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## Sunday 21 February 2021

### CBSE Class 11 Maths - MCQ and Online Tests - Unit 13 - Limits and Derivatives

#### CBSE Class 11 Maths  – MCQ and Online Tests – Unit 13 – Limits and Derivatives

Every year CBSE schools conducts Annual Assessment exams for 6,7,8,9,11th standards. These exams are very competitive to all the students. So our website provides online tests for all the 6,7,8,9,11th standard’s subjects. These tests are also very effective and useful for those who preparing for any competitive exams like Olympiad etc. It can boost their preparation level and confidence level by attempting these chapter wise online tests.

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#### CBSE Class 11 Maths  – MCQ and Online Tests – Unit 13 – Limits and Derivatives

Question 1.
Let f(x) = cos x, when x = 0 and f(x) = x + k, when x < 0 Find the value of k given that Limx?0 f(x) exists.
(a) 0
(b) 1
(c) -1
(d) None of these

Hint:
Given, Limx?0 f(x) exists
? Limx?0 – f(x) = Limx?0 + f(x)
? Limx?0 (x + k) = Limx?0 cos x
? k = cos 0
? k = 1

Question 2.
The value of Limx?0 (1/x) × sin-1 {2x/(1 + x²) is
(a) 0
(b) 1
(c) 2
(d) -2

Hint:
Given, Limx?0 (1/x) × sin-1 {2x/(1 + x²)
= Limx?0 (2 × tan-1 x)/x
= 2 × 1
= 2

Question 3.
The value of the limit Limx?0 {log(1 + ax)}/x is
(a) 0
(b) 1
(c) a
(d) 1/a

Hint:
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 4.
Limx?-1 [1 + x + x² + ……….+ x10] is
(a) 0
(b) 1
(c) -1
(d) 2

Hint:
Given, Limx?-1 [1 + x + x² + ……….+ x10]
= 1 + (-1) + (-1)² + ……….+ (-1)10
= 1 – 1 + 1 – ……. + 1
= 1

Question 5.
The value of Limx?01 (1/x) × sin-1 {2x/(1 + x²) is
(a) 0
(b) 1
(c) 2
(d) -2

Hint:
Given, Limx?0 (1/x) × sin-1 {2x/(1 + x²)
= Limx?0 (2× tan-1 x)/x
= 2 × 1
= 2

Question 6.
Limx?0 log(1 – x) is equals to
(a) 0
(b) 1
(c) 1/2
(d) None of these

Hint:
We know that
log(1 – x) = -x – x²/2 – x³/3 – ……..
Now,
Limx?0 log(1 – x) = Limx?0 {-x – x²/2 – x³/3 – ……..}
? Limx?0 log(1 – x) = Limx?0 {-x} – Limx?0 {x²/2} – Limx?0 {x³/3} – ……..
? Limx?0 log(1 – x) = 0

Question 7.
Limx?0 {(ax – bx)/ x} is equal to
(a) log a
(b) log b
(c) log (a/b)
(d) log (a×b)

Hint:
Given, Limx?0 {(ax – bx)/ x}
= Limx?0 {(ax – bx – 1 + 1)/ x}
= Limx?0 {(ax – 1) – (bx – 1)}/ x
= Limx?0 {(ax – 1)/x – (bx – 1)/x}
= Limx?0 (ax – 1)/x – Limx?0 (bx – 1)/x
= log a – log b
= log (a/b)

Question 8.
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: (d) x × tan x × sec x + sec x
Hint:
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 9.
Limy?8 {(x + 6)/(x + 1)}(x+4) equals
(a) e
(b) e³
(c) e5
(d) e6

Hint:
Given, Limy?8 {(x + 6)/(x + 1)}(x + 4)
= Limy?8 {1 + 5/(x + 1)}(x + 4)
= eLimy?8 5(x + 4)/(x + 1)
= eLimy?8 5(1 + 4/x)/(1 + 1/x)
= e5(1 + 4/8)/(1 + 1/8)
= e5/(1 + 0)
= e5

Question 10.
The derivative of [1+(1/x)] /[1-(1/x)] is
(a) 1/(x-1)²
(b) -1/(x-1)²
(c) 2/(x-1)²
(d) -2/(x-1)²

Hint:
Let y = [1+(1/x)] /[1-(1/x)]
then dy/dx = [{1-(1/x)}*(-1/x²)]/[{1+(1/x)}*(1/x²)]
= (1/x²) [(1/x) -1 – 1 – (1/x)]/[1-(1/x)]²
= [-2/x²]/[(x-1)/x]²
= -2/(x-1)²

Question 11.
The value of the limit Limx?0 (cos x)cot2 x is
(a) 1
(b) e
(c) e1/2
(d) e-1/2

Hint:
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 12.
The value of limit Limx?0 {sin (a + x) – sin (a – x)}/x is
(a) 0
(b) 1
(c) 2 cos a
(d) 2 sin a

Hint:
Given, Limx?0 {sin (a + x) – sin (a – x)}/x
= Limx?0 {2 × cos a × sin x}/x
= 2 × cos a × Limx?0 sin x/x
= 2 cos a

Question 13.
The value of Limn?8 {1² + 2² + 3² + …… + n²}/n³ is
(a) 0
(b) 1
(c) -1
(d) n

Hint:
Given, Limn?8 {1² + 2² + 3² + …… + n²}/n³
= Limn?8 [{n×(n + 1)×(2n + 1)}/6]/{n(n + 1)/2}²
= Limn?8 [{n×n×n ×(1 + 1/n)×(2 + 1/n)}/6]/{n × n ×(1 + 1/n)/2}²
= Limn?8 [{n³ ×(1 + 1/n)×(2 + 1/n)}/6]/{n² ×(1 + 1/n)/2}²
= Limn?8 [{(1 + 1/n)×(2 + 1/n)}/6]/[n4 × {(1 + 1/n)/2}²]
? Limn?8 [{(1 + 1/n)×(2 + 1/n)}/6]/[n × {(1 + 1/n)/2}²]
= [{(1 + 1/8)×(2 + 1/8)}/6]/[8×{(1 + 1/8)/2}²
= [{(1 + 0)×(2 + 0)}/6]/8 {since 1/8 = 0}
= {(1 × 2)/6}/8
= (2/6)/8
= (1/3)/8
= 0
So, Limn?8 {1² + 2² + 3² + …… + n²}/n³ = 0

Question 14.
The value of Limn?8 (sin x/x) is
(a) 0
(b) 1
(c) -1
(d) None of these

Hint:
Limn?8 (sin x/x) = Limy?0 {y × sin (1/y)} = 0

Question 15.
The value of Limx?0 ax is
(a) 0
(b) 1
(c) 1/2
(d) 3/2

Hint:
We know that
ax = 1 + x/1! × (log a) + x²/2! × (log a)² + x³/3! × (log a)³ + ………..
Now,
Limx?0 ax = Limx?0 {1 + x/1! × (log a) + x²/2! × (log a)² + x³/3! × (log a)³ + …}
? Limx?0 ax = Limx?0 1 + Limx?0 {x/1! × (log a)} + Limx?0 {x² /2! × (log a)²}+ ………
? Limx?0 ax = 1

Question 16.
If f(x) = (x + 1)/x then df(x)/dx is
(a) 1/x
(b) -1/x
(c) -1/x²
(d) 1/x²

Hint:
Given, f(x) = (x + 1)/x
Now, df(x)/dx = d{(x + 1)/x}/dx
= {1 × x – (x + 1)×1}/x²
= (x – x – 1)/x²
= -1/x²

Question 17.
Limx?0 (e – cos x)/x² is equals to
(a) 0
(b) 1
(c) 2/3
(d) 3/2

Hint:
Given, Limx?0 (e – cos x)/x²
= Limx?0 (e – cos x -1 + 1)/x²
= Limx?0 {(e – 1)/x² + (1 – cos x)}/x²
= Limx?0 {(e – 1)/x² + Limx?0 (1 – cos x)}/x²
= 1 + 1/2
= (2 + 1)/2
= 3/2

Question 18.
Limx?0 sin (ax)/bx is
(a) 0
(b) 1
(c) a/b
(d) b/a

Hint:
Given, Limx?0 sin (ax)/bx
= Limx?0 [{sin (ax)/ax} × (ax/bx)]
? (a/b) Limx?0 sin (ax)/ax
= a/b

Question 19.
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: (d) -x – x²/2 – x³/3 – ……..
Hint:
log(1 – x) = -x – x²/2 – x³/3 – ……..

Question 20.
If f(x) = x × sin(1/x), x ? 0, then Limx?0 f(x) is
(a) 1
(b) 0
(c) -1
(d) does not exist

Hint:
Given, f(x) = x × sin(1/x)
Now, Limx?0 f(x) = Limx?0 x × sin(1/x)
? Limx?0 f(x) = 0

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