Calculus – Limits & Derivatives Quiz - ICSE Class 11 Mathematics
Free practice quiz: Calculus – Limits & Derivatives (ICSE Class 11 Mathematics). 38 multiple-choice questions with answers and explanations. Play online, no sign-up needed.
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Evaluate the limit: $\lim_{x \to 0} \frac{\sin(ax)}{bx}$, where $a, b \neq 0$.
- $\frac{a}{b}$
- $ab$
- $1$
- $0$
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Answer: A. $\frac{a}{b}$
This is a standard limit. We can rewrite the expression as $\frac{a}{b} \cdot \frac{\sin(ax)}{ax}$. As $x \to 0$, $ax o 0$. We know that $\lim_{y \to 0} \frac{\sin(y)}{y} = 1$. Therefore, the limit is $\frac{a}{b} \cdot 1 = \frac{a}{b}$.
What is $\frac{d}{dx}(c)$ where $c$ is a constant?
- $0$
- $c$
- $1$
- $x$
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Answer: A. $0$
The derivative of a constant is always zero. This is because the rate of change of a constant value is zero.
Find the derivative of $f(x) = \sin^{-1}(x)$.
- $\frac{1}{\sqrt{1-x^2}}$
- $\frac{1}{1+x^2}$
- $\frac{1}{\sqrt{x^2-1}}$
- $-\frac{1}{\sqrt{1-x^2}}$
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Answer: A. $\frac{1}{\sqrt{1-x^2}}$
The derivative of the inverse sine function is a standard result: $\frac{d}{dx}(\sin^{-1}(x)) = \frac{1}{\sqrt{1-x^2}}$ for $|x| < 1$.
If $f(x) = \sec(x)$, find $f'(x)$.
- $\sec(x)\tan(x)$
- $\sec^2(x)$
- $-\csc(x)\cot(x)$
- $-\sec(x)\tan(x)$
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Answer: A. $\sec(x)\tan(x)$
The derivative of the secant function is a standard result: $\frac{d}{dx}(\sec(x)) = \sec(x)\tan(x)$. This can be derived using the quotient rule on $\frac{1}{\cos(x)}$.
Find the derivative of $f(x) = x^a$, where $a$ is a constant.
- $ax^{a-1}$
- $a x^a$
- $a x^{a-1}$
- $x^{a-1}$
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Answer: A. $ax^{a-1}$
This is the power rule for differentiation: $\frac{d}{dx}(x^n) = nx^{n-1}$. Here, the exponent is $a$, so the derivative is $a x^{a-1}$.