THEORY OF QUADRATIC EQUATION(XI) A- Z

Exercise -A

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1) Examine the nature of the roots of the following:

A) 9x²-24x +16= 0.      rational, equ

B) x²-3√5 x -1= 0.     Irrational, uneq

C) 7x²+ 8x +4= 0.             Imaginary

D) 16x²+40x +25= 0.      Rational, eq

E) qx²+px -q= 0.                Real, uneq

F) 7x²+4x -3= 0.          Rational, uneq

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

A) 3x²-2x +1= 0.                   2/3, 1/3

B) 2x²+x -1= 0.                     -1/2, -1/2

C) 3x²-1= 0.                                     0

D) (x-1)/(x+1)= (x-3)/2x.             0,3

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

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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

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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

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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

 

 

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.

 

TRIGONOMETRIC RATIOS OF PARTICULAR ANGLES (A - Z)

EXERCISE - A

1) 7 sin 30 cos 60

A) 5/4 B) 7/4 C) 9/4 D) none 

2) (sin 30 - sin 90 + 2 cos 0)/(tan 30. Tan 60)

A) 3/2 B) 5/2 C) 2/3 D) 1/2 E) none 

3) sin 45 + cos 45

A) 1/√2 B) √2 C) √3/2 D) 1 E) none 

4) 1/sin 30 - √3/cos 30

A) 2 B) 1/22 C) 0 D) 1 E) none 

5) (sin 30 + cos 30) - (sin 60 + cos 60)

A) 1 B) 0 C) 2 D) -1 E) none 

6) Sin60/cos²45 - 3 tan 30 +5 cos90

A) -107/20 B) 4 C) 2 D) 1 E) 0 

 

7) (sin² 45+ cos² 45)/tan²60

A) 1/3 B) 4/3 C) 2/3 D) 1 E) none 

8) 3 sin² 45+ 2 cos² 60

A) 5 B) 4 C) 2 D) 1 E) none 

9) sin² 45 - tan² 60 + cos²90.

A) 0 B) 5/2 C) 2 D) -5/2 E) none 

10) (4/3  sin² 45+ sin² 60 - 3 cos²60 + 3/4 tan²60 - 2 tan²45

A) 25/3 B) 25/36 C) 25/23 D) 1 E) none 

11) sin 60/cos² 45 - 3 tan 30 + 5 cos 90

A) 0 B) 4 C) 2 D) 1 E) none 

12) sin² 30  cos² 45 + 4 tan²30 + 1/2 sin² 90 - 2 cos 90+ 1/24

A) 0 B) 4 C) 2 D) 1 E) none 

13) 4 sin²x - 1= 0 and x is acute angle, find the value of x and hence find the value of 

A) cos²x + tan²x

a) 1 b) 2 c) 13/14 d) 13/12 e) none

B) cos 2x

a) 1 b) 2 c) 1/2 d) -1 e) -1/2

C) sin3x

a) 0 b) 1 c) -1 d) 2 e) -2

14) 4(sin⁴30 + cos⁴60)- 3(cos² 45 - sin² 90)

A) 0 B) 4 C) 2 D) 1 E) none 

 

15) sin x= 1/2 then the value of 3 cos x - 4 cos³x is

A) - 1 B) 0 C) 2 D) 1 E) none 

16) If x is an acute angle and 2 sinx =1, then the value of x and hence find the value of 4sin³x - 3 sinx 

a)0 b) 1 c) -1 d) 1/2 e) -1/2

17) If 2 sin(3x-15)=√3 then the value of sin²(2x+10) + tan²(x+5).

a) 1 b) 2 c) 3 d) 12/13 e) 13/12

18) If 2x is an acute angle and 2 sin 2x =√3 then x is

A) 30 B) 45 C) 60 D) 90

EXERCISE - B

1) √3 cosec 60 - sec 60

A) 0 B) 2 C) -2 D) 1 E) none 

2) 

EXERCISE - C

1) 2√2 cos 45 cos 60 + 2√3 sin 30 tan 60 - cos 0

A) 3 B) 4 C) 2 D) 1 E) none 

2)  (cos 0 + sin 45 + sin 30)(sin 90 - cos 45+ cos 60)

A) 7/4 B) 4/7 C) 7 D) 4 E) none 

3) 2 √2 cos 45 cos 60 + 2√3sin 30 -   cos 0

A) 0 B)  3 C) 2 D) 1 E) none 

4) cos² 45+ sin² 60 + sin²30

A) 5/2 B) 3/2 C) 2 D) 1 E) none 

5) cos² 30 cos² 45 + 4 sec²60 + 1/2 cos² 90 - 2 tan² 60

A) 0 B) 83 C) 83/8 D) 1 E) none 

6) cos² 30 + sin 30 + tan² 45

A) 9/4 B) 4 C) 2 D) 1 E) none 

7) cos90+ cos² 45 sin 30 tan² 45.

a) 0 b) 1 c) 1/2 d) 1/3 e) 1/4

EXERCISE- D 

1) Tan 30/cot 60

A) 1/√2 B) 1/√3 C) √3 D) 1 E) none 

2) Tan45/cosec30 + sec 60/cot 45 - 2 sin90/cos.

A) 1 B) 2 C) 1/2 D) -1/2 E) none 

3) tan²30 - 4 sin²45

A) 7/3 B) -5/3 C) -11/3 D) 1 E) none 

4) (1 - tan² 45)/(1+tan² 45) is

A) √3 B) 1/√3 C) 1/√2 D) 0 E) none 

5) 2tan30/(1+ tan²30).

a) 2 b) 3 c) √3 d) 2√3 e) 3√2

6) 4/3 tan² 30 + sin² 60 - 3 cos²60 + 3/4 tan²60 - 2 tan²45

A) -107/20 B) 25/36 C) 2 D) 1 E) 0  

7) 4/5  tan² 60 - 2/sin² 30 - 3/4  tan²30

A) -107/20 B) 4 C) 2 D) 1 E) none  

8) tan³60- 2 sin 60

a) 2 b) 3 c) 2√3 d) 3√2 e) none

9) If A= 30° then value of 2 tanA/(1- tan²A).

a) √3 b) 1/√3 c) -√3 d) -1/√3 e) none 

10) If tan A= √3 then cosec A is

A) 1/2 B) 2 C) √3/2 D) 2/√3 E) none 

11) If 3x is an acute angle and tan 3x - √3=0, then the value of x is

a) 10 b)20 c) 30 d) 40 e) 60

EXERCISE - E

1) 4/3 cot² 30 + 3 sin² 60 - 2cosec²60 -3/4 tan² 30.

A) 0 B) 10 C) 3 D) 10/3 E) none 

2) (√3+1)(3- cot30°)

a) 2 b) 3 c) √3 d) 2√3 e) none 

3) 

EXERCISE - F

1) If A= 10 then {3sin3A + 2cos(5A+ 10)}/{√3 tan3A - cosec(5A - 20)}.

a) 1 b) 2 c) 5 d) 5/2 e) -5/2

2) If A= 30 then the value of 2 sin A cos A is

A) 1/√2 B) √3/2 C) 1/2 D) 1 E) none 

3) If sec x sin x = 0, then cos x

A) 1/√2 B) 0 C) 1/2 D) 1 E) none 

4) If A= 60° and B= 30° then value of cosA cosB + sinA sinB.

a) 2√3 b) √3/2 c) 2/√3 d)√3 e) none 

5) If sin x= 1/2 and cos y= 1/2, then the value of (x+ y) is

A) 0° B) 30° C) 60° D) 90° E) none 

6) If ∆ABC is right angled at C, then the value of cos(A+ B) is

A) 1/2 B) 0 C) √3/2 D) 1 E) none 

7) If 40+ x is an acute angle and cos(40+ x)= sin30, then the value of x.

a) 0 b) 20 c)30 d) 45  e) 60 

INEQUALITIES (C)

* SOLVING AN INEQUATION:

It is the process of obtaining all the possible solution of an inequation.

* SOLUTION SET:

The set of all possible solutions of an inequation is known as its solution set.

=> SOLVING LINEAR INEQUATION IN ONE VARIABLE:

• RULE-1: Same number may be added to (or subtracted from) both sides of an inequation without changing the sign of inequality.

•RULE-2: Both sides of an inequation can be multiplied(or divided) by the same positive real number without changing the sign of inequality. However, the sign of inequality is reversed when both sides of an inequation are multiplied or divided by a negative number.

• RULE-3: Any term of an inequation may be taken to the other side with its sign changed without affecting the sign of inequality.

A linear inequation in one variable is of the form

 ax+b<0 or, ax+b≤0 or, ac+b> 0 or, ax+ b ≥ 0.

We follow the following STEP to solve a linear inequation in one variable.

• STEP-1: Obtain the linear inequation.

• STEP -2 : Collect all terms involving the variable on one side of the inequation and the constant terms on the other side.

• STEP -3: Simplified both sides of inequality in their simplest form to reduce the inequation in the form

ax <b, or ax≤b, or ax>b, or ax≥b

• STEP -4: Solve the inequation obtained in step - 3 by dividing both sides of the inequation by the coefficient of the variable.

• STEP -5: Write the solution set obtained in step -4 in the form of an interval on the real line.

                  EXERCISE - 1

              😀 ------------------ 😀

* Solve the following inequation:

1) 2x - 4≤ 0.                             (-∞, 2]

2) - 3x +12< 0.                          (4,∞)

3) 4x - 12 ≥ 0.                            [3,∞)

4) 12x < 50, when

a) x ∈ R.                              (-∞,25/6)

b) x ∈ Z.             {....,-3,-2,-1,0,1,2,3,4}

c) x ∈ N.                                {1,2,3,4}

5) - 4x > 30, when

a) x ∈ R.                             (-∞,-15/2)

b) x ∈ Z.                                {....,-9,-8}

c) x ∈ N.                                 null set

6) 7x +9> 30.                             (3,∞)

7) 4x -2 < 8, when

a) x ∈ R.                                 (-∞,5/2)

b) x ∈ Z.                        {....-2,-1,0,1,2}

c) x ∈ N.                                       {1,2}

8) 5x -3< 3x+1 when

a) x is a real number.               (-∞, 2)

b) x is integer number.          {..,-4,-3, -2, -1, 0,1}

c) x is a natural number.                {1}

8) 3x - 7 > x+1.                           (4,∞)

9) x+ 5 > 4x -10.                      (-∞, 5)

10) 3x+ 9 ≥ - x+19.                 (5/2,∞)

11) 3x+17≤ 2(1-x).                   (-∞, 3]

12) 2(2x+3)-10≤6(x-2).              [4,∞)

13) 2(3-x)≥ x/5 + 4.         (-∞, 10/11]

14) -(x-3)+4< 5- 2x.                  (-∞,-2)

15) (2x-3)/4 +9≥ 3+ 4x/3.                (-∞,63/10]

16) (3x-2)/5 ≤ (4x-3)/2.     [11/14,∞)

17) (5x-2)/3 -(7x-3)/5 > x/4.     (4,∞)

18) 1/2(3x/5 +4) ≥ (x- 6)/3.         (-∞,120]

19) 3(x- 2)/5 ≥ 5(2 - x)/3.          [2,∞)

20) x/5< (3x-2)/4 - (5x-3)/5.          (-∞,2/9)

21) 2(x-1)/5 ≤ 3(2+x)/7.       [-44,∞)

22) 5x/2 + 3x/4 ≥ 39/4.             [3,∞)

23) (x-1)/3 +4< (x -5)/5 - 2.   (-∞,50)

24) (2x+3)/4 - 3 < (x-4))3 - 2.     (-∞, - 13/2)

25) (5-2x)/3 < x/6 - 2.             (8,∞)

26) (4+2x)/3 ≥x/2 - 3.           (-26,∞)

27) (2x+3)/5 - 2 < 3(x-3)/5.     (-1,∞)

28) (x-2)≤(5x+8)/3.                  (-7,∞)

29) 1/(x -2)< 0.                         (-∞,2)

30) (x+1)/(x +2)≥ 1.                 (-∞,-2)

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

Type ::2

EQUATION OF THE FORM

* (ax+b)/(cx+d)> k, OR

* (ax+b)/(cx+d)≥ k, OR

* (ax+b)/(cx+d)< k, OR

* (ax+b)/(cx+d) ≤ k.

STEP -1: Obtain the inequation.

STEP-2:Transpose all terms on LHS

STEP-3: Simplify LHS of the inequation obtained in STEP-2 to obtain an inequation of the form 

(px+q)/(rx+s)> 0 OR

 (px+q)/(rx+s)≥0 OR

 (px+q)/(rx+s)< 0, OR

 (px+q)/(rx+s) ≤ 0

STEP-4: Make coefficient x positive in numerator and denominator if they are not.

STEP-5: Equate numerator and denominator separately to zero and obtain the values of x. These values of x are generally called critical points.

STEP-6: Plot the critical points obtained in STEP-5 on real line. These points will divide the real line in three regions.

STEP-7: In the right most region the expression on LHS of the inequation obtained in STEP-4 will be positive and in other regions it will be alternatively negative and positive. So, mark positive signs in the right most region and then mark alternatively negative and positive signs in other regions.

STEP-8: Select appropriate region on the basis of the sign of the inequation obtained in STEP-4 , Write these region in the form of interval to obtain the desired solution sets of the given inequation.

                 EXERCISE-2

                  -----------------

Solve the inequation of followings:

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

2) (x+3)/(x-2) ≤2.         (-∞,2)U(7,∞) 

3) (2x-3)/(3x-7) ≤2.           (-∞,3/2) U (7/3,∞) 

4) 3/(x-2) < 1.                (-∞,2)U(5,∞)

5) 1/(x-1) ≤2.            (-∞,2)U(3/2,∞)

6) (4x+3)/(2x-5) ≤2.         (-∞,5/2) U (33/8,∞) 

7) (5x-6)/(x+6) <1.                    (-6,3)

8) (5x+8)/(4-x) <2.    (-∞,0) U (4,∞) 

9) (x-1)/(x+3) >2.                      (-7,-3) 

10) (7x-5)/(8x+3)>4.             (-17/25, -3/8)

11) x/(x-5) > 1/2.        (-∞,-5)U(5,∞) 

12) (x-3)/(x+4) >0, x ∈R.      {x∈ R: x < -4}U {x ∈ R: x> 3}

13) (x+5)/(x-2) >0, x ∈R.    {x∈ R: x ≤ -5}U {x ∈ R: x> 2}

14) (2x+5)/(x+3) >1, x ∈R.       {x∈ R: x < -3} U {x∈ R: x > -2}

15) (x+7)/(x+4) >1, x ∈R.        {x∈ R: x > - 4} 

16) (x+4)/(x+6) >1, x ∈R.       {x∈ R: x < -6} 

17) 3/(x -2) >2, x ∈R.       {x∈ R: 2< x < 7/2}

18) (x-3)/(x+1) < 0, x ∈R.       {x∈ R: -1 < x < 3} 

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

TYPE -3:

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

* If a is a positive real number, then

1) |x| < a <=> -a<x< a i.e. x∈ (-a, a).

2) |x|≤a <=> -a≤x≤a i.e. x x∈ [-a, a].

3) | x|> a <=> x<-a or x> a.

4) | x|≥ a <=> x≤-a or x≥ a.

* Let r be a positive real number and a be a fixed real number. Then, 

1) |x - a| < r <=> a-r <x< a+r i.e. x∈ (a - r, a+ r).

2) |x -a|≤r <=> a-r ≤x≤a+r i.e.x∈ [a - r, a +r].

3) | x - a|> r <=> x<a -r or x> a +r

4) | x -a|≥ r <=> x≤-a- r or x≥ a+r.

               EXERCISE -3

                 ------------------

Solve the inequation of followings:

1) | x | < 5, x ∈R.        {x∈ R: -5< x < 5}

2) |x |≥ 5, x ∈R.       {x∈ R: x < -5} U {x∈ R: x ≥ 5}

3) |3x -2|≤ 1/2.            {x∈ R: x < -5}

4) | 4x - 5 | ≤ 1/3,         x∈[1/2,5/7}

5) | x - 2|≥5.                 (-∞,-3)U[7,∞)

6) |5 - 2x | ≤ 3, x ∈R.        {x∈ R: 1≤x< 4} 

7) | 3x - 7| > 4, x ∈R.     {x∈ R: x < 1} U {x∈ R: x > 11/3}

8) |2(3-x)/5| < 9/5, x ∈R.         {x∈ R: -3/2 < x < 15/2}

9) |x +1/3| > 8/3.                      (-∞,-3)U(7/3,∞)

10) |4 - x| + 1<3.                           (2,6)

11)|(3x-4)/2|≤ 5/12.   (19/18,29/18) 

12) |x -1|≤ 5, |x|≥ 2.       (-∞,-2]U[2,∞)

13) 1≤| x-2|≤ 3.                   [-1,1]U[3,5]

14) |x- 1|≤5, | x |≥2.       (-∞,-2]U[2,∞)

15) |x-2|/(x-2) > 0.                     (2,∞)

16) (|x| -1)/(|x| -2) ≥0, x∈R, x≠ ±2.        [-1,1]U(-∞,-2]U(2,∞)

17) -1/(|x| -2) ≥ 1, where x∈R, x≠ ±2.              [-2,-1]U[1,2)

18) |2/(x-4)|> 1, x ≠4.       (2,4)U(4,6)

19) (|x+3| + x)/(x+2) > 1.         (-5,-2) U(-1,∞)

20) 1/(|x| -3)< 1/2.     (-∞,-5) U(-3, 3) U(5,∞)

21) (|x +2| - x)/x < 2.    (-∞,2]U(1,∞)

22) |x -1| + |x-2|≥ 4.           (-∞,-1/2]U [7/2,∞)

23) |x-1|+ |x-2| + |x -3|≥6.   (-∞,0] U [4,∞).

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

              EXERCISE-4

               -----------------

1) 3x² - 7x + 4≤0

A) x> 0 B) x<0 C) all x D) no solution E) none

2) 3x² - 7x - 6 <0

A) -0.66< x < 3 B) x<-0.66 or x> 3 C) 3< x < 7 D) -2< x< 2 E) none

3) 3x² - 7x + 6 <0

A) 0.66< x < 3 B) -0.66<x<3 C) -1< x < 3 D) -0.66< x< 0.66 E) none

4) x² - 3x +5>0

A) x> 0  B) x< 0 C) both A and B D) - ∞< x< ∞ E) none 

5) x² - 14x - 15 <0

A) x < -1 B) 15< x  C) both A and B D) -1< x< 15 E) none

6) 2 - x - x² ≥ 0

A) - 2≤x ≤ 1 B) -2< x<2 C) x< -2 D) x> 1 E) none

7) |x² - 4x| < 5

A) -1≤ x ≤5 B) 1≤ x ≤5 C) -1< x ≤ 1 D) - 1< x< 5 E) none

8) |x² +x | <0

A) x< 0 B) x> 0 C) all values of x D) x> 5 E) none

9) |x² - 5x| < 6

A) - 1< x < 2 B) 3< x< 6  C) both A and B D) - 1< x< 6 E) none

10)  |x² - 2x| < x

A)  1< x < 3 B) -1< x< 3  C) 0< x <4 D) x > 3 E) none

11)  |x² - 2x - 3| < 3x - 3

A) 1< x < 3 B) -2< x< 5  C) x> 5 D) 2< x< 5 E) none

12)  |x² - 3x| + x - 2 < 0

A) (1 - √3) < x < (2+ √2) B) 0< x< 5 C) (1-√3),2-√2) D) 1< x< 4 E) none

13) x²- 7x +12< |x - 4|.

A) x < 2 B)  x> 4 C) 2< x <4 D) 2≤ x≤ 4 E) 1< x< 4

14) x²-  | 5x - 3| - x  < 2

A)  x >3+ 2√2 B) x< 3+2√2 C) x> - 5 D) - 5 < x< 3+2√2 E) none

15) |x - 6| > x²- 5x +9

A) 1 ≤ x < 3 B) 1< x< 3 C) 2< x< 5 D) -3< x< 1 E) none

16) |x - 6|< x²- 5 x +9 

A) x < 1 B) x > 3 C) 1< x< 3 D) both A and B E) x≥ 1 & x ≥ 3

17) |x - 2| ≤ 2x²- 9x +9 

A) (x > (4- √2)/2  B) x< (5+ 3√2)/2  C) both A and B D) x≥(4- √2)/2 E) none

18) 3x²- |x - 3| > 9 x - 2 

A) x < (4- √19)/3 B) x>(4+√19)/3 C) both A and B D) 2< x< 2 E) none

19) x²- |5x + 8| > 0

A) x < (5- √57)/2 B) x<(5+√57)/2 C) x> (5+ √57)/2  D) x> (5-√57)/2 E) both A and B

20) 3 |x - 3| + x² - 7> 0

A) x> -1 B) x< -1 C) x > 2 D) both B and C E) none

21) |x - 6| > |x²- 5x +9 |

A) x < 1 B) x> 3 C) 1< x < 3 D) Both A and B E) none

22) |x - 3| - 3 (|x +2| -5)< 0

A) -7<x < -2 B) x<-7 and x> 4 C) x< -2 and x> 3 D) any of these E) none of these

23) |x² - x - 8| > 2x 

A) x < 2√2 B) x< 3+ 3√5 C) x> 2+ 2√3 D) both A and C E) none

24) (x-1) √(x² - x - 2) ≥ 0

A) x≤ 2 B) x ≥ 2 C) x≤ - 2 D) x≥ 0 E) none

25) (x² -1) √(x² - x - 2) ≥ 0

A) x≤ -1 B) x ≥ -1 C) x≥2 D) A and C E) none

26) √{(x - 2)/(1- 2x)} ≥ -2

A) 0.5> x B) x > 2 C) both A and B D) 0.5 < x ≤ 2 E) none

27) √{(3x - 2)/(2- x)} > 1

A) 0 < x< 2 B) 0.75< x < 4C) 0.75< x < 2 D) 0< x < 4 E) none

28) √(3x - 10) > (6- x)

A) 4< x≤ 6 B) x< 4 or x > 6 C) x< 4 D) x > 8 E) 4< x < 8

29) √{(x² - 2x - 3)< 1

A) -1- √5< x < -3 B) 1≤ x <(√5-1) C) x > 1 D) both A and B E) none

30) √{ 1 - (x +2)/x²}< 2/3

A) -6/5 < x ≤ -1 or 2≤ x <3 B) -6/5 ≤ x < -1) C) 2≤ x < 3 D) -6/5 ≤ x < 3 E) none

31) 2 - √(x - 2)< x

A) x> 1 B) x≥ 1, x ≠ 2 C) x < 1 D) 1< x < 5 E) none

32) √{(x +18)< 2- x.

A) x≤ -18 B) x< -2 C) x > -2 D) -18≤ x < -2 E) none

33) x> √(24+ 5x(

A) x< 3 B) 3<x≤ 4.8 C) x ≥ 24/5 D) x> 8 E) none

34) √{(9x - 20)< x.

A) 4< x < 5 B) 20/9≤ x< 4 C) x > 5 D) both B and C E) none

35) √{(x +7 )< x.

A) x> 2 B)  x > √30/2 C) x> (1+ √29)/2 D) x> 1+ √29/2 E) none

36) √{(2x - 1)< x- 2

A) x< 5 B) x> 5 C) x > 5 or x < -5  D) 5< x < 15 E) none

37) √{(x +78)< x + 6

A) x 3 B) x> 3 or x< 2 C) x > 3 D) 3< x < 10 E) none

38) √{(5 - 2x)< 6x - 1

A) 0.5< x B) x< 2.5 C) 0.5< x < 2.5 D) x > 2.5 E) none

39) √{(x +61)< x + 5

A) x < 3 B) x> 3 or x <1 C) x > 3 D) 3<  x < 15 E) none

40) x < √(3 - x)

A) x> 1 B) x< 1 C) -2 <x < 1 D) -1 < x E) none

41) x + 3 < √(x + 33)

A) x> 3 B) x< 3 C) -4 <x < 3 D) -33 < x < 3 E) none

42) √(2x +14)> x +3

A) x< -7 B) -7≤ x< 1 C) x> 1 D) -7 < x< 1 E) none

43) x - 3 < √(x - 2)

A) 2≤x<(7+5√2)/2  B) 2≤ x C) x < (7+√5)/3 D) x ≤ 2 E) none

44) x + 2 < √(x + 14)

A) -14≤ x < 2 B) x> -14 C) x < 2 D) -1 < x< 2 E) none

45) x - 1 < √(7 - x)

A) x> 3 B) x< 3 C) -53 <x < 3 D) -103 < x < 3 E) none

46) √(3 - x) > x

A) x< 4 B) x > 5 C) x≤ 4 or x ≥ 5 D) 4 < x < 5 E) none

47) √(11 - 5x)> x - 1

A) x> 3, x < 5  B) -3 < x< 2 C) -25 <x < 2 D) x < 2  E) none

48) √(3 - x) > x

A) -2≤ x < 2 B) -2≤ x C) x < 2 D) x= -2 or x > 2 E) none 

** Find the largest integral x that satisfies the following::::

49) (x -2)/(x² -9) < 0

A) x=-4 B) x =-2 C) x= 3 D) none

50) 1/(x +1 ) - 2/(x²- x +1) < (1- 2x)/(x³ +1)

A) x= 1 B) x = 2 C) x= -1 D) none

51) (x +4)/(x² -9) - 2/(x +3) < 4x/(3x - x²).

A) x= 1 B) x = 2 C) x= -1 D) none

52) (4x +19)/(x + 5) < (4x-17)/(x - 3).

A) x= 1 B) x = 2 C) x= -1 D) none

53) (x +1)(x -3)²(x -5)(x -4)²(x -2) < 0

A) x= 1 B) x = 2 C) x= -1 D) none

** Solve the following inequalities:

54) (x -1)(3- x)(x -2)² > 0

A) 1<x < 3 B) 1< x < 3 but x ≠ 2 C) 0< x < 2 D) -1< x < 3 E) none 

55) (6x -5)/(4x+1) < 0

A) -1/4<x < 1 B) -1/2 < x < 1 C) -1< x < 1 D) -1/4 < x < 5/6 E) none

56) (2x -3)/(3x- 7) > 0

A) x < 3/2  B) 3/2< x < 7/3 C) x> 7/3 D) -1< x < 3 E) none

57) 3/(x -2) < 1

A) 2<x < 5 B) x < 2 C) x> 5 D) x< 2 or x > 5 E) none

58) 1/(x -1) ≤ 2

A) x < 1 B) x ≥ 1.5 C) -5< x < 1 D) both A and B E) none

59) (4x +3)/(2x-5) < 6

A) x < 2.5 B)  x < 33/8  C) x≥ 2.5  D) x < 2.5 or  x > 33/8 E) none

60) (5x -6)/(x + 6)< 1

A)-6 <x < 6 B) -6< x < 0 C) -6< x < 4 D) -1< x < 3 E) none

61) (5x +8)/(4- x)< 2

A) x<0 or x> 4  B) 0< x < 4 C) 0≤ x < 4 D) x > 4E) none

62) (x -1)/(x + 3) > 2

A) x<-7   B) x< -3 C) -7< x < -3 D) x > 4 E) none

63) (7x -5)/(8x + 3) > 4

A) -17/25 < x<-3/8   B) x> -17/25 C) 0< x < 3/8 D) -17/25 < x< 0 E) none

64) x/(x -5) > 1/2

A) -5 < x< 5  B) -5< x < 0 C) -5 ≤ x ≤ 5 D) x < -5 or x> 5 E) none

65) x ≤ 6/(x -5).

A)  x< -1  B) x> 5 C) -x < 6 D) x ≤ -1 or 5<  x ≤ 6 E) none

66) (30x -9)(x -2)≥ 25(x +2)

A) x< - 1.4 or x > 2  B) x< -1.4 or 2< x ≤ 2.6  C) x ≤ 1.4 or 2 < x ≤ 2.6 D) x < -5 or x> 5 E) none

67) 4/(x +2) > 3- x

A) -2 < x< -1 or x > 2  B) -2< x < 2 C) -2 < x < -1 D) 0< x < 3 E) none

68) x - 17 ≥ 60/x.

A) -x< -3  B)  x < 20 C) -3 ≤ x < 0 D) -3< x ≤ 0  or x≥ 20 E) none

69) √x² < x +1

A) x > 0.5  B) x> 0 C) all x D) x>-0. 5 E) none

** Find the smallest integral x satisfying the inequalities:::

70) (x -5)/(x²+ 5x -14) > 0.

A) x= -6  B) x = -3 C) x = -7 D) x = -5  E) none

71) x² - 5|x| + 6 < 0

A) -3 < x< -2  B) 2< x < 3 C) both A and B D) -3< x < 3  E) -3< x< 3 F) none

72) x² - |x| -2 ≥ 0

A) -2 < x< 2  B) x≤ -2 or x≥ 2 C) x< -2 or x> 1 D) -2< x < 1 E) none

** Solve the inequalities:

73) (x -1)(3- x)(x -2)²> 0

A) 1< x < 2 B) -1< x < 3 C) -3<x < -1 D) 1< x < 3, x≠ 2 E) 1< x < 3

74) 0.5/(x - x² -1) < 0

A) x> 0 B) x ≤ 0 C) x ≥ 0 D) x < 0  E) for all real x

75) (x² - 5x +6)/(x²+x +1)< 0

A) x < 2 B) x > 3 C) 2 <x < 3 D) x< 2 or  x > 3, x≠ 2 E) none

76) (x² +2x -3)/(x²+1)< 0

A) x < -3 B) -7<x <- 3 C) -3 <x < 2 D) -7< x< 1  E) none

77) (x -1)(x +2)²/(-1 - x)< 0

A) x < -2 B) x <-1 or x> 1 C) x<-2 and x ≠ 2 D) x< -2 or  x > 1, x≠ 2 E) none

78) (x² +4x +4)/(2x²-x -1)> 0

A) x < -2 B) x > 1 C) x≠ 2 D) all of the above  E) none

79) x⁴ - 5x² + 4 < 0

A) -2< x < 1 B) -2< x < 2 C) -2 <x < -1 or 1 < x < 2 D) 1< x < 2  E) none

80) x⁴ - 2x² - 63 ≤ 0

A) x≤ -3 or x≥ 3 B) -3≤ x ≤0 C) 0≤x ≤ 3 D) -3≤ x ≤3  E) none

81) (5x -1)/(x²+3)< 1

A) x< 4 B) 1< x < 4 C) x< 1 or x > 4 D) 1< x < 3 E) none

82) (x -2)/(x²+ 1) < -1/2

A) -3< x< 3 B)  x < -3 C) -3< x< 1< 6 D) -3 < x < 2 E) 1< x < 6

83) (x +1)/(x - 1)² < 1

A) x > 3 or x is negative B) x> 3 C) x > 3 or -23< x < 0 D) x is negative and x > 2 E) x< 3

84) (x² - 7x +12)/(2x²+4x +5)> 0

A) x< 3 or x> 4 B) 3< x < 4 C) 4< x< 24  D) 0< x < 3 E) 0< x < 4

85) (x²+ 6x -7)/(x²+1)≤ 2

A) x is negative B) x≥  0  C) x > 0 or x< 0 D) always E) never

86) (x⁴+x²+1)/(x²-4x - 5)< 0

A) x<-1 or x> 5 B) -1< x < 5 C)  x > 5 D) -5< x < -1 E) -5< x < 6

87) (1+ 3x²)/(2x²- 21x +40)< 0

A) 0< x<8  B) 2.5 < x < 8 C) -8< x < 8 D) 3 < x < 8 E) none

88) (1 - x²)/(x²- 5x +6)< 0

A) x< 2  B) x> 3 C) both A and B D) 2 < x < 3 E) none

89) (x⁴+x²+1)/(x²-4x - 5)> 0

A) -1< x< 5 B)  x <-1 or x>  5 C) x≤ -1 or x> 5 D) - 1 < x < 1 E) 1 < x < 5

90) (1- 2x - 3x²)/(3x - x² - 5)>  0

A) x<-1 or x> 1/3  B) x < -1 or x= 1/3  C) -1 < x < 13  D) x <1/3 E) none

91) (x² - 5x+7)/(-2x²+ 3x +2)>  0.

A) x>0. 5 B) x> 0.5  C) -0.5< x < 5 D) - 0.5< x < 2 E) 0.5< x < 2

92) (2x²- 3x- 459)/(x²+ 1)> 0

A) x> -20 B) x < 0 C)  x <-20 D) -20< x < 20  E) 0 < x < 20

93) (x²- 1)/(x²+ x +1)< 1

A) x> -2 B) x> 2 C)  -2< x < 2 D) x <2 E) none

94) 

MEASUREMENT OF ANGLE (C)

 MEASUREMENT OF ANGLE

A) Express the following angles in radians 

1) 36°

A) π/2 B) π/3 C) π/4 D) π)5

2) 120°

A) π/2 B) 2π/3 C) 3π/4 D) π/5

3) 225° 

A) π/2 B) π/3 C) 5π/4 D) π/5

4) 330°

A) 11π/2 B) 11π/3 C) 11π/6 D) π/5

5) 400°

A) π/20 B) 20π/9 C) π/4 D) π/5

6) 7°30'

A) π/24 B) π/3 C) π/4 D) π)/5

7) 22°22'22"

A) π/2 B) π/3 C) π/4 D) π/5 E) n

B) Express the following angles in circular measure.

8) 25ᵍ

A) π/2 B) π/3 C) π/4 D) π/n E) n

9) 22ᵍ 12′ 12′′

A) π/2 B) π/3 C) π/4 D) π/5 E) n

10) 12ᵍ 50′ 30′′ 

A) π/2 B) π/3 C) π/4 D) π/5 E) n

C) Express the following angles in degree 

11) 5π/12

A) 75° B) 648° C) 47°43'38" D) 229 E) n

12) 18π/5

A) 75° B) 648° C) 47°43'38" D) 229 E) n

13) 5/16

A) 75° B) 648° C) 47°43'38" D) 229 E) n

14) -4

A) 75° B) 648° C) 47°43'38" D) 229 E) n

15) The greatest of a triangle are in AP and the greatest angle is double the least. Find all the angles in degrees

A) 40°,60°, 80° B) C) 47°,43°,90° D) 50°, 60°,70° E) n

16) The greatest of a triangle are in AP and the greatest angle is double the least. Find all the angles in degrees.

A) π/3, 2π/9, 4π/9 B) π/2, 2π/9, 4π/9  C) 4π/3, 5π/9, 4π/9  D) n

17) The difference between the two acute angles of a right angled triangle is π/5, find these angles in degrees 

A) 63°,27° B) C) 43°,47° D) 50°, 40° E) n

18) The difference between the two acute angles of a right angled triangle is π/5, find these angles in circular measure

A) 7π/20, 3π/20 B) π/2, 2π/9  C) 4π/3, 5π/9  D) n

19) Find the radius of a circle in which a central angles of 45° intercepts an arc of length 33cm (take π= 22/7)

A) 32 B) 42 C) 52 D) 62 E) none

20) Find the length of an arc of a circle of radius 14cm which subtends an angle of 36° at the centre.

A) 6.8 B) 7.8 C) 8.8 D) 9.8 E) n

21) If the arcs of the same length in two circles subtends angles 75° and 120° at the centre, find the ratio of their radii.

A) 8:5 B) 5:8 C) 4:7 D) 8:9 E) n 

22) Find the degree measure of the angle subtended at the centre of a circle of diameter 60cm by an arc of length 16.5cm.

A) 31°30' B) 41°30' C) 51°30' D) 61°30" E) n

23) In a circle of diameter 30cm, the length of a chord is 15cm. Find the length of the minor arc of the chord 

A)12cm B) 21.7cm C) 15.72cm D) n

24) Find the angle in radians as well as in degrees through which a pendulum swings if its length is 45cm and its tio describes an arc of length 11cm.

A) 11/45, 14° B) 13/45, 45° C) 11/45, 15° D) n

25) The large hand of a clock is 42cm long. How many centimetres does its extremity move in 20mins.

A) 8cm B) 88cm C) 89cm D) n

26) A wheel makes 180 revolutions in 1 minute. Through how many radians does it turns in 1 second?

A) π B) 3π C) 5π D) 6π E) n

27) A train is moving on a circular curve of radius 1500m at the rate of 66km per hour. Through what angle has it turned in 10 seconds

A) 5° B) 6° C) 7° D) 8° E) n

28) The angle A of a parallelogram ABCD is 45º. find the other angle in radian.         ∠B=∠D=3πᶜ/4, ∠c=π/4

29) The angles of a triangle are in the ratio 1:4: 5 . find them in centensimal systemt.      20,80,,100

30) The Vertical angle of an isosceles triangle is three times the sum of of the base angles. find the angles in sexagesimal system.                                        20°30',22°30',135°

31) The length of a big hand of a clock is 20cm. How much will its extremity move in 20 minutes ?find also the area swept by the hand in that time.           27.94m,139.68cm²

32) The circumference of the dial of a clock is a 44cm. How far does the tip of the big hand move from 3 p.m. to 5-15 p.m.

A) 88cm B) 99cm C) 11cm D) 22cm E) n

33) Find in sexagesimal measure of each angle of a regular Pentagon.

A) 108° B)72° C) 76° D) 90° E) n

34) The sum of two angles is equal to 112°. If the number of grade in one is one is twice the number of degrees in the other, find the angle in grades.                         80, 44.44

35) Two circles are such that two arcs of equal lengh subtend 18° and 24°at the respective centre. if the radius of the the first circle is 3 cm, find that of the other.              2.25

36) Two circle are such that their radius are in the ratio 2:3. Find the ratio of the length of to arc which subtend which subtend at the centre of the respective circle angles in the ratio 4:5.           8:15

37) The angles of a quadrilateral are in AP and the ratio of the least angle in degrees and the greatest angle in Radian IS 90:π. Find the least angle in degreess.           60°

38) The distance of the sun from the earth is 149.5x 10⁶ Km and the angle subtended by the sun at a point on the earth is half a degree. find the approximate diameter of the Sun. (π=22/7).      1305159 km

39) A wire in the form of a square of each side 4cm is the bent to form a circular arc of radius 6cm. Find the angle made by that are at the centre.(π=22/7).              152.7°

40) A boy in a Merry Go Round of a diameter 10 m is revolving about the centre at a speed of 100m per hour. Find in degrees the angle turned by the boy in one minute.     19.1°

MIXTURE (C)

                    MIXTURE

                     + + ** ++

1) One vessel A contains mixture of milk and water in the Proportion 4:5 and in another vessel B they are mixed in the Proportion 5:1. In what Proportion should quantities be taken from the two vessels so as to form a mixture in which milk and water will be in the Proportion of 5:4.

2) A cask is completely filled with spirit. Four litres are drawn off and four litres of water is added. The operation is repeated for second time. If now the ratio of spirit and water in the cask is 4:5, find the capacity of the cask.

3) A mixture of 30 litres contains milk and water in the ratio of 7:3; how much water must be added to the mixture so that the ratio of milk to water may be 3:7.

4) A mixture of 30 litres contains milk and water in the ratio 7:3. How much mixture must be replaced by water to make the

ratio 3:7

5) Water and milk are mixed in the ratio 2:7 in one vessel and in the ratio 2:9 in another vessel. Determine the volume of the mixture in the second vessel that should be mixed with 225c.c. of the mixture in the first vessel to form a new mixture containing milk and water in the ratio 4:1.

6) Two mixture contains milk and honey with ratios of 7:2 and 5:1. In what ratio these two mixtures should be mixed so that the resulting mixture may contain milk and honey in the ratio 9:2.

7) Two vessels contain mixture of milk and water in the Proportion 2:3 and 4:3 respectively. In what Proportion should these two mixtures be mixed so as to form a new mixture containing equal proportions of milk and water.

8) Three equal glasses are filled with mixture of spirit and water, the Proportion of spirit to water being as 3:1, 5:3, 9:7. The context of three glasses are poured into a bigger vessel. What is the Proportion of spirit and water in the final mixture.

9) Two varieties rice at ₹10 per kg and ₹12 per kg are mixed together in the ratio 1:2. Find the average price of resulting mixture.

A) 10.33 B) 11.33 C) 12 D) 13.33 E) n

10) On combining two groups of students having 30 and 40 marks respectively in an exam, the resultant group has an average score of 34. Find the ratio of the number of students in the first group to the number of students in the second group.

A) 2/3 B) 3/2 C) 3/4 D) 4/3 E) n

11) On mixing 2 classes of students having average marks 25 and 40 respectively, the overall average obtained is 30 marks. find

i) the ratio of student in the classes.

A) 1:2 B) 2:1 C) 3:2 D) 2:3 E) n

ii) the number of students in the first class if the second class had 30 students.

A) 30 B) 50 C) 60 D) 80 E) n

12) 4 kg rice at ₹5 per kg is mixed with 8kg of rice at ₹6 per kg. Find the average price of the mixture.

A) ₹5 B) 5.66₹ C) ₹ 6 D) ₹ 6.32 E) n

13) 5kg of rice at₹6 per kg is mixed with 4 kg of rice to get a mixture costing ₹7 per kg. Find the price of the costlier rice.

A) ₹6.25  B) ₹7.25 C) ₹8.25 D) ₹ 9.25 E) n

14) On mixing two classes of students having average marks 25 and 40 respectively, the overall average obtained is 30 marks. Find

i) the ratio in which the classes were mixed.

A) 2:1 B) 3:1 C) 1:2 D) 1:3 E) none

ii) the number of students in the first class if the second class had 30 students.

A) 50 B) 60 C) 79 D) 80 E) n

15) 4kg of rice at₹5 per kg is mixed with 8 kg of rice at ₹6 per kg. Find the average price of the mixture.

A) 5.66 B) 5 C) 6.66 D) 7.66 E) n

16) 5 kg price at ₹6 per kg is mixed with 4 kg of rice to get a mixture coting ₹7 per kg. Find the price of the costlier rice.

A) 6.25 B) 7.25 C) 8.25 D) 9 D) n

17) a man buys 40 kg of rice at ₹20/kg and 60 kg of rice at ₹30/kg. Find the average price.

A) 23/kg B) 25/kg C) 26/kg D) 27/kg E) n

18) Pradeep mixes two mixtures of milk and water. He mixes 40 litres of the first containing 20% water and 60 litres of the second containing 30% water. Find the percentage of water in the final mixture.

A) 24% B) 25% C) 26% D) 27% E) n

19) Two classes are combined to form a larger class. The first class having 40 students scored an average of 20 marks on a test while the second having 60 students scored an average of 30 marks on the same test. What was the average score of the combined class on the test.

A) 24 B) 25 C) 26 D) 27 E) n

20) A trader earns a profit of 20% on 40% of his goods sold, while he earns a profit of 30% on 60% of his goods sold. Find his percentage profit on the whole.

A) 24 B) 25 C) 26 D) 27 E) n 

21) A car travels 20 km/h for 40 minutes and 30 km/h for 60 minutes. Find the average speed of the car for the journey.

A) 24 B) 25 C) 26 D) 27 E) n

22) 40% of the revenues of a school came from the junior classes while 60% of the revenues of the school came from the senior classes. If the school raises its fees by 20% for the junior classes and by 30% for the senior classes, find the percentage increase in the revenues of the school.

A) 24 B) 25 C) 26 D) 27 E) n

23) A merchant Mixes 3kgs of tea costing ₹50 per kg and 5kgs of another variety costing ₹34 per kg. At what price per kg should sell the mixture so as to make a profit of 10% on the cost price?

A) 34 B) 44 C) 54 D) 64 E) n

24) If 5 kg of salt consisting of 5/kg and 3 kg of salt consting ₹ 4/kg are mixed, find the average cost of the mixture per kilogram.

A)₹ 4.5 B) ₹4.625 C) ₹4.75 D) ₹ 4.125 E) ₹4.25

25) A vessel A contains milk and water in the ratio 4:5 and the vessel B contains the same in the ratio 5:1. In what proportion should the quantities to be taken from A and B to form a mixture in which milk and water are in the ratio of 5:4 ?

A)5:2 B) 2:5 C) 3:5 D) 5:3  E) n

26) Two types of oils having the rates of  ₹4/kg and ₹5/kg respectively are mixed in order to produce a mixture having the rate of ₹4.60/kg. What should be the amount of the second type of oil if the amount of the first type of the oil in the mixture is 40 kg 

A)75kg B) 50kg C) 60kg D) 40kg E) n

27) A container has a capacity of 80 litres and is full of milk, 5 litres of milk is taken out of it and the container is filled with water. This process is repeated 4 times. How much milk is left in the container?

A) 61.8 B) 62.5 C) 72.3 D) 83.5 E) n

28) how many kilograms of sugar worth ₹ 3.60 per kg should be mixed with 8 kg of sugar worth ₹4.20 per kg. such that by selling the mixture at ₹4.40 per kg, there may be a gain of 10%

A) 6 B) 3 C) 2 D) 5 E) 4kg

29) A mixture of 125 gallons of wine and water contains 20% water. How much water must be added to the mixture in order to increase the percentage of water to 25% of the new mixture?

A) 10gals B) 8.5gals C) 8gals D) 6.66gals E) 8.33gals

30) Ram lends ₹3600 on simple interest to Harsh for a period of 5 years. He lends a part of the amount at 4% interest and the rest at 6% and receives ₹960 as the amount of interest. How much money did he lend on 4% intrest rate?

A) 2800 B) 2100 C) 2400 D) 1200 E) 1000

31) A mixture of 40 litres of milk and water cont610% water. How much water must be added to make 20% in the new mixture.

A) 3 B) 5 D) 7 D) 10 E) n

32) 400 students took the exam. 60% of the boys and 80% of the girls cleared the cut off in the examination. If the total percentage of student qualifying is 65%, how many girls appeared in the examination 

A)100 B) 120 C)150 D) 300 E) 175

33) A man purchased a cow and a calf for ₹1300. He sold the calf at a profit of 20% in the cow at the profit of 25%, in this way, his total profit was 300/13% . Find the cost price of the cow.

A) 1100 B) 600 C) 500 D) 400 E) 800

33) The average salary per head of all employees of a company is ₹600. The average salary of 120 officers is₹4000. If the average salary per head of the rest of employees is ₹ 560, find the total number of workers in the company.

A) 1020 B) 1032 C) 1050 D) 1068 E) 1080 

34) In what ratio should silver at ₹5 per gram be mixed with silver at ₹4.75 per gram to obtain a mixed variety of silver worth ₹4.92 per gram?

A) 8:1 B) 8:23 C) 8:15 D) 8:17 E) n

35) In what ratio must water be added to the milk to gain 25/2% by selling the diluted milk at the cost price?

A) 1:8 B) 8:1 C) 2:3 D) 4:5 E) 5:6

36) A dishonest milkman purchased milk at 10 per litre and mixed of water in it. By selling the mixture at the rate of ₹10 per litre earns profit of 25%. The quantity of the amount of the mixture that he had was?

A) 15 B) 20  C) 25 D) 30 E) n

37) A cistern contains 50 litres of water. 5 litres of water is taken out of it and replaced by wine. the process is repeated again. Find the proportion of wine and water in the resulting mixture

A) 1:4 B) 41:50 C) 19: 81 D) 81:19 E) 50:41 

38) A container has a capacity of 20 gallons and is full of spirit. 4 gallons of spirit is drawn and the container is again filled with water. this process is repeated 5 times. Find out how much spirit is left in the resulting mixture finally

A) 257 B) 6257 C) 564 D) 2130 E) n

39) A vessel is full of refined oil. 1/4 of the refined oil is taken out and the vessel is filled with mustard oil. If the process is repeated 4 times and 10 litres of refined oil is finally left in the vessel, what is the capacity of the vessel?

A) 33 B) 2460/81 C) 2560/81 D) 30 E) n

40) how many kilograms of sugar worth ₹6.50 per kg must be mixed with 80kg of sugar worth ₹8.00 per kg so that the mixture is worth ₹7.50 per kg

A) 40 B) 60 C) 80 D) 100 E) n

41) A sum of ₹39 was divided among 45 boys and girls. Each girls gets 50 paise,.whereas a boy gets one rupee. Find the number of boys and girls.

A) 12,33 B) 33,12 C) 15, 60 D) 60,15 E) n

42) In what ratio should qualities of  coffee powder having the rates of ₹47 per kg and ₹ 32 per kg be mixed in order to get a mixture that would have a rate of 37 per kg

A) 1: 2 B) 2:1 C) 1:3 D) 3:1 E) 4:1

43) An alloy of two metals weighing 18gms is worth ₹87 but if their composition weights be interchanged, it would worth ₹78.60. if the price of one metal be ₹6.70 per gm. Find the weight of the other metal in the mixture.

A) 8 B) 10 C) 22 D) 15 E) n

44) A butler stole wine from a butt of sherry which contained 32% spirit and then replaced what he stole by wine containing 18% spirit. The butt was then of 24% strength only. How much butt did he steal ?

A) 4/7 B) 3/7 C) 2/7 D) 6 D) n

45) In what ratio should water be mixed with soda costing ₹12 per litres so as to make a profit of 25% by selling the dilute liquid at 13.75 per litre ?

A) 10:1 B) 11:1 C) 1:11 D) 12:1 E) 1:12

46) A sum of ₹36.90 is made up of 90 coins that are either 20 paise coins or 50 paise coins. Find out how many 20 paise coins are there in the total amount ?

A) 47 B) 43 C) 27 D) 63 E) 70

47) A  dishonest grocer professes to sale pure butter at cost price but he mixes it with adulterated fat and thereby gains 25%. Find the percentage of adulterated fat in the mixture assuming that adulterated fat is really available.

A) 20% B) 25% C) 33.33% D) 40% E)  35% 

48) A mixture of 70 litres of alcohol and water contains 10% of water. How much water must be added to the above mixture to make the water 12.5% of the resulting mixture ?

A)1 lt. B) 1.5 litres C) 2 lt D)  2.5 lt E) 3 lt

49) A mixture of 20 litres of brandy and water content 10% water. How much water should be added to it to increase the percentage of water 25% ?

A) 2lt B) 3lt C) 2.5lt D) 4lt E) 5 lt

50) A merchant purchased two qualities a pulse rate of 200 per quintal and ₹ 260 per quintal. In 52 quintal of the second quality, how much pulse of first quality should be mixed so that by selling the resulting mixture at ₹300 per quintal, he gains a profit of 25%?

A) 100 B) 104 C) 26  D) none

51) A man buys milk at 8.5 per litre and dilutes it with water. He sells the mixture at the same rate and thus gains 11.11%. Find the quantity of water mixed by him in every litres of milk.

A)0.111 lt B) 0.909 lt C) 0.1 lt D)0.125 lt

52) There are two mixtures of honey and water the quantity of honey in them being 25% and 75% of the mixture. If two gallons of the first are mixed with three gallons of the second, what will be the ratio honey to water in the new mixture?

A) 11:2 B) 11: 9 C) 9:11 D) 2:11

53) There are two kinds of alloys of tin and copper. The first alloy contains tin and copper such that 93.33% of it is tin. In the second alloy there is 86.66% tin. What weight of the first alloy should be mixed some weight of the second alloy so to make a 50 kg of mass contains 90% of tin?

A) 15kg b) 30kg c) 20 kg d) 25kg

54) Two containers of equal capacity are full of mixture of oil and water. In the first, the ratio of oil to water is 4:7 and in the second it is 7:11. Now both the mixture and mixed in a bigger container. What is the resulting ratio of oil to water?

A) 149:247 B) 247:149 C) 143: 241 D) 241:143

55) Two vessels contain spirit and water mixed respectively in the ratio 1:3 and 3:5. Find the ratio in which these are to be mixed to get a new mixture in which the ratio of spirit to water is 1:2.

A) 2:1 B) 3:1 c) 1:2 d) 1:3 

56) The price of pen and a pencil is ₹35. The pen was sold at a 20% profit and the Pencil at a 10% loss. If in the transaction a man gains ₹4, how much is cost price of the pen ?

A) 10 b) 25 c) 20 D) none 

57) A person purchased a cupboard and a cot for ₹18000. He sold the cupboard at a profit of 20% and the cot at a profit of 30%. If his Total profit was 25.833%, find the cost price of the cupboard.

A) 10500 B) 12000 c) 7500 d) 10000

58) A vessel is of a mixture of kerosene and petrol in which there is 18% kerosene. 8 liters are drawn off and then the vessel is filled with petrol. If the kerosene is now 15%, how much does the vessel hold ?

A) 40 litres b) 32 c) 36 litres d) 48 

59) two solutions of 90% 97% purity are mixed resulting in 21 l of mixture of 94% of purity. how much is the quantity of the first solution in the resulting mixture?

A) 15 litres b) 12 litres c) 9 liter d) 6

60) in the Singapore zoo,  there are deers and there are ducks. If the heads are counted, there are 180, while the legs are 448. What will be the number of deers in the zoo?

A) 136 b) 68 c) 44 d) 22

61) A bonus of ₹985000 was divided among 300 workers of factory. Each male worker get 5000 rupees and each family workers gets 2500 rupees. Find the number of male workers in the factory.

A) 253 b) 47 c) 94 d) 206 

62) What will be the ratio of petrol and kerosene in the final solution formed by mixing petrol and kerosene that are present in 3 vessels in the ratio 4:1, 5:2,  6:1 respectively?

A) 166 :22 b) 83 :22 c) 83 :44  d) n

63) A mixture worth ₹ 3.25 a kg is formed by mixing two types of flour, 1 costing of ₹ 3.1 0 per kg while the other ₹3.60 per kg, In what proportion must they have been mixed ?

A) 3:7 b) 7:10 c) 10:3 d) 7:3 

64) A gain percentage of 20 is made by selling a mixture of two types of ghee at ₹480 per kg. If the type costing 610 per kg was mixed with 126 kg of the other, how many kilograms of the former was mixed?

A)139 B) 34.5 c) 69 d) cannot be determined

65) In what proportion must water be mixed with milk so as to gain 20% by selling the mixture at the cost price of the milk ?(assume that water is free available)

A) 1:4 b) 1:5 c) 1:6 d) 1:12

66) A bartender stole champagne from a bottle that contained 50% of spirit and he replaced what he had stolen with champagne having 20% spirit . The bottle then contained only 25% spirit. How much of the bottle did he steal?

A) 80% b) 83.3 c) 85.71% d) 88.88% 

67) A bag contains a total 105 coins of ₹1.50p and 25p to denominations. Find the total number of coins of ₹one if there are a total of 50.5 rupees in the bag and it is known that the number of 25 paise coins are 133.33% more than the number of 1 rupees coins.

A) 56  b) 25 c) 24 d) none

68) A man possessing ₹6800, lent a part of it at 10% simple interest and the remaining at 7.5% simple interest. His total income after 7/2 years was ₹ 1904. Find the sum lent at 10% rates.

A) 1260 b) 1700 C) 1360 d) none

69) A man decides to travel 80 kilometres in 8 hours partly by foot and partly on a bicycle, his speed on foot being 8 km/hr and that on bicycle being 16 km/hr. What distance would he travel on foot?

A) 20 b) 30 c) 48 d) 60 

70) Two vessels contain a mixture of spirit and water. In the first vessel the ratio of spirit to water is 8:3 and in the second vessel the ratio is 5:1. A 35 litres cask is filled from these vessels so as to contain a mixture of spirit and water in the ratio 4:1. How many litres are taken from the first vessel ?

A) 11 b) 22 c) 16.5 d) 17.5

71) Rakshit bird 19 eraser for rupees 1025 more for each wise Razer them for each brownie razor what could be the price of a white eraser and how many white eraser producer he have but 65 865 12:55 855 10 after paying all your beans you find that you have Rupees 7.20 in your pocket you are equal number of 55 and 10 piece coins but no other coins no rainy other currency notes how many coins do you have 82427 30 Suresh Kumar went to the market with rupees 100 if he buy 3 pens and 6 pencil heuses of all his money on the other any few by three pencils and 6 would fall sort by 20% if you wants to buy equal number Parde work questions what is the amount of money he would save if you had to buy 3 pegs and 3 pencils 50 25 7540 Abdul goes to the market to buy bananas if he can burger in the price but doesn't by 2 he can buy 3 doesn't banana Institute of two doesn't with the money has how much money does he have Rupees 612 1824 inches 3 bananas and 4 apples cost 153 bananas and three apples how much did I pay 10 8:15 cannot be determine John Word 5 mangoes in orange together rupees 1234 00 km about school is to be built for hundred students up down B and 30 students up down A the expenditure on transport age 1.2 0 per kilometre per person if the total expenditure and transport by all 13 students is to be at the small as possible then the school should be built at 33 km from Townhall 33 km from town B Town a town b person who has certain amount with he he goes to the market you can buy 50 oranges are 40 mangoes heritance 10% of the amount for taxi fare and by 20 mangoes and the banana balance the Purchase oranges number of oranges chicken purchase 36 40 15 20 72 hands cost Rupees 96.7 then what does each hand cost were the number at or not divisible or written in illegal hand 3.435.31516.2 class this students are divided into three groups A B and C of 1525 students is the group D what is the average weight of the student in group D more than the average weight of a more than the average weight of she less than the average weight of C cannot be determine student from group B which of the following will be true the average weight of the both groups increases average decreases the average weight of the class remains the same cannot be to my all the students of the class the total weight of P and seastwise the total wet a b the average weight of these greater than the average weight of a the average weight of all the groups remains the same even if a number of students are shifted from one group two another the bridge marks of student in 10 papers are 80 if the highest and the lowest score are not considered the average is 81 it is highest score in 92 and find the lowest 55 662 cannot be to mind a shipping clerk has five boxes of different but unknown weights each wedding less than 100 kg the clockwid in the boxes in pair the words obtain are 102 and 13 10718 120 and 121 kg what is the weight of the having heaviest box 6062 64 cannot be determine the total expenses of a boarding house are partly fix and partly brain linearly with the number of borrowed the average expenses for border is 700 when there are 25 words and Rs 600 when there are 50 border what is the average expense for border when they are 100 border 5:5585 45 70 yearly payment to a servant is 90 + 1 Turban the servant leaves the job after 9 months and receive is 65 and the term then find the price of the turbo 10 15 7.5 0 cannot be determine a leather factory locks the profit margin is 20 on a standard bag and rupees 30 on a Deluxe bag every bag time required hours bags machinen machinery standard back 46 Deluxe 500 10 the total time available on machine A700 hours non machine beast 1250 hours among the following products plans which one made the machine availability constants and maximum the profit standard 75 best Deluxe 80 bags standard 10050 algebra test the average score of class 10 is 83 the average class score of class Y 76 the average score of class Jet is 85 what is the average score of class 6yz at 1.58.583 cannot be determine the order 4 and some additional Park Avenue shirts the price the order was executed it was found that the number of the two brands have been interchange this increase the bill by 40% the ratio of the number of Arrow shirts to the number of Arc Avenue shirts in the original order was 13 14 12 15 3 groups of companies data Birla and Reliance announces the average of the annual profit for all your sins there establishment the average profit of Tata is 75000 the average profit of Billa is 64000 the average profit of Reliance is 73000 the average profit of all results of Tata and Birla together is 70000 the average profit award is a bitland Reliance together 9000 aproximately what is the average profit for all the three groups 7000086

River water consist of 0.5% salt made Lake water consist of 1% Sir Three water consist of 5% of salt in it this three are mixed in the ratio 4:2 and boiled such that concentration of salt in the club eating water from the solution from Hyderabad 55% 45% 75% 50% 60% agrocel mix 3 varieties of Dal consisting of 25 3033 respectively packaging in the ratio 1 decided to say the mixture at rupees 35 per kg and other browser mixes the three variety of dull equal propose save the mixture of 35 kg 300 then find the ratio of 3 variety respectivali because that is brought is mixed up 8615 question what is the ratio of the first process there at 10 best solution capacity 10 litre the first battle content 10% pure milk the third has 100% pure milk 1 l from the first battle 2 l from the second basal and so on uncle 10 litre from the 10th what is the percentage of your name 55% 60% 33% 80% 70% is taken labour problem the 10th percentage of the solution will be pure milk grocer wanted to mixi varieties flower consisting him 15-1821 so that the resultant price of the mixture is 20.90 when sold at 10% was mixed the first one the last variety in the ratio 1 ratio the same 10% profit when the sold at the same price of 20.90 12 13 14 23 24 there are two bases A and B both containing universalism of 30% concentration I add some pure whenever to a to bring the concentration in the Basil b i take out some quantity of the solution and replace it with an equilate quantity of your University A and B if the quantity of initial solution in A and B are in the Rati 12 23 25 in a drum pull up your milk 15th of the milk is drawn a place to the water 15th of the solution is drawn and replace to the water disapporation is performed 5 times all together find the capacity of the drum the difference in the quantity of your mail can the solution wrong out one Drone from the third timing bronze is an alloy of Copper and Tin in the ratio 5 rational lawyer components in consist of these elements in the ratio 3 a new alloy is on the by making bronze and bra such that competition the popular such that contribution of copper in the new alloy equals interest by what percentage is the quantity of zinc present more than the quantity of teenager new alloy 750% 6650 in Homeopathy School of Medicine potential to indicate one part of your medicine mixed with 100 parts of alcohol and so on what is the ratio of pure medicine to the total mixture 2 unit substances to our mixed with one unit of potential one and three units 300 in the world questions important parts of the alcohol medicine potential 2 is 1 part of your medicine 100 parts of the alcohol medicine population so one then find the answer of the above question 3 21 23 Rajat the chemical analyst has got two solution of ferric sulphate in water the concentration of ferric sulphatever 60% then one solution and 50% another solution mixed up the solution and continuous solution the concentration of the new solution is 52% water was added to 100 l of the solution to obtain the solution of 40% concentration what fraction 14 15 413 16 alcohol 10 litres of alcohol and operation is defined as taking out 5 letters of what is represent In The vessel in 10 l water to it what is the ratio of alcohol to water after 2 operation 14 23 16 32 50 the contents of a 750 bottle filled with methanel of 75% concentration along with that of methanol are amted into a vessel with the present concentration of methanol In The vessel is equals to 60% then find the volume of the second bottle 1125 1215 875 150