MCQ : Application of derivative

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| + bx2 + 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 y2 = 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) = xa 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) = (a2 -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 x2 + b y2 = 1 and a1 x2 + b1 y2 = 1 intersect orthogonally then

(A)1/a – 1/b = 1/a1 + 1/b1

(B)1/a + 1/b = 1/a1 + 1/b1

(C)1/a + 1/b = 1/a1 + 1/b1

(D) none of these

Q:9. If the equation x5 – 10a3 x2 + b4 x + c5 = 0 has three equal roots,  then

(A) 2b2 –10 a3b2+c5 = 0

(B) 6a5 + c5 = 0

(C) 2c5 –10 a3b2+ b4c5 = 0

(D) b4 = 15 a5

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) = x3 + bx2 + cx + d and 0 < b2 < 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! + x2/2! + x3/3! …………+ xn/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 + a2x + 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)= x2e-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 x3-ax2 + 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 = 3x1 logx and y = xx -1 is

(A) 1

(B) 1/2

(C) 0

(D) 1/3

Q:20. The equation 8x3 – ax2 + bx –1 = 0 has three real roots in G.P. If λ1 ≤ a ≤ λ2 , then ordered  pair  (λ1,  λ2) 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)