Magnus Academy (Maths Specialist): November 2021

Thursday, 25 November 2021

STRAIGHT LINE IN SPACE

* ON FINDING THE VECTOR EQUATION OF A LINE SATISFYING THE GIVEN CONDITION AND REDUCING IT CARTESIAN FORM:

=> Formula to be used:

1) r= a + ¥b

2) r= a+ ¥(b - a)

3) The Cartesian form of a straight line passing through a fixed point (x₁, y₁,z₁) and having direction Ratios proportional a, b, c is :(x- x₁)/a = (y- y₁/b= (z- z₁)/c

4) The Cartesian equation of a line passing through two given points (x₁, y₁,z₁) and (x₂, y₂, z₂) is: (x- x₁)/(x₂ - x₁)= (y- y₁)/(y₂ - y₁)= (z- z₁)/((z₂ - z₁).

EXERCISE- 1

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1) Find the vector equation of a line which passes through the point with position vector 2i- j+ 4k and is in the direction of i+ j- 2k. Also, reduce it to Cartesian form. r=(2i-j+4k)+ ¥(i+j- 2k). (x-2)/1= (y+1)/1= (z -4)/-2

2) Find the vector equation of the line A(3,4,7) and B(1,-1,6). Find also, its cartesian equations. r=(3i +4j -7k)+ ¥(-2i -5j +13k). (x-3)/-2= (y-4)/-5= (z +7)/13

3) The point A(4,5,10), B(2,3,4) and C(1,2,-1) are three vertices of a parallelogram ABCD. Find vector and cartesian equations for the side AB and BC and find the coordinates of D. r=(4i + 5j+10k)+ ¥(I+j +3k). (x-4)/1= (y-5)/1= (z -10)/3. And r=(2i +3j+4k)+ €(I+j+5k). (x-2)/1= (y-3)/1= (z -4)/5. And (3,4,5)

4) Find the vector equation of a line passing through a point with position vector 2i- j+ k, and parallel to the line joining the point - i + 4j+ k and i+ 2j + 2k. Also, find the cartesian equivalent of this equation. r=(2i-j+k)+ ¥(2i - 2j+k). (x-2)/2= (y+1)/-2= (z - 1)/1.

5) Find the cartesian equation of a line passing through the point A(2,-1,3) and B(4,2,1). Also, reduce it to vector form. r=(2i-j+3k)+ ¥(2i+ 3j- 2k). (x-2)/2= (y+1)/3= (z - 3)/-2

6) The cartesian equations of a lines are 6x - 2= 3y +1= 2z -2. Find its direction Ratios and also find vector equation of the line. r=(i/3 -j/3 +k)+ ¥(i+ 2j +3k). (x-1/3)/1= (y+1/3)/2= (z -4)/3.

7) Find the direction cosines of the line (x-2)/2= (2y -5)/-3, z= -1 Also, find the vector equation of the line. 4/5, -3/5, 0, r=(2i +5j/2 - k)+ ¥(2i- 3j/2 +0k).

8) show that the points whose position vectors 5i +5k, 2i+j +3k and -4i + 3j - k are collinear.

9) If the points A(-1,3,2), B(-4,2,-2) and C(5,5, m) are collinear, find the value of m. 10

10) Find the point on the line (x+2)/3= (y+1)/3= (z -3)/2 at a distance of 3√2 from the point (1, 2, 3). (-2,-1,3) and (56/17, 43/17, 111/17)

11) find the vector and cartesian equations of the line through the point (5,2,-4) and which is parallel to the vector 3i+2j- 8k. r=(5i +2j-4k)+ ¥(3i+ 2j- 8k). (x-5)/3= (y-2)/2= (z +4)/-8

12) Find the vector equation of the line passing through the points (-1,0,2) and (3,4,6). r=(-i + 2k)+ ¥(4i+ 4j + 4k).

13) find the vector equation of a line which is parallel to the vector 2i - j + 3k and which passes through the point (5, -2,4). Also, reduce it to cartesian form. r=(5i- 2j- 4k)+ ¥(2i -j +3k). (x-5)/2= (y+2)/-1= (z -4)/3.

14) A line passes through the point with position vector 2i - 3j + 4k and is in the direction of 3i+ 4j - 5k. find the equation of the line in vector and cartesian form. r=(2i- 3j+4k)+ ¥(3i+ 4j- 5k). (x-2)/3= (y+3)/4= (z -4)/-5

15) ABCD is a parallelogram. The position vectors of the point A, B and C are respectively, r= 4i + 5j - 10k, 2i - 3j + 4k, and - i+2j+ k. Find the vector equation of the line BD. Also, reduced it to Cartesian form.

r=(2i- 3j+4k)+ ¥(i - 13j + 17k). (x-2)/1= (y+3)/-13= (z -4)/17

16) find in vector form as well as in cartesian form, the equation of the line passing through the point A(1,2,-1) and B(2,-1,1). r=(i + 2j -k)+ ¥(i - j + 2k). (x-2)/1= (y-2)/-1= (z +1)/2

17) Find the vector equation for the line which passes through the point (1,2,3) and parallel to the vector i- 2j + 3k. Reduce the corresponding equation in cartesian form. r=(i +2 j+3k)+ ¥(i - 2j + 3k). (x-1)/1= (y -2)/-2= (z - 3)/3

18) Find the vector equation of a line passing through (2,-1,1) and parallel to the line whose equations are (x-3)/2= (y+1)/7= (z -2)/-3. r=(2i-j+k)+ ¥(2i+ 7j- 3k).

19) The Cartesian equation of a lines are (x-5)/3= (y+4)/7= (z - 6)/2. Find a vector equation for the line. r=(5i-4j+6k)+ ¥(3i+7j + 2k).

20) find the Cartesian equation of a line passing through (1,-1,2) and parallel to the line whose equation are (x-4)/1= (y-1)/3= (z + 1)/-2. Also, reduce the equation obtained in a vector form. r=(i-j+2k)+ ¥(i+2j- 2k). (x-2)/1= (y+1)/2= (z -2)/-2

21) Find the direction cosines of the line (4-x)/2= y/6= (1- z)/3. Also, reduce it to vector form. 2/7, 6/7, -3/7; r=(4i- 0j+k)+ ¥(2i+ 6j- 3k)

22) The Cartesian equation of a line are x= ay +b, z= cy + d. Find its direction Ratios and reduce it to vector form. a,1,c; r=(bi 0j+dk)+ ¥(ai + j + ck)

23) Find the vector equation of a line passing through the point with position vector i - 2j - 3k and parallel to the line joining the points with position vector i- j+4k) and 2i+j + 2k). and Also, find the Cartesian equivalent of this equation. r=(i- 2j -3k)+ ¥(i+2j- 2k). (x-1)/1= (y+2)/3= (z +3)/-2

24) Find the point on the line (x+2)/3= (y+1)/2= (z -3)/2 at a distance of 5 units from the point P(1,3,3). (4,3,7),(-2,-1,3)

25) Show that the points whose position vectors are 2i +3j+4k, i+ 2j + 3k) and 7i+ 9k are collinear.

26) find the cartesian and vector equations of a line which passes through the point (1,2,3) and is parallel to the line (-x-2)/1= (y+3)/7= (2z -6)/3. r=(i +2j+3k)+ ¥(-2i+ 14j +3k). (x-1)/-2= (y-2)/14= (z -3)/3

27) The Cartesian equations of a line are 3x+1= 6y -2= 1 - z. Find the fixed point through which it passes, its direction Ratios and Also, its vector equation. (-1/3, 1/3, 1); 2, 1, -6; r= -i/3 + j/3 +k)+ ¥(2i +j- 6k)

++++++++++()+++(++(((+(+(((++++

ANGLE BETWEEN TWO LINES

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₁₁₂₂

VECTOR FORM:

let the vector equations of the lines be r= a₁ + ¥b₁ and r= a₂ + ¥ b₂. These two lines are parallel to the vector b₁ and b₂ respectively. Therefore, angle between these two lines is equal to the angle between b₁ and b₂. Thus, if ¢ is the angle between the given lines, then

Cos ¢= b₁. b₂/(|b₁|.| b₂|)

* CONDITION of perpendicularity:

If the lines b₁ and b₂ are perpendicular. Than, b₁. b₂= 0

* CONDITION of parallelism:

If the lines are parallel, then b₁ and b₂ are parallel.

b₁ = K b₂ for some scalar K.

CARTESIAN FORM:

(x - x₁)/a₁ = (y- y₁)/b₁=(z- z₁)/c₁.. (1)

And

(x- x₂)/a₂= (y- y₂)/b₂ =(z - z₂)/c₂..(2)

*Direction Ratios of line (1) are proportional to a₁, b₁, c₁.

So m₁= Vector parallel to line (1)= a₁i + b₁j + c₁k

* Direction Ratios of line (2) are proportional to a₂, b₂, c₂.

So m₂= Vector parallel to line (2)= a₂i+ b₂j + c₂k

Let ¢ be the angle between (1) and (2). Then, ¢ is also the angle between m₁, m₂.

Cos ¢= m₁. m₂/(|m₁|. |m₂|) OR

cos ¢= (a₁a₂+ b₁b₂+ c₁c₂)/{√(a₁²b₁²+c₁²) √(a₂²+b₂²+ c₂²)}

* CONDITION of perpendicularity:

If the lines are perpendicular, then

m₁. m₂ = 0 => a₁a₂+ b₁b+ c₁c₂ = 0

* CONDITION of parallelism:

If the lines are parallel, then m₁ and m₂ are parallel.

Then m₁ = K m₂ for some scalar K.

=> a₁/a₂ = b₁/b₂= c₁/c₂.

ON FINDING THE ANGLE BETWEEN TWO LINES:

FORMULA TO BE USED:

cos ¢= b₁. b₂/(|b₁|.| b₂|) OR

cos ¢= (a₁a₂+ b₁b₂+ c₁c₂)/{√(a₁²b₁²+c₁²) √(a₂²+b₂²+ c₂²)}

ON FINDING THE EQUATION OF A LINE PARALLEL TO A GIVEN LINE AND PASSING THROUGH A GIVEN POINT:

Formula to be used:

1) r = a + Kb

2) (x - x₁)/a = (y - y₁)/b = (z - z₁)/c

ON FINDING THE EQUATION OF A LINE PASSING THROUGH A GIVEN POINT AND PERPENDICULAR TO TWO GIVEN LINES:

Result to be used:

A line passing through a point having position vector $ and perpendicular to the lins r= a₁+ Kb₁ and r= a₂+ M b₂ is parallel to the vector b₁ x b₂. So, its vector equation is r= $ + K(b₁ x b₂)

STEP- 1 Obtain the point through which the line passes. Let its position vector be $.

STEP-2 Obtain the vectors parallel to the two given lines. Let the vectors be b₁ and b₂

STEP- 3 Obtain b₁ x b₂

STEP-4 The vector equation of the required line is r = $ + K(b₁ x b₂).

EXERCISE-2

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1) Find the angle between lines r= 3i+ 2j- 4k + K(i+ 2j + 2k) and r= 5j- 2k + M(3i+ 2j + 6k). cos ¢= (19/21)

2) Find the angle between the lines (x-2)/3 = (y+1)/-2, z= 2 and (x-1)/1 = (2y+3)/3, (z+ 5)/2. π/2

3) Prove that the line x= ay + b, z= cy + d and x= a'y + b' , z= c' y + d' are perpendicular if aa' + cc' +1= 0.

4) Find the angle between two lines whose direction Ratios proportional to 1, 1, 2 and (√3, -1),((- √3 -1), 4) π/3

5) Find the equation of a line passing through a point (2, -1, 3) and parallel to the line r= (i +j) + M(2i+ j - 2k). r= (2i - j +3k) + M(2i+ j -2k).

6) Find the equation of a line passing through (1, -1,0) and parallel to the line (x-2)/3 = (2y+1)/2 = (5-z)/1. (x-1)/3 = (y+1)/1, (z -0)/-1.

7) Find the cartesian equation of the line passing through the point (-1,3,-2) and perpendicular to the lines x/1 =y/2= z/3 and (x+2)/-3= (y -1)/2 = (z-1)/5. (x+1)/2= (y -3)/-7 = (z+2)/4.

8) A line passes through (2,-1,3) and is perpendicular to the line r= (i+ j - k)+ K(2i - 2j + k) and r= (2i- j - 3k)+ M(i + 2j + 2k). Obtain its equation. r= (2i- j +3k)+ M(2i +j - 2k), Where M= - 3K.

9) Find the value of P so that the lines. L ₁: (1-x)/3 =(7y- 14)/2P =(z-3)/2 and L₂: (7- 7x)/3P = (y -5)/1 = (6 - z)/5 are at right angle.

Also, find the equation of a line passing through the point (3,2, -4) and parallel to line L₁. 70/11, (x -3)/-3 = (y-2)/20/11 = (z+4)/2

10) Show that the three lines with direction cosines (12/13, -3/13, -4/13),(4/13, 12/13, 3/13),( 3/13, -4/13, 12)13) are mutually perpendicular.

11) Show that the line through the points (1,-1,2) and (3,4,-2) is perpendicular to the through the points (0,3,2) and (3,5,6).

12) Show that the line through the points (4,7,8) and (2,3,4) is parallel to the line through the points (-1,-2,1) and (1,2,5).

13) Find the cartesian equation of the line which passes through the point (-2,4,-5) and parallel to the line given by (x+3)/3 = (y-4)/5 = (z+8)/6. (x+2)/3 = (y-4)/5 = (z+5)/6.

14) Show that the lines are (x -5)/7 = (y +2)/-5 = z/1 and x/1 = y/2 = z/3 are perpendicular to each other.

15) Show that the line joining the origin to the point (2,1,1) is perpendicular to the line determined by the point (3,5,-1) and (4,3,-1).

16) Find the equation of a line parallel to x-axis and passing through the origin. x/1 = y/0 = z/0

17) Find the angle between the following pairs of lines:

A) r= (4i- j)+ K(i + 2j - 2k) and r= (i- j +2k)+ M(2i + 4j - 4k). 0

B) r= (3i + 2j - 4k)+ K(i + 2j + 2k) and r= (5j - 2k)+ M(3i + 2j +6k). Cos ¢=(19/21)

C) r= K(i + j +2k) and r= 2j + M{(√3 -1)i - (√3+1)j + 4k). π/3

18) Find the angle between the following pairs of lines:

A) (x +4)/3 = (y -1)/5 = (z-3)/4. And (x +1)/1 = (y -4)/1 = (z -5)/2. Cos ¢= (8/5 √3)

B) (x -1)/2 = (y -2)/3 = (z-3)/-3. And (x -3)/-1 = (y -5)/8 = (z -1)/4. Cos ¢= (10/9 √22)

C) (5 -x)/-2 = (y +3)/1 = (1-z)/3 and x/3 = (1- y)/-2 = (z +5)/-1. Cos ¢= (11/14)

D) (x -2)/3 = (y +3)/-2 , z = 5. And (x +1)/1 = (2y -3)/3 = (z -5)/2. π/2

E) (x -5)/1 = (2y +6)/-2 = (z-3)/1. And (x -2)/3 = (y +1)/4 = (z - 6)/5. π/2

F) (-x +2)/-2 = (y -1)/7 = (z+3)/-3. And (x +2)/-1 = (2y -8)/4 = (z -5)/4. π/2

19) Find the angle between the pairs of the lines with direction Ratios proportional to

A) 5, -12,13 & -3,4,5. Cos ¢= (1/65)

B) 2, 2,1 and 4, 1, 8. Cos ¢= (2/3)

C) 1,2,-2 and -2,2,1. π/2

D) a, b, c and b-c, c-a, a-b. π/2

20) Find the angle between two lines, one of which has direction 2, 2,1 while the other one is obtained by joining the points (3,1,4) and (7,2,12). Cos ¢= (2/3)

21) Find the equation of the line passing through the point (1,2,-4) and parallel to the line (x-3)/4 = (y-5)/2 = (z+1)/3. (x-)/4 = (y-2)/2 = (z+4)/3.

22) find the equation of the line passing through the point (-1,2,1) and parallel to the line (2x-1)/4 = (3y+5)/2 = (2-z)/3. (x+1)/2 = (y-2)/2/3 = (z-1)/-3.

23) Find the equation of the line passing through the point (2,-1,3) and parallel to the line r=(i - 2j + k) + K(2i + 3j - 5k). r=(2i - j + 3k) + K(2i + 3j - 5k).

24) Find the equation of the line passing through the point (2,1,3) and perpendicular to the lines (x-1)/1 = (y-2)/2= (z-3)/3 and x/-3 = y/2 = z/3. (x-2)/2 = (y-2)/-7 = (z-3)/4.

25) Find the equation of the line passing through the point i + j - 3k and perpendicular to the lines r=i + ¥(2i + j - 3k) and r=(2i + j -k) + M(i + j + k). r=(i +j +3 k) + ¥(4i - 5j +k)

26) find the equation of the line passing through the point (1,-1,1) and perpendicular to the lines joining the points (4,3,2), (1,-1,0) and (1,2,-1), (2,1,1). (x-1)/10 = (y+1)/-4 = (z- 1)/-7

27) Determine the equations of the line passing through the point (1,2,-4) and perpendicular to the lines (x-8)/8 = (y +9)/-16 = (z- 10)/7 and (x- 15)/3 = (y-29)/8 = (z-5)/-5.

28) Show that the lines (x-5)/7 = (y+2)/-5 = z/1 and x/1 = y/2 =z/3 are perpendicular to each other.

29) Find the vector equation of the line passing through the point (2,-1,-1) which is parallel to the line 6x -2= 3y+ 1= 2z -2. r=(2i -j - k) + ¥(i + 2j + 3k).

30) If the lines (x-1)/-3 = (y-2)/2P = (z -3)/2 and (x-1)/3P = (y-1)/1 = (z - 6)/-5 are perpendicular, find the value of P. -10/7

31) If the coordinates of the points A, B, C, D be (1,2,3),(4,5,7),(-4,3,-6) and (2,9,2) respectively, then find the angle between the lines AB and CD. 0

32) Find the value of P so that the following lines are perpendicular to each other. (x-5)/(5P+2) = (2-y)/5 = (1-z)/-1, x/1 = (2y +1)/4P = (1-z)/-3. 1

33) Find the direction cosines of the line (x +2)/2= (2y-7)/6 = (5-z)/6. also, find the vector equation of the line through the point (-1,2,3) and parallel to the given line. 2/7, 3/7,-6/7; (x+1)/2 = (y-2)/3 = (z -3)/-6

Saturday, 20 November 2021

3-D. Direction Cosines and Direction Ratios

EXERCISE-1

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1) Find the distance between the points P(-2,4,1) and (1,2,-5). 7

2) Prove by using distance formula that the points P(1,2,3), Q(-1,-1,-1) and R(3,5,7) are collinear.

3) Determine the point on XY- plane which is equidistant from three points A(2,0,3), B(0,3,2) and (0,0,1). (3,2,0)

4) Show that the point A(0,1,2), B(2,-1,3) and C(1,-3,1) are vertices of an isosceles right angle triangle.

5) Find the locus of the point of point which is equidistant from the points A(0,2,3) and B(2,-2,1). x - 2y - z+ 1= 0

6) Find the coordinates of a point equidistant from the four points O(0, 0,0), A(a,0,0), B(0,b,0) and C(0, 0, c). (a/2,b/2,c/2)

7) find the coordinates of the point which divides the join of P(2,1,4) and Q(4,3,2) in the ratio 2:3

A) internally. (14/5,3/5,16/5)

B) externally. (-2,-9,8)

8) Find the ratio in which the line joining the point (1,2,3) and (-3,4,-5) is divided by the xy plane. also, find the coordinates of the point of division. 3:5, (-1/2,11/4,0)

9) find the ratio in which join the A(2,1,5) and B(3,4,3) is divided by the plane 2x+ 2y- 2z= 1. Also, find the coordinates of the point of division. 5:7, (29)12,9/4,25/6)

10) Using Section formula, prove that three points A(-2,3,5), B(1,2,3) and C(7,0, -1) are collinear.

11) The midpoints of the sides of a triangle are (1,5,-1), (0,4,-2) and (2,3,4). Find its vertices. (1,2,3),(3,4,5),(-1,6,-7)

12) Given that P(3,2,-4), Q(5,4,-6) and R(9,8,-10) are collinear. Find the ratio in which Q divides PR. 1:2

13) Find the coordinates of the points which trisect the line segment AB, given that A(2,1,-3) and B(5,-8,3). (4,-5,1)

14) Find the coordinates of the foot of the perpendicular drawn from the point A(1,2,1) to the line joining B(1,4,6) and (5,4,4). (3,4,5)

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** NOTE:

1) DIRECTION COSINES:

The direction Cosines of a line are defined as the direction Cosines of any vector whose support is the given line.

Direction Cosines are either cos a, cos b, cos c OR -cos a, -cos b, -cos c.

Therefore, if l, m, n are Direction Cosines of a line, then - l, - m, - n are also its Direction Cosines and always have l²+ m² + n²= 1.

If A(x₁, y₁, z₁) and B (x₂, y₂, z₂) are two points on a line L, then its Direction Cosines are:

(x₂ - x₁)/AB , (y₂ - y₁)/AB, (z₂ - z₁)/AB OR

(x₁- x₂)/AB, (y₁-y₂)/AB, (z₁ - z₂)/AB

2) DIRECTION RATIOS:

The direction Ratios of a line are proportional to the direction Ratios of any vector whose support is the given line.

If A(x₁, y₁, z₁) and B (x₂, y₂, z₂) are two points on a line, then its Direction Ratios are proportional to:

x₂ - x₁, y₂ - y₁ , z₂ - z₁.

3) ANGLE BETWEEN TWO VECTORS:

It is defined as the angle between two vectors parallel to them. So, the results derived for vectors will also be applicable to lines. ₁₂ ₁ₓ ₁₂

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EXERCISE-2

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1) Using vector method: prove that the points A(3,-2,4), B(1,1,1) and C(-1,4,2) are collinear.

2) Find the distance between the points A and B with position vectors i - j and 2i+ j + 2k. 3

3) Find the angle between the vectors with direction ratios proportional to 4, -3, 5 & 3,4,5. π/3

4) find the angle between the lines whose direction ratios are proportional to 4, -3, 5 & 3, 4, 5. π/3

5) P(6,3,2), Q(5, 1,4) and R(3, 3,5) are the vertices of a triangle PQR. Find ang.PQR. π/2

6) Find the coordinates of the foot of the perpendicular drawn from the point A(1,2,1) to the line joining B(1,4,6) and C(5,4,4). (3,4,5)

7) find the direction cosines of the line which is perpendicular to the lines with direction cosines proportional to 1, -2 2 and 0,2,1. 2/3,-1/3,2/3

8) find the direction cosines of the sides of the triangle whose vertices are (3,5,-4), (-1,1,2) and (-5, -5,-2) and also find the angles of the triangle, what types of triangle it is ? Isosceles obtuse angled triangle

9) find the angle between the lines whose direction cosines are given by the equations 3l+ m + 5n= 0, 6mn- 2nl+ 5lm = 0. Cos⁻¹ (-1/6)

10) Find the direction cosines of the two lines which are connected by the relations. l -5m + 3n= 0 and 7l²- 3n²= 0. ±-1/6, ±1/6, ± 2/√6

EXERCISE -3

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1) If a line makes angles of 90°, 60°and 30°with the positive direction of x, y and z-Axis. respectively, find the direction cosines. 0, 1/2, √3/2

2) If a line and direction ratios 2, -1, 2, determine its direction cosines. 2/3,-1/3,-2/3

3) find the direction cosines of the line passing through two points (-2, 4, -5) and (1,2, 3). 3/√77, -2/√77, 8√77

4) Using direction ratios shows that the points A(2, 3, -4) and B(1, -2, 3) and C(3,8, -11) are collinear.

5) Find the direction cosines of the sides of the triangle whose vertices are (3,5,-4), (-1,1,2) and (-5,-5,-2). 2/√17, 2√17, -3/√17, ; 2/√17, 3/√17, 2/√17; 4/√42, 5/√42, -1/√42

6) Find the angle between the vectors with direction ratios proportional to 1,-2,1 and 4,3,2. π/2

7) find the angle between the vectors whose direction cosines are proportional to 2, 3, -6 and 3,-4,5. Cos⁻¹{-(18√2)/35}

8) Find the acute angle between the lines whose direction ratios are proportional to 2:3:6 and 1:2:2. Cos⁻¹(20/21)

9) Show that the point (2,3,4), (-1,-2,1), (5,8,7) are collinear.

10) Show that the line through the points (4,7,8) and (2,3,4) is parallel to the line through the points (-1,-2,1) and (1,2,5).

11) show that the line through the points (1,-1,2) and (3,4,-2) is perpendicular to the line through the points (0,3,2) and (3,5,6).

12) show that the line joining the origin to the point (2,1,1) is perpendicular to the line determined by the points (3,5,-1) and (4,3,-1).

13) Find the angle between the lines whose direction ratios are proportional to a, b, c and b- c, c - a, a - b. π/2

14) If the coordinates of the points A, B, C , D are (1,2,3), (4,5,7), (-4,3,-6) and (2,9,2), then find the angle between AB and CD. 0

15) Find the direction cosines of the lines, connected by the relations; l+ m+ n= 0 and 2lm + 2ln - mn= 0. ±1/√6, ±1/√6, ± -2/56; ±-1/√6, ±2/√6, ±1/√6.

16) find the angle between the lines whose direction cosines are given by equations:

A) l+ m+ n= 0 and l² + m² - n²= 0

B) 2l- m+ 2n= 0 and mn + ln + ml= 0.

C) l+ 2m+ 3n= 0 and 3lm - 4ln + mn= 0.

D) 2l+ 2m- n= 0 and lm +ln + mn= 0. (π/3, π/2, π/2, π/2)

Tuesday, 9 November 2021

Basic Probability (2)

TYPE -1

Exercise -1

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

1) A coin is tossed once. Find the probability of getting a head. 1/2

2) Two coins are tossed once. find the probability of:

A) getting two heads. 1/4

B) getting at least one head. 3/4

C) getting no head. 1/4

D) getting 1 head and 1 tail. 1/2

3) Three unbiased coins are tossed once. What is the probability of getting

A) all heads. 1/8

B) Two heads. 3/8

C) one head. 3/8

D) at least one head. 7/8

E) at least two heads. 1/2

4) A dice is tossed once. What is the probability of getting:

A) the number 4 ? 1/6

B) an even number. 1/2

C) a number less than 5. 2/3

D) a number greater than 4. 1/3

E) the number 8. 0

F) a number less than 8. 1

5) In a single throw of two dice, find the probability of obtaining a total of eight. 5/36

6) two dice are thrown simultaneously. find the probability of getting:

A) a doublet. 1/6

B) an even number as the sum. 1/2

C) A prime number as sum. 5/12

D) a multiple of 3 as the sum. 1/3

E) a total of at least 10. 1/6

F) a doublet of even numbers. 1/12

G) a multiple of 2 on one die and a multiple of 3 on other die. 11/36

7) 20 cards are numbered from 1 to 20. One card is then drawn at random what is the probability that the number of the card drawn is:

A) a prime number. 2/5

B) an odd number. 1/2

C) a multiple of 5 ? 1/5

D) not divisible by 3 ? 7/10

8) From a well shuffled deck of 52 cards, a card is drawn at random. find the probability of getting:

A) an Ace. 1/4

B) a heart. 1/4

C) an eight of Hearts. 1/52

D) a club. 1/4

E) a red card. 1/2

F) a face card. 3/13

G) a diamond. 1/4

H) a Jack. 1/13

I) a black card. 1/2

9) From a well shuffled deck of 52 cards, a card is drawn at random. Find the probability that the card is drawn is:

A) red and a king. 1/26

B) either red or a king. 7/13

10) a bag contains 9 red and 12 white balls. One ball is drawn at random, find the probability that the ball drawn is red. 3/7

11) If the probability of the occurrence of a certain event M is 3/11, find

A) the odds in favour of its occurrence. 3:8

B) the odds against its occurrence. 8:3

12) The odds in favour of occurrence of an event is 5:12. Find the probability of the occurrence of this event. 5/17

13) If 3/10 is the probability of that an event will happen, what is the probability that will not happen? 7/10

14) three dice are thrown together. find the probability of getting a total of at least 6. 103/108

15) if the odds in favour of an event be 3/5, find the probability of the occurrence of the event. 3/8

16) Two dice are thrown. find

A) the odd in favour of getting the sum 5,. 1:8

B) the odds against getting the sum 6. 31:5

17) a card is drawn from a well shuffled Deck of 52 cards. find

A) the odds in favour of getting a face card,. 3:10

B) the odds against the getting a spade. 3:1

18) Two cards are drawn at random from a pack of 52 cards. What is the probability that both the drawn cards are Aces? 1/221

19) a coin is tossed once. find the probability of getting a tail. 1/2

20) a dice is thrown. find the probability of:

A) getting a 5. 1/6

B) getting a 2 or a 3. 1/3

C) getting an odd number. 1/2

D) getting a prime number. 1/2

E) getting a multiple of 3. 1/3

F) getting a number between 3 and 6. 1/3

21) In a single throw of two dice, find the probability of

A) getting a sum less than 6. 5/18

B) getting a doublet of odd numbers. 1/12

C) getting the sum as prime number. 5/12

22) in a single throw of two dice, find

A) P(an odd number of the first die and a 6 on the second). 1/12

B) P(a number greater than 3 on each die). 1/4

C) P(a total of 10). 1/12

D) P(a total greater than 8). 5/18

E) P(a total of 9 or 11). 1/6

23) a bag contains 4 white and 5 black balls. A ball is drawn at random from the bag. find the probability that the ball drawn is white. 4/9

24) An urn contains 9 red, 7 white and 4 Black balls. A ball is drawn at random. find the probability that the ball drawn is :

A) red. 9/20

B) white. 7/20

C) red or white. 4/5

D) white or black. 11/20

E) not white. 13/20

25) in a lottery, there are 10 prizes and 25 blank. find the probability of getting a prize. 2/7

26) if there are two children in a family, find the probability that there is at least 1 boy in the family. 3/4

27) three unbiased coins are tossed once. find the probability of getting

A) exactly 2 tails. 3/8

B) exactly one tail. 3/8

C) at most two tails. 7/8

D) at least two tails. 1/2

E) atmost 2 tails or atleast 2 heads. 7/8

28) In a single throw of two dice, determine the probability of not getting the same number on the tow dice. 5/6

29) If a letter is chosen at random from the English alphabet, find the probability that the letter chosen is

A) a vowel. 5/26

B) a constant. 21/26

30) a card is drawn at random from a well shuffled pack of 52 cards. What is the probability that the card a number greater than 3 and less then 10 ? 6/13

31) tickets numbered from 1 to 12 are mixed up together and then a ticket is withdrawn at random. find the probability that the ticket has a number which is a multiple of 2 or 3. 2/3

32) What is the probability that an ordinary year has 53 Tuesday? 1/7

33) what is the probability that a leap year has 53 Sundays? 2/7

33) what is the probability that in a group of two people, both will have the same birthday, assuming that there are 365 days in a year and no one has/her birthday on 29 th February? 1/365

34) which of the following cannot be probability of occurrence of an event ?

A) 0 B) -3/4 C) 3/4 D) 4/3

35) If 7/10 is the probability of occurrence of an event, what is the probability that it does not occur ? 3/10

36) The odds in favour of the occurrence of an event are 8:13. find the probability that the event will occur. 8/21

37) The odds in favour of the occurrence of an event are 8:13. find the probability that will occur. 7/11

38) If 5/14 is the probability of occurrence of an event, find

A) the odds in favour of its occurrence. 5:9

B) the odds against its occurrence. 9:5

39) Two dice are thrown, find

A) the odds in favour of getting the sum 6. 5:31

B) the odds against getting the sum 7. 5:1

40) A combination lock on a suitcase has 3 wheels, each labelled with 9 digits from 1 to 9. If an opening combination is a particular sequence of three digits with no repeats, what is the probability of a person guessing the right combination ? 1/504

41) In a lottery, a person chooses six different numbers of random from 1 to 20. If these six numbers match with the six numbers already fixed by the lottery committee, he wins the prize. What is the probability that he wins the prize in the game? 1/38760

42) In a single throw of three dice, find the probability of getting

A) a total of five. 1/36

B) a total of atmost 5. 5/108

Exercise--2

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

1) A coin is tossed. If head comes up, a die is thrown but if tail comes up, the coin is tossed again. find the probability of obtaining:

A) two tails. 1/8

B) head and number 6. 1/8

C) head and an even number. 3/8

2) A letter is chosen at random from the word ' ASSASSINATION' find the probability that the letter is

A) a vowel. 6/13

B) a consonant. 7/13

3) In a lottery, a person choses six natural numbers at random from 1 to 20, and if these six numbers match with the six numbers already fixed by the lottery committee, he wins the prize. what is the probability of winning the prize in the game ? 1/38760

4) On her vacation Veena visits four cities A,B,C,D in a random order, what is the probability that she visits.

A) A Before B ? 1/2

B) A before B and B before C? 1/6

C) A first and B last ? 1/12

D) A either first or second ? 1/4

5) A die die has two faces each with number '1' three faces each with number '2' and one face with number '3'. If die is rolled once determine:

A) P(2). 1/2

B) P(1 or 3). 1/2

C) P(not 3). 5/6

6) If 4-digit numbers greater than or equal to 5000 a randomly formed from the digit 0, 1,3,4 and 7, what is the probability of forming number divisible by 5 when

A) the digits may be repeated. 2/5

B) the repetition of digits is not allowed. 3/8

7) A urn contains 9 red, 7 white and 4 Black balls. If two balls are drawn at random, find the probability that:

A) both the ball are red, 18/95

B) one ball is white. 91/190

C) the balls are of the same colour.

63/190

D) one is white and other is red.

63/190

8) A box contains 10 red marbles, 20 blue marbles and 30 green marbles. 5 marbles are drawn from the box, what is the probability that

A) all will be blue. ²⁰C₅/⁶⁰C₅

B) at least one will be green. 1 - ³⁰C₅/⁶⁰C₅

9) In a lottery 10000 tickets are sold and 10 equal prizes are awarded. what is the probability of not getting a prize if you buy

A) one ticket. 999/1000

B) two ticket. ⁹⁹⁹⁰C₂/¹⁰⁰⁰⁰C₂

C) 10 tickets. ⁹⁹⁹⁰C₁₀/¹⁰⁰⁰⁰C₁₀

10) The number of lock of suitcase a four wheels, each labelled with 10 digits (from 0 to 9). The lock opens with a sequence of 4 digits with no repeats. what is the probability of getting the right sequence to open the suitcase. 1/5040

11) Three letters are dictated to three persons and an envelop is addressed to each of them, the letters are inserted into the envelopes at random so that each envelope contains exactly one letter. find the probability that at least one letter is in its proper envelope. 2/3

12) Out of 100 students, two sections of 40 and 60 students are formed. if you and our friends are among the hundred students, what is the probability that

A) you both enter the same section ? 17/33

B) you both enter the different sections? 16/33

13) four cards are drawn at random from a pack of 52 playing cards. find the probability of getting.

A) all the four cards of the same suit. 198/20825

B) all the four cards are the same number. 13/270725

C) one card from each suit. 2197/20825

D) two red cards and two black cards. 325/833

E) all cards are of same colour. 92/833

F) all face cards. 99/54145

14) A word consists of 9 letters, 5 consonants and 4 vowels. Three letters are chosen at random. what is the probability that more than one will be selected? 17/42

15) 4 persons are to be chosen at random from a group of 3 men, 2 women and 4 children. find the probability of selecting :

A) 1 man, 1 woman and 2 children. 2/7

B) exactly 2 children. 10/21

C) 2 women. 1/6

16) A box contains 10 bulbs, of which just three are defective. If a random sample of five bulbs are drawn, find the probability that the sample contains:

A) exactly one defective. 5/12

B) exactly two defective. 5/12

C) No defective bulbs. 1/12

17) A bag contains tickets numbered 1 to 30. Three tickets are drawn at random from the bag. What is the probability that the maximum number of the selected tickets exceed 25? 88/203

18) 12 balls are distributed among 3 Boxes, find the probability that the first box will contain 3 balls. 110x2⁹/3¹²

19) find the probability that birth days of six different persons will fall in exactly two calendar months. 341/12⁵

20) 5 marbles are drawn from a bag which contains 7 blue marbles and 4 black marbles. what is the probability that:

A) all will be blue? 1/22

B) 3 will be blue and 2 black? 5/11

21) find the probability that when a hand of 7 cards is dealt from a well shuffled deck of 52 cards, it contains

A) all 4 kings. 1/7735

B) exactly Three Kings. 9/1547

C) at least three kings. 46/7735

22) In a single throw of dice, determine the probability of getting

A) a total of 5. 1/36

B) a total of at most 5. 5/108

C) a total of atleast 5. 53/54

23) three dice are thrown simultaneously. find the probability that:

A) all of them show the same face. 1/36

B) all show distinct faces. 5/9

C) two of them show the same face. 5/12

24) what is the probability that in a group of

A) two people, both will have the same birthday? 1/365

B) 3 people, at least two will have the same birthday ? 364x363/365

25) The letters of the word SOCIETY are placed at random in a row. what is the probability that three vowels comes together? 1/7

26) Find the probability that in a random arrangement of the letters of the word UNIVERSITY the two I's come together. 1/5

27) A digit number is formed by the digits 1,2,3,4,5 without repetition. find the probability that the number is divisible by 4. 1/5

28) Out of 9 standing students in a college, there are 4 boys and 5 girls. A team of four students is to be selected for a quiz programme. find the probability that two are boys and two are girls. 10/21

29) In a lottery of 12 microwave ovens, there are 3 defective units. A person has ordered 4 of these units and since each is identically packed, the selection will be random. What is the probability that

A) all 4 units are good. 14/55

B) exactly 3 units are good. 28/55

C) at least 2 units are good. 54/55

30) There are 4 letters and 4 addressed envelopes. find the probability that all the letters are displaced in right envelopes. 23/24

31) the odd in favour of an event are 3:5. find the probability of occurrence of this event. 3/8

32) A fair coin with 1 marked on one face and 6 on the other and a fair die are both tossed, find the probability that the sum of the numbers that turns up is

A) 3. 1/12

B) 12. 1/12

33) In a relay race there are five teams A,B,C,D, E.

A) what is the probability that A, B and C finish first, second and third respectively. 1/60

B) what is the probability that A, B and C are first to finish (in any order). 1/10

34) a card is drawn from an ordinary pack of 52 cards and a gambler bets that, it is a spade or an ace. What are the odds against his winning this bet. 9:4

35) In a single throw of three dice, find the probability of getting a total of 17 or 18. 1/54

36) Three coins are tossed together. find the probability of getting:

A) exactly 2 heads. 3/8

B) at least two heads. 1/2

C) at least one head and one tail. 3/4

37) what is the probability that a leap year has 53 Sundays and 53 Monday . 1/7

38) A and B throw a pair of dice. If A throws 9, find B's chance of throwing a higher number. 1/6

39) In a single throw of three dice, find the probability of getting the same number on all the three dice. 1/36

40) In shuffling a pack of 52 playing cards, four are accidently dropped, find the chance that the missing cards should be one from each suit. 2197/20825

41) ticket numbered from 1 to 20 are mixed up together and then a ticket is drawn at random. what is the probability that the tickets has a number which is a multiple of 3 and 7 ? 2/5

42) A bag contains 7 white, 5 black and 4 red balls. If two balls are drawn at random, find the probability that:

A) both the balls are white. 7/40

B) one ball is black and the other red. 1/6

C) both the balls are of the same colour. 37/120

43) A bag contains 6red, 4 white and 8 blue balls. If three balls are drawn at random, find the probability that:

A) one is red and two are white. 3/68

B) two are blue and one is red. 7/34

C) one is red. 33/68

44) Five cards are drawn from a pack of 52 cards. What is chance that these 5 will contain

A) just one ace. 3243/10829

B) atleast one ace?

45) The face cards are removed from a full pack, Out of the remaining 40 cards, 4 are drawn at random. What is the probability that they belong to different suits ? 1000/9139

46) A box contains 100 bulbs, 20 of which are defective. 10 bulbs are selected for inspection. find the probability that:

A) all 10 a defective. ²⁰C₁₀/¹⁰⁰C₁₀

B) all 10 are good. ⁸⁰C₁₀/¹⁰⁰C₁₀

C) at least one is defective. 1 - ⁸⁰C₁₀/¹⁰⁰C₁₀

D) none is defective. ⁸⁰C₁₀/¹⁰⁰C₁₀

47) find the probability that in a random arrangement of the letters of the word SOCIAL vowels come together. 1/5

48) The letters of the word FORTUNATES are arranged at random in a row. what is the dance that the two 'T' come together. 1/5

49) The letters of the word CLIFTON are placed at random in a row. What is the probability that two vowels come together? 2/7

50) find the probability that in a random arrangement of the letters of the word UNIVERSITY. the two 'I's do not come together. 4/5

51) If odds in favour of an event be 2 : 3, find the probability of occurrence of this event. 2/5

52) If odds against an event is 7:9, find the probability of non-occurrence of this event. 7/16

53) Two balls are drawn at random from a bag contains 2 white, 3 red, 5 green and 4 Black balls, one by one without, replacement find the probability that both the balls are of different colours. 0.78

54) Two unbiased dice are thrown. find the probability that:

A) neither a doublet nor a total of 8 will appear. 13/18

B) the sum of the numbers obtained on the two dice is neither a multiple of 2 nor a multiple of 3. 1/3

55) A bag contains 8 red, 3 white and 9 blue balls. If three balls are drawn at random, determine the probability that

A) all the three balls are Blue Balls.

7/95

B) all the balls of different colours.

18/95

56) A bag contains 5 red, 6 white and 7 black balls. two balls are drawn at random. what is the probability that both balls are red or both are black ? 31/153

57) If a letter is chosen at random from the English alphabet, find the probability that the letter is

A) a vowel. 5/26

B) a constant. 21/26

58) In a lottery, a person chooses 6 different numbers at random from 1 to 20, and if these 6 numbers match with six numbers already fixed by the lottery committee, he wins the prize. what is the probability of winning the prize in the game? 1/38760

59) 20 cards numbered from 1 to 20. one card is drawn at random. what is the probability that the number on the card is:

A) a multiple of 4 ? 1/4

B) not a multiple of 4 ? 3/4

C) odd ? 1/2

D) greater than 12? 2/5

E) divisible by 5 ? 1/5

F) not a multiple of 6 ? 17/20

60) Two dice are thrown. Find the odds in favour of getting the sum

A) 4. 1:11

B) 5 . 1:8

C) what is the odds against getting the sum 6? 31:5

61) what are the odds in favour of getting a spade if the card drawn from a well shuffled deck of cards ?

What are the odds in favour of getting a king ? 1:3, 1:12

62) a box contains 10 red Marbles, 20 blue Marbles and 30 green Marbles. 5 marbles are drawn at random. From the box, what is the probability that:

A) all are blue? 34/11977

B) at least one is green? 4367/4484

63) A box contains 6 red marbles numbered 1 through 6 and 4 white marbles numbered from 12 through 15. Find the probability that a marble drawn is

A) white. 2/5

B) white and odd number. 1/5

C) even numbered. 1/2

D) red or even numbered. 4/5

64) A class consists of 10 boys and 8 girls. Three students are selected at random. what is the probability that selected group has

A) all boys ? 5/34

B) all girls ? 7/102

C) 1 boy 2 girls ? 35/102

D) at least one girl ? 29/34

E) at most 1 girl? 10/17

65) Five cards are drawn from a well shuffled pack of 52 cards. find the probability that all the five cards are hearts. 33/66640

66) a bag contains tickets numbered from 1 to 20. Two tickets are drawn. find the probability that

A) both the tickets are prime numbers and them. 14/95

B) on one there is a prime number and on the other there is a multiple of 4. 4/19

67) An urn contains 7 white, 5 black balls and 3 red balls. two balls are drawn at random. find the probability that:

A) both the balls are red. 1/35

B) one ball is red and the other is black? 1/7

C) one ball is white. 8/15

68) A committee of two person is selected from 2 men and 2 women. what is the probability that the committee will have

A) no man? 1/6

B) one man ? 2/3

C) two man ?

69) There are 4 men and 6 women on the City Councils. If one Council member is selected for a committee at random, how likely is that it is a woman ? 3/5

TYPE-2

Exercise -1

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

1) If A and B are two events associated with a random experiment such that P(A)= 0.35, P(A or B)= 0.85 and P(A and B)= 0.15, find P(B). 0.65

2) Two dice are tossed together. Find the probability of getting a doublet or a total of 6. 2/9

3) In a single throw of two dice, find the probability that neither a doublet nor a total of 10 will appear. 7/9

4) A natural number is chosen at random from among the first 500. What is the probability that the number so chosen is divisible by 3 or 5 ? 233/500

5) A card is drawn at random from a well shuffled deck of 52 cards. find the probability of its being a spade or a king. 4/13

6) two cards are drawn at random from a well shuffled pack of 52 cards. What is the probability that either both are red or both are kings. 55/221

7) A box contains 100 bolts and 50 nuts. It is given that 50% bolts and 50% nuts are rusted. Two objects are selected from the box at random. find the probability that either both are bolts or both rusted. 0.58

8) If A and B are two events such that P(A)= 0.5, P(B)= 0.4 and P(A and B)= 0.1. Find

A) P(A or B). 0.7

B) P(A but not B). 0.4

C) P(B but not A). 0.2

D) P(neither A nor B). 0.3

9) the probability that at least one of the event A and B occurs is 0.6. if the probability of simultaneous occurrence of A and B is 0.2, find P(A') + P(B'). 1.2

10) The probability of the occurrence of two events A and B are 0.25 and 0.50 respectively. The probability of their simultaneously occurrence is 0.14. find the probability that neither A nor B occurs. 0.39

11) A card is drawn from a deck of 52 cards. find the probability of getting a king or heart or a red card.

7/13

12) If A and B are two events associated with the random experiment for a which P(A)= 0.60, P(A or B)= 0.85 and P(A and B) =0.42, find P(B). 0.67

13) let A and B be two events associated with random experiment for which P(A)= 0.4, P(B)= 0.5 and P(A or B)= 0.6. find P(A and B). 0.3

14) In a random experiment, let A and B be events such that P(A or B)= 0.7, P(A and B)= 0.3 and P(A')= 0.4. Find P(B). 0.4

15) if A and B are two events associated with the random experiment such that P(A)= 0.25, P(B)= 0.4 and P(A or B)= 0.5, Find the values of

A) P(A and B). 0.15

B) P(A and B'). 0.1

16) If A and B be two events associated with a random experiment such that P(A)=0.3, P(B)= 0.2 and P(A or B)=0.1. find

A) P(A' or B). 0.1

B) P(A or B'). 0.2

17) if A and B are two mutually exclusive events such that P(A)= 1/2 and P(B)=1/3. Find P(A or B). 5/6

18) let A and B to B mutually exclusive events of a random experiment such that P(not A)= 0.65 and P(A or B)=0.65, find P(B). 0.3

19) A, B, C are three mutually exclusive and exhaustive events associated with a random experiment. If P(B)=3/2 P(A) and P(C)=12P(B), find P(A). 4/13

20) the probability that a company executive will travel by plane is 2/5 and that he will travel by train is 1/3. find the probability of his traveling by plane or train. 11/15

21) From a well shuffled pack of 52 cards, a card is drawn at random. find the probability of its being a king or queen. 2/13

22) from a well shuffled pack of cards, a card is drawn at random. find the probability of its being either a Queen or a heart. 4/13

23) A card is drawn at random from a well shuffled deck of 52 cards. find the probability of its being a spade or a king. 4/13

24) A number is chosen from number 1 to 100. Find the probability of its being divisible by 4 or 6. 33/100

25) A die is thrown twice. What is the probability that at least one of the two throws comes up with the number 4 ? 11/36

26) Two dice are tossed once. find the probability of getting an even number on the first die or a total of eight. 5/9

27) two dice are thrown together. what is the probability that sum of the numbers on the two faces is neither divisible by 3 nor by 4 ? 4/9

28) In a class, 30% of the students offered mathematics, 20% offered chemistry and 10% offered both. if a student is selected at random. find the probability that he has offered mathematics or chemistry. 2/5

29) the probability that Ahmed passes and English is 2/3 and the probability that he passes in Hindi is 5/9. if the probability of his passing both the subjects is 2/5, find the probability that he will pass in at least 1 of the subjects. 37/45

30) the probability that a person will get an electrification contract is 2/5 and the probability that he will not get a plumbing contract is 4/7. if the probability of getting at least one contract is 2/3, what is the prob. that he will get both ? 17/105

31) The probability that a patient visiting a dentist will have a tooth extracted is 0.06, the probability that he will have a cavity filled is 0.2, and the probability that he will have a tooth extracted or a cavity filled is 0.23. what is the probability that he will have a tooth extracted as well as a cavity filled? 0.03

32) In a town of 6000 people, 1200 are over 50 years old and 2000 are females. it is known that 30% of the females are over 50 years. what is the probability that a randomly chosen individual from the town is either female or over 50 years ?

13/30

Sunday, 7 November 2021

Compound interest (C)

1) Find the compound interest on Rs 25000 at 10% per annum for 3 years. ₹8275

2) A man borrowed ₹25000 from a finance company at 20% per annum compounded half yearly. What amount of money will discharge his debts after 3/2 years. ₹33275

3) Find the compound interest on ₹15625 at 16% per per annum, for 9 months when compounded quarterly. ₹1951

4) Find the compound interest on ₹14000 for 2 years at 5% per annum. ₹

5) Find the compound interest on 5000 for 2 years when the rate of interest is 10% per annum.

6) Salma borrowed from Mahila Samiti a sum of ₹625 to purchase a sewing machine. if the rate of interest is 4% per annum, what is the compound interest that she has to pay after 2 years ?

7) A Man borrows ₹2000 to install a hand pump in his diary. if the rate of interest is 4% per annum.calculate the compound interest that he pay after 3 years.

8) Find out the compound interest on ₹ 32000 for 1 year 6 months at 10% per annum when the interest is payable half yearly.

9) A man puts ₹2000 in a bank in a fixed deposit account for a year. The bank pays compound interest at 8% per year, interest being payable half-yearly. Find the amount of money to the man's credit man's credit after one year.

10) Compute the compound interest on ₹5000 for 3/2 years at 16% per annum, compounded half yearly.

11) Compute the compound interest on ₹ 8000 for 1 year at 20% per annum compounded quarterl.

12) Find out the compound interest on ₹6400 for 9 months at 10% per annum. If the interest is interest is payable quarterly.

13) Mrs. Malhotra deposited ₹ 10000 in a bank for six months. if the bank pays the bank pays compound interest at 12% per annum reckoned quarterly, find the amount to be received by her on maturity.

USING FORMULA:

A= P(1+ R/100)ᵀ And C. I= A - P Where ,

A= Amount,

P= Principal

R= Rate %

T= Time,

C. I = Compound Interest

It becomes very easy to calculate the compound interest....

1) Find the amount and the compound interest on ₹24000 for 3 years at 10% per annum. 7944

2) Find the compound interest on ₹4960 for three years at 25/% per annum. ₹81

3) Karuna borrowed 57600 from Life Insurance Corporation against her policy at 25/4% per annum to build a house. Find the amount that she pays to the LIC after 3/2 years if the Interest is calculated half yearly. ₹69089.06

4) A man deposited Rs32768 in a bank, where the Interest is credited quarterly. If the rate of interest be 25/2% per annum, How much amount will he receive after 9 months? 35937

5) Find the amount of 12500 for 2 years compounded annually, the rate of interest being 15% for the first year and 16% for the second year. 16675

6) Find the compound interest on ₹ 8000 at 15% per annum for 7/3 years. 3109

7) Find Amount:

A) Principal = ₹3000, rate= 5% p.a, Time= 2 years.

B) Principal = ₹5000, rate= 10 paise per rupee p.a, Time= 2 years.

C) Principal = ₹625, rate= 4% p.a, Time= 2 years.

D) Principal = ₹2000, rate=4 paise per rupee p.a, Time=3 years.

E) Principal = ₹8000, rate= 15% p.a, Time= 3 years.

8) Find the compound interest on ₹ 8000 for 3/2 years at 10% per annum, interest being payable half yearly.

9) How much would a sum of ₹ 16000 amount to in 2 years time at 10% per annum compound interest, interest being payable half-yearly ?

10) Ram lent ₹40000 to Shyam for a shop at 10% compound interest compounded half yearly. Find the compound interest paid by Shyam after 6 months.

11) Find the compound interest on ₹1000, at the rate of 8% per annum for 3/2 years when interest is compounded half yearly.

12) Salma borrowed ₹64000 from a bank for 3/2 years at the rate 10% per annum. Find the amount paid by her after 3/2 years, If interest is compounded half-yearly.

13) Find the compound interest on ₹16000 for 1 year at the rate of 20% per annum, If the interest is compounded quarterly.

14) A man bought a refrigerator for ₹8000 on credit. The rate of interest for the first year is 5% and the second year is 15% 15% year is 15%. How much will it cost him if he pays the amount after 2 years.

15) What will ₹125000 amount to at the rate of 6% , if the interest is calculated after every four months?

16) Find the compound interest on 62500 where the rate of interest is 2% for the first year year , 3% for the second year and 4% for 3rd year.

17) Find the compound interest on ₹6400 at 35/2 % per annum for 2years.

18) Find the compound interest on 12800 at 15/2% per annum for 3 years.

19) Find the amount when a sum of ₹ 5120 is lent out out at the compound interest at 25/2% per annum for 11/5 years.

20) Find the amount of ₹ 31250 for 5/2 years compounded annually, rate of interest being 12% per annum.

21) What sum will becomes ₹5408 after 2 years at 4% per annum when the interest is compounded annually? 5000

22) The difference between compound interest and simple interest for 2 years at 5% per annum at certain sum of money is ₹ 2.50. Find the sum. ₹1000

23) Find the annual rate of compound interest at which ₹8000 will become ₹10648 after 3 years. 10% p.a

24) After what time will 5400 yields ₹ 1373.76 as compound interest at 12% per annum? 2yrs.

25) Find the principal, if the compound (compounded annually) at the rate of 10% per annum for 3 years is 331.

26) Rachna borrowed a certain sum at the rate of 15% per annum. if she paid at the end of 2 years ₹1290 as interest compounded annually. Find the sum she borrowed.

27) What sum will become ₹ 4913 in 3/2 years, if the rate of interest is 25/2% compounded half yearly.

28) The interest on a sum of Rs 2000 is being compounded annually at the rate of 4% per annum. Find the period for which the compound rest is 163.20

29) In what time will ₹64000 amount to ₹68921 at 5% per annum interest being compounded half yearly?

30) In how much time would ₹5000 amount ₹6655 at 10% per annum compound interest ?

31) Mamta invested ₹1000 in a finance company and received ₹1331 after 3 years. Find the rate of interest percent per annum compounded annually.

32) Arjun purchased Rahul Patra ₹1000, sum of which will fetch him ₹2000 after 5 year. Find the rate of interest is compounded half yearly.(given ¹⁰√2= 1.072)

33) At what rate percent per annum, compound interest will ₹ 1000 amount to ₹1331 in 3 years.

34) The difference between the compound interest and the simple interest on a certain sum of money at 10% per annum for 2 years is ₹500. Find the sum when the interest is compounded annually. 50000

35) The difference between the compound interest and the simple interest on a certain sum sum at 15% per annum for 3 years is ₹283.50 Find the sum.

36) Find the compound interest at the rate of 5% for 3 years on that principal which in 3 years at the rate of 5% per annum gives ₹ 1200 as simple interest. 1261

37) The present population of a city is ₹800000. if the rate of growth is 5% per year, find its population after 3 years. 926100

38) The population of a town in the year 1991 was 140000. If the increase during the day next three year be estimated at 5% , 6% and 5% respectively. What will be its population in the year 1994 ? 1636111

39) the population of a certain city is 125000. if the annual birth rate is is 3.3% and the annual death rate is 1.3 % . calculate the population after 3 years.

40) The bacteria in a culture grows by 10% in the first hour, decreases by 10% in the second hour and again increases by 10% in the third hour. if the count of bacteria in the sample is 1210000, what will be the count of bacteria after 3 hours. 1317690

41) A machinery plant costing ₹800000 depreciates in value by 15% annually. Find its value after 3 years. 491300

42) Rahim bought a new car for ₹95000. After using it for two years he sold it out. If its value depreciated 20% and 15% annually respectively, what did he get for the car ? 64600

43) The value of a residential flat constructed at a cost of 100000 is depreciating at 10% per annum. What will be its value 3 years after construction?

44) Mr. Cerian purchased a boat for ₹16000. If the cost of that boat is deprecating at the rate of 5% per annum, calculate its value after 2 years.

45) The Nagarpalika of a certain city started a campaigns to kill stray dogs which numbered 1250 in the City. As a result the population of stray digs started decreasing at the rate of 20% per month. Calculate the number of stray dogs in the City three months after the campaign started.

46) 8000 blood donors were registered with a charitable hospital. Some Student Organisation started mobilising people for this noble cause. As a result the number of donors registered increased at the rate of 20% per half year. Find the total number of new registrant during 3/2 years.

47) the cost of a TV set was quoted as ₹15000 at the beginning of 1996. after a few months, the price was hiked by 5%. Because of decrease in demand, the cost was reduced by 4%in 1997. However, because of some taxes in 1998 budget the price was again hiked by 5%. What is the cost of the TV set at the end of 1998?

48) In the month of January, the railway police caught 4000 ticketless travellers. In February , the number rose by 5%. However, due to constant visit by the police the number reduced by 5% in March and in April it further reduces by 10%. What is the total number of ticketless travelers caught in the month of April ?

49) the population of Pakistan in 1995 was 7.95 x 10⁷. After three years the population became 8.65 x 10⁷. Find the annual rate of increase, if it is given that ³√(8.65/7.95)= 1.02853.

50) The population of a city is 1500000. It increases by 9% in the first year, by 10% in the second year and by 11% in the third year. What will the population be at the end of 3 years.

51) The population of a town increases by 4% every year. It's present population is 540800. What was it 2 years ago ?

52) The population of a town is 64000. If the annual birth rate is 10.7% and the annual death rate in 3.2%, calculate the population after 3 years.

53) Due to migration to cities, the population of a village decreases at the rate of 4% per annum. If its present population is 11520, what it was 2 years ago ?

Binomial Theorem

EXERCISE -1

1) EXPAND:

a) (3x+2y)⁴. 81x⁴+216x³y+ 216x²y²+ 96xy³+ 16y⁴

b) (2x-3y)⁴. 16x⁴-96x³y+ 216x²y³- 216xy²+ 81y⁴

c) (x+ 1/y)¹¹. x¹¹+ 11x¹⁰/y + 55x⁹/y² + 165x⁸/y³ + 330x⁷/y⁴ + 462x⁶/y⁵ + 462x⁵/y⁶ + 330x⁴/y⁷ + 165x³/y⁸ + 55x²/y⁹ + 11x/y¹⁰ + 1/y¹¹

d) (√x + √y)¹⁰. x⁵+ 10x⁹⁾²y¹⁾²+ 45x⁴y+ 120x⁷⁾²y³⁾² + 210x³y²+252⁵⁾² y⁵⁾² + 210x²y³ + 120x³⁾²y⁷⁾²+ 45xy⁴ + 10x¹⁾²y⁹⁾² + y⁵

e) (x + 1/x)⁵. x⁵+5x³+ 10x+ 10/x+ 5/x³+ 1/x⁵

f) (x - 1/x)⁶. x⁶-6x⁴+15x²- 20+ 15/x²- 6/x⁴+ 1/x⁶

g) (2x/3 - 3/2x)⁶. 64x⁶/729 - 32x⁴/27 + 20x²/3 - 20+ 135/4x² - 243/8x⁴+ 729/64x⁶

h) (x² - 2/x)⁷. x¹⁴ -14x¹¹ + 84x⁸ - 280x⁵ + 560x² - 672/x + 448/x⁴ - 128/x⁷

I) (1+x)⁵. 1+ 5x + 10x²+ 10x³+ 5x⁴ + x⁵

j) (1- 3x)⁷. 1- 21x+ 189x² - 945x³ + 2835x⁴ - 5103x⁵ + 5103x⁶- 2187x⁷

k) (a² - 2bc)⁵. a¹⁰-10a⁸bc+ 40a⁶b²c²- 80a⁴b³c³+80a²b⁴c⁴-32b⁵c⁵

L) (x³ + 2/x²)⁵. x¹⁵+ 10x¹⁰+ 40x⁵ + 80+ 80/x⁵ + 32/x¹⁰

m) (1+ 2x- 3x²)⁵. 1+ 10x+ 25x²- 40x³ - 190x⁴+ 92x⁵+ 570x⁶- 360x⁷ - 675x⁸ + 810x⁹ - 243x¹⁰

2) EVALUATE::

a) (√2+1)⁶+(√2-1)⁶. 198

b) (2+√3)⁷+(2 - √3)⁷. 9884

c) (√3+1)⁵ +(√3 - 1)⁵ 88√3

d) (x+ √(x²-1)⁶+ (x- √(x²-1)⁶. 64x⁶ - 96x⁴ + 36x² - 2.

e) (2a+b)⁶ - 6b(2a+b)⁵+ 15b²(2a+b)⁴ - 20b³(2a+b)³+15b⁴(2a+b)²- 6b⁵(2a+b) + b⁶. 64a⁶

f) (1+2√x)⁵+(1-2√x)⁵. 2(1+40x+80x²)

g) (3+√2)⁵-(3-√2)⁵. 1178√2

h) (√3+1)⁵-(√3-1)⁵. 152

I) (96)³. 884736

j) (102)⁵. 11040808032

k) (101)⁴. 104060401

l) (98)⁵. 9039207968

m) (994)⁴ 976215137296

n) (1.01)⁵. 1.0510100501

o) (1001)⁵. 1005010010005001

3) FIND:

a) 7th term of (4x/5 + 5/2x)⁸. 4375/x⁴

b) 10th term of (a/b - 2b/a²)¹². -366080b⁵/a¹⁴

c) 16th term of (√x - √y)¹⁷. -136xy¹⁵⁾²

d) 4th term of (x/y - y/x)¹⁰. -120x⁴/y⁴

e) 5th term of (x²+ 3/x³)¹⁰. 17010

f) 19th term of (2√x - √y)²⁰. 760xy⁹

g) 11th term of (2x - 1/x²)²⁵. ²⁵C₁₀(2¹⁵/x⁵)

h) 7th term in (3x² - 1/x³)¹⁰. 17010/x¹⁰

I) 7th term in (4x/5 + 5/2x)⁸. 4375/x⁴

j) 4th term in (x+ 2/x)⁹. 672x³

4) Find the coefficient of:

a) x⁷ in (x²+ 1/x)¹¹. 462

b) x² in (3x- 1/x)⁶. 1215

c) x¹⁸ in (x² - 3a/x)¹⁵. 110565a⁴

d) x¹⁰ in (x² - 2)¹¹. 29568

e) x⁶ in (3x² - 1/3x)⁹. 378

f) x¹⁰ in (2x² -1/x)²⁰. ²⁰C₁₀ 2¹⁰

g) x⁷ in (x - 1/x²). -⁴⁰C₁₁

h) 1/x¹⁵ in (3x² - a/3x³)¹⁰. -40a⁷/27

I) x⁹ in (x² - 1/3x)⁹. -28/9

j) 1/x² in (2x³ - 1/x²)⁶. 60

k) x¹² in (ax⁴ - bx)⁹. 9ab⁸

l) x³² in (x⁴ - 1/x³)¹⁵. 1365

m) 1/x in (2x² - 1/x)¹⁰. -960

n) 1/x¹¹ in (x² - 1/x³)¹². -792

5) Find the term and coefficient independent of x in the expansion of::

a) (x+1/x)¹⁰. 252

b) (x² +1/x)¹². 495

c) (2x+1/3x²)⁹. 1792/9

d) (x - 1/x)¹². 924

e) (x² - 2/x³)¹⁵. 320320

f) (3x²/2 -1/3x)⁹. 7/18

g) (9x²-1/3x)¹². 495

h) (√x - √c/√x)¹⁰. -252√c⁵

I) (2x²- 3/x³)²⁵. ²⁵C₁₀(2¹⁵× 3¹⁰)

j) {√(x/3)+3/2x²)¹⁰. -3003×3¹⁰×2⁵

6) Find the middle term in the expansion of:

a) (3 + x)⁶. 540x³

b) (2x/3 - 3/2x)²⁰. ²⁰C₁₀

c) (1 - x²/2)¹⁴. -429x¹⁴/16

d) (a/x + bx)¹². 924a⁶b⁶

e) (x² - 2/x)¹⁰. -8064x⁵

f) (a/3 + 9b)⁸. 5670a⁴b⁴

g) (x/a - a/x)¹⁰. -252

h) (3x/ - x³/6)⁹. 189x¹⁷/8, -21x¹⁹/16

i) (2x² - 1/x)⁷. -560x⁵, 280x²

j) (3x - 2/x²)¹⁵. -6435×3⁸×2⁷/x⁶, 6437×3⁷×2⁸/x⁹

k) (x⁴ - 1/x³)¹¹. -462x⁹, 462x²

I) (x - 1/x)¹⁰. -252

m) (2x - x²/4)⁹. 63x¹³/4, -63x¹⁴/32

7) a) Find the 5th term from the end in the expansion of (x - 1/x)¹². 495x⁴

b) Find the 4th term from the end in the expansion of (4x/5 - 5/2x)⁹. 10500/x³

c) (x⁴ + 1/x³)¹⁵, 4th term from the end. 455x³⁹ or 455/x²⁴

d) (3/x² - x³/6)³, 4th term from the end. 35x⁶/48

e) (2x - 1/x²)²⁵, 11th term from the end. - ²⁵C₁₅ 2¹⁰/x ²⁰

f) 11th term from end (2x - 1/x²)²⁵. ²⁵C₁₅ (2¹⁰/x²⁰)

g) 5th term from the end of (3x -1/x²)¹⁰. 17010/x⁸

h) 4th term from the end in (x+2/x)⁹. 5376/x³

I) 4th term from the end of (4x/5 - 5/2x)⁹. 10500/x³

j) 7th term from the end in (2x² - 3/2x)⁸. 4032x¹⁰

EXERCISE -2

1) Using Binomial theorem, prove

A) 6ⁿ - 5n leaves remainder 1 when divided by 25.

B) 2³ⁿ - 7n -1 is divisible by 49, where n belongs to N.

C) 3ⁿ⁺² - 8n - 9 is divisible by 64, where n belongs to N.

D) 3³ⁿ - 26n -1 is divisible by 676.

E) 9ⁿ⁺¹ - 8n - 9 is divisible by 64, where n belongs to N.

2) Show that the middle term in the expansion of

A) (1+ x)²ⁿ is :

{1.3.5....(2n -1) 2ⁿ. xⁿ}/n!

B) (x+ 1/x)²ⁿ is :

{1.3.5....(2n -1) 2ⁿ}/n!

3) Find the general term in the expansion

a) (x² - y)⁶.

4) Find n, if the ratio of the fifth term from the beginning to the fifth term from the end in the expansion (⁴√2 + 1/⁴√3)ⁿ is √6: 1. 10

5) If 17th and 18th term in the expansion of (2+ a)⁵⁰ are equal, then find a.

6) If the fourth term in the expansion of (ax + 1/x))ⁿ is 5/2, then find the value of a and n. 1/2, 6

7) Find the value of a so that the term independent of x in (√x + a/x²)¹⁰ is 405. ±3

8) Find a positive value of n for which the Coefficient of x² in the expansion of (1+ x)ⁿ is 6. 4

9) Find the coefficient of x⁷ in (ax² + 1/bx)¹¹ and 1/x⁷ in (ax - 1/bx²)¹¹. 11C5 a⁶/b⁵

10)A) Prove that the coefficient of (1+ x)²ⁿ is equal to the sum of the coefficient of middle terms in the expansion of (1+ x)²ⁿ⁻¹.

B) Find the value of k for which the coefficient of the middle term in (1+ Kx)⁴ and (1- Kx)⁶ are equal. -3/10

11) The sum of the coefficient of first three terms in the expansion of (x - 3/x²)ⁿ, x≠ 0, n being a natural number, is 559. Find the term of the expansion containing x³. 12

12) If the coefficient of (2r+4)th and (r -2)th terms in the expansion of (1+ x)¹⁸ are equal, find r. 6

13) If the coefficient of (2r+1)th and (r +2)th terms in the expansion of (1+ x)⁴³ are equal, find r. 14

14) If the coefficient of (r - 5)th and (2r -1)th terms in the expansion of (1+ x)³⁴ are equal, find r. 14

15) The co-efficients of 5th, 6th, 7th terms in the expansion (1+ x)ⁿ are in AP, find n. 7 or 14

16) The co-efficients of 2nd, 3rd, 4th terms in the expansion (1+ x)²ⁿ are in AP, show that 2n² - 9n +7= 0.

17) If the coefficient of (r - 1)th, r th and (r +1)th terms in the expansion of (1+ x)ⁿ are in the ratio 1:3:5 then find n, r. 7, 3

18) The 3rd, 4th, and 5th terms in the expansion of (x + a)ⁿ are respectively 84, 280 and 560, find the value of x, a, n. 1, 7, 2

19) If the coefficient of three consecutive terms in the expansion of (1+ x)ⁿ be 76, 95 and 76, find n. 8