(Quadratic Equations 2

MANOJ CHAUHAN SIR(IIT-DELHI)

EX. SR. FACULTY (BANSAL CLASSES)

QUADRATIC EQUATIONS

EXERCISE–I

Q.1 A quadratic polynomial f (x) = x2 _{+ ax + b is formed with one of its zeros being}

3 2 3 3 4   where a and b are integers. Also g (x) = x4 _{+ 2x}3 _{– 10x}2 _{+ 4x – 10 is a biquadratic polynomial such that}

          3 2 3 3 4

g  = c 3 where c and d are also integers. Find the values of a, b, c and d.d

Q.2 If (x2 _{ x cos(A + B) + 1) is a factor of the expression,}

2x4 _{+ 4x}3 _{sin A sin B  x}2_{(cos 2A + cos 2B) + 4x cos A cos B  2. Then find the other factor.}

Q.3 , are the roots of the equation K(x2 _{–x) + x+ 5 = 0. If K}

1 & K2 are the two values of K for

which the roots , are connected by the relation (/)+(/) = 4/5. Find the value of (K₁/K₂)+(K₂/K₁).

Q.4 If the quadratic equations, x2 _{+ bx + c = 0 and bx}2 _{+ cx + 1 = 0 have a common root then prove that}

either b + c + 1 = 0 or b2 _{+ c}2 _{+ 1 = bc + b + c.}

Q.5 Let a, b be arbitrary real numbers. Find the smallest natural number 'b' for which the equation x2 _{+ 2(a + b)x + (a – b + 8) = 0 has unequal real roots for all a  R.}

Q.6 Find the range of values of a, such that f (x) = ax a x a x x 2 2 2 1 9 4 8 32       ( ) is always negative.

Q.7 Find the product of uncommon real roots of the two polynomials P(x) = x4 + 2x3 – 8x2 – 6x + 15 and Q(x) = x3 _{+ 4x}2 _{– x – 10.}

Q.8 Let the quadratic equation x2 _{+ 3x – k = 0 has roots a, b and x}2 _{+ 3x – 10 = 0 has roots c, d such that}

modulus of difference of the roots of the first equation is equal to twice the modulus of the difference of the roots of the second equation. If the value of 'k' can be expressed as rational number in the lowest form as m n then find the value of (m + n).

Q.9 When y2 _{+ my + 2 is divided by (y – 1) then the quotient is f (y) and the remainder is R}

1. When

y2 _{+ my + 2 is divided by (y + 1) then quotient is g (y) and the remainder is R}

2. If R1 = R2 then find the

value of m.

Q.10 Find the value of m for which the quadratic equations x2 _{– 11x + m = 0 and x}2 _{– 14x + 2m = 0 may have}

common root.

Q.11 If the quadratic equations x2 _{+ bx+ ca = 0 & x}2 _{+ cx+ ab = 0 have a common root, prove that}

the equation containing their other root is x2 _{+ ax+ bc = 0.}

Q.12 If by eleminating x between the equation x²+ ax+ b = 0 & xy+ l(x+y)+ m = 0, a quadratic in y is formed whose roots are the same as those of the original quadratic in x. Then prove either a = 2l & b = m or b+m = al.

Q.13(a) If ,  are the roots of the quadratic equation ax2_{+bx+c = 0 then which of the following expressions}

(iii) f (, ) = ln  

(iv) f (, ) = cos ( – )

(b) If ,  are the roots of the equation x2 _{– px+ q = 0, then find the quadratic equation the roots of}

which are (2 _{ }2_)(3 _{ }3_{) & }3 _2 _{+ }2 _3_.

Q.14 Let P(x) = 4x2 + 6x + 4 and Q(y) = 4y2 – 12y + 25. Find the unique pair of real numbers (x, y) that satisfy P(x) · Q(y) = 28.

Q.15 Find a quadratic equation whose sum and product of the roots are the values of the expressions (cosec 10° – 3 sec10°) and (0.5 cosec10° – 2 sin70°) respectively. Also express the roots of this quadratic in terms of tangent of an angle lying in





Q.16 Find the product of the real roots of the equation, x2 _{+ 18x + 30 = 2 x}2_18x45

Q.17 We call 'p' a good number if the inequality

1 x x 3 x 2 x 2 2 2    

 p is satisfied for any real x. Find the smallest integral good number.

Q.18 Find the values of ‘a’ for which 3< [(x2 _+ax2)/(x2 _{+x+1)] < 2 is valid for all real x.}

Q.19 If the range of the function f (x) =

3 x 2 x b ax x 2 2    

is [–5, 4], a, b  N, then find the value of (a2 _{+ b}2_).

Q.20 Suppose a, b, c  I such that greatest common divisor of x2 _{+ ax + b and x}2 _{+ bx + c is (x + 1) and}

the least common multiple of x2 _{+ ax + b and x}2 _{+ bx + c is (x}3 _{– 4x}2 _{+ x + 6). Find the value of}

(a + b + c).

Q.21 Let a, b, c and m  R+. Find the range of m (independent of a, b and c) for which atleast one of the following equations.               0 bm ax cx and 0 am cx bx 0 cm bx ax 2 2 2

have real roots.

Q.22 If a & b are positive numbers, prove that the equation

b x 1 a x 1 x 1   

 = 0 has two real roots, one between a/3 & 2a/3 and the other between –2b/3 & –b/3.

Q.23 If the roots of x2 _{ ax+ b = 0 are real & differ by a quantity which is less than c(c > 0), prove that}

b lies between (1/4)(a2 _{ c}2_{) & (1/4)a}2_.

Q.24 Find all real numbers x such that, 2

1 x 1 x        + 2 1 x 1 1        = x.

Q.25 Find the minimum value of 3 3 3 6 6 6 x 1 x x 1 x 2 x 1 x x 1 x                          for x > 0.

ANSWER KEY

EXERCISE–I

Q.10 0 or 24 Q.13 (a) (ii) and (iv) ; (b) x2 _{ p(p}4 _{ 5p}2_{q + 5q}2_{)x+ p}2_q2_(p2 _{ 4q)(p}2 _{q) = 0}

Q.14        2 3 , 4 3 Q.15 x2 _{– 4x + 1 = 0; = tan }       12 ;  = tan       12 5 Q.16 20 Q.17 4 Q.18 2 < a < 1 Q.19 277 Q.20 – 6 Q.21 m  _      4 1 , 0 _{Q.24 x =} 2 1 5  Q.25 y_min = 6