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 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 **