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Continuity and Differentiability Set-3 MCQ Online Practice Test | Class 12 | Quiz Tweek

Continuity and Differentiability Set-3

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

Continuity and differentiability are core ideas in Class 12 Mathematics because they connect limits, smoothness, and the behavior of functions near a point. Board exams and competitive exams heavily use these concepts to test (i) existence of derivatives at a point, (ii) conditions for continuity, (iii) behavior of composite/absolute value functions, and (iv) how growth/oscillation (like $x^a\sin\frac1x$ type functions) affects limits. Mastery of these results helps you solve a wide variety of MCQs and higher-level JEE/NEET-style reasoning questions quickly and accurately.

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Q1. Let $f$ be defined by $f(x)=x^{2}\sin\!\left(\frac{1}{x}\right)$ for $x\neq 0$ and $f(0)=0$ . Which of the following describes the behaviour of $f$ at $x=0$ ?

(A)

$f$ is continuous at $0$ but not differentiable at $0$ .

(B)

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

(C)

$f$ is not continuous at $0$ .

(D)

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

Q2. Let $f(x)=(x^{2}+kx)\sin\!\left(\frac{1}{x}\right)$ for $x\neq 0$ and $f(0)=0$ . For which real value(s) of $k$ is $f$ differentiable at $0$ ?

(A)

$k=1$

(B)

$k\in\mathbb{R}$ (all real $k$ )

(C)

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

(D)

$k=0$

Q3. Let $\alpha>0$ and define $f(x)=x^{\alpha}\sin\!\left(\frac{1}{x}\right)$ for $x\neq 0$ , $f(0)=0$ . For which values of $\alpha$ is $f$ differentiable at $0$ ?

(A)

$\alpha>1$

(B)

$\alpha>0$

(C)

$\alpha\ge 1$

(D)

$\alpha>2$

Q4. Assertion (A): If $f$ is differentiable at $c$ , then the function $g(x)=|f(x)|$ is differentiable at $c$ .
Reason (R): If $f(c)\neq 0$ then $g$ is differentiable at $c$ ; if $f(c)=0$ differentiability of $g$ may fail.

(A)

Both A and R are true and R is the correct explanation of A.

(B)

Both A and R are true but R is not the correct explanation of A.

(C)

A is true and R is false.

(D)

A is false and R is true.

Q5. Define $f(x)=x^{2}\sin\!\left(\frac{1}{x}\right)$ and $g(x)=x^{3}\sin\!\left(\frac{1}{x}\right)$ for $x\neq 0$ , with $f(0)=g(0)=0$ . Which statement is correct about differentiability at $0$ and continuity of derivatives at $0$ ?

(A)

Neither $f$ nor $g$ is differentiable at $0$ .

(B)

Both $f$ and $g$ are differentiable at $0$ , and $g'$ is continuous at $0$ while $f'$ is not continuous at $0$ .

(C)

Both $f$ and $g$ are differentiable at $0$ and both $f'$ and $g'$ are continuous at $0$ .

(D)

$f$ is differentiable at $0$ and $f'$ is continuous at $0$ , but $g$ is not differentiable at $0$ .

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

(A)

$f'(0)$ does not exist

(B)

$0$

(C)

$1$

(D)

$\displaystyle \lim_{x\to0}\frac{\sin(1/x)}{x}$

Q7. Let $f(x)=|x|^{\alpha}\sin\frac{1}{x}$ for $x\neq0$ and $f(0)=0$ . For which real $\alpha$ is $f$ differentiable at $0$ ?

(A)

$\alpha>1$

(B)

$\alpha\ge1$

(C)

$\alpha>0$

(D)

$\alpha\in\mathbb{Z},\ \alpha\ge2$

Q8. Define $f$ by $f(x)=\dfrac{e^{kx}-1}{x}$ for $x\neq0$ and $f(0)=2$ . For which value of $k$ is $f$ differentiable at $0$ ?

(A)

$k=0$

(B)

$k=1$

(C)

no such $k$

(D)

$k=2$

Q9. Let $f$ be one-to-one and differentiable in a neighbourhood of $a$ , and suppose $f'(a)=0$ . Which statement is necessarily true?

(A)

$f^{-1}$ is differentiable at $f(a)$ with derivative $\dfrac{1}{f'(a)}$

(B)

$f$ has a local extremum at $a$

(C)

$f^{-1}$ is not differentiable at $f(a)$

(D)

$f$ is not monotone in any neighbourhood of $a$

Q10. Let $f(x)=x^{2}\sin\frac{1}{x^{2}}$ for $x\neq0$ and $f(0)=0$ . Which of the following is correct about $f$ at $0$ ?

(A)

$f'(0)$ does not exist

(B)

$f'(0)$ exists but $f'$ is not continuous at $0$

(C)

$f'$ is continuous at $0$

(D)

$f'$ is bounded on some neighbourhood of $0$

Q11. Using the definition of derivative, find $f'(0)$ for $f(x)=\sqrt{1+x}$ .

(A)

$0$

(B)

$1$

(C)

$\dfrac{1}{2}$

(D)

$-\dfrac{1}{2}$

Q12. For which real values of $p$ is the function $f(x)=|x|^{p}$ differentiable at $x=0$ ?

(A)

$p>1$

(B)

$p\ge 1$

(C)

$0<p<1$

(D)

$p>2$

Q13. Let $f$ be differentiable on $\mathbb{R}$ and satisfy $f(x+y)=f(x)+f(y)+xy(x+y)$ for all real $x,y$ . Then $f(x)$ must be equal to

(A)

$k\,x+\dfrac{x^{3}}{2}$ for some constant $k$

(B)

$k\,x+\dfrac{x^{3}}{6}$ for some constant $k$

(C)

$\dfrac{x^{3}}{3}$

(D)

$k\,x+\dfrac{x^{3}}{3}$ for some constant $k$

Q14. Let $f$ be differentiable on $\mathbb{R}$ and suppose $f'(q)=0$ for every rational number $q$ . Which of the following is correct?

(A)

$f$ is constant on $\mathbb{Q}$ but need not be constant on $\mathbb{R}$

(B)

$f$ is constant on $\mathbb{R}$

(C)

$f'$ may be nonzero at some irrational points

(D)

$f'(x)=0$ for every irrational $x$ as well

Q15. Let $f(x)=x^{a}\sin\!\left(\tfrac{1}{x}\right)$ for $x\ne0$ and $f(0)=0$ , where $a\in\mathbb{R}$ . For which values of $a$ are the following true?
(i) $f$ is continuous at $0$ ;
(ii) $f$ is differentiable at $0$ ;
(iii) $f'$ is continuous at $0$ ?

(A)

(i) $a>0$ ; (ii) $a>1$ ; (iii) $a>2$

(B)

(i) $a\ge 0$ ; (ii) $a>1$ ; (iii) $a\ge 2$

(C)

(i) $a>1$ ; (ii) $a>2$ ; (iii) $a>3$

(D)

(i) $a>0$ ; (ii) $a\ge 1$ ; (iii) $a>2$

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