10 Question MCQ (including Assertion)

Question 1 of 10

Assertion (A): The function \(f(x)=e^x\) has no local maximum or minimum.
Reason (R): The derivative of the function, \(f^{\prime}(x)=e^x\), is never equal to zero.

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.

Select one option. Answers are shown after the test.

Question 2 of 10

Assertion (A): The function \(f(x)=x+\frac{1}{x}\) has a local minimum at \(x=1\).
Reason \((\mathrm{R})\) : \(f^{\prime}(1)=0\) and \(f^{\prime \prime}(1)>0\).

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.

Select one option. Answers are shown after the test.

Question 3 of 10

NCERT Exemplar

Let the \(\mathrm{f}: \mathbf{R} \rightarrow \mathbf{R}\) be defined by \(\mathrm{f}(\mathrm{x})=2 \mathrm{x}+\cos \mathrm{x}\), then f :

A has a minimum at \(\mathrm{x}=\pi\) B has a maximum, at \(\mathrm{x}=0\) C is a decreasing function D is an increasing function

Select one option. Answers are shown after the test.

Question 4 of 10

NCERT Exemplar

If \(y=x^4-10\) and if \(x\) changes from 2 to 1.99 , what is the change in \(y\)

A 0.32 B 0.032 C 5.68 D 5.968

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Question 5 of 10

NCERT Exemplar

Finding the shortest distance from a point to a curve is an application of:

A The first derivative test for increasing/decreasing intervals. B Finding the absolute minimum of a distance function C The chain rule for related rates. D Finding points of inflection.

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Question 6 of 10

NCERT Exemplar

The maximum value of \(\left(\frac{1}{x}\right)^x\) is:

A e B \(\mathrm{e}^{\mathrm{e}}\) C \(e^{\frac{1}{e}}\) D \(\frac{1}{e}^{\frac{1}{e}}\)

Select one option. Answers are shown after the test.

Question 7 of 10

NCERT Exemplar

\(\mathrm{f}(\mathrm{x})=\mathrm{x}^{\mathrm{x}}\) has a stationary point at

A \(\mathrm{x}=\mathrm{e}\) B \(x=\frac{1}{e}\) C \(x=1\) D \(x=\sqrt{e}\)

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Question 8 of 10

NCERT Exemplar

The sides of an equilateral triangle are increasing at the rate of 2 \(\mathrm{cm} / \mathrm{sec}\). The rate at which the area increases, when side is 10 cm is:

A \(10 \mathrm{~cm}^2 / \mathrm{s}\) B \(3 \mathrm{~cm}^2 / \mathrm{s}\) C \(10 \sqrt{\mathbf{3}} \mathrm{~cm}^2 / \mathrm{s}\) D \(\frac{10}{3} \mathrm{~cm}^2 / \mathrm{s}\)

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Question 9 of 10

NCERT Exemplar

If the curve a \(y+x^2=7\) and \(x^3=y\), cut orthogonally at ( 1, 1), then the value of \(a\) is:

A 1 B 0 C -6 D 6

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Question 10 of 10

NCERT Exemplar

The equation of normal to the curve \(3 x^2-y^2=8\) which is parallel to the line \(x+3 y=8\) is

A \(3 x-y=8\) B \(3 x+y+8=0\) C \(x+3 y \pm 8=0\) D \(x+3 y=0\)

Select one option. Answers are shown after the test.

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10 Question MCQ (including Assertion)