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

A function \(f(x)\) represents the temperature of a chemical reaction over time. If \(f^{\prime}(t)>\theta\) and \(f^{\prime}(t)<0\) for a specific time interval, it means:

A The temperature is rising at an increasing rate. B The temperature is falling at a decreasing rate. C The temperature is rising, but the rate of increase is slowing down. D The temperature is falling at an increasing rate.

Select one option. Answers are shown after the test.

Question 4 of 10

A company wants to design a cylindrical can with a fixed volume that uses the least amount of material. This means they need to:

A Maximize the volume. B Minimize the surface area. C Maximize the height of the can. D Minimize the radius of the can.

Select one option. Answers are shown after the test.

Question 5 of 10

NCERT Exemplar

For a function \(f(x)\) on a closed interval \([a, b]\), the absolute maximum value can occur:

A Only where \(f^{\prime}(x)=0\). B Only at the endpoints a or b . C Either at a point where \(f^{\prime}(x)=0\) or at an endpoint. D Only where the function is increasing.

Select one option. Answers are shown after the test.

Question 6 of 10

NCERT Exemplar

The slope of the tangent line to the curve \(y=\mathrm{x}^2-3 x\) at \(x=1\) represents:

A The average rate of change of the function. B The maximum value of the function. C The instantaneous rate of change of the function at \(x=1\). D The point where the function crosses the \(x\)-axis.

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Question 7 of 10

NCERT Exemplar

The two curves \(x^3-3 x y^2+2=0\) and \(3 x^2 y-y^3=2\)

A touch each other B cut at right angle C cut at an angle \(\frac{\pi}{3}\) D cut at an angle \(\frac{\pi}{4}\)

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Question 8 of 10

NCERT Exemplar

If a function \(f(x)\) is decreasing on \((0,2)\) and increasing on \((2,5)\), then at \(x=2\) the function has a :

A Local maximum. B Point of inflection. C Local minimum. D Discontinuity.

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Question 9 of 10

NCERT Exemplar

The function \(f(x)=2 x^3-3 x^2-12 x+4\), has

A two points of local maximum B two points of local minimum C one maxima and one minima D no maxima or minima

Select one option. Answers are shown after the test.

Question 10 of 10

NCERT Exemplar

The interval on which the function \(f(x)=2 x^3+9 x^2+12 x-1\) is decreasing is:

A \([-1, \infty)\) B \([-2,-1]\) C \((-\infty,-2]\) D \([-1,1]\)

Select one option. Answers are shown after the test.

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10 Question MCQ (including Assertion)