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Continuity and Differentiability Set-2 MCQ Online Practice Test | Class 12 | Quiz Tweek

Continuity and Differentiability Set-2

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

In “Continuity and Differentiability,” you learn how to test whether functions behave smoothly at points (continuity) and how to find/justify slopes at a point (differentiability). These ideas are heavily used in limits, tangent/normal problems, inverse functions, and in handling absolute value and piecewise definitions—making this chapter essential for both board exams and competitive tests like JEE/NEET.

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Q1. 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)

Does not exist

(D)

$-1$

Q2. Let $f(x)=ax^2+bx+1$ for $x\le 1$ and $f(x)=\ln(x+1)$ for $x>1$ . If $f$ is differentiable at $x=1$ , then $(a,b)=$

(A)

$\left(\dfrac{1}{2}-\ln 2,\;2\ln 2-\dfrac{1}{2}\right)$

(B)

$\left(1-\ln 2,\;\ln 2-1\right)$

(C)

$\left(\dfrac{3}{2}-\ln 2,\;\dfrac{1}{2}-2\ln 2\right)$

(D)

$\left(\dfrac{3}{2}-\ln 2,\;2\ln 2-\dfrac{5}{2}\right)$

Q3. For the curve $x^2y+xy^2=6$ , the equation of the tangent at the point $(1,2)$ is

(A)

$y-2=-\dfrac{8}{5}(x-1)$

(B)

$y-2=-\dfrac{8}{3}(x-1)$

(C)

$y-2=-\dfrac{3}{5}(x-1)$

(D)

$y-2=\dfrac{8}{5}(x-1)$

Q4. Let $f$ be differentiable at $a$ with $f(a)=0$ . Consider the statements:
A: If $f'(a)=0$ then $g(x)=|f(x)|$ is differentiable at $a$ .
R: If there exists a neighbourhood of $a$ on which $f$ does not change sign, then $g(x)=|f(x)|$ is differentiable at $a$ and $g'(a)=\pm f'(a)$ accordingly. Which one is correct?

(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 but R is false.

(D)

A is false but R is true.

Q5. Let $f$ be differentiable on $\mathbb{R}$ , strictly increasing and bijective, with $f(0)=0$ and $f'(0)=0$ . About $g=f^{-1}$ at $0$ which statement is correct?

(A)

$g$ is differentiable at $0$ and $g'(0)$ is finite.

(B)

$g$ is differentiable at $0$ and has horizontal tangent, i.e. $g'(0)=0$ .

(C)

$g$ is not differentiable at $0$ (no finite derivative).

(D)

Differentiability of $g$ at $0$ depends on higher order behaviour of $f$ near $0$ .

Q6. Que51: Let the function be $f(x)=x|x|$ for all real $x$ . Then $f$ is differentiable at $x=0$ and $f'(0)=$ ?

(A)

( $1$ )

(B)

( $-1$ )

(C)

( $0$ )

(D)

(Derivative does not exist)

Q7. Que52: Let

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

Which of the following is true?

(A)

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

(B)

( $f$ is differentiable at $0$ and $f'$ is continuous at $0$ )

(C)

( $f$ is not differentiable at $0$ )

(D)

( $f$ is differentiable at $0$ and $f'$ is unbounded near $0$ )

Q8. Que53: Define

\begin{aligned} f(x)=x^2,&\quad x\in\mathbb{Q}<br> f(x)=x,&\quad x\notin\mathbb{Q} \end{aligned}

Which statement about $f$ at $x=0$ is correct?

(A)

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

(B)

( $f$ is discontinuous at $0$ )

(C)

( $f$ is differentiable at $0$ with $f'(0)=0$ )

(D)

( $f$ is continuous at $0$ but not differentiable at $0$ )

Q9. Que54: For real $k$ , consider $f(x)=|x|^k$ . For which values of $k$ is $f$ differentiable at $x=0$ ?

(A)

( $k>0$ )

(B)

( $k>1$ )

(C)

( $k\ge 1$ )

(D)

( Only for positive even integers $k$ )

Q10. Que55: Let

\begin{aligned} f(x)=\frac{\sin(x+a)}{x},&\quad x\ne0<br> f(0)=b \end{aligned}

For which real constants $a,b$ is $f$ differentiable at $0$ ?

(A)

( $a=n\pi,\; b=(-1)^n$ for some integer $n$ )

(B)

( $a=\frac{\pi}{2}+n\pi,\; b=0$ for some integer $n$ )

(C)

( Any real $a$ with $b=1$ )

(D)

( No choice of $a,b$ makes $f$ differentiable at $0$ )

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