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Limits and Derivatives Class 11 MCQs Questions with Answers
Solving the MCQ Questions of Limits and Derivatives Class 11 with answers can help you understand the concepts better.
Question 1.
The expansion of log(1 – x) is
(a) x – x² /2 + x³ /3 – ……..
(b) x + x² /2 + x³ /3 + ……..
(c) -x + x² /2 – x³ /3 + ……..
(d) -x – x² /2 – x³ /3 – ……..
Answer
Answer: (d) -x – x² /2 – x³ /3 – ……..
log(1 – x) = -x – x² /2 – x³ /3 – ……..
Question 2.
The value of Limx→a (a × sin x – x × sin a)/(ax² – xa²) is
(a) = (a × cos a + sin a)/a²
(b) = (a × cos a – sin a)/a²
(c) = (a × cos a + sin a)/a
(d) = (a × cos a – sin a)/a
Answer
Answer: (b) = (a × cos a – sin a)/a²
Given,
Limx→a (a × sin x – x × sin a)/(ax² – xa²)
When we put x = a in the expression, we get 0/0 form.
Now apply L. Hospital rule, we get
Limx→a (a × cos x – sin a)/(2ax – a²)
= (a × cos a – sin a)/(2a × a – a²)
= (a × cos a – sin a)/(2a² – a²)
= (a × cos a – sin a)/a²
So, Limx→a (a × sin x – x × sin a)/(ax² – xa²) = (a × cos a – sin a)/a²
Question 3.
Limx→-1 [1 + x + x² + ……….+ x10] is
(a) 0
(b) 1
(c) -1
(d) 2
Answer
Answer: (b) 1
Given, Limx→-1 [1 + x + x² + ……….+ x10]
= 1 + (-1) + (-1)² + ……….+ (-1)10
= 1 – 1 + 1 – ……. + 1
= 1
Question 4.
The value of the limit Limx→0 {log(1 + ax)}/x is
(a) 0
(b) 1
(c) a
(d) 1/a
Answer
Answer: (c) a
Given, Limx→0 {log(1 + ax)}/x
= Limx→0 {ax – (ax)² /2 + (ax)³ /3 – (ax)4 /4 + …….}/x
= Limx→0 {ax – a² x² /2 + a³ x³ /3 – a4 x4 /4 + …….}/x
= Limx→0 {a – a² x /2 + a³ x² /3 – a4 x³ /4 + …….}
= a – 0
= a
Question 5.
The value of the limit Limx→0 (cos x)cot² x is
(a) 1
(b) e
(c) e1/2
(d) e-1/2
Answer
Answer: (d) e-1/2
Given, Limx→0 (cos x)cot² x
= Limx→0 (1 + cos x – 1)cot² x
= eLimx→0 (cos x – 1) × cot² x
= eLimx→0 (cos x – 1)/tan² x
= e-1/2
Question 6.
Then value of Limx→1 (1 + log x – x)}/(1 – 2x + x²) is
(a) 0
(b) 1
(c) 1/2
(d) -1/2
Answer
Answer: (d) -1/2
Given, Limx→1 (1 + log x – x)}/(1 – 2x + x²)
= Limx→1 (1/x – 1)}/(-2 + 2x) {Using L. Hospital Rule}
= Limx→1 (1 – x)/{2x(x – 1)}
= Limx→1 (-1/2x)
= -1/2
Question 7.
The value of limy→0 {(x + y) × sec (x + y) – x × sec x}/y is
(a) x × tan x × sec x
(b) x × tan x × sec x + x × sec x
(c) tan x × sec x + sec x
(d) x × tan x × sec x + sec x
Answer
Answer: (d) x × tan x × sec x + sec x
Given, limy→0 {(x + y) × sec (x + y) – x × sec x}/y
= limy→0 {x sec (x + y) + y sec (x + y) – x × sec x}/y
= limy→0 [x{ sec (x + y) – sec x} + y sec (x + y)]/y
= limy→0 x{ sec (x + y) – sec x}/y + limy→0 {y sec (x + y)}/y
= limy→0 x{1/cos (x + y) – 1/cos x}/y + limy→0 {y sec (x + y)}/y
= limy→0 [{cos x – cos (x + y)} × x/{y × cos (x + y) × cos x}] + limy→0 {y sec (x + y)}/y
= limy→0 [{2sin (x + y/2) × sin(y/2)} × 2x/{2y × cos (x + y) × cos x}] + limy→0 {y sec (x + y)}/y
= limy→0 {sin (x + y/2) × limy→0 {sin(y/2)/(2y/2)} × limy→0 { x/{y × cos (x + y) × cos x}] + sec x
= sin x × 1 × x/cos² x + sec x
= x × tan x × sec x + sec x
So, limy→0 {(x + y) × sec (x + y) – x × sec x}/y = x × tan x × sec x + sec x
Question 8.
Limx→0 (ex² – cos x)/x² is equals to
(a) 0
(b) 1
(c) 2/3
(d) 3/2
Answer
Answer: (d) 3/2
Given, Limx→0 (ex² – cos x)/x²
= Limx→0 (ex² – cos x – 1 + 1)/x²
= Limx→0 {(ex² – 1)/x² + (1 – cos x)}/x²
= Limx→0 {(ex² – 1)/x² + Limx→0 (1 – cos x)}/x²
= 1 + 1/2
= (2 + 1)/2
= 3/2
Question 9.
The expansion of ax is
(a) ax = 1 + x/1! × (log a) + x² /2! × (log a)² + x³ /3! × (log a)³ + ………..
(b) ax = 1 – x/1! × (log a) + x² /2! × (log a)² – x³ /3! × (log a)³ + ………..
(c) ax = 1 + x/1 × (log a) + x² /2 × (log a)² + x³ /3 × (log a)³ + ………..
(d) ax = 1 – x/1 × (log a) + x² /2 × (log a)² – x³ /3 × (log a)³ + ………..
Answer
Answer: (a) ax = 1 + x/1! × (log a) + x² /2! × (log a)² + x³ /3! × (log a)³ + ………..
ax = 1 + x/1! × (log a) + x² /2! × (log a)² + x³ /3! × (log a)³ + ………..
Question 10.
The value of the limit Limn→0 (1 + an)b/n is
(a) ea
(b) eb
(c) eab
(d) ea/b
Answer
Answer: (c) eab
Given, Limn→0 (1 + an)b/n
= eLimn→0(an × b/n)
= eLimn→0(ab)
= eab
Question 11.
The value of Limx→0 cos x/(1 + sin x) is
(a) 0
(b) -1
(c) 1
(d) None of these
Answer
Answer: (c) 1
Given, Limx→0 cos x/(1 + sin x)
= cos 0/(1 + sin 0)
= 1/(1 + 0)
= 1/1
= 1
Question 12.
Lim tanx→π/4 tan 2x × tan(π/4 – x) is
(a) 0
(b) 1
(c) 1/2
(d) 3/2
Answer
Answer: (c) 1/2
Given, Lim tanx→π/4 tan 2x × tan(π/4 – x)
= Lim tanh→0 tan 2(π/4 – x) × tan(-h)
= Lim tanh→0 -cot 2h/(-cot h)
= Lim tanh→0 tan h/tan 2h
= (1/2) × Lim tanh→0 (tan h/h)/(2h/tan 2h)
= (1/2) × {Lim tanh→0 (tan h/h)}/{Lim tanh→0 (2h/tan 2h)}
= (1/2) × 1
= 1/2
Question 13.
Limx→2 (x³ – 6x² + 11x – 6)/(x² – 6x + 8) =
(a) 0
(b) 1
(c) 1/2
(d) Limit does not exist
Answer
Answer: (c) 1/2
When x = 2, the expression
(x³ – 6x² + 11x – 6)/(x² – 6x + 8) assumes the form 0/0
Now,
Limx→2 (x³ – 6x² + 11x – 6)/(x² – 6x + 8) = Limx→2 {(x – 1) × (x – 2) × (x – 3)}/{(x – 2) × (x – 4)}
= Limx→2 {(x – 1) × (x – 3)}/(x – 4)
= {(2 – 1) × (2 – 3)}/(2 – 4)
= 1/2
Question 14.
The value of the limit Limx→2 (x – 2)/√(2 – x) is
(a) 0
(b) 1
(c) -1
(d) 2
Answer
Answer: (a) 0
Given, Limx→2 (x – 2)/√(2 – x)
= Limx→2 -(2 – x)/√(2 – x)
= Limx→2 -{√(2 – x) × √(2 – x)}/√(2 – x)
= Limx→2 -√(2 – x)
= -√(2 – 2)
= 0
Question 15.
The derivative of the function f(x) = 3x³ – 2x³ + 5x – 1 at x = -1 is
(a) 0
(b) 1
(c) -18
(d) 18
Answer
Answer: (d) 18
Given, function f(x) = 3x³ – 2x² + 5x – 1
Differentiate w.r.t. x, we get
df(x)/dx = 3 × 3 × x² – 2 × 2 × x + 5
⇒ df(x)/dx = 9x² – 4x + 5
⇒ {df(x)/dx}x =-1 = 9 × (-1)² – 4 × (-1) + 5
⇒ {df(x)/dx}x =-1 = 9 + 4 + 5
⇒ {df(x)/dx}x =-1 = 18
Question 16.
Limx→0 sin²(x/3)/x² is equals to
(a) 1/2
(b) 1/3
(c) 1/4
(d) 1/9
Answer
Answer: (d) 1/9
Given, Limx→0 sin² (x/3)/ x²
= Limx→0 [sin² (x/3)/ (x/3)² × {(x/3)² /x²}]
= Limx→0 [{sin (x/3)/ (x/3)}² × {(x² /9)/x²}]
= 1 × 1/9
= 1/9
Question 17.
The expansion of ax is
(a) ax = 1 + x/1! × (log a) + x² /2! × (log a)² + x³ /3! × (log a)³ + ………..
(b) ax = 1 – x/1! × (log a) + x² /2! × (log a)² – x³ /3! × (log a)³ + ………..
(c) ax = 1 + x/1 × (log a) + x² /2 × (log a)² + x³ /3 × (log a)³ + ………..
(d) ax = 1 – x/1 × (log a) + x² /2 × (log a)² – x³ /3 × (log a)³ + ………..
Answer
Answer: (a) ax = 1 + x/1! × (log a) + x² /2! × (log a)² + x³ /3! × (log a)³ + ………..
ax = 1 + x/1! × (log a) + x² /2! × (log a)² + x³ /3! × (log a)³ + ………..
Question 18.
Differentiation of cos √x with respect to x is
(a) sin x /2√x
(b) -sin x /2√x
(c) sin √x /2√x
(d) -sin √x /2√x
Answer
Answer: (d) -sin √x /2√x
Let y = cos √x
Put u = √x
du/dx = 1/2√x
Now, y = cos u
dy/du = -sin u
dy/dx = (dy/du) × (du/dx)
= -sin u × (1/2√x)
= -sin √x /2√x
Question 19.
Differentiation of log(sin x) is
(a) cosec x
(b) cot x
(c) sin x
(d) cos x
Answer
Answer: (b) cot x
Let y = log(sin x)
Again let u = sin x
du/dx = cos x
Now, y = log u
dy/du = 1/u = 1/sin x
Now, dy/dx = (dy/du) × (du/dx)
⇒ dy/dx = (1/sin x) × cos x
⇒ dy/dx = cos x/sin x
⇒ dy/dx = cot x
Question 20.
Limx→∞ {(x + 5)/(x + 1)}x equals
(a) e²
(b) e4
(c) e6
(d) e8
Answer
Answer: (c) e6
Given, Limx→∞ {(x + 5)/(x + 1)}x
= Limx→∞ {1 + 6/(x + 1)}x
= eLimx→∞6x/(x + 1)
= eLimx→∞ 6/(1 + 1/x)
= e6/(1 + 1/∞)
= e6/(1 + 0)
= e6
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