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Continuity and Differentiability Set-6 MCQ Online Practice Test | Class 12 | Quiz Tweek

Continuity and Differentiability Set-6

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

The chapter on Continuity and Differentiability is essential because it forms the foundation for applying limits, derivative definitions, and powerful theorems like the Mean Value Theorem (MVT) and Rolle’s theorem. These ideas frequently appear in board exams for continuity/differentiability at points, and in competitive exams for more subtle functions involving oscillations, absolute values, and parameter-based conditions—where understanding exact definitions and theorem requirements is crucial.

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Q1. Let $f$ be continuous on $[0,2]$ and differentiable on $(0,2)$ with $f(0)=3$ and $f(2)=7$ . Which of the following is guaranteed by the Mean Value Theorem?

(A)

There exists $c\in(0,2)$ such that $f'(c)=1$

(B)

There exists $c\in(0,2)$ such that $f'(c)=2$

(C)

There exists $c\in(0,2)$ such that $f'(c)=4$

(D)

There exists $c\in(0,2)$ such that $f''(c)=0$

Q2. Let $f$ be defined by

f(x)=\begin{aligned} &\dfrac{\sin(ax)}{x}\quad \text{for } x\ne 0 \\ &b\quad \text{for } x=0 \end{aligned}

where $a,b\in\mathbb{R}$ . For which values of $a$ and $b$ is $f$ differentiable at $0$ ?

(A)

$a=0$ and $b=0$

(B)

$b=\dfrac{a}{2}$ with $a\ne 0$

(C)

$b=0$ for every real $a$

(D)

$b=a$ for every real $a$

Q3. Let $f(x)=|x|^{p}\sin\!\frac{1}{x}$ for $x\ne 0$ and $f(0)=0$ , where $p\in\mathbb{R}$ . For which values of $p$ is $f$ differentiable at $0$ ?

(A)

$p>2$

(B)

$p\ge 1$

(C)

$p>1$

(D)

$p>0$

Q4. Suppose $f$ is differentiable on $\mathbb{R}$ and $f'(x)=0$ for every rational $x$ . Which of the following statements must hold?

(A)

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

(B)

$f'(x)=0$ only at rational points

(C)

$f$ is constant only on the set of rational numbers

(D)

$f'$ can have a jump discontinuity at some irrational point

Q5. Let

f(x)=\begin{aligned} &x^{2}\sin\!\frac{1}{x}\quad \text{for } x\ne 0\\ &0\quad \text{for } x=0 \end{aligned}

Which of the following is correct about $f$ and $f'$ ?

(A)

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

(B)

$f$ is differentiable everywhere but $f'$ is not continuous at $0$

(C)

$f$ is not differentiable at $0$

(D)

$f'$ is unbounded in every neighbourhood of $0$

Q6. Let $f(x)=\dfrac{\sin x}{x}$ for $x\ne 0$ and $f(0)=1$ . Which of the following is true about $f$ at $x=0$ ?

(A)

$f$ is continuous at $0$ but 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 discontinuous at $0$

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

(A)

continuous iff $k=0$ , and $f'(0)=0$

(B)

continuous iff $k=0$ , and $f'(0)=\tfrac{1}{2}$

(C)

continuous iff $k=\tfrac{1}{2}$ , and $f'(0)=\tfrac{1}{2}$

(D)

continuous iff $k=0$ , and $f'(0)$ does not exist

Q8. Let $f(x)=x^{2}\sin\!\frac{1}{x}$ for $x\ne 0$ and $f(0)=0$ . Which statement is correct?

(A)

$f$ is differentiable for all $x$ and $f'$ is not continuous at $0$

(B)

$f$ is not differentiable at $0$

(C)

$f$ is differentiable for all $x$ and $f'$ is continuous at $0$

(D)

$f$ is continuous at $0$ but not differentiable at infinitely many points arbitrarily close to $0$

Q9. Define $f(x)=x^{2}\sin\!\frac{1}{x^{2}}$ for $x\ne 0$ and $f(0)=0$ . Which of the following describes the behaviour of $f$ and $f'$ near $0$ ?

(A)

$f$ is not differentiable at $0$

(B)

$f'$ exists everywhere and is bounded in some neighbourhood of $0$

(C)

$f'$ exists everywhere and is continuous at $0$

(D)

$f'(0)=0$ , $f'$ exists for all $x$ , but $f'$ is unbounded in every neighbourhood of $0$

Q10. Let $f_{\alpha}(x)=|x|^{\alpha}$ with real $\alpha>0$ . Which statement about differentiability at $0$ is correct?

(A)

$f_{\alpha}$ is differentiable at $0$ iff $\alpha\ge 1$ , and $f_{\alpha}''(0)$ exists iff $\alpha>2$

(B)

$f_{\alpha}$ is differentiable at $0$ iff $\alpha>1$ , and $f_{\alpha}''(0)$ exists iff $\alpha\ge 2$

(C)

$f_{\alpha}$ is differentiable at $0$ for every $\alpha>0$ , and $f_{\alpha}''(0)$ exists iff $\alpha>2$

(D)

$f_{\alpha}$ is differentiable at $0$ iff $\alpha>1$ , and $f_{\alpha}''(0)$ exists only for $\alpha>2$ (with value $0$ )

Q11. Let $f(x)=x|x|$ . Which of the following is true about differentiability 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 $f''(0)$ exists.

Q12. Let $f(x)=x^2\sin\!\tfrac{1}{x}$ for $x\neq0$ and $f(0)=0$ . Which of the following statements about $f$ at $0$ is correct?
(i) $f$ is continuous at $0$ .
(ii) $f$ is differentiable at $0$ .
(iii) $f'$ is continuous at $0$ .

(A)

(i) and (ii) only

(B)

(i), (ii) and (iii) all true

(C)

Only (i) is true

(D)

(ii) and (iii) only

Q13. For which value of $k$ is the function

f(x)=\begin{cases}kx^2+1,&x\le1,\$$2pt]x^2+2x,&x>1,\end{cases}

differentiable at $x=1$ ?

(A)

$k=1$

(B)

$k=0$

(C)

No real $k$ makes $f$ differentiable at $1$

(D)

$k=2$

Q14. Consider $f(x)=x^3$ defined on $\mathbb{R}$ . Which statement is correct regarding $f$ and its inverse $f^{-1}$ ?

(A)

$f$ is differentiable on $\mathbb{R}$ and $f^{-1}$ is differentiable on $\mathbb{R}$ .

(B)

$f$ is differentiable on $\mathbb{R}$ and strictly increasing (hence invertible), but $f^{-1}$ is not differentiable at $0$ .

(C)

$f$ is differentiable on $\mathbb{R}$ and $f^{-1}$ is differentiable at $0$ with $(f^{-1})'(0)=0$ .

(D)

$f$ is differentiable on $\mathbb{R}$ but $f$ is not invertible.

Q15. Let $f$ be differentiable on $\mathbb{R}$ and suppose $f'(x)=0$ for every rational $x$ . Which conclusion must hold?

(A)

$f$ is constant on the set of rational numbers but may be non-constant on irrationals.

(B)

$f'$ is nonzero at some irrational points, so $f$ need not be constant.

(C)

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

(D)

$f'$ must be continuous on $\mathbb{R}$ (hence identically zero only if continuous).

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

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