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Application of Derivatives Set-2 MCQ Online Practice Test | Class 12 | Quiz Tweek

Application of Derivatives Set-2

Practice Important MCQs for Application of Derivatives — Mathematics, Class 12.

The chapter “Application of Derivatives” is crucial for CBSE Class 12 as it builds on the idea of derivative as an instantaneous rate of change and uses it to solve real problems: maxima-minima, tangents/normals, monotonicity, convexity, and optimization. These concepts also appear frequently in competitive exams because they test both computation and the ability to reason using derivative properties (like Darboux’s theorem and convexity).

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Q1. Use linear approximation (tangent at $x_0=27$ ) to estimate the value of $\sqrt[3]{27.3}$ .

(A)

$3+\dfrac{1}{30}$

(B)

$3+\dfrac{1}{90}$

(C)

$3-\dfrac{1}{90}$

(D)

$3+\dfrac{1}{9}$

Q2. Find the point(s) on the curve $y=x^2+1$ that are nearest to the point $(0,3)$ .

(A)

$(0,1)$

(B)

$(\pm 1,\,2)$

(C)

$\left(\pm\sqrt{\tfrac{3}{2}},\,\tfrac{3}{2}\right)$

(D)

$\left(\pm\sqrt{\tfrac{3}{2}},\,\tfrac{5}{2}\right)$

Q3. A rectangle has one vertex at the origin and the opposite vertex on the curve $y=12-x^2$ in the first quadrant. Find the width and height of the rectangle with maximum area.

(A)

width $2$ , height $8$

(B)

width $\sqrt{12}$ , height $0$

(C)

width $3$ , height $3$

(D)

width $1$ , height $11$

Q4. Consider the statements
(A) If $f$ is differentiable on $(a,b)$ and $f'(x)\neq 0$ for all $x\in(a,b)$ , then $f$ is one-to-one on $(a,b)$ .
(R) The derivative $f'$ has the intermediate value (Darboux) property; hence if $f'$ never vanishes on $(a,b)$ it cannot change sign there, so $f$ is monotonic on $(a,b)$ .

(A)

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

(B)

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

(C)

Both (A) and (R) are true and (R) is a correct explanation of (A).

(D)

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

Q5. For $p>0$ define $f(x)=|x|^p$ . For which values of $p$ is $f$ differentiable at $x=0$ and convex on $\mathbb{R}$ ?

(A)

$0<p<1$

(B)

$p>1$

(C)

$p\ge 1$

(D)

all $p>0$

Q6. Find the equation of the tangent to the curve $y=\ln\bigl(x^{2}+1\bigr)$ at the point where $x=1$ .

(A)

$y = x + \ln 2$

(B)

$y = x - 1 + \ln 2$

(C)

$y = x - \ln 2$

(D)

$y = \ln(2x)$

Q7. Find all real values of the parameter $a$ for which the tangent to the curve $y = x^{3}-3ax+a$ at $x=a$ passes through the origin.

(A)

$a=0$ only

(B)

$a=\pm\dfrac{1}{\sqrt{2}}$

(C)

$a=0,\;\dfrac{1}{2}$

(D)

$a=0,\;\pm\dfrac{1}{\sqrt{2}}$

Q8. Let $f(x)=\ln\bigl(x^{2}+1\bigr)$ . For which real value(s) of $a$ does the normal to the curve at the point $(a,f(a))$ pass through the origin?

(A)

$a=0$

(B)

No real value of $a$

(C)

$a=\pm 1$

(D)

$a=\pm\dfrac{1}{\sqrt{2}}$

Q9. Assertion (A): If $f$ is twice differentiable on $\mathbb{R}$ and $f''(x)>0$ for all $x\in\mathbb{R}$ , then the equation $f(x)=0$ has at most two distinct real roots.
Reason (R): If $f''(x)>0$ for all $x$ , then $f$ must be a quadratic polynomial with positive leading coefficient.

(A)

Both A and R are true and R is a correct explanation for A.

(B)

Both A and R are true but R is not a correct explanation for A.

(C)

A is true but R is false.

(D)

A is false but R is true.

Q10. Let

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

Determine which of the following is true about $g$ at $0$ .

(A)

$g$ is not differentiable at $0$ .

(B)

$g$ is differentiable at $0$ but $g'$ is not continuous at $0$ .

(C)

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

(D)

$g$ is differentiable everywhere and $g'$ is unbounded in every neighbourhood of $0$ .

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