(Quadratic Equations 2

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MANOJ CHAUHAN SIR(IIT-DELHI) EX. SR. FACULTY (BANSAL CLASSES)

QUADRATIC EQUATIONS EXERCISE–I Q.1

43 3 where a and 2 3 b are integers. Also g (x) = x4 + 2x3 – 10x2 + 4x – 10 is a biquadratic polynomial such that

A quadratic polynomial f (x) = x2 + ax + b is formed with one of its zeros being

43 3    =  c 3  d where c and d are also integers. Find the values of a, b, c and d. g   2  3  

Q.2

If (x2  x cos (A + B) + 1) is a factor of the expression, 2x4 + 4x3 sin A sin B  x2(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 K1 & 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 bx2 + cx + 1 = 0 have a common root then prove that either b + c + 1 = 0 or b2 + c2 + 1 = b c + 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

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

Q.7

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

Q.8

Let the quadratic equation x2 + 3x – k = 0 has roots a, b and x2 + 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). When y2 + my + 2 is divided by (y – 1) then the quotient is f (y) and the remainder is R1. When y2 + my + 2 is divided by (y + 1) then quotient is g (y) and the remainder is R2. If R1 = R2 then find the value of m.

Q.9

Q.10

Find the value of m for which the quadratic equations x2 – 11x + m = 0 and x2 – 14x + 2m = 0 may have common root.

Q.11

If the quadratic equations x2 + bx + ca = 0 & x2 + 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 in ,  will denote the symmetric functions of roots. Give proper reasoning. (i) f (, ) = 2 –  (ii) f (, ) = 2 + 2 ETOOS Academy Pvt. Ltd. : F-106, Road No. 2, Indraprastha Industrial Area, End of Evergreen Motors 2 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005)

(iii)

f (, ) = ln

 

f (, ) = cos ( – )

(iv)

(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 0,  2 . 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

2x 2  2x  3  p is satisfied for any real x. Find the smallest x2  x 1

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.

x 2  ax  b Q.19 If the range of the function f (x) = 2 is [–5, 4], a, b  N, then find the value of (a2 + b2). x  2x  3 Q.20

Suppose a, b, c  I such that greatest common divisor of x2 + ax + b and x2 + bx + c is (x + 1) and the least common multiple of x2 + ax + b and x2 + bx + c is (x3 – 4x2 + 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.

ax 2  bx  cm  0   bx 2  cx  am  0  have real roots. and cx 2  ax  bm  0 1 1 1   = 0 has two real roots, x xa xb one between a/3 & 2a/3 and the other between – 2b/3 & – b/3.

Q.22

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

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  c2) & (1/4)a2. 1

Q.24

1

1 2 1 2 Find all real numbers x such that,  x   + 1   = x. x   x

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6

Q.25

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

ANSWER KEY EXERCISE–I a = 2, b = – 11, c = 4, d = – 1

Q.5

5

Q.10

0 or 24

Q.14

 3 3  ,   4 2

Q.15

  5  x2 – 4x + 1 = 0; = tan   ;  = tan    12   12 

Q.16

20

Q.18  2 < a < 1

Q.19

277

 1 m   0,   4

Q.24

x=

Q.6

Q.2

2x2 + 2x cos (A  B)  2

Q.1

Q.3 254

1  a    ,  Q.7 6 Q.8 191 Q.9 0 2  Q.13 (a) (ii) and (iv) ; (b) x2  p(p4  5p2q + 5q2) x + p2q2(p2  4q) (p2  q) = 0

Q.20

–6

Q.21

Q.17

4

5 1 2

Q.25 ymin = 6

ETOOS Academy Pvt. Ltd. : F-106, Road No. 2, Indraprastha Industrial Area, End of Evergreen Motors 4 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005)

ETOOS Academy Pvt. Ltd. : F-106, Road No. 2, Indraprastha Industrial Area, End of Evergreen Motors 5 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005)