MATHEMATICS SCHOLEIO

MATHS (IIT-JEE)
QUADRATIC & LOGARITHMS WORK SHEET–1

Single Correct
Q.1 If 1, 2, 3, ….., 2019 are distinct zeroes of the polynomial P(x) = x2019 + 19x2018 + 1

 1 
and f(x) be another polynomial whose degree is 2019 and satisfying f  i  0
 i 
361f (1)
for i = 1, 2, 3, ………., 2910. Then the value of is
f (1)
(A) 237 (B) 341 (C) 327 (D) 289

Q.2 The number of ‘n’ such that logn (32013) is an integer is, n  N

(A) 8 (B) 7 (C) 1 (D) 3

Q.3 Let P(x) = 0 be fifth degree polynomial equation with integer coefficient that has atleast

one integral root. If P(2) = 13 and P(10) = 5. Then the number of integral values of x such

that P(x) can be equal to be zero is :

(A) 0 (B) 1 (C) 2 (D) 3

Q.4 Let , ,  and  are roots of x4 + x3 – 3x2 + 8 = 0 then the value of

1 1 1 1
2
 2  2  2 is equal to :
  6  8   6  8   6  8   6  8

3 3 3
(A) (B) (C) 0 (D)
5 5 10

1 1
Q.5 If x, y, z are the roots of t3 – 2t2 +
2
t – 4 = 0 then the value of  xy  z  1 equals
(A) 1/9 (B) 2/9 (C) –1/9 (D) –2/9
Q.6 If f(x) is a polynomial of degree 4 with leading coefficient 1 and
 f (5)  f (1) 
f(1) = 1, f(2) = 2, f(3) = 3 then the value of   equals ([ . ] denotes GIF)
 f (0)  f (4) 
(A) 0 (B) 5 (C) 7 (D) 9

One or More than one correct

Q.7 Let f : R  R, f(x) = x3 + 3x2 – x + 1 and  is a real root of f(x) = 0. Which of following is

INCORRECT?

(A) sin–1(sin[]) + cos–1(cos[]) = 8 –  (where [·] denotes greatest integer function)

(B) y = f(x) is strictly decreasing function

(C) f(x) = 0 has 3 distinct real roots

(D) sin–1 (sin[]) + cos–1(cos[]) =  (where [·] denotes greatest integer function)

Q.8 Which of the following is/are CORRECT?

(A) If x2 + ax + bc = 0 and x2 + bx + ca = 0 (a  b) have a common root, then their other

roots satisfy the equation x2 + cx + ab = 0.

(B) If the equations x3 – mx2 – 4 = 0 and x3 + mx + 2 = 0 (m  R) have one common root,

then the value of ‘m’ is equal to ‘–3’.

(C) If there exists atleast one common ‘x’ which satisfies the equation x2 + 3x + 5 = 0 and
ax2 + bx + c = 0; a, b, c  N. The minimum value of a + b + c = 9.
(D) If a, b, c be the sides of ABC and equations ax2 + bx + c = 0 and 5x2 + 12x + 13 = 0
have a common root, then C = 90°.

Q.9 The given equation is 3x4 – 2x3 + 4x2 – 4x + 12 = 0. Now, which of the following option

is/are CORRECT?

(A) The given equation has all its roots are real and distinct

(B) The given equation has 2 roots real and pair of imaginary roots

(C) The given equation has no real roots

(D) The sum of real parts of imaginary roots of the equation is 2/3
Q.10 If p(x) is polynomial with rational coefficient so that for all |x|  1;

  x  3  3x 2 
p(x)  p   , then
 2 
 

(A) p(x) = a0 + a1(3x – 4x3)1 + a2(3x – 4x3)2 + …….. + (3x – 4x3)k k(x) where k(x) is a

polynomial with rational coefficient

(B) p'(x) = a1  '(x) + 2a2(x) '(x) + ……….... + k[(x)]k–1 '(x) k(x + [(x)]k k'(x)

where f(x) is a polynomial with rational coefficient and (x) = 3x – 4x3

(C) can ' t be predicted

(D) none of these

Q.11 If P(x), Q(x) are polynomials with rational coefficients of least possible degree such that

P(x)x3 + P(x)x2 + Q(x)x + 2Q(x) = 1 then which is/are correct

1 1 1
(A) P(1) = 2 (B) P(0) = – (C) Q(1) = (D) Q(0) =
4 2 2

Q.12 The equation x4 – 2ax2 + x + a2 – a = 0, has four distinct real roots if

 1
(A) a  [0, 2] (B) a  [1, 3] (C) a   0,  (D) a  [2, 4]
 2

Q.13 Let f(x) = ax2 + bx + c, a  0, a, b, c  R, f(x) = 0 has two real and distinct roots  and .

If f(x + k) + f(x) = 0(k,   R) has exactly one root between  and  then which of the

following is/are correct?

(A) f( + k) f( + k) < 2

(B) k does not depend upon the value of 

(C) k can take any real value

(D) 0  | ak |   where is discriminant of f(x) = 0

Q.14 If roots of equation ax2 + bx + c = 0, are distinct and less then unity. Then which of the

following is/are true?

9a  6b  4c 9a  3b  c
(A) is positive (B) is positive
25a  10b  4c a bc

9a  6b  4c c
(C) is positive (D) is positive
25a  10b  4c 4a  2b  c

Q.15 P(x) = 0 is a cubic polynomial equation with integral coefficients that has atleast one

integral root. If P(2) = 5 and P(4) = 3 then identify correct option?

(A) Sum of all possible integral roots is 11

(B) Number of possible values of integral roots are 3

(C) Number of possible integral positive roots are 2

(D) All possible real roots are odd

Q.16 The equation x5 + 5x4 – x3 + ( – 4)x2 – (8 + 3)x +  – 2 = 0 is to be considered.

Let  = p such that the given equation has exactly one root independent of  and  = q

such that the given equation has exactly two roots independent of . Which of the

following is/are correct ? ([.] denotes G.I.F, ,   R).

p p
(A)    3 (B)    4 (C) [p + q] = –16 (D) [p + q] = –15
q  q 

Q.17 a + b + c = 6, ab + bc + ca = 11, (a, b, c  R). Then which of the following is/are true?

([.] denotes G.I.F)

(A) Minimum value of [a] is 1 (B) maximum value of [b] is 3

(C) Minimum value of [abc] is 5 (D) maximum value of [abc] is 6

Q.18 Let quadratic equations.
(p – 1)x2 – (p2 + 2)x + (p2 + 2p) = 0 & (q – 1)x2 – (q2 + 2)x + (q2 + 2q) = 0 where
p, q are distinct natural numbers, have a common root; then
(A) p.q = 6

(B) p.q = 7
(C) Minimum value of  x  p  x  q  x  0 is 2

(D) Number of integral solutions of x  p  x  q  2 x  0 is 3

Q.19 let f(x) = x3 + px2 + qx + r, where p, q and r are integers, f(0) and f(–1) are odd
integers. Which of the following is/are CORRECT?
(A) f(1) is an even integer

(B) f(1) is an odd integer

(C) f(x) = 0 has three distinct integer roots

(D) f(x) = 0 cannot have three integer roots

8 2
Q.20 If |f(x)| = |ax2 + bx + c|  1 for |x|  1 (a, b, c)  R and a + 2b2 is maximum, then
3
which of the following is/are true?
c 1
(A) b = 0 (B) |a + c| = 1 (C) ac = 1 (D) 
a 2

Q.21 Let 1 <  < 2 <  < 3 and f(x) = (x + ) (x – ) + 2 then select the correct option(s) :
(A) equation f(x) = 0 has real and distinct roots

(B) equation 5 | f (x) |  1 has 4 solutions

(C) equation 3 | f (x) |  25 can have 4 solutions

(D) equation f (| x |)  1 can have 4 solutions

Numerical Type

Q.22 If 1, 2, 3, 4, are the roots of x4 + 2x3 + bx2 + cx + d = 0 such that 1 – 3 = 4 – 2,
then b – c is equal to
Q.23 If the quadratic equation a1 x2 – a2 x + a3 = 0 where a1, a2, a3  N has two distinct real

roots belonging to the interval (1, 2), then the least value of ‘a1 ’ is

Q.24 The graph of the function f(x) = x10 + 9x9 + 7x8 + a7x7 ……….... + a1x + a0 intersects

the line y = b at the points B1, B2, B3, …….…., B10 (from left to right) and the line

y = c at the points C1, C2, C3, ……..….., C10 (from left to right). Let P be a point on

the line y = c to the right of the point C10. if b = 5 and c = 3, then find the sum

10
 cot   Bn Cn P  .
n 1

Q.25 If k1 and k2 (k1 > k2) are two non-zero integral values of ‘k’ for which the cubic

equation x3 + 3x2 + k = 0 has all integers roots, then the value of k1 – k2 is equal to

Q.26 The real numbers ,  satisfy the equation

3 – 32 + 5 – 17 = 0; 3 – 32 + 5 + 11 = 0. Then  +  is equal to

Q.27 Let f(x) = ax2 + bx + c, (a < b) and f(x)  0  x  R. if the minimum value of

abc
is k then value of k is
ba

Q.28 Suppose that the polynomial x2 + ax + b has the property such that if s is a root,

then s2 – 6 is a root. What is the largest possible value of a + b?

Q.29 Let a, b and c be the roots of the polynomial x3 – 3x2 – 4x + 5. Compute

a 4  b4 b 4  c4 c4  a 4
  .
ab bc ca

Q.30 Find the sum of the coefficient of the polynomial P(x) = x4 – 29x3 + ax2 + bx + c,

given that P(5) = 11, P(11) = 17, and P(17) = 23.

Q.31 Find the sum of all real solutions for x to the equation

2
2
 2x  3) (x  2x  3)
(x 2  2x  3)(x  2012.

Q.32 If p and q are zeroes of the polynomial f(x) = x2 – 3px + p2 – q, p, q  R – {0} and

m
minimum value of f(f(f(x))) is m then find the value of   .
5

Q.33 Given, the equation x3 – 75x + a = 0 has three integral roots ,  and  (where   I)

| | || |  |
then   is equal to
 4 

Q.34 Find maximum value of 8.27log 6 x  27.8log6 x  x 3 ; x > 0

x
1
Q.35 Number of solutions of log 1 x   
16  16 

ANSWER KEY
Qus. 1 2 3 4 5 6 7 8 9 10

Ans. C A B A
D D B ABC ABCD CD
Qus. 11 12 13 14 15 16 17 18 19 20

Ans. B,C,D B,D ABD

BCD BD ABD BCD ABD ABCD
Qus. 21 22 23 24 25 26 27 28 29 30

Ans. 1 5 0 8
AB 2 3 8 869/7 -3193
Qus. 31 32 33 34 35

Ans. 5 216 3
-2 7