1. Let f(x) = x^{2} + bx + c, where b, c ∈ R. If f(x) is a factor of both x^{4} + 6x^{2} + 25 and 3x^{4} + 4x^{2} + 28x + 5, then the least value of f(x) is

(A) 2

(B) 3

(C) 5/2

(D) 4

2. Let a, b, c be the sides of a triangle. No two of them are equal and λ ∈ R. If the roots of the equation x^{2} + 2(a + b+ c) x + 3λ (ab + bc + ca) = 0 are real, then

(A) λ < 4/3

(B) λ > 5/3

(C) λ ∈(1/3 , 5/3)

(D) λ ∈(4/3 , 5/3)

3. Let f(x) = x^{2} + ax + b be a quadratic polynomial in which a and b are integers. If for a given integer n, f(n) f(n + 1) = f(m) for some integer m, then the value of m is

(A) n(a + b) + ab

(B) n^{2} + an + b

(C) n(n + 1) + an + b

(D) n^{2} + n + a + b

4. If the equations x^{2} + ax + b=0 and x^{2} + bx + a = 0 have exactly one common root, then the numerical value of a + b is

(A) 1

(B) –1

(C) 0

(D) none of these

5. The number of ordered pairs of positive integers x, y such that x^{2} + 3y and y^{2} + 3x are both perfect squares is

(A) 2

(B) 3

(C) 4

(D) 5

6. For the equations x^{2} + bx + c = 0 and 2x^{2} + (b + 1)x + c + 1 = 0 select the correct alternative

(A) both the equations can have integral roots

(B) both the equations can’t have integral roots simultaneously

(C) none of the equations can have integral roots

(D) nothing can be said

7. If x^{2} +ax +b is an integer for every integer x then

(A) ‘ a ‘ is always an integer but ‘ b ‘ need not be an integer.

(B) ‘ b ‘ is always an integer but ‘ a ‘ need not be an integer.

(C) a + b is always an integer.

(D) none of these.

8. If a , b , c be the sides of ΔABC and equations ax^{2} + bx + c=0 and 5x^{2} + 12x + 13=0 have a common root, then ∠C is

(A) 60°

(B) 90°

(C) 120°

(D) 45°

9. The equation x^{2} + nx + m = 0, n, m ∈ I, can not have

(A) integral roots

(B) non-integral rational roots

(B) irrational roots

(D) complex roots

10. If $\large log_{3x+5}(ax^2 + 8x + 2) > 2 $ then x lies in the interval

(A) (-4/3 , -20/11)

(B) (-4/3 , -23/22)

(C) (-5/3 , -23/22)

(D) None of these

__ANSWER:__

**1. (D) 2. (A) 3. (C) 4. (B) 5. (B) 6. (B) 7. (C) 8. (B) 9. (B) 10. (B) **

11. The number of ordered pairs (a, b) such that the equations ax + by = 1 and xλ + yλ = 50 have all solutions integral is

(A) 72

(B) 66

(C) 84

(D) 36

12. If the roots of the equation (a^{2} + b^{2}) x^{2} + 2x (ac + bd) + c^{2} + d^{2} = 0, are real, then these are equal. This statement is (a, b, c, d ∈ R)

(A) true

(B) false

(C) can’t say

(D) none of these

13. If equation x^{2} – (2 + m)x + (m^{2} – 4m + 4) = 0 has coincident roots then

(A) m = 0, 1

(B) m = 2/3 , 1

(C) m = 0, 2

(D) m = 2/3 , 6

14. A root of the equation, sinx + x – 1 = 0, lies in the interval

(A) (0, π/2)

(B) (- π/2, 0)

(C) (π/2, π)

(D) ( -π, -π/2)

15. If p, q, r ∈ R and the quadratic equation px^{2} + qx + r = 0 has no real root, then

(A) p(p + q + r) < 0

(B) p(p – 2q + 4r)

(C) p(p + 4q + 2r) < 0

(D) None of these

16. If x^{2} – 4x + log_{1/2} a = 0 does not have two distinct real roots, then maximum value of a is

(A) 1/4

(B) 1/ 16

(C) –1/4

(D) none of these

17. The least value of |a| for which tan q and cot q are the roots of the equation x^{2} + ax + b = 0 is

(A) 2

(B) 1

(C) 1/2

(D) 0

18. If the equation x^{3} – 3ax^{2} + 3bx – c = 0 has positive and distinct roots, then

(A) a^{2} > b

(B) ab > c

(C) a^{3} > c

(D) a^{3} > b^{2} > c

19. The value of a for which exactly one root of the equation e^{a}x^{2} – e^{2a}x + e^{a} – 1 = 0 lies between 1 and 2 are given by

(A) $\large ln(\frac{5-\sqrt{13}}{4}) < a < ln(\frac{5+\sqrt{13}}{4})$

(B) 0 < a < 100

(C)

(D) None of these

20. If α, β be the roots of x^{2} – a(x – 1) – b = 0, then the value of $\large \frac{1}{\alpha^2 – a \alpha} + \frac{1}{\beta^2 – a \beta} + \frac{2}{a+b}$ is

(A) 4/(a+b)

(B) 1/(a+b)

(C) 0

(D) 1

__ANSWER:__

**11. B 12. A 13. D 14. A 15. B 16. B 17. A 18. A 19. D 20. C **

21. Consider the equation x^{3} – nx + 1 =0, n ∈ N , n ≥ 3 . Then

(A) Equation has atleast one rational root .

(B) Equation has exactly one rational root.

(C) Equation has all real roots belonging to (0, 1).

(D) Equation has no rational root.

22. If sina, sin b and cosa are in GP, then roots of x^{2} + 2xcotβ + 1 = 0 are always

(A) equal

(B) real

(C) imaginary

(D) greater than 1

23. If a, b ∈ (0, 2) and the equation $\frac{x^2 + 5}{2} = x-2 cos(ax+b)$ has at least one solution then a + b is

(A) 1

(B) 2

(C) e

(D) π

24. Let P(x) and Q(x) be two polynomials. If f(x) = P(x^{4}) + xQ(x^{4}) is divisible by x^{2} +1, then (A) P(x) is divisible by (x-1)

(B) Q(x) is divisible by (x-1)

(C) f(x) is divisible by (x-1)

(D) all of them

25. If α, β be the roots of 4x^{2} – 16x + λ = 0, λ ∈ R such that 1 < α < 2 and 2 < β < 3, then the number of integral solutions of λ is

(A) 5

(B) 6

(C) 3

(D) 2

26. The solution of the equation |x + 1|^{2} – 2|x + 2| – 26 = 0 is

(A) ±7

(B) –7, √29

(C) ±√29

(D) –7, 29

27. The roots of the equation (c^{2} –ab)x^{2} – 2(a^{2} –bc)x + (b^{2} – ac) =0 are equal then

(A) a^{2} + b^{3} + c^{3} = 3abc or a = 0

(B) a + b + c = 0

(C) a^{2} + b^{3} + c^{3} = 3abc or a = 1

(D) none of these

28. If (λ^{2} + λ – 2)x^{2} + (λ + 2)x < 1, x ∈ R, then λ belongs to the interval

(A) (-2, 1)

(B) (-2 , 2/5)

(C) (2/5 , 1)

(D) None of these

29. If expression x^{2} – 4cx + b^{2} > 0 ∀ x ∈ R and a^{2} + c^{2} < ab, then range of the function $\frac{x+a}{x^2 + bx+ c^2}$ is

(A) (- ∞, 0)

(B) (0, ∞)

(C) (- ∞, ∞)

(D) None of these

30. If for all real values of x , $ \frac{4x^2 + 1}{64x^2 – 32x sin\alpha + 29} > \frac{1}{32}$

then α lies in the interval

(A) (0, π/3)

(B) (π/3 , 2π/3)

(C) (4π/3 , 5π/3)

(D) none of these

__ANSWER:__

**21. A 22. B 23. D 24. D 25. C 26. B 27. A 28. D 29. C 30. A **