Class 11 Mathematics Limits And Derivatives MCQs Set D

Question. Evaluate lim_x→0 Sin² 2x/x² .
(a) 4
(b) – 4
(c) sin x
(d) cos x
Answer: a

Question. If a and b are fixed non-zero constants, then the derivative of (ax + b)ⁿ is
(a) n(ax + b)^{n – 1}
(b) na(ax + b)^{n – 1}
(c) nb(ax + b)^{n – 1}
(d) nab(ax + b)^{n – 1}
Answer: b

Question. If f (x) = x sin x, then f ‘ (π/2) is equal to
(a) 0
(b) 1
(c) – 1
(d) 1/2
Answer: b

Question. The value of lim_x→0 x³ cot x / 1− cos x is
(a) 1
(b) –2
(c) 2
(d) 0
Answer: c

Question. If x³ + y³ = 3xy, then value of dy/dx is
(a) x−y²/x² − y
(b) x² − y/x−y²
(c) x² − y²/x−y
(d) x−y/x² − y²
Answer: b

Question. If sin y = x sin (a + y), then value of dy/dx is
(a) sin(a+y)/sin a
(b) sin² (a+y)/sina
(c) sin² (a+y)
(d) cos² (a+y)/sin a
Answer: b

Question. What is the value of lim _x→0x sin 5x/sin²4x ?
(a) 0
(b) 5/4 
(c) 5/16 
(d) 25/4
Answer: c

Question. If y = ax^{n +1} + bx ^{– n} , then value of x² d²y/dx² is
(a) n (n + 1) y
(b) n² (n + 1) y
(c) 2n (n – 1) y
(d) None of these
Answer: a

Question. If f (x) = x + | x |/x, then the value of lim_x→0 f (x) is
(a) 0
(b) 2
(c) does not exist
(d) None of these
Answer: c

Question. lim_x→0 2sin²3x/x² is equal to :
(a) 12
(b) 18
(c) 0
(d) 6
Answer: b

Question. Value of lim_x→0 sin 4x/sin 2x is
(a) 1
(b) 2
(c) 4
(d) None of these
Answer: b

Question. If lim_x→0 sin(2+x)−sin(2−x)/x is equal to p cos q, then sum of p and q is
(a) 2
(b) 1
(c) 3
(d) 4
Answer: d

Question. lim_x→0 sin 2x/x is equal to
(a) 2
(b) 0
(c) 1
(d) 3
Answer: a

Question. Derivative of the function f (x) = 7x^–3 is
(a) 21x^–4
(b) –21x^–4
(c) 21x⁴
(d) –21x⁴
Answer: b

Question. The derivative of the function f (x) = 3x at x = 2 is
(a) 0
(b) 1
(c) 2
(d) 3
Answer: d

Question. If a is a fixed non-zero constant, then the derivative of sin(x+a)/cosx is
(a) cosa / cos²x
(b) − cosa / cos²x
(c) sina / cos²x
(d) − sina / cos²x
Answer: a

Question. If lim_x→a[f (x)/g(x)] exists, then which one of the following correct ?
(a) Both lim_x→a f(x) and lim_x→a g(x) must exist
(b) lim_x→af (x) need not exist but lim_x→a g(x) must exist
(c) Both lim_x→a f (x) and lim_x→a g(x) need not exist
(d) lim_x→a f (x) must exist but lim_x→a g(x) need not exist
Answer: a

Question. Consider the function h (x) = x²−4/x−2 ,x≠2 Then,
I. h (2) is not defined.
II. lim_x→2h(x) = 4.
Which of the following is/are true?
(a) Both I and II are true
(b) Only I is true
(c) Only II is true
(d) Both I and II are false
Answer: a

Question. Which of the following is/are true?
I. The derivative of – x is –1.
II. The derivative of (–x)^–1 is 1/x²
(a) Both I and II are true
(b) Only I is true
(c) Only II is true
(d) Both I and II are false
Answer: a

Question. Which of the following is/are true?
I. The derivative of y = 2x−3/4 is 2.
II. The derivative of y = (5x³ + 3x – 1) (x – 1) is 20x³ + 15x² + 6x – 4
(a) Both I and II are true
(b) Only I is true
(c) Only II is true
(d) Both I and II are false
Answer: b

Question. Which of the following is/are true?
I. The derivative of f (x) = sin 2x is 2(cos² x – sin² x).
II. The derivative of g (x) = cot x is – cosec² x.
(a) Both I and II are true
(b) Only I is true
(c) Only II is true
(d) Both I and II are false
Answer: a

Question. Which of the following is/are true?
I. The derivative of f(x) = 1 + x + x² + … + x⁵⁰ at x = 1 is 1250.
II. The derivative of f (x) = x+1/x is 1/x² .
(a) Both I and II are true
(b) Only I is true
(c) Only II is true
(d) Both I and II are false
Answer: d