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Continuity and Differentiability Set-5 MCQ Online Practice Test | Class 12 | Quiz Tweek

Continuity and Differentiability Set-5

Practice Important MCQs for Continuity and Differentiability — Mathematics, Class 12.

The chapter “Continuity and Differentiability” forms the foundation for many board and competitive-exam problems because it links limits, continuity, and local linearization via derivatives. Mastery of these ideas helps you quickly identify when functions are continuous/differentiable, compute parameters using standard limits/series, and handle piecewise and absolute-value types where one-sided limits and corner behavior matter.

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Q1. Let $f$ be defined by

\begin{aligned} \frac{e^x-1-x}{x^2}, &\quad x\neq 0 <br> k, &\quad x=0 \end{aligned}

. Find $k$ so that $f$ is continuous at $0$ .

(A)

$k=0$

(B)

$k=\dfrac{1}{2}$

(C)

$k=1$

(D)

No real $k$ makes $f$ continuous at $0$

Q2. Let $f$ be defined by

\begin{aligned} \sin x + kx, &\quad x\le 0 <br> 1-\cos x, &\quad x>0 \end{aligned}

. For which real $k$ is $f$ differentiable at $0$ ?

(A)

$k=0$

(B)

$k=1$

(C)

$k=-2$

(D)

$k=-1$

Q3. Let $f$ be defined by

\begin{aligned} \dfrac{x-\sin x}{x^3}, &\quad x\neq 0 <br> k, &\quad x=0 \end{aligned}

. Determine $k$ for which $f$ is continuous and differentiable at $0$ .

(A)

$k=\dfrac{1}{6}$ and $f$ is differentiable at $0$

(B)

$k=\dfrac{1}{6}$ but $f'$ does not exist at $0$

(C)

$k=0$

(D)

$k=\dfrac{1}{2}$

Q4. Assertion (A): If $f$ is differentiable at $0$ and $f(0)=0$ , then $g(x)=|f(x)|$ is differentiable at $0$ .
Reason (R): Since $f$ is differentiable at $0$ , $f(x)=f'(0)\,x+o(x)$ as $x\to 0$ , hence $|f(x)|=|f'(0)\,x+o(x)|$ which is differentiable at $0$ .

(A)

Both (A) and (R) are true and (R) explains (A).

(B)

Both (A) and (R) are true but (R) does not explain (A).

(C)

Both (A) and (R) are false.

(D)

(A) is false but (R) is true.

Q5. Let $f$ be defined by

\begin{aligned} x^\alpha\sin\frac{1}{x}, &\quad x\neq 0 <br> 0, &\quad x=0 \end{aligned}

for a real parameter $\alpha$ . Which of the following is true?

(A)

$f$ is differentiable at $0$ iff $\alpha\ge 1$ , and $f'$ is continuous at $0$ iff $\alpha\ge 2$ .

(B)

$f$ is differentiable at $0$ for $\alpha>1$ , and $f'$ is continuous at $0$ precisely when $\alpha>2$ .

(C)

$f$ is differentiable at $0$ only if $\alpha>2$ , and $f'$ is continuous only if $\alpha>3$ .

(D)

$f$ is differentiable at $0$ for all $\alpha>0$ , and $f'$ is continuous at $0$ for all $\alpha>1$ .

Q6. Let $f(x)=x^2\sin\!\frac{1}{x}$ for $x\neq 0$ and $f(0)=0$ . Then $f'(0)=$ ?

(A)

$1$

(B)

$0$

(C)

$-1$

(D)

Does not exist

Q7. Let $f(x)=\dfrac{1-\cos x}{x}$ for $x\neq 0$ and $f(0)=k$ . For which value of $k$ is $f$ differentiable at $0$ and what is $f'(0)$ ?

(A)

$k=\dfrac{1}{2},\ f'(0)=\dfrac{1}{2}$

(B)

$k=0,\ f'(0)=0$

(C)

No such $k$ ; $f$ cannot be differentiable at $0$

(D)

$k=0,\ f'(0)=\dfrac{1}{2}$

Q8. Let $f(x)=|x|^{k}\sin\!\frac{1}{x}$ for $x\neq 0$ and $f(0)=0$ , where $k\in\mathbb{R}$ . For which real values of $k$ is $f$ differentiable at $0$ ?

(A)

$k>1$

(B)

$k\ge 1$

(C)

$k>0$

(D)

All real $k$

Q9. Assertion (A): Let $f$ be continuous at $x=a$ and differentiable on $(a-\delta,a)\cup(a,a+\delta)$ for some $\delta>0$ . If $\displaystyle\lim_{x\to a}f'(x)=L$ (finite), then $f$ is differentiable at $a$ and $f'(a)=L$ .
Reason (R): For each $x\neq a$ in the neighbourhood there exists $c$ between $x$ and $a$ (by the Mean Value Theorem) such that $\dfrac{f(x)-f(a)}{x-a}=f'(c)$ . As $x\to a$ we have $c\to a$ , so the difference quotient tends to $L$ .

(A)

Both (A) and (R) are true and (R) explains (A)

(B)

Both (A) and (R) are true but (R) does not explain (A)

(C)

(A) is true and (R) is false

(D)

(A) is false and (R) is true

Q10. Assertion (A): Let $f$ be differentiable at $x=a$ with $f(a)=0$ and $f'(a)\neq 0$ . Then $g(x)=|f(x)|$ is not differentiable at $x=a$ .
Reason (R): Since $f(x)=f'(a)(x-a)+o(x-a)$ as $x\to a$ , we get $|f(x)|=|f'(a)|\,|x-a|+o(|x-a|)$ ; the leading term $|f'(a)|\,|x-a|$ has a corner at $x=a$ , so $g$ is not differentiable there.

(A)

Both (A) and (R) are true but (R) does not explain (A)

(B)

(A) is true and (R) is false

(C)

Both (A) and (R) are true and (R) explains (A)

(D)

(A) is false and (R) is true

Q11. Let

f(x)=\begin{aligned} x^2\sin\frac{1}{x}, &\quad x\ne0\\ 0, &\quad x=0 \end{aligned}

Then which of the following is true about $f$ at $x=0$ ?

(A)

$f$ is not differentiable at $0$ .

(B)

$f$ is differentiable at $0$ and $f'(0)=1$ .

(C)

$f$ is differentiable at $0$ and $f'(0)=0$ .

(D)

$f$ is differentiable at $0$ and $f'$ is continuous at $0$ .

Q12. Suppose $f$ is twice differentiable on $[0,2]$ with $f(0)=0,\;f(1)=1,\;f(2)=8$ . Then there exists $c\in(0,2)$ such that

(A)

$f''(c)=6$ .

(B)

$f''(c)=3$ .

(C)

$f''(c)=12$ .

(D)

no such $c$ is guaranteed to exist.

Q13. Let

f(x)=\begin{aligned} \frac{x^3-p}{\,x-1\,}, &\quad x\ne1\\ q, &\quad x=1 \end{aligned}

For which pair $(p,q)$ is $f$ differentiable at $x=1$ ?

(A)

$(p,q)=(1,1)$ .

(B)

$(p,q)=(0,1)$ .

(C)

$(p,q)=(1,0)$ .

(D)

$(p,q)=(1,3)$ .

Q14. Let

f(x)=\begin{aligned} |x|^\alpha\sin\frac{1}{x}, &\quad x\ne0\\ 0, &\quad x=0 \end{aligned}\quad(\alpha\in\mathbb{R}).

Which of the following correctly describes the sharp conditions on $\alpha$ for (i) $f$ to be differentiable at $0$ and (ii) $f'$ to be continuous at $0$ ?

(A)

(i) $\alpha>2$ , (ii) $\alpha>3$ .

(B)

(i) $\alpha>1$ , (ii) $\alpha>2$ .

(C)

(i) $\alpha\ge1$ , (ii) $\alpha\ge2$ .

(D)

(i) only integer $\alpha>1$ , (ii) only integer $\alpha>2$ .

Q15. Let

f(x)=\begin{aligned} x^2\sin\frac{1}{x^2}, &\quad x\ne0\\ 0, &\quad x=0 \end{aligned}

About $f$ which one of the following is true?

(A)

$f$ is differentiable at $0$ with $f'(0)=0$ , but $f'$ is unbounded in every neighbourhood of $0$ (hence $f'$ is not continuous at $0$ ).

(B)

$f$ is not differentiable at $0$ .

(C)

$f$ is differentiable at $0$ and $f'$ is bounded in a neighbourhood of $0$ .

(D)

$f$ is differentiable at $0$ and $f'$ is continuous at $0$ .

...and 5 more challenging questions available in the interactive simulator.

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