z in equation are n₁, n₂ , n₃ respectively which are proportional to the direction ratios of vector n normal to the plane.

Thus, the coefficient of x, y and z in the Cartesian equation of a plane are proportional to the direction ratios of normal to the plane.

For Example:

The direction ratios of a vector normal to the plane 2x+ y - 2z - 5 = 0 are proportional to 2, 1, -2 and hence a vector normal to the plane is 2i + j - 2k.

EXERCISE - C

1) Find the vector equation of a plane passing through a point having

A) position vector 2i + 3j - 4k and Perpendicular to the vector 2i- j + 2k. Also, reduce it to Cartesian form. r. (2i - j + 2k)= -7. 2x- y +2z = -7

B) position vector 2i - j +k and Perpendicular to the vector 4i+ 2 j 3 k. Also, reduce it to Cartesian form. r. (4i + 2j - 3k)= 3.

2) Find the cartesian form of equation of a plane whose vector equation is

A) r. (12i -3j +4k)+5= 0. 12x - 3y + 4z +5= 0

B) r. (- i + j +2k)= 9. -x + y +2z = 9

3) Find the vector equation of each one of the following:

A) 2x - y +2z = 8. r. (2i - j +2k)=8

B) x +y -z = 5. r. (i +j -k)= 5

C) x +y = 3. r. (i + j)= 3

4) Find the vector and Cartesian equations of a plane passing through the

A) point (1,-1,1) and normal to the line joining the points (1,2,5) and (-1,3,1). r. (2i - j +4k)=7, 2x - y +4z = 7

B) point (3,-3,1) and normal to the line joining the points (3,4,-1) and (2,-1,5). r. (-i - 5j +6k)=18, x + 5y -6z = -18

5) a) The foot of the perpendicular drawn from the origin to the plane is (4,-2,-5). Find the equation of the plane. r. (4i - 2j -5k)=45, 4x -2y -5z = 45.

B) The coordinates of the foot of the perpendicular drawn from the origin to a plane is (12,-4,3). Find the equation of the plane. 12x -4y +3z = 169.

6) A) If the line drawn from the point (-2, -1,-3) meets a plane at right angle at the point (1,-3,3), find the equation of the plane. r. (3i - 2j +6k)=27

B) If the line drawn from the point (4, -1,2) meets a plane at right angle at the point (-10,5,4), find the equation of the plane. 7x - 3y - z+ 89= 0.

7)A) Find the equation of the plane which bisects the line segment joining the points A(2,3,4) and B(4, 5, 8) at right angles. r. (i +j +2k)=19, x+ y+ 2z= 19.

B) Find the equation of the plane which bisects the line segment joining the points A(12,3) and B(3,4, 5) at right angles. x+ y+ z= 9.

C) Find the equation of the plane passing through the point (1,-1,2) having 2,3,2 as direction ratios of normal to the plane. r. (2i +3j +2k)=3, 2x+ 3y+ 2z= 3.

D) Find the equation of the plane passing through the point (2,3,1) given that the direction ratios of normal to the plane are proportional to 5,3,2. 5x+ 3y+ 2z= 21

8) Show that the normals to the following of planes are perpendicular to each other:

A) x - y+ z= 2 and 3x+ 2y- z= -4

B) r. (2i -j +3k)=5, r . (2i - 2j -2k)=5.

Incomplete

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* The vector equation of a plane normal to unit vector n and at a distance d from the origin= r. n = d.

* If l, m, n are direction of the normal to a given plane which is at a distance p from the origin, then the equation of the plane is lx + my + nz= p.

Note-1

The equation r.n= d is normal form if n is a unit vector and in such a case d on the right hand side denotes the distance of the plane from the origin. If n is not a vector, then to reduce the equation r. n = d to normal form divide both sides by |n| to obtain:

r. n/|n|= d/|n|

=> r. n= d/|n|

In order to reduce the Cartesian equation ax+ by + cz+ d= 0 of a plane to normal form,

Use following steps

Step-1: Keep terms containing x,y and z on LHS and shift the constant term d on the RHS.

Step-2 : If the constant term on RHS is not positive make it positive by multiplying both sides by -1.

Step-3: Divide each term on two sides by

√(a²+ b²+ c²)= √(coef. of x)²+ (Coeff. of y)²+ (Coeff. z)².

The coefficient of x,y and z in the equation so obtained will be the direction cosines of the normal to the plane and the RHS will be the distance of the plane from the origin.

REMARK: The Cartesian equations of the normal to the plane lx + my+ nz= p drawn from the origin are x/l = y/m = z/n and the coordinates of the foot N of the perpendicular drawn from the origin O are given x/l = y/m = z/n= p i.e., (lp, mp, np)

EXERCISE -D

1)a) Find the vector equation of a plane at a distance of 5 units from the origin and has i as the unit vector normal to it. r. i= 5

b) Find the vector equation of a plane at a distance of 3 units from the origin and has k as the unit vector normal to it. r. k = 5

2)a) find the vector equation of a plane which is at a distance of 8 units from the origin and which is normal to the vector to 2i + j +2k. r.(2i+ j+2k)= 24

b) find the vector equation of a plane which is at a distance of 5 units from the origin and which is normal to the vector to i -2 j -2k. r.(i/3- 2j/3-2k/3)= 5

3) a) Reduce the equation r.(3i- 4j+ 12k)= 5 to the Normal form and hence, find the length of the perpendicular from the origin to the plane. r.(3i/13- 4j/13+ 12k/13)= 5/13, 5/13

b) Reduce the equation r.(i- 2j+ 2k)= -6 to the Normal form and hence, find the length of the perpendicular from the origin to the plane. r.(-i/3+4j/3- 2k/3)= 2, 2

4) a) Reduce the equation of the plane x - 2y - 2z =12 to the normal form and hence find the length of the perpendicular from the origin to the plane. Also, find the direction cosines of the normal to the plane. x/2 - 2y/3 - 2z/3= 4, 4, 1/3,-2/3,-2/3

b) Reduce the equation of the plane 2x - 3y - 6z =14 to the normal form and hence find the length of the perpendicular from the origin to the plane. Also, find the direction cosines of the normal to the plane. 3, 2/7, -3/7, -6/7

5) a) Find the vector equation are plane which is at a distance of 6 units from the origin and has 2,-1,2 as the direction ratios of a normal to it. Also, find the coordinates of the foot of the normal drawn from the origin. r.(2i/3 - j/3+ 2k/3)= 6, (4,-2,4)

b) Find the vector equation are plane which is at a distance of 5 units from the origin and has 12,-3,4 as the direction ratios of a normal to it. Also, find the coordinates of the foot of the normal drawn from the origin. 12x/13- 3y/13 + 4z/13= 5

6) Find the co-ordinate of the foot of the perpendicular drawn from the origin to the plane 2x - 3y + 4z - 6= 0. (12/29, -18/29, 24/29)

7) Find the direction cosines of the perpendicular from the origin to the plane. r.(2i - 3j- 6k)+5= 0, -2/7, 3/7, 6/7

8) Find the equation of the plane passing through the point (-1,2,1) and the perpendicular to the line joining the points (-3,1,2) and (2,3,4). Find also the perpendicular distance of the origin from this plane. r.(5i/√33+ 2j/√33 + 2k/√33)= 1/√33, 1/√33

9) Write the normal form of the equation of the plane 2x - 3y +6z +14= 0. -2x/7 + 3y/7 -6z/7 = 2

10) Write a unit normal vector to the plane x + 2y +3z -6= 0. i/√14 + 2j/√14 + 3k/√14 = 2

11) a) Find the equation of the plane which is at a distance of 3√3 units from the origin and the normal to which is equally inclined with the coordinates axes. x +y +z = 9

b) Find the vector equation of the plane which is at a distance of 6/√29 units from the origin and the normal from the origin is 2i -3j + 4k. Also, find its Cartesian form. r.(2i/√29- 3j/√29 + 4k/√29)= 6/√29, 2x -3y + 4z = 6

12) Find the equation of the plane passing through the point (1,2,1) and perpendicular to the line joining the points (1,4,2) and (2,3,5). Find also the perpendicular distance of the origin from this plane. x -y +3z -2= 0, -2/√11

13) Find the distance of the plane 2x -3x +4z -6= 0 from the origin. 6/√29

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Vector Equation Of a Plane Passing Through 3 Given Points

n= AB x AC = (b-a) x(c -a)

ax b + bxc + cxa

Exercise -E

1) Find the vector equation of the plane passing through the points

A) A (2,2,-1), B( 3,4,2) and C(7,0,6). also, find the cartesian equation of the plane. r.(5i + 2j- 3k)= 17, 5x+2y-3z= 17

B) A (1,1,1), B(1,-1,1) and C(-7,-3,-5). r.(3i -4j- 4k)+1= 0

C) A (2,5-3), B(-2,-3,5) and C(5,3,-3). r.(2i +3j+4k)= 7

D) A (1,1,-1), B(6,4,-5) and C(-4,-2,3). 5a -3b-4c= 0

E) 3i +4j+2k, 2i -2j-k, and 7i + 6k. r. (9i +2j-7k)= 21

F) i +j-2k, 2i -j+k, and i +2j+ k. r. (9i + 3j- k)=14

2) Find the vector equation of the plane passing through the points A(a,0,0), B(0,b,0) and C(0,0,c). Reduce it to normal from. If plane ABC is at a distance p from the origin, prove that 1/p²= 1/a²+ 1/b²+ 1/c².

________________( )_________________

Angle Between Two Planes

Definition: The angle between two planes is defined as the angle between their normals.

VECTOR FORM

The angle ¢ between the planes r. n₁ = d₁ and r. n₂= d₂ is

Cos¢= (n₁. n₂)/(|n₁| |n₂|)

CARTESIAN FORM

The angle ¢ between the planes a₁x+ b₁y + c₁z + d₁=0 and a₂x+ b₂y+ c₂z+ d₂= 0 is

Cos¢ =(a₁a₂ + b₁b₂+ c₁c₂)/{√(a₁²+ b₁²+ c₁²) √(a₂²+ b₂²+ c₂²)

Condition of perpendicularity:

* n₁. n₂= 0

* a₁a₂+ b₁b₂+ c₁c₂= 0

Condition of parallelism:

* a₁/a₂ = b₁/b₂= c₁/c₂

ON FINDING A PLANE PASSING THROUGH A GIVEN POINT AND PERPENDICULAR TO TWO GIVEN PLANES

The normal to the plane passing through a point having position vector a and perpendicular to the planes r. n₁= d₁ and r. n₂ = d₂ is perpendicular to the vector n₁ and n₂. So it is parallel to n₁ x n₂. We may use the following steps to find the plane passing through a given point and perpendicular to two planes.

Step 1: Obtain the position vector of the given point say, a

Step 2: Obtain the normal vector to two planes. Let the normal vector be n₁ and n₂.

Step 3: Compute n=n₁ x n₂. Clearly, n is normal to the required plane.

Step 4: write the equation of the desired plane as

(r-a).(n₁ x n₂)= 0 OR

r (n₁ xn₂)= a. (n₁ xn₂) OR

[r n₁n₂ ]=[a n₁n₂]

ON FINDING A PLANE PASSING THROUGH TWO GIVEN POINTS PERPENDICULAR TO A GIVEN PLANE

The normal to the plane passing through the two points P and Q having their position vectors a and b respectively and perpendicular to the plane r. n₁ = d₁ is perpendicular to the vectors PQ and n₁. So, the normal vector n to the plane is parallel to the vector PQ x n₁. Thus, use the following step to find the plane passing through two given points and perpendicular to a given plane.

Step 1: Obtain the position vector of the given points. Let the positions vectors of the given points P and Q be a and b respectively.

Step 2: Obtain the equation of the plane perpendicular to the required to the required plane. Let its equation be r. n₁= d.

Step-3: Let n be the normal vector to the required plane. Then, n is perpendicular to both n₁ and PQ.

So, Compute n= n₁ x PQ.

Step-4: Write the equation of the required plane as (r - a). n = 0 OR

r. n = a. n

EXERCISE -F

1) Find the angle between the planes

A) r. (2i- j+ k)= 6 and r. (i + j+ 2k)=5. π/3

B) r. (2i- 3j+ 4k)= 1 and r. (-i + j)=4. cos⁻¹(5/√58)

C) r. (2i- j+ 2k)= 6 and r. (3i + 6j- 2k)=9. cos⁻¹(-4/21)

D) r. (2i+ 3j- 6k)= 5 and r. (i -2 j+ 2k)=9 cos⁻¹(16/21)

2) Find the angle between the planes

A) x+ y+ 2z= 9 and 2 x- y+ z= 15. π/3

B) 2x- y+ z= 4 and x+ y+ 2z= 3. π/3

C) x+ y- 2z= 3 and 2 x- 2y+ z= 5. cos⁻¹(2/3√6)

D) x- y+ z= 5 and x+ 2y+ z= 9. π/2

E) 2x- 3y+ 4z= 1 and -x+ y= 4. cos⁻¹(5/√58)

F) 2x+ y- 2z= 5 and 3 x- 6y- 2z= 7. cos⁻¹(4/21)

3) Show that the planes

A) 2x+ 6y+ 6z= 7 and 3x+ 4y- 5z= 8

B) x-2y+ 4z= 10 and 18x+ 17y+ 4z= 49

C) r. (2i- j+k)= 5 and r. (-i - j+ k)=3 are at right angles.

4) Determine the value of λ for which the following planes are perpendicular to each other.

a) r. (2i- j+ λk)= 5 and r. (3i + 2j+ 2k)=4. -2

b) r. (i + 2j+ 3k)= 7 and r. (λi + 2j - 7k)= 26. 17

c) 2x - 4y + 3z =5 and x + 2y + λz = 5. 2

d) 3x - 6y - 2z =7 and 2x + y - λz = 5. 0

5) Find the equation of the plane

a) passing through the point (1,1,-1) and Perpendicular to the plane x + 2y + 3z -7=0 and 2x - 3y + 4z =0. 17x + 2y - 7z =26

b) passing through the point (-1,-1,2) and Perpendicular to the plane 3x + 2y - 3z -1=0 and 5x - 4y + z =5. 5x + 9y + 11z =8

c) passing through the point (1,-3,-2) and Perpendicular to the plane x + 2y + 2z -5=0 and 3x + 3y + 2z =8. 2x - 4y + 3z =8

d) passing through the origin and Perpendicular to each of the plane x + 2y - z = 1 and 3x - 4y + z =5. x + 2y - 3z +3 =0

6) Find the equation of the plane passing through the point (1,-1,2) and (2,-2,2) and which is Perpendicular to the plane 6x - 2y + 2z -9=0. x + y - 2z +4=0

7) Find the equation of the plane passing through the point (2,1,-1) and (-1,3,4) and Perpendicular to the plane x - 2y + 4z = 10, also that the plane thus obtained contains the line r= - i+ 3j + 4k + λ(3i - 2j - 5k). 18x +17y + 4z =49.

8) Find the equation of the plane passing through the point (2,2,1) and (9,3,6) and which is Perpendicular to the plane 2x + 6y + 6z = 1. 3x + 4y - 5z =9

9) Find the equation of the plane passing through the points (-1,1,1) and (1, -1,1) and Perpendicular to the plane x + 2y + 2z =5. 2x + 2y - 3z + 3=0

10) Find the equation of the plane passing through the point (3,4,2) and (7,0,6) and is Perpendicular to the plane 2x - 5y = 15. 5x + 2y - 3z -17 =0

11) Find the equation of the plane with intercept 3 on the y-axis and parallel to ZOX plane. y=3

12) Find the equation of the plane that contains the point (passing through the point (1,-1,2) and is perpendicular to each of the planes 2x +3y - 2z - 5=0 and x + 2y - 3z =8. 5x - 4y - z= 7

13) Find the equation of the plane passing through (a,b,c) and parallel to the plane r.(i + j + k)=2. x + y + z = a+ b+ c

14) Find the equation of the plane passing through the point (-1,3,2) ) and perpendicular to each of the planes x + 2y + 3z =5 and 3 x + 3y +z=0. 7x - 8y + 3z +25=0

15) Find the vector equation of the plane passing through the point (2,1,-1) and (-1,3,4) and which is Perpendicular to the plane x - 2y + 4z = 10. 18x + 17y +14z =49

EXERCISE - G

1) Find the vector equation of the following plane in scalar product form:

a) r= (i - j)+ λ(i + j + k)+ μ(i - 2j+ 3k). r.(5i - 2j - 3k)= 7

b) r= (2i - k)+ λi + μ(i - 2j - k). r.(j - 2k)= 2

c) r= (1+ s - t)i + (2- s) j + (3- 2s + 2t)k. r.(2i + k)= 5

d) r= (i + j)+ λ(i + 2j - k)+ μ(-i + j - 2k). r.(-i +j +k)= 0

e) r= (i - j)+ λ(i + j + k)+ μ(4i - 2j+ 3k). r.(5i + j - 6k)= 4

2) Find the Cartesian form of the equation of the plane

a) r= (s - 2t) i+ (3- t) j + (2s + t)k. 2x - 5y - z = -15

b) r= (i - j)+ s(-i + j + 2k)+ t(i + 2j+ k). x - y + z= 2

c) r= (1+ s + t)i + (2- s + t)+ (3 - 2s + 2t)k. 2y - z = 1

3) Find the vector equation of the plane in non-parametric form:

a) r= (1+ s - t) i+ (2- s) j + (3- 2s + 2t)k. r.(2i + 0j + k)= 5

b) r= (λ - 2μ)i + (3- μ) j + (2λ + μ)k. r.(2i -5j - k)+15=0

c) r= (2i + 2j - k)+ λ(i + 2j + 3k)+ μ(5i - 2j+ 7k). r.(5i + 2j - 3k)= 17

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EXERCISE - H

1) Find the equation of a plane through the intersection of the planes r.(i + 3j - k)= 5 and r.(2i - j + k)=3 passing through the point (2, 1, -2). r.(3i + 2j + 0k)=8

2) Find the equation of the plane containing the line of intersection of the plane x + y + z -6=0 and 2x + 3y + 4z +5=0 and passing through the point (1,1,1). 20x + 23y + 26z -69=0

3) Find the direction ratios of the normal to the plane passing through the point (2,1,3) and the line of intersection of the plane x + 2y + z=3 and 2x - y - z =5. 13,6,1

4) Find the equation of the plane which is perpendicular to the plane 5x + 3y + 6z +8=0 and which contains the line of intersection of the plane x+ 2y + 3z -4=0 and 2x+ y - z +5 =0. 51x+ 15y - 50z +173=0

5) Find the equation of the plane through the intersection of r.(2i - 3j + 4k)=1 and r.(i -j) + 4 = 0 and perpendicular to r.(2i - j + k) + 8= 0. r.(-5i + 2j + 12k)=47

6) Find the equation of the plane passing through the intersection of the planes 2x - 3y + z - 4=0 and x - y + z +1= 0 and perpendicular to the plane x + 2y - 3z +6=0. x - 5y - 3z - 23= 0

7) Find the cartesian as well as vector equations of the planes through the intersection of the planes r.(2i + 6j) + 12= 0 and r.(3i - j + 4k)=0 which are at a unit distance from the origin. r.(2i + j + 2k) + 3=0 and r.(-i + 2j -2k)+ 3=0

8) The plane lx + my =0 is rotated through an angle α about its line of intersection with the plane z= 0. prove that the equation of the plane in its new position is lx + my ± √(l²+ m² tanα)z = 0.

9) The plane x - 2y + 3z =0 is rotated through a right angle about the line of intersection with the plane 2x + 3y - 4z - 5 =0, find the equation of the plane in its new position. 22x + 5y - 4z -35=0

10) Find the equation of the plane which is parallel to 2x - 3y + z=0 and which passes through (1,-1,2). 2x - 3y + z=7

11) Find the equation of the plane through (3,4,-1) which is parallel to the plane r.(2i - 3j + 5k)+ 2= 0. r.(2i - 3j + 5k)+ 11= 0.

12) Find the question of the plane passing through the line of intersection of the planes 2x -7y + 4z -3=0, 3x - 5y + 4z +11=0 and the point (-2,1,3). 15x -47y + 28z=7

13) Find the equation of plane through the point 2i + j - k. and passing through the line of intersection of the plane r.(i + 3j - k)= 0 and r.(j + 2k)= 0. r.(i + 9j + 11k)= 0

14) Find the equation of the plane passing through the line of intersection of the planes 2x - y =0 and 3z - y=0 and perpendicular to the plane 4x + 5y - 3z =8. 28x -17y +9z = 0

15) Find the equation of the plane which contains the line of intersection of the planes x+ 2y + 3z -4=0 and 2x + y - z+5=0 and which is perpendicular to the plane 5x + 3y - 6z +8=0. 33x + 45y + 50z - 41=0

16) Find the equation of the plane through the line of intersection of the planes x+ 2y + 3z +4=0 and x - y + z +3=0 and passing through the origin. x - 10y - 5z =0

17) Find the vector equation (in scalar product form) of the plane containing the line of intersection of the planes x - 3y + 2z -5=0 and 2x - y + 3z -1=0 and passing through (1,-2,3). r.(i + 7j)+ 13= 0.

18) Find the equation of the the plane which is perpendicular to the plane 5x + 3y + 6z +8=0 and which contains the line of intersection of the planes x + 2y + 3z -4=0, 2x + y - z +5=0. 51x + 15y - 50z + 173=0

19) Find the equation of the plane through the line of intersection of the planes r.(i + 3j)+ 6= 0 and r.(3i - j - 4k)= 0, which is at a distance from the origin. r.(-2i +4j + 4k)+ 6= 0 or r.(4i + 2j - 4k)+ 6= 0

20) Find the equation of the plane passing through the intersection of the planes 2x + 3y - z +1=0 and x + y - 2z +3=0 and perpendicular to the plane 3x - y - 2z -4=0. 7x + 13y + 4z -9=0

21) Find the equation the plane passing through the intersection of the planes 2x +3y - z +1=0, and x +y - 2z +3 =0 and perpendicular to the plane 3x - y - 2z -4=0. 7x + 13y +4z -9=0

22) Find the equation of the plane that contains the line of intersection of the planes r.(i + 2j + 3k) -4=0 and r.(2i + j - k)+5=0 and which is perpendicular to the plane r.(5i + 3j - 6k)+8=0. 33x + 45y + 50z -41=0

23) Find the equation of the plane passing through the intersection of the planes r.(i + j + k)- 6 =0 and r.(2i + 3j + 4k)+5=0 and the point (1,1,1). r.(20i + 22 j +26 k)= 69

24) Find equation of the plane passing through the intersection of the planes r.(2i + j + 3k) =7, r.(2i + 5j +3 k)=9 and the point (2,1,3). r.(2i -13 j +3 k)=0

25) Find the equation of the plane through the intersection of the planes 3x - y + 2z =4 and x + y + z =2 and the point (2,2,1). 7x - 5y + 4z =8

26) Find the vector equation of the plane through the line of intersection of the Planes x+ y + z =1 and 2x + 3y + 4z =5 which is perpendicular to the plane x - y + z=0. r.(i - k)+2=0

27) Find the equation of the plane passing through (a, b, c) and parallel to the plane r.(i + j + k)=2. x+ y+ z = a+ b+ c

EXERCISE - I

1) Find the distance of the point 2i + j - k from the plane r. (i - 2j + 4k)= 9. 13/√21

2) Find the distance of the point (2,1,0) from the plane 2x + y + 2z +5=0. 10/3

3) Prove that if a plane has an interceptive a,b,c and is at a distance of p units from the origin, then 1/a²+ 1/b²+ 1/c²= 1/p²

4) Show that the points i - j + 3k and 3(i + j + k) are equidistant from the plane r. (5i + 2j + k)+ 9 =0 and lie on opposite sides of it.

5) Find the equations of the Planes parallel to the plane x - 2y + 2z -3=0 which is at a unit distance from the point (1,2,3). x - 2y + 2z =0 and x - 2y + 2z - 6=0

6) if the points (1,1, M) and (-3.0.1) be equidistant from the plane r.(3i + 4j -12k)+ 13= 0, find the value of M. 1 or 7/3

7) Find the distance between the points P(6,5,9) and the plane determined the points A(3,- 1, 2), B(5,2,4) and C(-1, -1, 6). 6/√34

8) Find the equation of a plane through the point P(6,5, 9) and parallel to the plane determined by the point A(3, - 1,2) , B(5, 2, 4) and C(-1,-1,6). Also , find the distance of this plane from the point A. 3x - 4y + 3z = 25, 6/√34

7) Two systems of rectangular axes have the same origin. If a plane cuts them at distance of a,b,c and a', b', c' respectively prove that 1/a²+ 1/b²+ 1/c²= 1/a'² + 1/b'² + 1/c'².

8) A variable plane which remains at a constant distance 3p from the origin cut co-ordinate axes at A, B, C. Show that the locus of the centroid of triangle ABC is 1/x²+ 1/y²+ 1/z²= 1/p².

9) A variable plane is at the constant distance p from the origin and meets the coordinate axes in A, B, C. Show that the locus of the centroid of the tetrahedron OABC is 1/x²+ 1/y²+ 1/z²= 16/p².

10) If a variable plane at a constant distance p from the origin meets the coordinate axis in pointr A, B and C respectively. Through these points , planes are drawn parallel to the coordinate planes. Show that the locus of the point of intersection is 1/x²+ 1/y²+ 1/z²= 1/p².

11) Find the distance of the point 2i - j - 4k from the plane r.(3i - 4j +12k)- 9=0. 47/13

12) Show that the points i - j +3k and 3i + 3j +3k are equidistant from the plane r.(5i + 2j - 7k)+9=0.

13) Find the distance of the point (2,3,-5) from the plane x + 2y - 2z -9=0. 3

14) Find the equation of the planes parallel to the plane x + 2y - 2z + 8=0 which are at distance of 2 units from the point (2,1,1). x+ 2y - 2z +4=0 or x + 2y - 2z -8=0

15) Show that the points (1,1,1) and (-3,0,1) are equidistant from the plane 3x + 4y - 12z +13=0.

16) Find the equations of the planes parallel to the plane x - 2y + 2z -3=0 and which are at a unit distance from the point (1,1,1). x - 2y + 2z +2=0, x - 2y + 2z -4=0

17) Find the distance of the point (2,3,5) from the xy plane. 5

18) Find the distance of the point (3,3,3) from the plane r.(5i + 2j - 7k)+o=0. 9/√78

19) if the product of the distance of the point (1, 1,1) from the origin and the plane x - y + z + λ= 0 be 5, find the value of λ. 4

20) Find an equation for the set of all points that are equidistant from the Planes 3x - 4y + 12z = 6 and 4x + 3z =7. 36x + 20y - 21z =61, 67x -20y +99z =121

21) Find the distance between the point (7,2,4) and the plane determined by the points A(2,5,-3), B(-2,-3,5) and (5,3,-3). √29

22) A plane makes intersects -6, 3, 4 respectively on the coordinate axes. Find the length of the perpendicular from the origin on it. 12/√29

EXERCISE - J

1) Find the distance between the parallel planes x + y - z +4=0 and x + y - z +5=0. 1/√3

2) Find the distance between the parallel planes 2x - y +2z +3 =0 and 4x -2y +4 z +5=0. 1/6

3) Find the distance between the parallel planes r.(2i - 3j + 6k)=5 and r.(6i - 9j + 18k) + 20 =0. 5/3

4) Find the distance between the parallel planes 2x - y + 3z - 4=0 and 6x - 3y + 9z + 13 =0. 25/3√14

5) Find the equation of the plane passing through the point (3,4,-1) and is parallel to the plane 2x -3y +5z +7 =0. Also, find the distance between the two planes. 2x -3y +5 z +11=0, 4/√38

6) Find the equation of the plane mid-parallel to the plane 2x -2 y +z +3 =0 and 2x - 2y + z +9 =0. 2x - 2y + z +6=0

7) Find the distance between the planes r.(i + 2j + 3k)+ 7=0 and r.(2i + 4j + 6k) + 7 =0. 7/√56

Exercise - K

1) Reduce in symmetrical form, the equation of the line x - y + 2z =5, 3x + y + z =6. (4x-11)/-3= (4y+9)/5= (z -0)/1

2) Reduce in symmetrical form ,the equation of the line x = ay + b, z= cy + d. (x-b)/a = (y-0)/1= (z -d)/c

3) Find the angle between the lines x - 2y + z = 0= x + 2y - 2z and x+ 2y + z =0 = 3x + 9y + 5z. Cos⁻¹(8/√406)

4) Find the angles between the line r= (i+ 2j - k)+ λ(i - j + k) and the plane r.(2i - j + k)= 4. sin⁻¹(2√2/3)

5) Find the angle between the line (x+ 1)/3= (y - 1)/2 = (z -2)/4 and the plane 2x + y - 3z +4=0. sin⁻¹(-4/√406)

6) If the line r= (i- 2j + k)+ λ(2i + j + 2k) is parallel to the plane r. (3i- 2j + mk) = 14, find the value of m. -2

7) Show that the line whose vector equation is r= (2i- 2j +3k)+ λ(i - j + 4k) is parallel to the plane whose vector equations is r. (i+ 5j + k)= 5. Also, find the distance between them. 10/√27

8) Find the vector equation of the line passing through the point with position vector (2i- 3j - 5k) and perpendicular to the plane r. (6i - 3j + 5k) +2=0. r= (2i - 3j - 5k)+ λ(6i - 3j + 5k)

9) Find the equation of the line passing through the point (3, 0, 1) and parallel to the planes x + 2y=0 and 3y - z= 0. (x -3)/-2 = (y - 0)/1 = (z -1)/3

10) Find the equation of the plane passing through the line of intersection of the planes 2x + y - z =3, 5x - 3y + 4z +9=0 and parallel to the line (x -1)/2 = (y - 3)/4 = (z -5)/5. 7x + 9y - 10z -27 =0

11) Find the equation of the plane passing through the intersection of the Planes r. (i + j + k) -1 =0 and r. (2i + 3j -k) +4=0 and parallel to x-axis. r. (- j/2 + 3k/2) = 3 or r. (- j + 3k) =6

12) Find the equation of the plane through the point (1, 0, -1), (3, 2, 2) and parallel to the line (x -1)/1 = (y - 1)/-2 = (z -2)/3. 4x - y - 2z -6=0

13) State when the line r= a + λb is parallel to the plane r.n = d. Show that the line r= (i + j + λ(2i + j + 4k) is parallel to the plane r. (- 2i + k) = 5. Also, find the distance between the line and the plane. 7/√5

14) Find the equation of the plane passing through the intersection of the Planes 4x - y + z =10 and x + y - z = 4 and parallel to the line with direction ratios proportional to 2, 1,1. Find also the perpendicular distance of point (1,1,1) from this plane. 5y - 5z -6=0, 3√2/5

15) Find the equation of the plane passing through the point A(1, 2, 1) and perpendicular to the line joining the points P(1, 4, 2) and Q(2,3,5). Also, find the distance of this plane from the line (x +3)/2 = (y - 5)/-1 = (z -7)/-1. x - y +3z -2 =0, √11

16) Find an equation for the line that passes through the point (2, 3, 1) and is parallel to the of intersection of the planes x + 2y - 3z = 4 and x - 2y +z = 0. (x -2)/1 = (y - 3)/1 = (z -1)/1.

17) Find the plane passing through the point (4,-1,2) and parallel to the lines (x +2)/3 = (y - 2)/-1 = (z +1)/2 and (x -2)/1 = (y - 3)/2 = (z -4)/3. x + y - z -1= 0

18) Find the angle between the line r= (2i + 3j + 9k)+ λ(2i + 3j + 4k) and the plane r. (i + j + k) = 5. sin⁻¹(3√3/29)

19) Find the angle between and the line (x -1)/1 = (y - 2)/-1 = (z +1)/1 and the plane 2x + y - z = 4. 0

20) Find the angle between the line joining the points (3,-4,-2) and (12,2,0) and the plane 3x - y + z =1. sin⁻¹(23/11√11)

21) The line r= i + λ(2i - mj - 3k) is parallel to the plane r. (mi + 3j + k) = 4. find m. -3

22) Show that the line whose vector equation is r= (2i + 5j + 7k)+ λ(i + 3j + 4k) is parallel to the plane whose vector equation is r. (i + j - k) = 7. Also, find the distance between them. 7/√3

23) Find the vector equation of the line through the origin which is perpendicular to the plane r. (i + 2j + 3k) = 3. r= λ(i - 2j + 3k)

24) Find the equation the plane through (2, 3, -4) and (1,-1,3) and parallel to x-axis. 7y + 4z -5=0

25) Find the equation of a plane passing through the points (0, 0, 0) and (3, -1,2) and parallel to the line (x -4)/1 = (y + 3)/-4 = (z +1)/7. x - 19y - 11z = 0

26) Find the vector and cartesian equations of the line passing through (1, 2, 3) and parallel to the Planes r. (i - j + 2k) = 5 and r. (3i + j + 2k) = 6. r= (i + 2j + 3k)+ λ(-3i + 5j + 4k) ; (x -1)/-3 = (y -2)/5 = (z -3)/4

27) Prove that the line of section of the plane 5x + 2y - 4z +2=0 and 2x + 8y + 2z -1=0 is parallel to the plane 4x - 2y - 5z - 2=0.

28) Find the vector equation of the line passing through the point (1, -1, 2) and perpendicular to the plane 2x - y + 3z - 5 =0. r= (i - j + 2k)+ λ(2i - j + 3k)

29) Find the equation the plane through the points (2, 2, -1) and (3, 4, 2) and parallel to the line whose direction ratios are 7, 0, 6. 12x +15y - 14z - 68=0

30) Find the angle between the line (x -2)/3 = (y + 1)/- 1 = (z -3)/2 and the plane 3x + 4y + z + 5= 0. sin⁻¹√(7/52)

31) Find the equation the plane passing through the intersection of the Planes x - 2y + z =1 and 2x + y + z = 8 and parallel to the line with direction ratios proportional to 1, 2, 1. Find also the perpendicular distance of (1, 1, 1) from the plane. 9x - 8y + 7z = 21, 13/√194

32) State when the line r= a + λb is parallel to the plane r.n = d. Show that the line r= (i +j )+ λ(3i - j + 2k) is parallel to the plane r. (2i +k) = 3. Also , find the distance between the line and the plane. b.n =0, 1/√5

33) Show that the plane whose vector equation r. (i +2 j -k) =1 and the line whose vector equation is r= (-i +j + k)+ λ(2i + j + 4k) are parallel. Also , find the distance between them. 1/√6

34) Find the equation of the plane through the intersection of the plane 3x - 4y + 5z =10 and 2x + 2y - 3z = 4 and parallel to the line x = 2y = 3z. x - 20y + 27z = 14

35) Find the vector and cartesian forms of the equation of the plane passing through the point (1,2,-4) and parallel to the lines r= (i +2j -4k)+ λ(2i + 3j + 6k) and r= (i -3j + 5k)+ μ(i + j - k). Also, find the distance of the point (9,-8,-10) from the plane thus obtained. r. (-9i + 8j - k)= 11, 9x + 8y - z= 11, √146

36) Find the equation of the plane passing through the point (3,4,1) and (0,1,0) and parallel to the line (x +3)/2= (y -3)/7 = (z -2)/5. 8x - 13y + 15z +13=0

37) Find the coordinates of the point where the line (x -2)/3 = (y +1)/4 = (z -2)/2 interesects the plane x - y + z -5=0. Also, find the angle between the line and the plane. (2,-1,2), sin⁻¹(1/√87)

38) Find the vector equation of the line passing through (1, 2, 3) and perpendicular to the plane r. (i + 2j - 5k) -9 = 0. r= (i +2j + 3k)+ λ(i + 2j - 5k)

39) Find the angle between the line (x +1)/2 = y/3 = (z -3)/6 and the plane 10x + 2y -11z =3. sin⁻¹(8/21)

40) Find the vector equation of the line passing through (1,,2,3) and parallel to the planes r (i - j + 2k) = 5 and r(3i + j + k) = 6. r= (i +2j + 3k)+ λ(-3i + 5j +4k)

41) Find the value of λ such that the line (x -2)/6 = (y-1)/λ = (z +5)/-4 is perpendicular to the plane 3x - y - 2z = 7. -2

42) Find the equation of the plane passing through the points (-1,2,0), (2,2,-1) and parallel to the line (x -1)/1 = (2y +1)/2 = (z +1)/-1. r= (i +2j + 3k)+ λ(-3i + 5j +4k)

EXERCISE - L

1) Find the coordinates of the point where the line through the points

A) A(3,4,1) and B (5,1,6) crosses the XY-plane. (13/5,23/5,0)

B) A(5,1,6) and B (3,4,1) crosses the yz-plane. (0,17/2,-13/2)

C) A(5,1,6) and B (3,4,1) crosses the zx plane. (17/3,0,23/3)

2) Find the coordinates of the point where the line through (3,-4,-5) and (2,-3,1) crosses the plane 2x+y+z= 7. (1,-2,7)

3) Find the distance between the point with position vector -i - 5j - 10k and the point of intersection of the line (x-2)/3= (y+1)/4 = (z-2)/12 with the plane x - y +z = 5. 13

4) Find the distance between the point with position vector (-1, - 5 - 10) from the point of intersection of the line r= (2i - j+ 2k)+ K(3i+ 4j+ 2k) and the plane r.(i - j+ k)= 5. 13

5) Find the distance between the point with position vector (2,12,5) from the point of intersection of the line r= (2i - 4j+ 2k)+ K(3i+ 4j+ 2k) and the plane r. (i - 2j+ k)= 0. 13

6) Find the distance between the point (1,-2,3) from the plane x- y +z= 5 measured parallel to the line whose direction cosines are proportional to 2,3,-6. 7/6

7) Find the distance between the point (-1,-5,-10) from the point of intersection of the joining the points A(2,-1,2) and B(5,3,4) with the plane x- y +z= 5. 13

8) Find the distance between the point (3,4,4) from the point, where the line joining the points A(3,-4,-5) and B(2,-3,1) intersect the plane 2x +y +z= 7. 7

CONDITION FOR A LINE TO LIE IN A PLANE:

Vector form:

If the line r= a + ¥b lies in the plane r. n= d, then

A) b.n= 0

B) a.n = d.

Cartesian form:

If the line (x-x₁)/l= (y -y₁)/m= (z -z₁)/n lies in the plane ax+ by+ cz + d= 0, then

A) ax₁ + by₁ + cz₁+ d= 0

B) al + bm+ cn = 0.

CONDITION OF PERPENDICULARITY OF TWO LINES AND EQUATION OF THE PLANE CONTAINING THEM

Vector form:

If the lines r= a₁ +¥b₁ and r= a₂ +€b₂ are coplanar, then

r₁. (b₁ x b₂= a₂. (b₁ x b₂)

[r b₁ b₂ ]= [a₂ b₁ b₂]

And the equation of the plane containing them is

r. (b₁ x b₂) = a₁. (b₁ x b₂)

r. (b₁ x b₂)= a₂. (b₁x b₂)

Cartesian form:

If the line (x- x₁)/l₁ = (y- y₁)/m₁ = (z- z₁)/n₁ and (x- x₂)/l₂ = (y- y₂)/m₂ = (z- z₂)/n₂ are coplanar, then

Mode of

(x₂- x₁) (y₂- y₁) (z₂- z₁)

l₁ m₁ n₁ = 0

l₂ m₂ n₂

And the equation of the plane containing them is

(x- x₁) (y- y₁) (z- z₁)

l₁ m₁ n₁ = 0

l₂ m₂ n₂

(x-x₂) (y- y₂) (z - z₂)

l₁ m₁ n₁ = 0

l₂ m₂ n₂

EXERCISE-M

1)a) Show that the lines r= (i+ j- k)+ ¢(3i - j) and r= (4i- k)+ ¥(2i +3k) are coplanar. Also, find the plane containing these two lines. . r. (3i + 9j - 2k)= 14

b) Show that the lines r= (2i- 3k)+ ¢(i +2j+3k) and r= (2i+ 6j + 3k)+ ¥(2i+ 3j+ 4k) are coplanar . Also , find the equation of the plane containing them. r. (i -2j +k)= -7

c) show that the lines (x+1)/-3 = (y- 3)/2 = (z+2)/1 and x/1 = (y- 7)/-3 = (z+7)/2 are coplanar. Also, find the plane containing these two lines. x+ y+ z= 0

d) Show that the lines (x+1)/3 = (y+ 3)/5 = (z+5)/7 and (x-2)/1 = (y- 4)/4 = (z-6)/7 are coplanar. Also, find the plane containing these two lines. x- 2 y+ z= 0

2) Find the equation of the plane containing the line (x+1)/-3 = (y- 3)/2 = (z+2)/1and the point (0,7,-7) and show that the line x/1 = (y- 7)/-3 = (z+7)/2 also lies in the same plane. x+ y+ z= 0

3) Find the equation of the plane which contains two parallel lines (x-4)/1 = (y- 3)/-4 = (z-2)/5 and (x-3)/1 = (y+2)/-4 = z/5. 11x - y - 3z = 35

4) Show that the lines (x+4)/3 = (y+6)/5 = (z-2)/-2 and 3x-2y+ z= 0= 2x+ 3y+ 4z - 4 intersect. Find the equation of the plane in which they lie and also their point of intersection. (2,4,-3), 45x- 17y+ 25z+53= 0

5) a) Show that the plane whose vector equation is r. (i+2j- k)= 3 contains the line whose vector equation is r= (i +j)+ ¥(2i+ j + 4k).

6) Find the equation of the plane determined by the intersection of the lines (x+3)/3 = y/-2 = (z-7)/6 and (x+6)/1 = (y+5)/-3 = (z-1)/2. 2x - z +13=0

7) Show that the lines (x- a +d)/(a - δ)= (y - a)/α = (z - a - d)/(α+ δ) and (x - b + c)/(β- γ) = (y - b)/β = (z - b - c)/(β+ γ).

8) Find the vector equation of the plane that contains the lines r= (i+ j)+ ¥(i +2j - k) and r= (i+ j)+ ¢(-i + j -2k). Also , find the length of the perpendicular drawn from the point (2,1,4) to the plane thus obtained. r.(-i + j +k)= 0, √3

9) Find the equation of the plane passing through the point (0,7,-7) and containing the line (x+1)/-3 = (y- 3)/2 = (z+2)/1. x+ y + z=0

10) Show that the plane whose vector equation is r.(i + 2j - k)= 3 contains the line whose vector equation is r= (i+ j)+ λ(2i+ j + 4k).

11) Find the vector and cartesian equations of the plane containing the two lines r= (2i + j - 3k + λ(i+ 2j + 5k) and r= (3i + 3j + 2k) + μ(3i- 2j + 5k). r. (10i + 5j - 4k)= 37, 10x + 5y - 4z = 37

12) If 4x+ 4y + λz= 0 is the equation of the plane through the origin that contains the line (x-1)/2= (y+1)/3 = z/4, find the value of λ. 5

13) If the lines (x-1)/2 = (y+1)/3 = (z-1)/4 and (x-3)/1 = (y- k)/2 = z/1 intersect, then find the value of k, and hence find the equation of the plane containing these lines. 5x - 2y - z =6

14) Find the equation of the plane determined by the intersection of the lines (x+3)/3 = y/-2 = (z-7)/6 and (x+6)/1 = (y+5)/-3 = (z-1)/2. 2x - z +13=0

15) Find the vector equation of the plane passing through the points (3,4,2) and (7,0,6) and perpendicular to the plane 2x - 5y -15=0. Also, show that the plane thus obtained contains the line r= i + 3j - 2k + λ(i - j + k). r(5i + 2j - 3k)=17

16) If the lines (x-1)/-3 = (y-2)/-2k = (z-3)/2 and (x-1)/k = (y-2)/1 = (z-3)/5 are perpendicular, find the values of k and hence find the equation of the plane containing these lines. 2, -22x + 19y + 5z =31

17) Find the coordinates of the point where the line (x-2)/3 = (y+1)/4 = (z-2)/2 intersect the plane x - y + z -5=0. Also, find the angle between the line and the plane. (2,-1,2), sin⁻¹(1/√87)

18) Find the vector equation of the plane passing through three points with position vectors i+ j - 2k, 2i - j + k and I+ 2j + k. Also, find the coordinates of the point of intersection of this plane and the line r= 3i - j - k + λ(2i - 2j + k). r(9i + 3j - k)= 14, (1,1,-2)

19) Show that the lines (5-x)/-4 = (y-7)/4 = (z+3)/-5 and (x-8)/7 = (2y-8)/2 = (z-5)/3 are coplanar .

20) Find the equation of a plane which passes through the point (3,2,0) and contains the line (x- 3)/1 = (y - 6)/5 = (z- 4)/4. x - y + z -1=0

EXERCISE - N

Saturday, 15 April 2023

Theory of Quadratic equations (C)

Exercise -A

-----------------

* Examine the nature of the roots of the following:

1) 9x²-24x +16= 0.

a) rational b) equal c) irrational d) unequal e) a and b

2) x²-3√5 x -1= 0.

a) rational b) equal c) irrational d) unequal e) a and b f) c and d

3) 7x²+ 8x +4= 0.

a) rational b) equal c) irrational d) unequal e) Imaginary

4) 16x²+40x +25= 0.

a) rational b) equal c) irrational d) unequal e) a and b f) imaginary

5) qx²+px -q= 0.

a) rational b) equal c) irrational d) unequal e) a and b f) Real, unequal

6) 7x²+4x -3= 0. Rational, uneq

a) rational b) equal c) irrational d) unequal e) a and c f) imaginary

EXERCISE - B

Find the sum and Product of the roots of the following:

1) 3x²-2x +1= 0.

a) 2/3 b) 1/3 c) both a and b d) none

2) 2x²+x -1= 0.

a) -1/2 b) -1/2 c) both a and b d) none

3) 3x²-1= 0.

a) 0 b) 1 c) -1 d) none

4) (x-1)/(x+1)= (x-3)/2x.

a) 0 b) 3 c) both a and b d) none

3) Form the equation whose roots are:

A) 3, -8. x²+5x -24= 0

B) 2/7,5/7. 49x²-49x +10= 0

C) 1/2, 3/2. 4x²-8x +3= 0

D) 2+√3, 2- √3. x²-4x +1= 0

E) p+q, p - q. x²-2px +p¹- q²= 0

F) 2+3i, 2- 3i. x²-4x +14= 0

G) 2 - √5. x²-4x -1= 0

H) √5. x²-5= 0

I) 3 - 2i. x²-6x +13= 0

J) 2i. x² +4= 0

4) A) If m and n are the roots of the equation 2x²-5x +1= 0, find the value of

a) m²+ n². 3/2

b) m³+ n³. 95/8

c) m² - n². ±5√17/4

B) If p, q are the roots of 2x²-5x +7= 0, find the values of:

a) 1/p + 1/q.

b) p/q + q/p.

c) p²/q + q²/p. -85/28

C) If m, n are the roots of 2x²+x +7= 0 find the values of (1+ m/n)(1+ n/m). 1/14

D) If p, q are the roots of ax²+bx +c= 0, find the values of:

a) p²+ q². (b²-2ac)/a²

b) (p - q)². (b⁴-4ac)/a²

c) p²q+ q²p. -bc/a²

d) p³+ q³. -(3abc -b³)/a³

e) p³q + q³p. c(b²-2ac)/a³

f) p²/q + q²/p. (3abc - b³)/a²c

E) If the roots of 3x²-6x +4= 0 are m and n, find the value of (m/n + n/m) + 2(1/m + 1/n) + 3mn. 8

F) If m, n are the roots of the equation ax² + bx + c= 0. Find the values of:

a) (1+ m+ m²)(1- n+ n²). (a²+b² + c² + ab - ca +bc)/a²

b) m⁴+ n⁴. (b⁴-4ab²c +2a²c²)/a⁴

c) 1/m⁴ + 1/n⁴. (b²-4a²c² +2a²c²)/a⁴

d) m⁶+ n⁶. (b⁶+9a²b²c² -6ab⁴c - 2a³c³)/a⁶

5) A) If m and n are the roots of the equation x²-4x +11= 0, find the equation whose roots are

a) m+2, n+2. x²- 8x +23= 0

b) 1/m, 1/n. 11x²+ 4x +1= 0

c) m/n, n/m. 11x²+6x +11= 0

B) If p, q are the roots of 2x²-6x +3= 0, form the equation whose roots are p+ 1/q and n+ 1/m. 6x²-30x+25= 0.

C) If m and n are the roots of the equation x²- 4x +11= 0. Find the equation whose roots are:

a) m+2, n+2. x²- 8x +23= 0

b) 1/m, 1/n. 11x²+ 4x +1= 0

c) m/n, n/m. 11x² +6x +11= 0

D) If p, q are the roots of 2x²-6x +3= 0, form the equation whose roots are p+ 1/q and n+ 1/m. 6x²-30x +25= 0

6) a) Prove that the roots of the equation (x-a)(x-b)= p² are always real.

b) Prove that the roots of 3x²+22x +7= 0 can not be imaginary.

c) Find the sum and Product of the roots of x²- 12x +23= 0 and hence determine the square of the difference of the roots. 12, 23, 52

d) The sum and the product of the roots of a quadratic equation are 12 and -27 respectively. Find the equation. x²- 12x -27= 0

e) For what value of m the product of the roots of the equation mx² - 5x + (m+4)= 0 is 3 ? 2

f) For what value of k will the sum of the roots of the equations x²- 2(k+3)x +21k +7 = 0. 15

g) Find the value of m if the product of the roots of the equation x² + 21x +(m+8) = 0 be 13. 5

h) Determine the value of p, so that the roots of the equation px² - (3p+2)x +(5p -2)= 0 are equal. P+2, -2/11

i) Determine the value of m if the difference between the roots of the equation 2x²- 12x +m+ 2= 0 be 2. 14

j) Determine the values of p and q, so that the roots of the equation x² + px +q= 0 are p and q. (1,2) or (0,0)

k) If the equation x²+ 2(m+2)x +9m = 0 has equal roots, find m. 4,1

l) For what values of m will roots of the equation x²- (5+ 2m)x +(10+ 2m) = 0 be

i) equal in magnitude but opposite in sign. 5/2

ii) reciprocal. -9/2

m) For what value/s of m will be equation x²- 2(5+2m)x +3(7+10m) = 0 have

i) equal roots. 2 or 1/2

ii) reciprocal roots - 2/3

n) For what values of m will the sum of the roots of the equation 2x²- 12x +m+ 2 = 0 be equal to twice their product. 4

o) The roots m, n of the equation x² + Kx +12= 0 are such that m - n= 1, find k. ±7

p) Find the values of p for which the equation x² - px +p+ 3 = 0 has

A) coincide roots. 6, -2

B) real distinct roots. p< -3, p> 6

C) one positive and negative root. P < - 3.

7) If m and n are the roots of the equation x² - 4x +11= 0, find the equations whose roots are

a) m+2, n+2. x²- 8x +23=0

b) 1/m and 1/n. 11x²+ 4x +1=0

c) m/n and n/m. 11x²+ 6x +11= 0

8) If m, n are the roots of the equation x² - px +q = 0, find the equation whose roots are:

a) m² and n². x²- (p²-2q)x+ q²= 0

b) m/n and n/m. qx²-(p²-2q)x+q= 0

c) m+ 1/n and n + 1/m. qx² - p(q+1)x +(q+1)²= 0

d) 2m - n and 2n - m. x²- px+ 9q -2p²= 0

e) m²/n and n²/m. qx² - (p³- 3pq)x + q²= 0

f) 1/(m+n) and (1/m +1/n). pqx² - (p² + q)x + p = 0

9) If m, n are the roots of the equation 2x²- 6x +3= 0, form the equation whose roots are m+ 1/n and n+ 1/m. 6x²- 30x+25=0

10) If m, n are the roots of the equation ax²+ bx + c= 0, form the equation whose roots are (m+ n)² and (m - n)² a⁴x²- 2a(b² - 2ac)x+ b²(b² - 4ac) =0

11) If m, n are the roots of the equation 4x²- 8x +3= 0, form the equation whose roots are 1/(m+n)² and 1/(m- n)² 4x²- 5x+1=0

12) If m, n are the roots of the equation 2x²- 3x +1= 0, form the equation whose roots are m/(2n+3) and n/(2m+3). 40x²- 14x+1=0

13) If m, n are the roots of the equation 2x²- 6x +2= 0, form the equation whose roots are (1-m)/(1+ m) and (1- n)/(1+n). 5x²- 1=0

14)a) Find the equation whose roots are the reciprocal of the roots of the equation x² + px + q= 0. qx²- px +1=0.

b) Find the equation whose roots are the reciprocal of the roots of the equation 2x² + 3x + 7= 0. 7x²+3x +2=0.

c) Find the equation whose roots are squres of the roots of the equation x² + 3x + 2= 0. x²- 5x +4 =0

Exercise -2

------------------

1) If the difference of roots of x² - px +q = 0 be unity, prove p² + 4p² = (1+ 2q)².

2) If the difference of roots of ax² + bx + c= 0 be 2, prove b²= 4a(a+c).

3) If the difference of roots of x² + px +q = 0 be k, prove p² =4q+ k².

4) If one of root of x² - px +q = 0 be twice the other, prove 2p² = 9q.

5) If one of root of ax² + bx +c= 0 be four times the other, prove b² =25ac

6) If one of root of x² - px +q = 0 be thrice the other, prove 3p² = 16q.

7) If one of root of x² + px +q = 0 be r times the other, prove +r+1)² q = rp².

8) If the roots of (b²-ab)(2x -a) = (x² - ax)(2b - a) are equal in magnitude but opposite in sign, show that a² = 2b².

9)a) If the roots of lx² + mx + m= 0 be in the ratio p : q, show that √(p/q) + √(q/p) + √m/l = 0.

b) If the roots of px² + qx + q= 0 be in the ratio m : n, show that √(m/n) + √(n/n) + √q/p = 0.

10) If the roots of x² + px + q= 0 be in the ratio m : n, show that mnp²= q(m+ n)².

11) If the roots of ax² + bx + c= 0 be in the ratio 4 : 5 , show that 20b² = 81ac.

12) If one root of x² + px + q= 0 be square of the other, show that p³ - q(3p -1)+ q² = 0.

13) If the ratio of the roots of x² + bx + c= 0 be equal to the ratio of the roots x² + px + q= 0, show that b²q = p²c.

14) If the sum of the roots of x² + px + q= 0 be three times their difference, show that 2p² = 9q.

15) If k be the ratio of the two roots of ax² + bx + c= 0 show that (k+1)²a c = kb².

16) Prove that If the ratios of the roots of x² - 2px + q²= 0 and x² - 2lx + m²= 0 be equal, show that p²m² = q²l².

17) If m and n are the roots of x² + x -1 = 0, prove m² = n+ 2.

18) The ratio of the roots of ax²+ bx + c = 0 is 3: 4. Prove 12b² = 49 ac.

19) If one root of ax²+ bx + c= 0 be the square of the other, show b³+ a²c + ac² = 3abc.

20) If the difference between the roots of ax²+ bx + c = 0 be equal to the difference between the roots of px² + qx + r= 0, show that p²(b² - 4ac) = a²(q² - 4qr).

Exercise - 3

-------------------

1) Find the value of k for which 3x² + 2kx + 2= 0 and 2x² + 3x - 2= 0 may have a common root. 7/2, -11/4

2) Find the value of m for which x²- 5x + 6= 0 and x² + mx +3= 0 may have a common root. -4, 7/2

3) Find the value of k for which x²- kx + 21= 0 and x² - 3kx +35= 0 may have a common root. ±4

4) If the equation x²+ p₁x+ q₁ = 0 and x² + p₂x+ q₂ = 0 have a common root, prove that it is either (p₁q₂ - p₂q₁)/(q₁ - q₂) or (q₁ - q₂)/(p₁ - p₂).

5) prove that if x²+ px +q= 0 and x² + qx + p = 0 have a common root, then either p= q or p+ q +1= 0.

6) If the equation x² - 5x + 6= 0 and x² + mx + 3 = 0 have a common root, find the value of m. -7/2, -4

7) If the equation ax² + bx + c= 0 and bx² + cx + a= 0 have a common root, prove that, a³+ b³+ c³ = 3abc.

MIXED PROBLEM

*******************

1) If the roots of the equation (m - n) x² + (n -1)x + l = m are equal, show that l, m and n are in AP.

2) If the sum of the roots of the equation px² + qx + r = 0 is equal to the sum of the squares of their reciprocals, show that qr², rp², pq² are in AP.

3) The roots of the equation px² - 2(p +2)x + 3p= 0 are m, n. If m - n = 2, calculate the values of m, n and p. (-1,-3,-2/3) or (3,1,2)

4) Find the condition that the roots of the equation ax² + bx + c = 0 may differ by 5. b² - 4ac = 25a²

A) SHORT ANSWER TYPE:

1) If m, n are the roots of the equation x²+ x+1= 0, then find the value of m⁴+ n⁴+ 1/mn. 0

2) For what value of p(≠0) sum of the root of px²+2x+3p= 0 is equal to their product? -2/3

3) Form a quadratic equation whose one root is 2-√5. x²-4x-1=0

4) If 2 +i√3 is a root of x²+ px+q= 0, find p and q. -4,7

5) If one root of 2x²- 5x+k= 0 be double the other, find k. 25/9

6) If one root of x²+ (2-i)x - c= 0 be i. Find the value of c and other root of the equation. 2i, -2

7) Form a quadratic equation whose one root is 2 - 3i. x²-4x+13=0

8) If the roots of the equation qx²+ px+ q= 0 are imaginary, find the nature of the roots of the equation px²-4qx+ p=0. Real, unequal

9) If one root of x²+ px+8= 0 is 4 and two roots of x²+ px+q= 0 are equal, find q. 9

10) Construct a quadratic in x such that AM of its roots is A and GM is G. x²-2Ax+ G²= 0

11) if 5p²- 7p+4= 0 and 5q²- 7q+4= 0, but p≠ q, find pq. 4/5

12) if the equation x²+px+6= 0 and x²+4x+4=0 have a common root, find p. 5

13) if x is a real, show that the expression is always positive. Find its minimum value and the value of x for which it will be minimum. 14/5, 4/5

14) If c, d are the roots of (x-a)(x-b) - K= 0 show that a, b are the roots of (x- c)(x- d)+ K= 0.

15) If the roots of the equation x²- 4x - log₂a=0 are real, find the minimum value of a. 1/16

16) Given that m, n are the roots of x² -(a -2)x - a+1= 0. If a be real, Find the least value of m²+n². 1

17) If m, n are the roots of x²- 4x+5 = 0, form an equation whose roots are m/n +1 and n/m +1. 5x²-16x+16=0

B) CHOOSE THE CORRECT:

1) The sum of their reciprocals of the roots of 4x²+3x+7= 0 is

A) 7/4 B) -7/4 C) -3/7 D) 3/7

2) If one root of 5x²-6x+K= 0 be reciprocal of the other, then

A) K= 6 B) K= 5 C) K= -5 D) K= 1/5

3) If x be real, the maximum value of 5+ 4x- 4x² will be

A) 5 B) 6 C) 1 D) 2

4) The roots of x²+ 2(3m,+5)x+ 2(9m²+25) = 0 will be real if

A)m>5/3 B)m=5/3 C)m<5/3 D) m=0

5) The equation (4-n)x²+(2n+4)x +8n +1= 0 has equal integral roots, if

A) n= 0 B) n=1 C) n=3 D) none

6) The equation whose roots are reciprocal of the roots of ax²+ bx+c= 0, is

A) bx²+ cx+a= 0 B)cx²+ bx+a= 0

C) bx²+ ax+c= 0 D) cx²+ ax+b= 0

7) The value of the expression (ax)²+ bx+c, for any real x, will be always positive, if

A) b²- 4ac>0 B) b² - 4ac< 0

C) b²- 4a²c> 0 D) b² - 4a²c< 0

8) The value of m for which the equation x²-x+m²= 0, has no real roots, can satisfy

A) m>1/2 B) m>-1/2 C) m<-1/2 D) m<1/2

9)If x be real and a> 0, the least value of ax²+ bx+c will be

A) -b/a B) -b/2a C) -(b²-4ac)/2a D) -(b²- 4ac)/4a

10) The roots of ax²+ bx+c= 0 will be both negative, if

A) a>0, b> 0, c< 0

B) a>0, c> 0 ,b< 0

C) a>0, b> 0, c>0

D) b>0, c> 0 a< 0

11) If a, b are the roots of x² -2x +2= 0, the least integer n(>0) for which aⁿ/bⁿ = 1, is

A) 2 B) 3 C) 4.D) none

C) GENERAL QUESTIONS:

1) If the roots of 2x²+ x+1= 0 are p and q, from an equation whose roots are p²/q and q²/p. 4x²-5x+2=0

2) the equation x² - c x+d= 0 and x²- ax+b= have one root common and the second equation has equal roots.

Prove that ac= 2(b+d).

3) If the roots x²+ 3x+4= 0 are m,n, form an equation whose roots are (m-n)² and (m+n)². x² - 2x -63= 0.

4) If the roots of x²- px+q=0 are in the ratio 2:3, show that 6p²=25q.

5) If the roots of ax²+ bx+c=0 are m, n, form an equation whose roots are 1/(m+n), and 1/m + 1/n. bcx²+ (ac+b²)x + ab= 0.

6) If m, n are the roots of ax²+ 2b x+c= 0 and m+ + K, n+ K those of Ax²+ 2Bx+C= 0, prove that (b²- ac)/(B² - AC)= (a/A)².

7) Show that if one root of ax²+ bx+c=0 be the square of the other, than b³ + a²c + ac²= 3abc.

8) If m, n are the roots of the equation x²+ px - q= 0 and a, b those of the equation x²+ px+q=0, prove that (m- a)(m - b)= (n- a)(n- b)= 2q.

9) If the ratio of the roots of ax²+ cx+c= 0 be p: q, show that, √(p/q) + √(q/p)+ √(c/d)= 0.

10) if m be a root of equation 4x²+ 2x-1=0, prove that its other root is 4m³ - 3m.

11) If the sum of the roots of 1/(x+p) + 1/(x+ q) = 1/r be equal to zero, show that the product of root is 1/2 (p²+ q²).

12) If a, b are the roots of x²+ px+1= 0 and c, d are the roots of x²+ qx+1=0, show that q²- p²= (A-- c)(b - c)(a+ d)(b+ d).

13) Show that if x is real, the expression (x²- bc)/(2x- b - c) has no real values between b and c.

14) If one root of the equation ax²+ bx+c= 0 be the cube of the Other, show that ac(a+ c)²= (b² - 2ac)².

15) If a²= 5a - 3, b² = 5b - 3 but a≠ b, then find the equation roots are a/b and b/a. 3x²- 19x+3= 0

16) the coefficient of x in x²+ px+q= 0 is misprinted 17 for 13 and the roots of the original equation. -3, -10

17) if b³ + a²c + ac²= 3abc, then what relation may exist between the roots of the equation ax²+ bx+c= 0 ? One root is the square of the other.

18) find the maximum and minimum value of: x/(x²-5x+9). 1, -1/11

19) If m, n are the roots of ax²+ 2bx+c= 0, form an equation, whose roots are mw + nw² and mw² + nw (w= omega). (ax - b)²= 3(ac - b²)

20) If √m ± √n denote the roots of x² - px+q= 0, show that the equation, whose roots are m± n is (4x - p²)²= (p² - 4q)².

21) prove that for all real value of x, the value of p²/(1+x) - q²/(1- x) is real.

22) if x be real, prove that 4(a - x)(x - a + √(a²+ b²)) can never be greater than (a²+ b²).

23) If the quadratics x²+ px+q=0 and x²+ qx+p= 0 have a common root, prove that their other roots will satisfy the equation x²+ x+pq = 0

24) Show that if a, b, c are real, the roots of the equation (b - c) x²+ (c - a)x+(a - b)= 0 are real and they are equal if a, b, c are in AP.

25) If the the roots of the equation ax²+ 2bx+b =0 are Complex, show that the roots of the equation bx²+ (b - c)x- (a+ c - b)= 0 are real and cannot be equal unless a =b =c.

26) If a, b, c are real, show that the roots of the equation 1/(x+a) + 1/(x+ b) + 1/(x- c) = 3/x are real.

27) Show that the equation (b - c)x²+ (c - a)x+(a - b)= 0, (c - a)x²+ (a - b)x+(b - c)= 0, have a common root, find it and the remaining roots of the equations. 1, (a-b)/(b- c) and (b-c)/(c-a)

28) Prove that the roots of the equation (a - b)x²+ 2(a + b - 2c)x++ 1= 0, are real or complex according as c does not or lie between a and b.

29) prove that if the equation ax²+ bx+ c= 0 and bx²+ cx+ a= 0 have a common root, then neither a+ b+ c= 0 or a= b= c.

30) If the equation ax+ by =1 and cx²+ dy² = 1 have only one solution, prove that, a²/c + b²/d = 1 and x= a/c, y= b/d.

31) if (a - K)x²+ b(b - K)y²+ (c - K)z²+ 2fyz+ 2gzx + 2hxy is a perfect square, show that a - gh/f = b - hf/g = c - fg/h = K

32) Prove that x²+ y²+ z² + 2ayz + 2bzx + 2cxy can be resolved into two rational factors if if a² + b² + c² - 2abc = 1.

33) find K so that the value of x given by K/2x = a/(x+ c) + b/(x- c) may be equal. If m, n are two values of K and l, p the corresponding values of x, show that m. n = (a - b)² and l² p²= c².

a+ b± 2√(ab)

MISCELLANEOUS-1

1) Prove that the roots of ax² + 2bx + c= 0 will be real and distinct if and only if the roots of (a+ c)(ax² + 2bx+ c)= 2(ac - b²)(x² + 1) are imaginary.

2) Form an equation whose roots are squares of the sum and the difference of the roots of the equation 2x² + 2(m+ n)+ m²+ n²= 0. x² 4mnx - (m² - n²)²= 0

3) Find the value of p if the equation 3x²- 2x + p= 0 and 6x²- 17x + 12= 0 have a common root. -15/4, -8/3

4) If the equation x²- ax + b= 0 and x²- cx + d= 0 have one root in common and second equation has equal roots, prove that ac= 2(b + d).

5) Find the values of the parameter k for which the roots of x² + 2(k - 1)x + k + 5= 0 are

A) opposite in sign. K∈(-∞,-5)

B) equal in magnitude but opposite in sign.

C) positive. K∈(-5, -1)

D) negative. K∈(4,∞)

E) one root is greater than 3 and other is smaller than 3. K∈(-∞,-8/7)

6) If m, n are the roots of the equation 6x² - 6x +1= 0 then prove that 1/2 (a+ bm + cm² + dm³)+ 1/2 (a+ bn + cn²+ dn³)= a/1+ b/2+ c/3 + d/4.

7) For what values of m ∈ R, both roots of equation x² - 6mx + 9m² - 2m +2= 0 exceed 3 ? M∈(11/9,∞)

8) If the roots of the equation ax² + bx + c= 0 be (k+1)/k and (k+2)/(k +1) show (a+ b+ c)² = b² - 4ac.

9) If m, n are the roots of the equation x² - p(x +1) - c= 0, then prove that (m² + 2m+1)/(m² + 2m+c) = (n² + 2n+1)/(n² + 2n+c).

10) The condition that the equation 1/x + 1/(x + b) = 1/m + 1/(m+ b) has real roots that are equal in magnitude but opposite in sign is.

A) b² = m² B) b² = m² C) 2b² = m² D) none

11) The value of a for which one root of the equation (a -5)x² - 2ax + (a - 4)= 0 is smaller than 1 and the other greater than 2 is

A) a∈(5, 24) B) a∈(20/3,∞)

C) a∈(5,∞). D) a∈(-∞,∞)

12) If m, n be the roots of ax²+ bx + c= 0 then the value of (am²+ c)/(am + b) + (an²+ c)/(an + b) is

A) b(b² - 2ac)/4a B) (b² - 2ac)/2a. C) b(b² - 2ac)/a²c. D) 0

13) solve:

a) (7y²+1)/(y² -1) - 4(y²-1)/(7y² +1)= -3

B) {x - x/(x+1)²} + 2x{x/(x+1)}= 3.

14) If m, n are the roots of ax² + by + cid = 0, find the equation whose roots are 1/m³, 1/n³.

15) If the equation x² - (2+ m)+ (m² - 4m + 4)= 0 in x has equal roots, then the value of m are

A) 2/3,1 B) 2/3,6 C) 0,1 D) 0,2

PROBLEM ON QUADRATIC EQUATION

EXERCISE--1

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1) Find two natural numbers which differ by 3 and the sum of the square is 117. 6 and 9

2) The sum of the squares of two consecutive natural numbers is 41, Find the numbers. 4, 5

3) Find the natural numbers which differ by 5 and the sum of whose squares is 97. 4, 9

4) The sum of a number and its reciprocal is 4.25. find the number. 4 or 1/4

5) two natural numbers differ by 3. Find the numbers, if the sum of their reciprocal is 7/10. 2,5

6) Divide 15 into two parts such that the sum of their reciprocal is 3/10. 5, 10

7) A can do a piece of work in 'x' days and B can do the same work in (x+16) days. If both working together can do it in 15 days. Find x. 24

8) the square of a number added to one-fifth of it, is equal to 26. find the number. 5 or -26/5

9) the sum of the squares of two consecutive positive even numbers is 52. find the numbers. 4, 6

10) find two consecutive positive odd numbers, the sum of the square is 74. 5 and 7

11) two numbers are in the ratio 3: 5. find the numbers; if the difference between their square is 144. 9, 15 or -9, -15

12) three positive numbers are in the ratio 1/2;1/3;1/4. Find the numbers, if the sum of their squares is 244. 12,8,6

13) divide 20 into two parts such that three times the square of one part exceeds the other part by 10. 3, 17

14) the sum of two number is 32 and their product is 175. Find the numbers. 7, 25

15) three consecutive natural numbers are such that the square of the middle number exceeds the difference of the squares of the other two by 60. Assume the middle number to be x and form a quadratic equation satisfying the above statement. Hence; find the three numbers. 9,10,11

16) The ages of two Sisters are 11 years in 14 years. In how many years time will the product of their ages be 304 ? 5 yrs

17) out of the three consecutive positive integers, the middle number is is P. If three times the square of the largest is greater than the sum of the squares of the other two numbers by 67; calculate the value of P. 5

18) Five times a certain whole number is equal to three less than twice the square of the number. Find the number. 3

18) The product of two consecutive integers is 56. Find the integers. 7 or 8 or -8 and -7

20) Find two consecutive numbers whose squares have the sum 85. 6,7 or -6,-7

21) The sum of two numbers is 48 and their product is 432. Find the numbers. 36,12

22) If an integer is added to its square, the sum is 90. Find the integer with the help of quadratic equation. -10,9

23) Find the whole number which when decreased by 20 is equal to 69 times the reciprocal of the number. 23

24) Find two consecutive natural numbers whose product is 20. 4,5

25) The sum of the squares of two consecutive odd positive integers is 290. Find them. 11,13

26) The sum of two numbers is 8 and 15 times the sum of their reciprocal is also 8. Find the numbers. 3,5

27) The sum of a number and its positive square root is 6/25. Find the numbers. 1/25

28) The sum of a number and its square is 63/4, find the numbers. 7/2 or -9/2

29) There are three consecutive integers such that the square of the first increased by the product of the other two gives 154. What are the integers. 8,9,10

30) The product of two successive integrals multiples of 5 is 300. Determine the multiples. 15,20, or -20,-15

31) The sum of the squares of two numbers is 233 and one of the numbers is 3 less than twice the other number. Find the numbers. 8,13

32) Find the consecutive even integers whose squares have the sum 340. 12,14

33) Find two consecutive even integers, the sum of whose squares is 164. 8,10

34) Find two natural numbers which differ by 3 and whose squares have the sum 117. 6,9

35) The sum of the squares of three consecutive natural numbers is 149. Find the numbers. 6,7,8

36) Divide 57 into two parts whose products is 782. 23, 34

37) Determine two consecutive multiples of 3 whose product is 270. 15, 18

38) A sum of a number and its reciprocal is 17/4. Find the number. 4 or 1/4

39) A two digit number is such that the product of its digits is 8. When 18 is substracted from the number, the digits interchange their places. Find the number. 42

40) A two digit number is such that the product of its digits is 12. When 36 is added to the number, the digits interchange their places. Find the number. 26

41) A two digit number is such that the product of its digits is 16. When 54 is substracted from the number, the digits interchanged their places. Find the number. 82

42) the product of the digits of a two digit number is 24. If its unit's digit exceeds twice its ten's digit by 2; find the number. 38.

43) Two numbers differ by 3 and their product is 504, find the numbers. 21, 24 or -24,-21

44) two numbers differ by 4 and their product is 192. find the numbers. 12, 16 or -16,-12

45) A two digit number is 4 times the sum of its digits and twice the product of its digits. find the number. 36

46) the sum of the squares of two positive integers is 208. If the square of the larger number is 18 times the smaller, find the numbers. 8, 12

47) A two digit number is such that the product of its digits is 18. When 63 substracted from the number, the digits interchange their places. find the number. 92

48) A two digit number is such that the product of the digits is 14. When 45 is added to the number, the digits are reversed. Find the number. 27

49) divide 16 into two parts such that the square of the larger part exceeds the the square of the smaller part by 64. 10,6

50) The sum of the square of two positive integers is 208. If the square of the larger number is 18 times is smaller number, find the numbers. 8 ,12

51) the difference of the squares of two numbers is 45. the square of the smaller number is four times the larger number. find the numbers. 9,6 or -6,-9

51) If the sum of n successive odd natural numbers starting from 3 is 48, find the value of n. 6

52) If the sum of the first n even natural numbers is 420, find the value of n. 20

53) The denominator of a fraction is one more than twice the numeritor. if the sum of the fraction and its reciprocal is 58/21, find the fraction. 3/7

54) A two digit number is 4 times the sum and the three times the product of its digits. find the number. 24

EXERCISE--2

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1) The sides of a right angled triangle containing the right angl are 4x cm and (2x-1) cm. If the area of the triangle is 30 cm²; calculate the length of its sides. 12, 5, 13

2) The hypotenuse of a right-angled triangle is 26cm the sum of other two sides is 34cm. Find the length of its sides. 10, 24, 26

3) The sides of a right-angled triangle are (x-1) cm, 3x cm and (3x+1) cm. Find:

A) value of x. 8

B) the length of its sides. 7,24,25

C) its area. 84

4) The hypotenuse of a right-angled triangle exceeds one side by 1cm and other side by 18cm, find the length of the sides of the triangle. 25,24,7

5) The length of a rectangular plot exceeds its breadth by 12cm and the area is 1260m². Find its length and breadth. 42, 30

6) The perimeter of a rectangular plot is 104m and its breadth is 640m². Find its length and breadth. 32, 20

7) A footpath of uniform width runs around the inside of a rectangular field 32m long and 24m wide. If the path occupies 208 m², find the width of the footpath. 2m

8) Two squares have sides x cm and (x+4)cm. The sum of their areas is 656cm². express this as an algebraic equation in x and solve the equation to find the sides of the squares. 16, 20

9) the length of a rectangular board exceeds its breadth by 8cm. if the length were decreased by 4 cm and the breadth doubled, the area of the board would be increased by 256 cm². find the length of the board. 24

10) An area is paved with square tiles of a certain size and the number required is 600. if the tires had been 1cm smaller each way. 864 would have been needed. Find the size of the larger tiles. 6cm

11) A farmer has 70m of fencing, with which he encloses three sides of a rectangle sheep pen; the fourth side being a wall. If the area of the pen is 600 m², find the length of its short side. 15m when longer side is 40m and 20m when longer side is 30.

12) A square lawn is bounded on three sides by a part 4m wide. If the area of the path is 7/8 that of the lawn, find the dimensions of the lawn. Each side is 16m

13) the area of a big room is 300 m². If the length were decreased by 5m and the breadth increased by 5m; the area would be unaltered. Find the length of the room. 20m

14) The hypotenuse of a right angled triangle is 13cm and the difference between the other two sides is 7cm. 5, 12

15) The length of a verandah is 3m more than its breadth. The numerical value of its area is equal to the numerical value of its perimeter.

A) taking x as the breadth of the verandah, write an equation in x that represents the above statement.

B) Solve the equation obtained in A above and hence find the dimensions of the verandah. 6,3

16) The perimeter of a rectangular field is 82m and its area is 400 m². find the breadth of the rectangle. 16m

17) the length of a hall is 5m more than its breadth. If the area of the floor of the hall is 84 m², what are the length and the breadth of the hall? 7, 12

18) the hypotenuse of a right angle triangle is 25cm. the difference between the lengths of the other two sides of the triangle is 5cm. Find the length of these sides. 15,20

19) the hypotenuse of a right angle triangle is 3√10 cm. If the smaller leg is tripled the longer leg doubled, new hypotenuse will be 9√5cm.how long are the legs of the triangle ? 3, 9 cm

20) Two squares have sides x cm and (x+4) cm. the sum of their area is 656 cm². find the sides of the squares. 16, 20

21) the area of a right angle triangle is 165 m². determine its base and altitude if the latter exceed the formed by 7m. 15, 22m

23) The hypotenuse of a right angle triangle is 6 m more than twice the shortest side. If the third side is 2m less than the hypotenuse, find the sides of the triangle. 10,26,24

24) The hypotenuse of right angled triangle is 3√5 cm. If smaller side is tried and the largest side is doubled, the new hypotenuse will be 15cm. Find the length of each side. 3, 6

25) Vikram wishes to fit three rods together in the shape of a right angle triangle. the hypotenuse is to be 2cm longer than the base and 4 cm longer than the altitude. what should be the length of the rods? 8, 6,10

26) The hypotenuse of a grassy land in the shape of a right triangle is 1 m more than twice the shortest side. If the third side is 7m more than the shortest side, find the sides of the grassy land. 8,17,15

27) if twice the area of a smaller square is subtracted from the area of a larger square, the result is 14cm². however, if twice the area of the larger square is added to three times the area of the smaller square, the result is 203 cm². Determine the sides of the square. 5,8cm

28) A farmer wishes to grow a 100m² rectangular vegetable garden. Since he has with him only 30m barbed wire, he fences three sides of the rectangular garden letting compound wall of his house act as the fourth side fence. Find the dimensions of his garden. 5x20 or 10x10.

29) the area of the right angle triangle is 600cm². If the base of the triangle exceeds the altitude by 10 cm, find the dimension of the triangle. 40, altitude 30cm

30) the perimeter of a rectangular field is 82n and its area is 400m². find the breadth of a rectangle. 25 or 16m

31) the length of the sides forming right angle of a right angle triangle are 5xcm and (3x-1)cm, if the area of the triangle is 60cm², find its hypotenuse. 20.518cm

32) the area of an isosceles triangle is 60 cm² and the length of each one of its equal side is 13cm.find its base. 24 or 10cm

33) the perimeter of a right angle triangle is 60 cm. Its hypotenuse is 25 cm. Find the area of the triangle. 150cm²

34) the length of a rectangle exceeds its width by 8cm and the area of the rectangle is 240 cm². find the dimension of the rectangle. 12, 20

35) the side of a square exceeds the side of the another square by 4cm and the sum of its areas of the two square is 400cm². find the dimensions of the squares. 12cm

36) there is a square field whose side is 44 m. A square flower bed is prepared in its Centre leaving a gravel path all round the flower bed. the total cost of laying the flower bed and gravelling in the path at ₹2.75 and ₹1.50 per square metre, respectively, is ₹4904. find the width of the gravel path. 2m

37) two squares have sides x cm and (x+4) cm. The sum of their areas is 656 cm². Find the sides of the squares. 13, 16

EXERCISE --3

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1) A man travels 200 km with a uniform speed of 'x' km/hr. the distance could have been covered in 2 hours less, had the speed been (x+5) km/hr. calculate the value of x. 20km/hr

2) The speed of a boat in still water is 8km/hr. It can go 15 km upstream and 22km downstream in 5 hours. Find the speed of the stream. 3 km/hr

3) the speed of an ordinary train is X kilometre per hour and that of an Express train is (x+20) km/hr.

A) find the time taken by train to convert 300 km. 300/x

B) if the ordinary train takes 2 hours more than the express train. Calculate speed of express train. 300/(x+25)

4) If the speed of a car is increased by 10 km/hr, it takes 18 minutes less to cover a distance of 36 km. find the speed of the car. 75 km/hr

5) If the speed of an aeroplane is reduced by 40km/hr. It takes 20 minutes more to cover 1200km. find the speed of the aeroplane. 400km/hr

6) A train covers a distance of 300 km between two stations at a speed x km/hr. Another train cover the same distance at a speed of (x-5) km/hr

A) find the time with each train takes to cover the distance between the stations. 300/x, 300/(x-5)

B) If the second train takes 3 hours more than the first train, find the speed of each train. 25, 20 km/hr

7) A girl goes to a friend's house, which is at a distance of 12km. She covers half of the distance at a speed of x km/hr, and the remaining at a speed of (x-2)km/hr. if she takes 2 hours 30 minutes to cover the whole distance find x. 4

8) A car made a run of 390 km in x hours. if the speed had been 4 km per hour more, it would have taken 2 hours less for the journey find x. 15

9) A passenger train takes 3 hours less for a journey of 360 km, if its speed is increased by 10 km/hr from its usual speed. What is the usual speed? 30 km/hr

10) A fast train takes one hour less than a slow train for a journey of 200 km. If the speed of the slow train is 10km/hr less than that of the fast train, find the speed of the two trains. 50, 40km/hr

11) A passenger train takes one hour less for a journey of 150 km if its speed is increased by 5 km/hr from its usual speed. Find the usual speed of the train. 25 km/hr

12) The time taken by a person to cover 150 km was 2.5 hrs more than the time taken in the return journey. If he returned at a speed of 10 km/hr more than the speed of going, what was the speed per hour in each direction? 20, 30km/hr

13) A plane left 40 minutes late due to had weather and in order to reach its destination, 1600 km away in time, it had to increase its speed by 400 km/hr from its usual speed. Find the usual speed of the plane. 800 km/hr.

14) A plane takes 1 hour less for a journey of 1200 km if its speed is increased by 100 km/hr from its usual speed. Find its usual speed. 300 km/hr.

15) A goods train leaves a station at 6 p.m. followed by an Express train which leaves at 8 p.m. and travel 39 km per hour faster than the goods train. The express train arrives at a station, 180 km away, 36 minutes before the good train. Assuming that the speeds of both the trains remain constant between the two stations; calculate their speeds. 36, 75

16) ) a passenger train takes 2 hours less for a journey of 300 km if its speed is increased by 5 km)hr from its usual speed. Find the usual speed of the train. 25km/hr

17) ) A train travels a distance of 300 km at constant speed. if the speed of the train is increased by 5km/hr, the journey would have taken 2 hours less. find the original speed of the train. 25km/hr

18) The speed of a boat in still water is 15 km per hour. it can go 30km upstream and return downstream to the original point in 4 hours 30 minutes. Find the speed of the stream. 5km/hr

19) A first train takes 3 hours less than a slow train for a journey of 600 km. If the speed of the slow train is 10km/hr less than that of the fast train. Find the speeds of the two trains. 40, 50

20) A plane left 30 minutes later than the schedule time and in order to reach its destination 1500km away in time it has to increase its speed by 250 km/hr from its usual speed. Find its usual speed. 750km/hr

21) In a flight of 600km, an aircraft slowed down due to bad weather. Its average speed for the trip was reduced by 200km/hr and the time of flight increased by 30 minutes. find the duration of flight. 1hr

22) Swati can row her boat at a speed of 5km/hr in still water. if it takes 1 hour more to row the boat 5.25 km upstream than to return downstream, find the speed of the stream. 2km/hr

EXERCISE --4

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1) A piece of cloth costs ₹35. If the piece were 4m longer and each metre costs ₹ one less, the cost would remain unchanged. How long is the piece ? 10m

2) By selling an article for ₹24, a trader losses as much as percent as the cost price of the article. calculate the cost price. 60 or 40

3) A trader bought a number of articles for ₹1200. Ten were damaged and he sold each of the rest at ₹2 more than what he paid for it, thus getting a profit of ₹60 on the whole transaction.

Taking the number of articles he bought as x, form an equation in x and solve it. 100

4) Mr Mehra sends his servant to the market to buy oranges worth ₹15. the servant having eaten three oranges on the way, Mr. Mehra pays 25 paise per orange more than the market price. Taking x to be the number of oranges which Mr. Mehra receives, form a quadratic equation in x. find the value of x. 12

5) A man bought an article for ₹x and sold it for ₹16. If his loss was x percent, find the cost price of the article. 20, or 80

6) A trader bought an article for ₹x and sold it for ₹52, thereby making a profit of (x-10) percent on his outlay. calculate the cost price, for trader, of the article. 40

7) by selling chair for ₹75, Mohan gained as much as percent as its cost. calculate the cost of a chair. 50

8) An employer finds that if he increases the weekly wages of each worker by ₹3 and employs one worker less, he reduces his weekly wages bill from ₹816 to ₹781. Taking the original weekly wage of each worker as ₹x; obtain an equation in x and then solve it to find the weekly wages of each worker. 68

9) a dealer sells an article for ₹24 and gains as much as percentage as the cost price of the article. find the cost price of the article. 20

10) If the list price of a toy is reduced by ₹2, a person can buy 2 toys more for ₹360. find the original price of the toy. ₹20

11) A piece of cloth costs ₹200. If the piece was 5m longer and each metre of cloth costs ₹2 less the cost of the piece would have remained unchanged. how long is the piece and what is the original rate per metre ? 20m, ₹10 p/m

12) A shopkeeper buys a number of books for ₹80. If he had bought four more books for the same amount, each book would have cost ₹1 less. how many books did buy? 16

13) If the price of a book is reduced by ₹5, a person can buy 5 more books for ₹300. find the original list price of the book. ₹20

14) A factory kept increasing its output by the same percentage every year. find the percentage if it is known that the output is doubled in the last two years. 100(-1+√2)

15) A dealer sells a toy for ₹24 and gains as much percent as the cost price of the toy. Find the cost price of the toy. ₹20

. EXERCISE --5

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1) Ashu is x years old while his mother Mrs Veena is x² years old. 5 years hence Mrs. Veena will be three times as old as Ashu. find their present ages. 5, 25

2) the sum of the ages of a man and his son is 45 years. five years ago, the product of their ages was four times the man's age at that time. find their present age. 37, 9

3) the product of Shikha's age 5 years ago and her age 8 years later is 30, Her age at both times being given in years. find her present age. 7yrs.

4) The product of Ramu's age (in years) five years ago and his age (in years) nine years later is 15. Determine Ramu's present age. 6

5) 1 years ago, A man was 8 times as old as his son. Now his age is equal to the square of his son's age. find their present ages. 7, 49

6) the product of Ramu's age(in years) 5 years ago with his age(in years) 9 years later is 15. find Ramu's present age. 6years.

7) the sum of the ages of a father and his son is 45 years. Five years ago, the product of their ages(in years) was 124. Determine their present ages. 36, 9

Exercise -6

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1) A takes 10 days less than the time taken by B to finish a piece of work. if both A and B together can finish the work in 12 days. find the time taken by B to finish the work. 30 days

2) If two pipes function simultaneously, a reservoir will be filled in 12 hours. one pipe fills the Reservoir 10 hours faster than the other. how many hours will the second pipe take to fill the reservoir? 30 hrs

3) A takes 6 days less than the time taken by B to finish a piece of work. if both A and B together can finish it in 4 days, find the time taken by B to finish the work. 12 days

4) A swimming pool is filled with three pipes with uniform flow. the first two pipes operating simultaneously, fill the pool in the same time during which the pool is filled by the third pipe alone. the second pipe fills the pool 5 hours faster than the first pipe and 4 hours slower than the third pipe. find the time required by each pipe to fill the pool separately. 15,10,6

5) two pipes running together can fill a cistern in 40/13 minutes. if one pipe 3 minutes more than the other to fill it, find the time in which each pipe would fill the cistern. 5,8

6) one pipe can fill a cistern in 3 hours less than the other. the two pipes together can fill a cistern in 6 hours 40 minutes. find the time that is each pipe will take to fill the cistern. 12,15

EXERCISE--7

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Miscellaneous

1) Some students plant picnic. The budget for food was ₹480. But 8 of these fail to go and this the cost of food for each member increased by ₹10. how many students attended the picnic? 16

2) ₹ 250 is divided equally among a certain number of children; if there were 25 children more, each would would received 50 paise less. find the number of children. 100

3) out of a group of swans 7/2 times the square root of the total number of playing on the shore of a pond. The two remaining ones are swinging in a water. find the total number of swans. 16

4) ₹900 were divided equally among a certain number of persons. Had there been 20 more persons each would have got ₹160 less. find the original number of persons. 25

5) some students planned a picnic. the budget for food was ₹500. but, 5 of them failed to go and thus the cost of food for each member increased by ₹5. how many students attended the picnic? 20

6) one fourth of a herd of Camels was seen in the forest. twice the square root of the herd had gone to mountains and the remaining 15 camels were seen on the bank of the river. find the total number of camels. 36

7) Out of a group of swans 7/2 times the square root of the number are playing on the Shore of a tank. The two remaining ones are playing, with amarous fight, in the water. what is the total number of the swans? 16

8) A chess board contains 64 equal squares and the area of each square is 6.25 cm². A border round the board is 2cm wide. find the length of the side of the chess board. 24cm

9) A person on tour has 360 for his expenses. if he extends his tour for 4 days, he has to cut down his daily expenses by ₹3. find the original duration of the tour. 20 days

10) ₹ 6500 were divided equally among a certain number of persons. Had there been 15 more persons, each would have got ₹30 less. find the original number of persons. 50

11) the Angry Arjun carried some arrows for fighting with the Bheesm. with half the arrows, he cut down the arrows thrown by Bheesm on him and with six other arrows he killed the rath driver of Bheesm. With one arrow each knocked down respectively the rath, flag and the bow of Bheesm. Finally, with one more than 4 times the square root of arrows he laid Bheesm unconscious on an arrow bed. find the total number of arrow Arjun had. 100

12) one fourth of a herd of Camels was seen in the forest. Twice the square root of the herd had gone mountains and the remaining 15 camels were seen on the bank of a river. find the total number of camels. 36.

13) A stone is thrown vertically downwards and the formula d= 16t²+ 4t gives the distance, d metres, that it falls in t seconds. How long does it take to fall 420 metres? 5 sec

EXERCISE-8

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1) 8x² -22x-21= 0. 7/2, -3/4

2) x² + 2√2 x -6= 0. -3√2, √2

3) x/(x+1) + (x+1)/x = 34/15, x≠ 0, x≠ -1. 3/2, -5/2

4) (x +3)/(x-2)-(1-x)/x= 17/4. 4, -2/9

5) 4/x - 3= 5/(2x+3). -2, 1

6) 2x/(x-3) + 1/(2x+3)+ (3x+9)/{(x-3)(2x+3)} =0. -1

7) x² - 2ax + a² - b²= 0. a-b, a+b

8) x² - 4ax + 4a² - b²= 0. 2a-b, 2a+b

9) 4x²-4ax +(a² - b²)= 0. (a-b)/2, (a+b)/2

10) 4x² - 4a²x +(a⁴ - b⁴)= 0. (a²-b²)/2, (a²+b²)/2

11) 4x² - 2(a²+b²)x + a²b²= 0. a²/2,b²/2

12) 9x² - 9(a+b)x + (2a²+5ab +2b²)= 0. (2a+b)/3, (a+2b)/3

13) x² + {a/(a+b) + (a+b)/a}x +1= 0. - a/(a+b), -(a+b)/a

14) x² + x -(a+1)(a+2)= 0. -(a+2), a+1

15) x² + 3x -(a² + a -2)= 0. -(a+2), a-1

16) 1/(a+b+x) = 1/a + 1/b + 1/x, a+ b≠ 0. -a, - b

17) x + 1/x = 626/25. 25, 1/25

18) (x-3)(x-4)= 34/(33)². 98/33, 133/33

19) a²x² - 3abx + 2b²= 0. 2b/a, b/a

20) 4x² + 4bx - (a² - b²)= 0. (a-b)/2, -(a+b)/2

21) ax² + (4a² - 3b)x - 12ab= 0. 3b/a, -4a

22) (x - 1/2)²= 4. 5/2, -3/2

23) (x+3)/(x+2) = (3x -7)/(2x-3). -1,5

24) 2x/(x-4) + (2x -5)/(x-3)=25/3. 6, 40/13

25) (x+3)/(x-2) - (1-x )/x= 17/4. 4, -2/9

26) (x-3)/(x+3) - (x +3)/(x-3)= 48/7, x≠ 3, x≠ -3. -4, 9/4

27) (x+2)/(x-1) - (x -1)/(x +1)= 5/6, x≠ 1, x≠ -1. 5, -1/5

28) (x-1)/(2x+1)+(2x+1)/(x-1) = 5/2, x≠-1/2. 1,-1

29) 3x² -14x-5= 0. 5, -1/3

30) mx²/n + n/m = 1 - 2x. -n±√(mn))/m

31) (x-a)/(x-b) + (x -b)/(x-a)= a/b + b/a. 0, a+ b

32) 1/{(x-1)(x-2)} + 1/{(x -2)(x-3)} + 1/{(x-3)(x-4)}= 1/6. -2,7

33) (x-5)(x-6) =25/(24)². 145/24, 119/24

34) 7x + 3/x = 178/5. 5, 3/35

35) a/(x-a)+ b/(x-b)= 2c/(x-c). 0, (2ab - bx- ac)/(a+b - 2c)

36) x² + 2ab = (2a+ b)x. 2a, b

37) (a + b)²x² - 4abx - (a-b)²= 0. 1, {(a-b)/(a+b)}²

38) a(x²+1)- x(a²+1)= 0. a, 1/a

39) x² - x - a(a+1)= 0. -a, a+1

40) x² + (a+ 1/a)x +1= 0. - a, -1/a

EXERCISE - 9

*********

A) Solve:

1) p²x²+ (p²-q²)x -q= 0. -1, q²/p²

2) 9x²- 9(a+b)x +(2a²+5ab+2b²)= 0.

(2a+b)/3, (a+3b)/3

3) abx²+(b²-ac)x -bc= 0. c/b, -b/a

4) (x-1)/(x+2) + (x-3)/(x-4) =10/3. (1±√297)/4

5) 1/(x+1) + 2/(x+2)= 4/(x+4), x≠-1,-2,-4. 2±2√3

B) Find the Discriminant:

1) x²- 4x +2= 0. 8

2) 3x² + 2x - 1= 0. 16

3) x²- 4x + a = 0. 16 - 4a

4) √3x²- 2√2x -2√3= 0. -3

5) x² + px +2q= 0. p² - 8q

6) (x-1)(2x-1)= 0. 1

C) Find the nature of the roots:

1) 2x²+x -1= 0. Real, distinct

2) x²- 4x +4= 0. Real, equal

3) x²+x +1= 0. Not real

4) 4x²- 4x +1= 0. Real, equal

5) 2x² +5x +5= 0. Not real

6) (x-1)(2x-5)= 0. Not real

7) 3x²/5- 2x/3 +1= 0. Not real

8) x² + 2√3 x -1= 0. Real, distinct

9) 3x²- 2√6x +2= 0. Real, equal

10) (x-2a)(x-2b)= 4ab. Real, distinct

11) 9a²b³x²- 24abcdx +16c²d³= 0, a≠ 0, b ≠0. Real, equal

12) 2(a²+b²)x²+2(a+b)x +2= 0. Not real

13) (b+c)x²- (a+b+c)x +a= 0. Real, distinct

D) Determine whether the given equations have real roots and if so, find the roots:

1) 9x²+7x -2= 0. 2/9, -1

2) 2x²+ 5√3 x +6= 0. -√3/2, 2√3

3) 3x²+ 2√5 x -5= 0. √5/3,-√5

4) x²+5x +5= 0. (-5±√5)/2

5) 6x²+x -2= 0. 1/2, -2/3

6) 25x²+20x +7= 0. Not real

7) 3a²x²+ 8abx +4b²= 0, a≠ 0. -2b/a, -2b/3a

E) Find the values of k for which the given equations has real and equal roots.

1) 2x²- 10x +k= 0. 25/2

2) 9x²+3kx +4= 0. ±4

3) 12x²+4kx +3= 0. ±3

4) 2x²+3x +k= 0. 9/8

5) 2x²- kx +k= 0. ±2√2

6) kx²- 5x +k= 0. ±5/2

7) x²+k(4x+k-1) +2= 0. 2/3,-1

8)x²-2x(3k+1) +7(3+2k)= 0. 2,-10/9

9) (k+1)x²- 2(k-1)x +1= 0. 0, 3

10) (k-12)x²+ 2(k-12)x +2= 0. 12,14

11) x²- 2(5+2k)x +3(7+10k)= 0. 2, 1/2

12) (3k+1)x²+ 2(k+1)x +k= 0. 1,-1/2

13) kx²+ kx +1= -4x²- x. 5,-3

14) (k+1)x²+2(k+3)x +(k+8)= 0. 1/3

15) x²- 2kx + 7k -12= 0. 4, 3

16) (k+1)x²- 2(3k+1)x + 8k+2= 0. 0,3

17) 5x²- 4x +2+k(4x²-2x-1)= 0. 1, -6/5

18) (4-k)x²+(2k+4)x +(8k+1)= 0. 0,3

19) (2k+1)x²+2(k+3)x +(k+5)= 0. (-5±√41)/2

20) 4x²- 2(k+1)x +(k+4)= 0. -3,5

21) k²x²- 2(2k-1)x +4= 0. 1/4

1) For what value of k, (4-k)x² +(2k+4)x + (8k+1)= 0, is a perfect square. 0, 3

2) If the roots of the equation (b- c)x² + (c-a)x + (a-b)= 0 are real, then show that 2b= a+ c.

3) If the roots of the equation (a²+ b²)x² - 2(ac + bd)x + (c²+ d²)= 0 are equal, then show that a/b= c/d

4) If the roots of the equation ax² + 2bx + c= 0 and bx² - 2 √(ac)x + b= 0 are simultaneously real, then show that b²= ac.

5) If p, q are real and p+q, then show that the roots of the equation (p- q)x² + 5(p +q)x 2 (p-q)= 0 are real and unequal.

6) If the roots of the equation (c² - ab)x² +2 (a²-bc)x + b² - ac= 0 are equal, then show that a= 0 or a³+ b³+ c³= 3abc.

7) If the equation (1+ m²)x² + 2mcx+ (c²-a²)= 0 are real, then show that c²= a²(1+ m²).

QUADRATIC EQUATIONS

" If a variable occurs in an equation with all positive integer powers and the highest power is two, then it is called a Quadratic Equation(in that variable)."
In other words, a second-degree polynomial in x equated to zero will be a quadratic equation, the coefficient of x² should not be zero.
The most general form of a quadratic equation is ax²+ bx + c= 0, where a≠ 0(and a, b, c are real)
Some examples of quadratic equations are:
1) x² - 5x + 6 = 0
2) x² - x - 6 = 0
3) 2x² + 3x - 2 = 0
4) 2x² + 5x - 3 = 0

Like a first degree equation in x has one value of x satisfying the equation, a quadratic equation in x will have TWO values of x that satisfy the equation. The values of x that satisfy the equation are called the ROOTS of the equation. These roots may be real or complex.
For the four quadratic equations given above, the roots are given below:
In (1) x= 2 and x= 3
In (2) x=- 2 and x= 3
In (3) x= 1/2 and x= -2
In (4) x= 1 and x= -3/2

In general, the roots of a quadratic question can be found out in two ways.
i) by factorizing the expression on the left hand side of the quadratic equation.
ii) by using the standard formula.

All the expressions may not be easy to factorise whereas applying the formula is simple and straight forward.

finding the roots by factorization if the quadratic equation ax² + bx + c= 0 can be written in the form of (x - m)(x - n) = 0, then the roots of the equation are m and n.

To find the roots of a quadratic equation, we should first write it in the form of (x - m)(x - n) = 0, i.e., the left hand side ax² + bx + c of the quadratic equation ax²+ bx + c= 0 should be factorised into two factors.
For the purpose, we should go through the following steps. We will understand these steps with the help of the equation x² - 5x + 6= 0 which is the first of the four quadratic equations we looked at as examples above.
* first write down b(the coefficient of x) as a sum of two quantities whose product is equal to ac.
In this case -5 has to be written as the sum of two quantities whose product is 6. We can write-- 5 as (-3)+ (-2) so that the product of (-3) and (-2) is equals to 6.
* Now rewrite the equation with 'bx' term split in the above manner.
In this case, the given equation can be written as x² - 3x - 2x+ 6= 0.
* Take the first to terms and rewrite them together after taking out the common factor between the two of them. Similarly, the third and the fourth terms should be rewritten after taking out the common factor between the two of them. In other words, You should ensure that what is left from the first and the second terms (after removing the common factor) is the same as that left from the third and the fourth term (after removing their common factor).
In this case, the equation can be written as x(x-3) - 2(x -3)= 0; Between the first and second terms as well as the third and fourth terms, we are left with (x-3) is a common factor.
* Rewrite the entire left hand side to get form (x - m)(x - n).
In this case, if we take out (x -3) as the common factor, we can rewrite the given equation as (x -3)(x -2)= 0.
* Now, m and n are the roots of the given quadratic equation.
=> for x² - 5x + 6= 0, the roots of the equation are 3 and 2.

For the other three quadratic equations given above as examples, let us see how to factorise the expression and get the roots.
For equation (2), i.e., x²- x -6= 0, the coefficient of x which is -1 can be written as (-3) + (+2) so that their product is -6 which is equals to ac (1 multiplied by -6). Then we can rewrite the equation as (x -3)(x +2)= 0 giving us the roots as 3 and -2.
For equation (3), i.e., 2x²+ 3x- 2= 0, the coefficient of x which is 3 can be written as (+4) + (-1) so that their product is -4 which is the value of ac (-2 multiplied by 2). Then we can rewrite the equation as (2x -1)(x +2)= 0 giving the roots as 1/2 and-2.
For equation (4), i e., 2x²+ x -3= 0, the coefficient of x which is 1 can be written as (+3)+ (-2) so that their product is -6 which is equal to ac (2 multiplied by -3). Then we can rewrite the given equation as (x -1)(2x +3)= 0 giving us the roots as 1 and -3/2.

Finding out the roots by using the formula
if the quadratic equation is ax² + bx + c= 0, then we can use the standard formula given below to find out the roots of the equation.
x= {-b ± √(b² - 4ac)}/2a.
The roots of four quadratic equations we can took as examples above can be taken and their roots found out by using the above formula. The student is advised to check it out for himself that the roots can be obtained by using this formula also.

Sum and product of roots of a quadratic equation
For the Quadratic Equation ax²+ bx+ c= 0, the sum of the roots and the product of the roots can be given by the following:
Sum of the roots= - b/a
Product of the roots= c/a
These two rules will be very helpful in solving problems on quadratic equation.

Nature Of The Roots

We mentioned already that the roots of the Quadratic Equation with real Coefficient can be real or complex. When the roots are real, they can be equal or unequal. All this will depend on the expression b² - 4ac. Since b² - 4ac determines the nature of the roots of the quadratic equation, it is called the DISCRIMINANT of the quadratic equation.
* if b² - 4ac > 0, then the roots of the quadratic equation will be real and distinct.
* if b² - 4ac = 0, the roots are real and equal.
* if b² - 4ac < 0, then the roots of the quadratic equation will be complex conjugates.
Thus we can write down the following about the nature of the roots of a quadratic equation when a, b, c are all rational.
* when b²- 4ac< 0, the roots are complex and unequal
* when b²- 4ac = 0 the roots are rational and equal
* when b² - 4ac > 0 and a perfect square, the roots are rational and unequal.
* When b²- 4ac > 0 but not a perfect square, the roots are irrational and unequal.

Whenever the roots of the Quadratic Equation are irrational, (a, b, c being rational) they will be of the form a+ √b and a - √b, i.e., whenever a+ √b is one root of a quadratic equation, then a - √b will be second root of the quadratic equation and vice versa.

Sign Of The Roots
We can comment on the signs of the roots, i.e., whether the roots are positive or negative, based on the sign of the sum of the roots and the product of the roots of the quadratic equation. The following table will make the clear relationship between the sum and the product of the roots and the signs of the roots themselves.
Signs of      Sign of       Sign of
Product      Sum of       the roots
Of the         the roots
Roots
+ve             + ve       Both the roots are positive
+ ve            - ve the roots are negative
- ve            + ve      the numerically larger root is positive and the other root is negative
- ve            - ve       the numerically larger root is negative and the other root is positive.

Constructing A Quadratic Equation

We can build a quadratic equation in the following cases:
* when the roots of the quadratic equation are given
* when the sum of the roots and the product of the roots of the quadratic equation are given.
* When the relation between the roots of the equation to be framed and the roots of another equation is given.
if the roots of the quadratic equation are given as m and n, the equation can be written as (x - m)(x - n)= 0 i.e., x² - x(m+ n)+ mn= 0.
if p is the sum of the roots of the quadratic equation and q is the product of the roots of the quadratic equation, then the equation can be written as x²- px + q= 0.

Constructing A New Quadratic Equation By Changing The Roots Of A Given Quadratic Equation

If we are given a quadratic equation, we can build a new quadratic equation by changing the roots of this equation in the manner specified to us.
For example, let us take a quadratic equation ax²+ bx + c= 0 and let its roots be m and n respectively. Then we can build new quadratic equations as per the following patterns:
i) A quadratic equation whose roots are the reciprocal of the given equation ax² + bx + c= 0, i.e., the roots are 1/m, and 1/n:
This can be obtained by substituting 1/x in place of x in the given equation given giving us cx²+ bx + a= 0, i.e., we get the equation required by inter-changing the coefficient of x² and the constant term.
ii) A quadratic equation whose roots are k more than the roots of the equation ax²+ bx+ c= 0, i.e., the roots are (m+ k) and (n + k).
This can be obtained by substituting (x - k) in place of x in the given equation.
iii) A quadratic equation whose roots are k less than the roots of the equation ax²+ bx + c= 0, i.e., the roots are (m - k) and (n - k).
This can be obtained by substituting (x + k) in place of x in the given equation.
iv) A quadratic equation whose roots are k times the roots of the equation ax²+ bx + c= 0, i.e., the roots are km and kn.
This can be obtained by substituting x/k in place of x in the given equation.
v) A quadratic equation whose roots are 1/k times the roots of the equation ax²+ bx+ c = 0, i.e., the roots are m/k and n/k
This can be obtained by substituting kx in place of x in the given equation.
An equation whose degree is 'n' will have n roots

Maximum Or Minimum Value Of A Quadratic Expression
An equation of the type ax²+ bx+ c= 0 is called a quadratic equation. An expression of the type ax²+ bx+ c is called a "quadratic expression". The quadratic expression ax²+ bx + c takes different values as x takes different values.
As x varies from -∞ to +∞, (i.e., when x is real) the quadratic expression ax²+ bx + c
i) has a minimum value whenever a> 0 (i.e., a is positive). The minimum value of the quadratic expression is (4ac - b²)/4a and it occurs at x= - b/2a.
ii) has a maximum value whenever a< 0 (i.e., a is negative). The maximum value of the Quadratic Expression is (4ac - b²)/4a and it occurs at x= - b/2a.

Equations Of Higher Degree

The index of the highest power of x in the equation is called degree of the equation. For example, if the highest power of x in the equation is x³, then the degree of the equation is said to be 3. An equation whose degree is 3 is called a cubic equation.

Existence Of A root

if f(x) is an nth degree polynomial in x and f(a) and f(b) have opposite signs, then there exists a root of the equation f(x)= 0, between a and b.

Number Of Roots

A linear equation has 1 root, a quadratic has 2 roots ( provided they are counted properly). For example x² = 0 has two roots, both of which are 0).
Similarly an nth degree equation has n roots, provided they are counted properly.
We know that if a is a root of f(x)= 0, then x - a is a factor of f(x).
If (x - a)ᵐ is a factor of f(x) but (x - a)ᵐ⁺¹ is not, then the root a should be counted m times. m is said to be the multiplicity of the root a. The root a is said to be a simple, double, triple or n-tuple root according to as m= 1, 2, 3 or n.
If we count each root as many times as it's multiplicity, we find that an nth degree equation has n roots.

Type Of roots
1) If all the coefficients of f(x) are real, and p + iq (where i= √-1) is a root of the equation f(x)= 0, then p - iq is also a root, i e , Complex roots occur as a conjugate pairs. Therefore, if the degree of an equation is odd, it has atleast 1 real root.
2) if the number of changes of sign in f(x) is p, then f(x)= 0 has at most p positive roots. The actual number of positive roots could be o, p -2, p - 4 .... i.e., the number of the positive roots is equal to the number of sign changes in f(x) or less than that by an even number.
Ex: f(x)= 6x³ - 6x² + 11x - z
Consider the changes in the signs of successive terms of f(x) .
+ - + -
There are three changes of sign in f(x), so f(x)= 0 has 3 or 1 positive roots.
3) If the number of changes in the signs of the terms of f(-x) is q, then f(x) = 0 has at most q negative roots. The actual number of negative roots could be q, q-2, q- 4,...., i e , the number of negative roots is equal to the number of sign changes in f(-x) or less than that by an even number.
f(x)= 2x⁵ + 3x⁴ + 5x³ + 6x² + 2x +1
=> f(-x)= -2x⁵+ 3x⁴ - 5x³ + 6x² - 2x +1
i e., There are 5 changes of sign in f(-x), so f(x)= 0 has 5, 3 or 1 negative roots.
Consider the equation f(x)= x⁴ + 4x³ + 6x +24= 0.
Since there is no change of sign in f(x) = 0, p= 0
f(x)= 0 does not have any positive real root.
f(-x) = x⁴ - 4x³ - 6x + 24
The number of changes of sign in f(x) is 2.
So f(x)= 0 has 2 or 0 negative roots.
So, the number of complex roots is 2 or 4.
NOTE: Rule (2) and (3) are known as Descarte's rule of sign.

EXERCISE -1

1) Find the roots of the equation x²+ 9x -10= 0.
A) 1 B) 1, -10 C) -10 D) 1, 10 E) n

2) Find the roots of the equation x² - 12x + 13= 0.
A) 1,13 B) -1, -13 C) 6+ √23, 6 - √23 D) 1, -13 E) n

3) Find the roots of the equation 4x²-17x +4 = 0.
A) 4 B) 1/4, 4 C) 1/4 D) -4, -1/4 E) n

4) Find the nature of the equation x²- 3x +1 = 0.
A) real B) unequal C) equal D) imaginary E) real and equal

5) Find the nature of the equation 5x²- x - 4 = 0.
A) real B) rational and unequal C) equal D) imaginary E) real and equal

6) Find the nature of the equation 2x² + 6x - 5 = 0.
A) real B) rational and unequal C) equal D) imaginary E) real and equal

7) If the sum of the roots of the equation kx²- 52x +24 = 0 is 13/6, find the product of its roots.
A) 2 B) 4 C) 24 D) 42 E) none

8) If the sum of the roots and product of the roots of the equation 13 and 30, find its roots
A) 10,3 B) -10,-3 C) 10,-3 D) -10,3 E) none

9) If the roots of the equation 6x²- 7x + b = 0 are reciprocal of each other. Find b.
A) 2 B) 4 C) 6 D) 8 E) none

10) The roots of the equation are a and - a. The product of its roots is -9. Form the equation in variable of x.
A) x²+9 B) x² - 9 C) 2x²+ 9 D) 2x² - 9 E) none

11) The roots of the equation x²- 12x + k= 0 is in the ratio 1: 2. find k
A) 2 B) 3 C) 32 D) 23 E) none

12) A quadratic equation has rational coefficients. One of its roots is 2+ √2. Find its other root.
A) 2 B) √2 C) 2+ √2 D) 2 - √2 E) n

13) I can buy 9 books less for ₹1050 if the price of each book goes up by ₹15. Find the original price and the number of books I could buy at that price.
A) 35 B) 50 C) either 35 or 50 D) neither 35 nor 50 E) n

* P and Q are the roots of the equation x²- 22x + 120 = 0. find the value of
14) P²+ Q²
A) 11 B) 60 C) 11/60 D) 60/11 E) none

15) 1/P + 1/Q
A) 2 B) 4 C) 6 D) 8 E) n

16) If √(x+9) + √(x+ 29)= 10. Find x
A) 7 B) 17 C) 27 D) 37 E) 47

17) 4ˣ⁺² + 4 ²ˣ⁺¹= 1280, find x
A) 2 B) 4 C) 6 D) 8 E) n

18) The minimum value of 2x²+ bx + c is known to be 15/2 and occurs at x= -5/2. Find the value of b and c.
A) 10,20 B) 20,10 C) 1, 2 D) 2,1 E) n

19) Find the number of positive and negative roots of the equation x³ - ax + b= 0 where a> 0 and b> 0
A) 0 or 1 positive and 1 negative B) 0 or 2 positive and 1 negative C) 1 positive and 1 negative D) 1 positive 2 negative E) n

20) If -1 and 2 are two of the roots of the equation x⁴- 3x³+ 2x²+ 2x - 4= 0, then find the other two roots.
A) 1+ i B) 1 - i C) 1± i D) none

21) Find a quadratic equation whose roots are 3 and 5
A) x²+ 7x+12= 0 B) x²- 7x+12= 0 C) x²+ 7x -12= 0 D) x²- 7x-12= 0

22) Find the value of discriminate of the equation 3x²+7x +2 = 0
A) 5 B) 6.25 C) 25 D) 43 E) none

23) Find the degree of the equation (x³- 3)²- 6x⁵= 0
A) 5 B) 6 C) 7 D) 9 E) n

24) How many roots (both real and complex) does (xⁿ - a)²= 0 have?

A) 2 B) n+1 C) 2n D) n

25) Find the signs of the roots of the equation x²+ x - 420= 0

A) both are positive B) both are negative C) The roots are of opposite signs with the numerically larger root being positive. D) The roots are of opposite signs with the numerically larger root being negative.

26) Construct a quadratic equation whose roots are 2 more than the roots of the equation x²+ 9x+ 10= 0,

A) x²+ 5x - 4 = 0 B) x²+ 13x+ 32 = 0 C) x²- 5x - 4= 0 D) x² -13x+ 32 = 0

27) Construct a quadratic equation whose roots are reciprocal of the roots of the equation 2x²+ 8x+ 5= 0.

A) 5x²+ 8x+ 12= 0 B) 8x²+ 5x+ 2= 0 C) 2x²+ 5x+ 8= 0 D) 8x²+ 2x+ 5= 0

28) The square of the sum of the roots of a quadratic equation E is 8 times the product of its roots. Find the value of the square of the sum of the roots divided by the product of the roots of the equation whose roots are reciprocal of those of E.

A) 8 B) 1/8 C) 1 D) 4

29) Construct a quadratic equation whose roots are one third of the roots of x²+ 6x+ 10= 0.

A) x²+ 18x+ 90= 0 B) x²+16x+ 80= 0 C) 9x²+ 18x+ 10= 0 D) x²+ 17x+ 90= 0

30) Find the maximum value of the quadratic expression -3x²+ 4x+ 5.

A) 19/3 B) 31/12 C) 3/19 D) -19/3

31) the quadratic expression ax²+ bx+ c has its maximum/minimum value at

A) -b/2a B) b/2 C) -2b/a D) 2b/a

32) the expression (4ac - b²)/4a represents the maximum/minimum value of the quadratic expression ax²+ bx+ c which of the following is true?

A) it represents the maximum value when a> 0 B) it represents the minimum value when a< 0. C) both A and B D) neither A nor B

33) The square of the sum of the sum of the roots of a quadratic equation is equals to four times the product of its roots, the roots are

A) complex conjugates B) equal C) conjugate surds D) unequal and rational

34) A quadratic equation in x has its roots as reciprocals of each other. The coefficient of x is twice the coefficient of x². Find the sum of the squares of its roots.

A) 5 B) 4 C) 3 D) 2

35) A quadratic equation in x has the sum of its roots as 19 and the product of its roots are the 90. Find the difference of its roots.

A) 9 B) 10 C) 1 D) √7739

36) Find the common root of x²+ 10x+ 24 = 0 and x²+ 14x+ 48= 0.

A) - 6 B) 6 C) -8 D) -4

37) the sum of the roots of a quadratic equation is 33 and the product of its roots is in 90. Find the sum of the square of its roots.

A) 909 B) 8034 C) 36 D) 729

EXERCISE -2

1) solve: x⁴ - 35x²+ 196= 0

A) ±√7,±2√7 B) ±√7,±7 C) ± 2√7,±√7 D)±7,±14 E) ±7,±14

2) solve: 2{3²⁽¹⁺ˣ⁾} - 4(3²⁺ˣ)+ 10= 0

A) -1, log₃(5/3) B) -1, log₃2 C) -1, 5/3 D) -1, log₃(3/5) E) -1, log₃5

3) Find the values of k for which (k+12)x² +(k+12)x -2= 0 has equal Roots

A) -20,-12 B) -20 C) 12 D) -20, 12 E) 12, -6

4) In a school, 5/2 times the square root of the total number of children play football. 14th of the total number of children play tennis. The remaining 28 children play Basketball. Find the total number of the children in the school.

A) 36 B) 16 C) 100 D) 144 E) 64

5) If (x²+1/x²) - 4(x+1/x) +23/4= 0 what can be the value of x+ 1/x

A) 3/2 B) 2 C) 7/2 D) 9/2 E) 5/2

6) If k is a natural number and (k²- 3k +2)(k² -7k+12)= 120, find k

A) 7 B) 6 C) 5 D) 9 E) 10

7) if the product of the roots of the equation x² -(R+7)x +2(2R-2)= 0 is thrice the sum of the roots, find R

A) 20 B) 25 C) 30 D) 23 E) 32

8) Both A and B were trying to solve a quadratic equation. A copied the coefficient of x wrongly and got the roots of the equation as 12 and 6. B copied the constant term wrongly and got the roots as 1 and 26. Find the roots of the correct equation.

A) 6,16 B) -6,-16 C) 24,3 D) -3,-24 E) 6 +16

9) If √(x² - 2x -3) + √(x² +5x -24)= √(x² +7x -30), then find x.

A) 4 B) 5 C) 6 D) 8 E) 10

10) if the roots of the equation (x- m)(x -n)+1= 0, m, n are integers, then which of the following must be true ?

A) m,n are two consecutive integers. B) m- n= 2 C) m- n= 0 D) either B or C E) none

11) The area of the playground is 153m². if the length of the playground is decreased by 4m and the breath is increased by 4 m, the playground becomes square. Find the side of the square

A) 11m B) 12m C) 13m D) 14m E) n

12) if the roots of the equation 3x² +17x +6= 0 are in the ratio p:q, compute √p/√q + √q/√p.

A) 17√2/6 B) 17/6√2 C) -17√2/6 D) -17/6√2 E) 17/6

13) if x+ y, find the maximum/ minimum possible value of x²+ y²

A) minimum, 8 B) maximum, 8 C) maximum ,16 D) minimum, 16 E) maximum, 12.

14) If the roots of the equation ax² + bx +c= 0 are m, n, find the equation whose roots are m², n²

A) a²x² + (b²- 2ca)x +c² = 0. B) a²x² - (b²+ 2ca)x +c² = 0. C) a²x² - (b²- 2ca)x +c² = 0. D) a²x² + b²+ 2cax +c² = 0. E) n

15) The sides of a right angled triangle are such that the sum of the lengths of the longest and that of the shortest side is twice the length of the remaining side. Find the longest side of the triangle if the longer of sides containing the right angles is 9cm more than half the hypotenuse.

A) 30cm B) 25cm C) 20cm D)15cm E) 35cm

16) The roots of the equation ax² + bx + c= 0 are k less than those of the equation px² + qx + r = 0. Find the equation whose roots are k more than those of px² + qx + r = 0.

A) ax² + bx + c= 0 B) a(x - 2k)² + b(x - 2k) + c= 0 C) a(x+ 2k)² + b(x +2k) + c= 0 D) a(x-k)² + b(x -k) + c= 0 E) n

17) If m, n are the roots of the equation x² - 11x + 24 = 0, find the value of 1/m - 1/n given that it is positive.

A) 5/24 B) 7/24 C) 1/24 D) 1/8 E) n

18) if one root of the equation x² -10 x + 16= 0 is half of one of the roots of x² - 4Rx + 8= 0. Find R such that both the equation have integral roots.

A) 1 B) 2 C) 3 D) 4 E) n

19) How many equations of the form x² + 4x + p = 0 exist such that the equation has real roots and p is a positive integer?

A) 2 B) 3 C) 4 D) 5 E) 6

20) if the roots of 2ᵐx²+ 8x+ 64ᵐ= 0 are real and equal, find m

A) 2/3 B) 1/2 C) 3/4 D) 4/5 E) n

21) In a set of 6 consecutive integers, the product of the first and the second Integers is equal to the sum of the third and the fifth integers. Also, the product of the first and the third Integers is equal to the sum of the fifth and the sixth Integers. Find the sixth integers

A) 7 B) 8 C) 10 D) 12 E) 15

22) Two software Professional Ranjan and Raman had 108 floppies between them. They sell them at different prices, but each receives the same sum. If Raman and sold his at Ranjan's price, he would have received ₹722 and if Ranjan had sold his at Raman's price, he would have received ₹578. How many floppies did Ranjan have

A) 51 B) 57 C) 68 D) 40 E) 47

23) Find positive integral value (s) of p such that the equation 2x² + 8x + p= 0 has rational roots.

A) 8 B) 4 C) 6 D) B or C E) A or C

24) The coefficients of the equation ax² + bx + c= 0 satisfy the condition 64ac= 15b². Find the condition satisfied by the coefficients of the equation whose roots are the reciprocal of the roots of the equation, ax² + bx + c = 0

A)15ac= 64b² B) 64ac= 15b² C) 15bc= 64a² D) 15ab= 64c² E) n

25) if the sum of the roots of the quadratic equation is 8, the sum of the squares of the roots must be

A) at least 24 B) at most 24 C) at least 32 D) at most 32 E) none

26) If m, n are the roots of the equation ax² + bx + c= 0 where c³+ abc + a³= 0, which of the following is it true ?

A) mn²= 1 or m²n= 1 B) mn³= 1 or m³n= 1 C) m= n² or m²= n D) m= n³ or m³= n E) m² = n³ or m³= n²

27) Two equations have a common root which is positive. The other roots of the equations satisfy x² 9x + 18 = 0. the product of the sums of the roots of the two equations is 40. Find the common root.

A) 1 B) 2 C) 3 D) 4 E) 5

28) Both the roots of each of the equations x² + px + q= 0 and x² + qx + p= 0 are real. Which of the following is the condition or combination of conditions for the two equations to have exactly one common root ? i) 1+p +q= 0 ii) p= q iii) p≠ -1/2

A) i B) ii C) iii D) i and ii E) i and iii

29) In a quadratic equation,(whose coefficient are not necessarily real) the constant term is not 0. The cube of the sum of the squares of its roots is equals to the square of the sum of the cubes of its roots. which of the following is true ?

A) both roots are real B) neither of the roots is a real C)at least one root is non real D) at least one root is real E) exactly one root is non real

30) The roots of x³+ px² + qx + r= 0 are consecutive positive integers. Which of the following can never be the value of q ?

A) 47 B) 11 C) 107 D) 27 E) 146

31) if the equations x³- 4x² + x + 6= 0 and x³- 3x² - 4x + k= 0 have a common root, which of the following could be a value of k?

A) -6 B) -3 C) 2 D) 6 E) 12

32) The sum and the product of the roots of a quadratic equation E are a and b respectively. Find the equation whose roots are the product of first root of E and the square of the second root of E, and the product of the second root of E and the square of the first root of E

A) x² - abx + b³= 0 B) x² + abx + b³= 0 C) x² + abx - b³= 0 D) x² - abx - b³= 0 E) n

33) if a and b are positive numbers, what is the nature of the roots of the equation (a+ b)x² + 2abx + (a+ b)³/16= ?

A) real and distinct B) real and equal C) non-real and equal D) non-real and distinct E) either B or D

34) if the equation x⁵+ 15x⁴+ 85x³+ 25x² + 274x + a - 119= 0 has exactly 5 negative roots, then the value of a can be

A) 100 B) 85 C) 120 D) 90 E) 115

35) The equation acx⁷ + bx⁴+ cx³+ dx²+ e= 0 has exactly two roots and a> 0, d< 0, then how many of the following statements could be true

A) b> 0, c> 0, e> 0 B) b< 0, c> 0, e> 0 C) b< 0, c> 0, e< 0 D) b> 0, c< 0, e< 0 E) b< 0, c< 0, e< 0

a) A b) B c) C d) D e) E

EXERCISE - 5

1) solve for K is a positive integer. (K+1)(k+2)(k+4)= 360

A) 2 B) 3 C) 4 D) 5 E) n

2) Find the equation whose roots are thrice the roots of the equation 2x²- 15x +18= 0

A) x²+ 45x +324= 0 B) 2x²- 45x +81= 0 C) x²+ 45x - 324= 0 D) 2x²- 45x +162 = 0 E) n

3) which of the following opinions represent/s a condition for the equation x²+ ax + b= 0 and x²+ bx +a = 0 to have an exactly 1 common root, given that the roots of the both the equations are real ?

A) a- b= 1 B) b- a= 1 C) 1+ a+b= 0 D) either A or B E) n

4) A person bought a certain number of oranges for ₹70, if the price of each orange was ₹2 less, he would have but 4 more oranges for the same amount. Find the number of oranges he bought originally.

A) 12 B) 10 C) 18 D) 15 E) n

5) the numerical value of the sum of the squares of the ages (in years) of Akash and Bhuvan is 1013. If Akash is one year younger to Bhuvan, find Bhuvan's age.

A) 21yrs B) 22yrs C) 23yrs D) 24yrs E) 25yrs

6) the equation x²- 2x - 8= 0 will have.

A) the numerically larger root as positive B) the numerically larger root as negative C) both roots are negative D) both the roots as positive E) none

7) Find the value of p in the equation x² + qx +p = 0, where one of the roots of the equation (2+ √3) and q and p are Integers.

A) 3 B) 2 C) -1 D) -2 E) 1

8) if a positive number is increased by three and then squared, the result is 23 more than the original number. Find the original number.

A) 1 B) 2 C) 3 D) 4 E) n

9) If k is a perfect square, the roots of the equation 4kx²+ 4 √k x - k = 0 are

A) always rational B) rational for only some of the values of K. C) always irrational D) always Complex E) none

10) Find the value of R, so that one of the roots of x²+ 6Rx +64 = 0 is the square of the other root.

A) 10/3 B) 8/3 C) 5/3 D) 7/3 E) n

11) Find the values of K for which the roots of x²+ x(14- k)- 14k +1 = 0 are equal Integers.

A) -11,-23 B) -12,-16 C) -13,-25 D) -11,-12 E) n

12) if the roots of 2x²+ (4m +1)x +2(2m -1) = 0 are reciprocals of each other, find m.

A) -1 B) 0 C) 1 D) 3/4 E) n

13) if the value of p in the equation x²+ 2(p+1)x + 2p = 0, is real roots of the equation are

A) rational and unequal B) irrational and unequal C) real and unequal D) real and equal E) n

14) the value of the equation 6x⁴- 6x³ -24x²- 6x + 6 = 0 are

A) 1 and (6±√10)/2 B) ±2 and (3±√5)/2 C) ±3 and (-6±√10)/2 D) ±4 and (6-√10)/2 E) -1 and (3±√5)/2

15) if the roots of the equation ax²+ bx + c = 0 are m and n,find the value of m/n + n/m - 2(1/m + 1/n) + 2mn

A) (b²+ 2ac)/ac B) (b²- 2ac)/ac C)(b²+ 4ac)/ac D) (b²- 4ac)/ac E) n

16) If the price of a book goes down by ₹20 per dozen, a person can purchase 50 dozen books more for ₹30000. Find the original price( in ₹) of each book.

A) 10 B) 12 C) 25/3 D) 53/6 E) 26/3

17) if 31 is split up into two parts such that the sum of the squares of the two parts is 481, find the difference between the two parts.

A) 7 B) 5 C) 3 D) 1 E) 9

18) Solve for x: x⁴ - 42x² + 216 = 0

A) ±√6,±6 B) ±2√6, ±6 C) ±3√6, ±6 D) ±4√6, ±6 E) ±6√6, ±6

19) 16(3²ˣ⁺¹) - 32(3ˣ)+ 4= 0

A) - log₃2, - log₃6 B) - log₃4, - log₃6 C) -1, - log₃12 D) - og₃6, - log₃8 E) -1, log₃6

A) in a class, eight students play Basketball. the remaining students, who represent 7 times the square root of the strength of the class, play football. Find the strength of the class.

A) 36 B) 16 C) 64 D) 100 E) 81

20) x²+ 1/x² - 2(x - 1/x) - 5/4 = 0, which of the following can be the value of x - 1/x ?

A) 7/2 B) 1/2 C) -1/2 D) -3/2 E) 5/2

21) If the roots of the equation 6x² - 7x + 2 = 0 differ by y, find y.

A) 1/3 B) 1/9 C) 1/6 D) 1/12 E) 1/2

22) solve for x: √(2x +3) + √(4x + 13) = 8

A) 2 B) 3 C) -3 D) -179 E) 179

23) find the minimum value of the expression 3x²- x - 6

A) -65/12 B) -73/12 C) -79/6 D) -85/12 E) -73/6

24) find the equation whose roots are twice the roots equation 3x²- 7x + 4 = 0.

A) 3x²- 14x + 8 = 0 B) 3x²- 14x + 16 = 0 C) 3x² +14x -16 = 0 D) 3x²+ 14x + 16 = 0 E) 3x²+ 14x - 8 = 0

25) the area we playground is 247 if its length is decreased by 2 and brief is increased by 4 at the becomes square find the side of the square the length of a triangle is 1 cm more than its diagonal is 29 what is the measure of the right angle of the side containing right angle triangle is 8 cm longer than the smaller of the sides the sum of the lens of the sides containing the right angle is 15 CM more than the length of the other side find the length in centimetre of the smallest side 18 12 24 16 20 roots of the equation 414 A and B find the roots of the equation are which of the following holds true P and Q we are trying to solve a quadratic equation of X wrongly and obtained 12 and 9 otherwood the constant term wrongly in the obtained 2016 the rules find the roots of the equation 180 1806 1806 1806 128 is any single digit Prime natural number how many question of the form and both real roots 1518 21 24 22 Sab Dekho the roots of the equation are less than those are the equation the roots of the equation are more than those are the equation which of the following the expression is a positive at and a negative at which of the following can be concluded the equation has all positive rules and the value of a could be 611 500 if 10th degree equation and it has three negative then which of the following can be number of science changes in 125 634