Class 11 Math - Limits & Derivatives - MERIT YARD
Class 11 Math - Limits & Derivatives - MERIT YARD
1 / 40What is the standard formula for the limit:
\( \lim_{x\to a} \frac{x^n - a^n}{x - a} \)?
A) \( na^{n+1} \)
B) \( nx^{n-1} \)
C) \( na^{n-1} \)
D) \( a n^{n-1} \)
2 / 40The derivative of any constant \( c \) with respect to \( x \) is:
A) \( 0 \)
B) \( 1 \)
C) \( c \)
D) \( x \)
3 / 40The derivative of \( x^n \) with respect to \( x \) is:
A) \( x^{n-1} \)
B) \( nx^n \)
C) \( \frac{x^{n+1}}{n+1} \)
D) \( nx^{n-1} \)
4 / 40Evaluate the standard limit:
\( \lim_{x\to 0} \frac{\sin x}{x} \)
A) \( 0 \)
B) \( 1 \)
C) \( -1 \)
D) \( \infty \)
5 / 40The derivative of \( \sin x \) with respect to \( x \) is:
A) \( -\sin x \)
B) \( -\cos x \)
C) \( \cos x \)
D) \( \sec x \)
6 / 40The derivative of \( \cos x \) with respect to \( x \) is:
A) \( \sin x \)
B) \( \sec x \)
C) \( -\csc x \)
D) \( -\sin x \)
7 / 40Evaluate the standard limit:
\( \lim_{x\to 0} \frac{1 - \cos x}{x} \)
A) \( 1 \)
B) \( 0 \)
C) \( -1 \)
D) \( \infty \)
8 / 40The derivative of \( x \) with respect to \( x \) is always:
A) \( 1 \)
B) \( 0 \)
C) \( x \)
D) \( x^2 \)
9 / 40The derivative of \( \tan x \) with respect to \( x \) is:
A) \( \cot x \)
B) \( -\sec^2 x \)
C) \( \sec^2 x \)
D) \( \csc^2 x \)
10 / 40The derivative of \( \sec x \) with respect to \( x \) is:
A) \( \sec^2 x \)
B) \( \tan x \)
C) \( -\sec x \tan x \)
D) \( \sec x \tan x \)
11 / 40The derivative of \( \csc x \) with respect to \( x \) is:
A) \( -\csc x \cot x \)
B) \( \csc x \cot x \)
C) \( -\csc^2 x \)
D) \( \cot^2 x \)
12 / 40The derivative of \( \cot x \) with respect to \( x \) is:
A) \( \csc^2 x \)
B) \( -\csc^2 x \)
C) \( -\sec^2 x \)
D) \( \tan^2 x \)
13 / 40Evaluate the simple limit:
\( \lim_{x\to 1} (x + 1) \)
A) \( 2 \)
B) \( 1 \)
C) \( 0 \)
D) \( 3 \)
14 / 40Evaluate the simple limit:
\( \lim_{x\to 2} x^2 \)
A) \( 2 \)
B) \( 6 \)
C) \( 4 \)
D) \( 8 \)
15 / 40The limit of a constant function \( \lim_{x\to a} c \) is always equal to:
A) \( a \)
B) \( 0 \)
C) \( x \)
D) \( c \)
16 / 40What is the correct Product Rule formula for derivative
\( (uv)' \)?
A) \( u'v - uv' \)
B) \( u'v + uv' \)
C) \( u'v' \)
D) \( \frac{u'v - uv'}{v^2} \)
17 / 40What is the correct Quotient Rule formula for derivative
\( \left(\frac{u}{v}\right)' \)?
A) \( \frac{u'v - uv'}{v^2} \)
B) \( \frac{u'v + uv'}{v^2} \)
C) \( \frac{uv' - u'v}{v^2} \)
D) \( \frac{u'}{v'} \)
18 / 40Which formula correctly represents the First Principle of derivative
\( f'(x) \)?
A) \( \lim_{h\to 0} \frac{f(x+h)+f(x)}{h} \)
B) \( \lim_{h\to 0} \frac{f(x)-f(x+h)}{h} \)
C) \( \lim_{h\to 0} \frac{f(x+h)-f(x)}{h} \)
D) \( \lim_{h\to 0} f(x+h) \)
19 / 40Find the derivative of:
\( x^2 \)
A) \( x \)
B) \( 2x \)
C) \( x^3 \)
D) \( 2 \)
20 / 40Find the derivative of:
\( x^3 \)
A) \( 3x \)
B) \( x^2 \)
C) \( 2x^3 \)
D) \( 3x^2 \)
21 / 40Evaluate the standard limit:
\( \lim_{x\to 0} \frac{\tan x}{x} \)
A) \( 0 \)
B) \( 1 \)
C) \( \infty \)
D) \( -1 \)
22 / 40Evaluate the simple limit:
\( \lim_{x\to 3} 5 \)
A) \( 5 \)
B) \( 3 \)
C) \( 15 \)
D) \( 0 \)
23 / 40Find the derivative of:
\( 5x \)
A) \( x \)
B) \( 0 \)
C) \( 5x^2 \)
D) \( 5 \)
24 / 40The limit of any polynomial \( P(x) \) as \( x \to a \) is always given by:
A) \( 0 \)
B) \( a \)
C) \( P(a) \)
D) \( 1 \)
25 / 40The derivative of \( cx \) (where \( c \) is a constant) is:
A) \( x \)
B) \( c \)
C) \( 0 \)
D) \( cx^2 \)
26 / 40Evaluate the simple limit:
\( \lim_{x\to 0} \sin x \)
A) \( 1 \)
B) \( -1 \)
C) \( \infty \)
D) \( 0 \)
27 / 40Evaluate the simple limit:
\( \lim_{x\to 0} \cos x \)
A) \( 1 \)
B) \( 0 \)
C) \( -1 \)
D) \( \pi \)
28 / 40If \( f(x) = \sin x + \cos x \).
Find its derivative \( f'(x) \).
A) \( \cos x + \sin x \)
B) \( -\cos x - \sin x \)
C) \( \cos x - \sin x \)
D) \( \sin x - \cos x \)
29 / 40If \( f(x) = x^2 + 1 \).
Find its derivative \( f'(x) \).
A) \( 2x \)
B) \( 2x + 1 \)
C) \( x \)
D) \( 2 \)
30 / 40The derivative of the sum of two functions
\( \frac{d}{dx}[f(x) + g(x)] \) is equal to:
A) \( f'(x) \cdot g'(x) \)
B) \( f'(x) - g'(x) \)
C) \( f'(x) + g'(x) \)
D) \( 0 \)
31 / 40The derivative of the difference of two functions
\( \frac{d}{dx}[f(x) - g(x)] \) is equal to:
A) \( f'(x) + g'(x) \)
B) \( f'(x) \cdot g'(x) \)
C) \( 0 \)
D) \( f'(x) - g'(x) \)
32 / 40Evaluate the limit using factorization:
\( \lim_{x\to 1} \frac{x^2 - 1}{x - 1} \)
A) \( 1 \)
B) \( 2 \)
C) \( 0 \)
D) Not defined
33 / 40Evaluate the limit using standard formula:
\( \lim_{x\to 2} \frac{x^3 - 8}{x - 2} \)
A) \( 4 \)
B) \( 8 \)
C) \( 12 \)
D) \( 0 \)
34 / 40Find the derivative of:
\( \frac{1}{x} \)
A) \( \frac{1}{x^2} \)
B) \( \log x \)
C) \( x^{-1} \)
D) \( -\frac{1}{x^2} \)
35 / 40Find the derivative of:
\( \sqrt{x} \)
A) \( \frac{1}{2\sqrt{x}} \)
B) \( \frac{1}{\sqrt{x}} \)
C) \( \sqrt{x} \)
D) \( 2\sqrt{x} \)
36 / 40Evaluate the limit using substitution:
\( \lim_{x\to 0} \frac{\sin 2x}{x} \)
A) \( 1 \)
B) \( 2 \)
C) \( 0 \)
D) \( 1/2 \)
37 / 40Evaluate the simple limit:
\( \lim_{x\to \pi/2} \sin x \)
A) \( 0 \)
B) \( -1 \)
C) \( \infty \)
D) \( 1 \)
38 / 40The derivative of \( c \cdot f(x) \) (where \( c \) is a constant) is:
A) \( 0 \)
B) \( c \cdot f'(x) \)
C) \( f'(x) \)
D) \( c + f'(x) \)
39 / 40Evaluate the simple limit by substitution:
\( \lim_{x\to 1} (3x^2 - 2) \)
A) \( 1 \)
B) \( 5 \)
C) \( 3 \)
D) \( 0 \)
40 / 40If \( f(x) = x \), find the value of its derivative \( f'(x) \) exactly at \( x = 2 \).
A) \( 2 \)
B) \( 0 \)
C) \( 1 \)
D) \( 4 \)
Test Analysis

Correct ✅ 0

Wrong ❌ 0

Unattempted ⚠️ 40

Accuracy 🎯 0%

Time Taken ⏱️ 00m 00s

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