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Application of Derivatives Set-5 MCQ Online Practice Test | Class 12 | Quiz Tweek

Application of Derivatives Set-5

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

The chapter “Application of Derivatives” is crucial because it directly trains you to solve real-life rate problems and optimization questions (maximum/minimum values, nearest points, minimum surface area, etc.) using derivatives. These concepts are frequently asked in CBSE board exams as well as competitive tests like JEE/NEET because they blend calculus with reasoning, and methodical derivative-based steps quickly lead to correct results.

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Q1. The radius of a circle is increasing at a constant rate of $0.1\ \mathrm{cm/s}$ . At the instant the radius is $10\ \mathrm{cm}$ , the rate of change of the area of the circle is

(A)

$\pi\ \mathrm{cm}^2\!/\mathrm{s}$

(B)

$2\pi\ \mathrm{cm}^2\!/\mathrm{s}$

(C)

$20\pi\ \mathrm{cm}^2\!/\mathrm{s}$

(D)

$0.2\pi\ \mathrm{cm}^2\!/\mathrm{s}$

Q2. Find the point(s) on the parabola $y=x^2$ that are closest to the point $(0,1)$ .

(A)

$(0,0)$

(B)

$(1,1)$

(C)

$\displaystyle\left(\pm\frac{1}{2},\frac{1}{4}\right)$

(D)

$\displaystyle\left(\pm\frac{1}{\sqrt{2}},\frac{1}{2}\right)$

Q3. For the function $f(x)=\dfrac{2x+1}{x+2}$ , $x\neq -2$ , which of the following is true?

(A)

$f$ is strictly increasing on its domain.

(B)

$f$ has a local maximum at $x=-2$ .

(C)

$f$ has a local extremum at $x=-1$ .

(D)

$f$ is decreasing for $x>-2$ and increasing for $x<-2$ .

Q4. For real parameter $a$ , the cubic equation $x^3-3ax+1=0$ has three distinct real roots precisely when

(A)

$a<0$

(B)

$0<a<2^{-2/3}$

(C)

$a>2^{-2/3}$

(D)

$a=2^{-2/3}$

Q5. For which values of the real parameter $a$ does the curve $y=\ln(1+x^2)-ax$ have two distinct critical points?

(A)

$|a|>1$

(B)

$|a|<1$

(C)

$a=\pm 1$

(D)

$a=0$

Q6. Use linearization of $f(x)=\sqrt{x}$ at $x=9$ to estimate $\sqrt{9.2}$ . Which of the following is the linear approximation?

(A)

$3.0329$

(B)

$3.0340$

(C)

$3.0333$

(D)

$3.0666$

Q7. For $x>0$ , find the point on the curve $y=x+\dfrac{1}{x}$ that is nearest to the origin.

(A)

$\displaystyle\left(2^{-1/4},\,2^{-1/4}+2^{1/4}\right)$

(B)

$\displaystyle\left(2^{1/4},\,2^{1/4}+2^{-1/4}\right)$

(C)

$\displaystyle\left(\dfrac{1}{\sqrt{2}},\,\sqrt{2}+\dfrac{1}{\sqrt{2}}\right)$

(D)

$(1,2)$

Q8. A closed cylindrical can of volume $1000\ \text{cm}^3$ is to be made so that its total surface area is minimum. The radius $r$ and height $h$ (in cm) that minimize the surface area are:

(A)

$\displaystyle r=\left(\dfrac{1000}{4\pi}\right)^{1/3},\quad h=\left(\dfrac{1000}{\pi}\right)^{1/3}$

(B)

$\displaystyle r=\left(\dfrac{250}{\pi}\right)^{1/3},\quad h=4\left(\dfrac{250}{\pi}\right)^{1/3}$

(C)

$\displaystyle r=\left(\dfrac{1000}{\pi}\right)^{1/3},\quad h=\left(\dfrac{1000}{\pi}\right)^{1/3}$

(D)

$\displaystyle r=\left(\dfrac{500}{\pi}\right)^{1/3},\quad h=2\left(\dfrac{500}{\pi}\right)^{1/3}$

Q9. Assertion (A): Let $f$ be thrice differentiable at $x_0$ . If $f''(x_0)=0$ and $f'''(x_0)\neq 0$ , then $x_0$ is a point of inflection of $f$ .
Reason (R): If $f'''(x_0)\neq 0$ then $f''(x)$ changes sign as $x$ passes through $x_0$ .

(A)

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

(B)

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

(C)

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

(D)

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

Q10. For which real values of the parameter $k$ does the cubic equation $x^3-3x+k=0$ have three distinct real roots?

(A)

$k\in(-\infty,-2)\cup(2,\infty)$

(B)

$k\in[-2,2]$

(C)

$k\in(-2,2)$

(D)

$k\in\mathbb{R}\setminus\{\pm 2\}$

Q11. Using linear approximation (linearization) of $f(x)=x^{1/3}$ at $x=8$ , approximate $\sqrt[3]{8.125}$ . Which of the following (rounded to four decimal places) is the linear approximation?

(A)

$2.0156$

(B)

$2.0125$

(C)

$2.0104$

(D)

$2.0078$

Q12. A wire of total length $20\ \text{m}$ is cut into two parts; one part is bent to form a square and the other to form a circle. To minimize the total enclosed area, what length of the wire should be used for the circle? (Exact value.)

(A)

$\dfrac{20\pi}{\pi+4}\ \text{m}$

(B)

$\dfrac{80}{\pi+4}\ \text{m}$

(C)

$\dfrac{20}{\pi+4}\ \text{m}$

(D)

$10\ \text{m}$

Q13. A ladder of length $10\ \text{m}$ leans against a vertical wall. The top is sliding downwards at the rate $0.5\ \text{m/s}$ when its height is $6\ \text{m}$ . At that instant, find (i) the speed of the foot of the ladder moving away from the wall and (ii) the rate of change of the area of the right triangle formed by the ladder, wall and ground.

(A)

$\displaystyle\left(\dfrac{3}{8}\ \text{m/s},\ \dfrac{7}{8}\ \text{m}^2\!/\text{s}\right)$

(B)

$\displaystyle\left(\dfrac{1}{2}\ \text{m/s},\ -\dfrac{1}{2}\ \text{m}^2\!/\text{s}\right)$

(C)

$\displaystyle\left(\dfrac{8}{3}\ \text{m/s},\ -\dfrac{7}{8}\ \text{m}^2\!/\text{s}\right)$

(D)

$\displaystyle\left(\dfrac{3}{8}\ \text{m/s},\ -\dfrac{7}{8}\ \text{m}^2\!/\text{s}\right)$

Q14. Assertion (A): Let $f$ be twice differentiable at $x=a$ . If $f'(a)=0$ and $f''(a)>0$ , then $f$ has a local minimum at $x=a$ .
Reason (R): If $g$ is differentiable and increasing on an interval, then $g'(x)\ge 0$ for every $x$ in that interval.

(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 but (R) is false.

(D)

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

Q15. For which real values of the parameter $k$ does the cubic equation $x^3-3kx+2k=0$ have three distinct real roots?

(A)

$k>1$

(B)

$0<k<1$

(C)

$k\le 0$

(D)

$k=1$

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