Q:1. Equation of normal to the curve where it cuts x – axis is

(A) x + y = 1

(B) x – y = 1

(C) x + y = 0

(D) None of these

Q:2. If y = a ln |x| + bx^{2} + x has its extreme values at x = -1 and x = 2 then P ≡ (a , b) is

(A) (2 , -1)

(B) (2 , -1/2)

(C) (-2 , 1/2)

(D) none of these

Q:3. f(x) = tan^{-1} (sinx + cosx) is an increasing function in

(A) (–π/2, 0)

(B) (0, π/2)

(C) (-π/4, π/4)

(D) none of these

Q:4. If the parabola y^{2} = 4x meets a circle with centre at (6,5) orthogonally, then possible point (s) of intersection can be;

(A) (4, 4)

(B) (9, 4)

(C) (2, √8)

(D) (3, 2)

Q:5. If f (x) = x^{a} log x and f(0) = 0, then the value of ‘ a ‘ for which Rolle’s theorem can be applied in [0, 1] is

(A) -2

(B) -1

(C) 0

(D) 1/2

Q:6. In [0, 1] Lagrange Mean Value theorem is NOT applicable to

(A)$ \large f(x) = \left\{\begin{array}{ll} \frac{1}{2}-x \; , x < 1/2 \\ (\frac{1}{2}-x)^2 \; , x \ge 1/2 \end{array} \right. $

(B) $ \large f(x) = \left\{\begin{array}{ll} \frac{sinx}{x} \; , x \ne 0 \\ 1 \; , x = 0 \end{array} \right. $

(C) f(x) = x|x|

(D) f(x) = |x|

Q:7. For what value of ‘a’ does the curve f(x) = (a^{2} -2a -2 ) + cosx is always strictly monotonic for all x ∈ R

(A) a ∈ R

(B) a > 0

(C) 1-√2 < a < 1 + √2

(D) None of these

Q:8. If the curves a x^{2} + b y^{2} = 1 and a_{1} x^{2} + b_{1} y^{2} = 1 intersect orthogonally then

(A)1/a – 1/b = 1/a_{1} + 1/b_{1}

(B)1/a + 1/b = 1/a_{1} + 1/b_{1}

(C)1/a + 1/b = 1/a_{1} + 1/b_{1}

(D) none of these

Q:9. If the equation x^{5} – 10a^{3} x^{2} + b^{4} x + c^{5} = 0 has three equal roots, then

(A) 2b^{2} –10 a^{3}b^{2}+c^{5} = 0

(B) 6a^{5} + c^{5} = 0

(C) 2c^{5} –10 a^{3}b^{2}+ b^{4}c^{5} = 0

(D) b^{4} = 15 a^{5}

Q:10. The function f is a differentiable function and satisfies the functional equation f(x) + f(y) = f(x + y) – xy – 1for every pair x, y of real numbers. If f(1) = 1, then the number of integers n ≠ 1 for which f(n) = n is

(A) 0

(B) 1

(C) 2

(D) 3

**Answer:**

1. (B) 2. (B) 3. (C) 4. (A) 5. (D)

6. (A) 7. (C) 8. (A) 9. (B) 10. (B)

Q:11. Let f(x) = x – sinx and g(x) = x – tanx where x∈ (0, π/2) . Then for these values of x ;

(A) f(x). g(x)>0

(B) f(x) g(x) <0

(C)f(x)/ g(x)>0

(D) none of these

Q:12. If f (x) = x^{3} + bx^{2} + cx + d and 0 < b^{2} < c, then in (-∞ , ∞)

(A) f (x) is a strictly increasing function

(B) f(x) has a local maxima

(C) f(x) is a strictly decreasing function

(D) f(x) is bounded

Q:13. If f(x) = 1 + x/1! + x^{2}/2! + x^{3}/3! …………+ x^{n}/n! , then f(x) = 0 (n is odd , n≥3)

(A) can’t have any real root

(B) can’t have any repeated root

(C) has one positive root

(D) none of these

Q:14. If f (x) = sin x + a^{2}x + b is an increasing function for all values of x, then

(A) a ∈ (-∞,-1)

(B) a ∈ R

(C) a ∈ (-1 , 1)

(D) none of these

Q:15. If f(x)= x^{2}e^{-x2/a2} is an increasing function then (for a > 0), x lies in the interval

(A) [a , 2a]

(B) (-∞, -a] ∪ [0 , a]

(C) (-a, 0)

(D) None of these

Q:16. Let f(x) = |x-1| + |x-2| + |x-3| + |x-4| ∀x ∈ R . Then

(A) x = 2 is the point of local minima

(B) x = 3 is the point of local minima

(C) x = 1 is the point of local minima

(D) none of these

Q:17. If a, b are real numbers such that x^{3}-ax^{2} + bx – 6 = 0 has its roots real and positive then minimum value of b is

(A) 1

(B) 2

(C) 3(36)^{1/3}

(D) None of these

Q:18. If f'(x) exists for all x∈R and g(x) = f(x) – (f(x))^{2} + (f(x))^{3} ∀ x ∈R , then

(A) g(x) is increasing whenever ‘ f ’ is increasing

(B)g(x) is increasing whenever ‘ f ’ is decreasing

(C)g(x) is decreasing whenever ‘ f ’ is increasing

(D)none of these

Q:19. Cosine of the angle of intersection of curves y = 3^{x}^{–}^{1} logx and y = x^{x} -1 is

(A) 1

(B) 1/2

(C) 0

(D) 1/3

Q:20. The equation 8x^{3} – ax^{2} + bx –1 = 0 has three real roots in G.P. If λ_{1} ≤ a ≤ λ_{2 }, then ordered pair (λ_{1}, λ_{2}) can be

(A) (-2, 2)

(B) (18, 12)

(C) (-10, -8)

(D) none of these

**Answer:**

11.( B) 12. (A) 13. (B) 14. (A) 15. (B)

16. (D) 17. (C) 18. (A) 19. (A) 20. (C)