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Application of Derivatives Set-3 MCQ Online Practice Test | Class 12 | Quiz Tweek

Application of Derivatives Set-3

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

In the chapter “Application of Derivatives,” we learn how derivatives help in solving real-world and exam-oriented problems such as finding tangents with given properties, maxima/minima of functions, optimization in geometry, and estimating function values using linear approximation. These ideas directly appear in CBSE board questions and are frequently used (in both conceptual and computation-heavy forms) in competitive exams like JEE and NEET.

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Q1. For the curve $y = x^3 - 6x^2 + 9x + 1$ , the x-coordinates where the tangent is horizontal are:

(A)

$x=1$

(B)

$x=1\text{ and }x=3$

(C)

$x=3$

(D)

$x=-1\text{ and }x=1$

Q2. A box is formed by cutting equal squares of side $x$ (in cm) from each corner of a $20\text{ cm}\times15\text{ cm}$ rectangular sheet and folding up the sides. The value of $x$ in $(0,7.5)$ that maximizes the volume is:

(A)

$x=\dfrac{5}{2}\text{ cm}$

(B)

$x=\dfrac{35+5\sqrt{13}}{6}\text{ cm}$

(C)

$x=\dfrac{6}{5}\text{ cm}$

(D)

$x=\dfrac{35-5\sqrt{13}}{6}\text{ cm}$

Q3. Find the coordinates of the point(s) on the curve $y=x^3-3x$ at which the tangent line passes through the point $(0,1)$ .

(A)

$\displaystyle\left(-\sqrt[3]{\tfrac{1}{2}},\; -\tfrac{1}{2}+3\sqrt[3]{\tfrac{1}{2}}\right)$

(B)

$\displaystyle\left(\sqrt[3]{\tfrac{1}{2}},\; \tfrac{1}{2}-3\sqrt[3]{\tfrac{1}{2}}\right)$

(C)

$\displaystyle(-1,2)\text{ and }(1,-2)$

(D)

No such point exists

Q4. Assertion (A): If $f$ is twice differentiable on $\mathbb{R}$ and $f''(x_0)=0$ , then $x_0$ is a point of inflection of $f$ .
Reason (R): If $f$ is twice differentiable on $\mathbb{R}$ and $x_0$ is a point of inflection of $f$ , then $f''(x_0)=0$ .

(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.

Q5. For the curve $y=\dfrac{x}{1+x^2}$ , the normal at a point $(a,\tfrac{a}{1+a^2})$ passes through the origin for which value(s) of $a$ ?

(A)

$a=\pm 1$

(B)

$a=\pm\sqrt{3}$

(C)

Only at $(0,0)$

(D)

At no point

Q6. Use linear approximation of $f(x)=x^{1/3}$ about $x=8$ to estimate the value of $(8.2)^{1/3}$ .

(A)

$2+\dfrac{1}{40}$

(B)

$2+\dfrac{1}{50}$

(C)

$2+\dfrac{1}{60}$

(D)

$2+\dfrac{1}{100}$

Q7. Let $f(x)=\dfrac{\ln x}{x}$ for $x>0$ . Using derivatives, find the point where $f$ attains its maximum and the maximum value.

(A)

$x=e,\ \text{maximum } \dfrac{1}{e}$

(B)

$x=1,\ \text{maximum } 0$

(C)

$x=\sqrt{e},\ \text{maximum } \dfrac{1}{2\sqrt{e}}$

(D)

$x=e^2,\ \text{maximum } \dfrac{2}{e^2}$

Q8. A rectangle is inscribed with its base on the x-axis in the right triangle with vertices $(0,0),(6,0),(0,8)$ so that the upper-right corner of the rectangle lies on the hypotenuse. Find the base and height of the rectangle which gives the maximum area.

(A)

base $4$ , height $\dfrac{8}{3}$

(B)

base $2$ , height $\dfrac{16}{3}$

(C)

base $6$ , height $0$

(D)

base $3$ , height $4$

Q9. For $x>0$ , consider $f(x)=x^x$ . Using derivatives, find the point at which $f$ attains its minimum and the minimum value.

(A)

$x=1,\ f_{\min}=1$

(B)

$x=\dfrac{1}{e},\ f_{\min}=\left(\dfrac{1}{e}\right)^{1/e}$

(C)

$x=e,\ f_{\min}=e^e$

(D)

$f$ has no minimum on $(0,\infty)$

Q10. For real $\lambda$ , determine the range of $\lambda$ for which the cubic equation $x^3-3x+\lambda=0$ has three distinct real roots.

(A)

$\lambda\in(-2,2)$

(B)

$\lambda\in(-\infty,-2]$

(C)

$\lambda\in[2,\infty)$

(D)

$\lambda\in[-2,2]$

Q11. Use linear approximation (differential) at $x=8$ to approximate $\sqrt[3]{8.1}$ .

(A)

$2.002$

(B)

$2.010$

(C)

$2.008333\ldots$

(D)

$2.00625$

Q12. At which point on the curve $y=\dfrac{x}{x^2+1}$ is the slope of the tangent maximum?

(A)

$(0,0)$

(B)

$(1,\tfrac{1}{2})$

(C)

$\big(\tfrac{1}{\sqrt3},\tfrac{\sqrt3}{4}\big)$

(D)

$(-1,-\tfrac{1}{2})$

Q13. Assertion (A): If $f$ is continuous on $[a,b]$ , differentiable on $(a,b)$ and $f'(x)>0$ for all $x\in(a,b)$ , then $f$ is convex on $[a,b]$ .
Reason (R): A function $f$ is convex on $[a,b]$ if and only if $f'$ is non-decreasing on $(a,b)$ .

(A)

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

(B)

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

(C)

A is true but R is false.

(D)

A is false but R is true.

Q14. For which value(s) of $k$ does the cubic equation $x^3-6x^2+11x-k=0$ have a repeated root?

(A)

$k=5$

(B)

$k=\dfrac{54\pm 2\sqrt{3}}{9}$

(C)

$k=6$

(D)

$k=\dfrac{7}{2}$

Q15. For which real values of $a$ does the equation $x^x=a$ (with $x>0$ ) have exactly two positive real solutions?

(A)

$0<a<e^{-1/e}$

(B)

$a>e^{-1/e}$

(C)

$e^{-1/e}<a<1$

(D)

$0<a<1$

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