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ISC Class 12 Continuity, Differentiability & Derivatives — practice questions

Explore 115 free Continuity, Differentiability & Derivatives questions in ISC Class 12 Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Continuity, Differentiability & Derivatives questions and exercises with answers

Showing questions 1–30 of 115.

The area of a circle is $A=\pi r^2$. The rate of change of the area with respect to the radius $r$, at $r=5$ cm, isRead the answer and worked solution →The volume of a cube is increasing at $9$ cm$^3$/s. When the edge length is $10$ cm, the surface area is increasing at the rate ofRead the answer and worked solution →The length $x$ of a rectangle decreases at $3$ cm/min and the width $y$ increases at $2$ cm/min. The rate of change of the perimeter isRead the answer and worked solution →The total revenue (in Rupees) from selling $x$ units is $R(x)=3x^2+36x+5$. The marginal revenue when $x=15$ isRead the answer and worked solution →The function $f(x)=x^2-4x+6$ is decreasing on the intervalRead the answer and worked solution →For $f(x)=4x^3-6x^2-72x+30$, the function is decreasing on the intervalRead the answer and worked solution →Which statement about $f(x)=x^3-3x^2+4x$ on $\mathbf{R}$ is correct?Read the answer and worked solution →The function $f(x)=\cos x$ is decreasing on the intervalRead the answer and worked solution →The function $f(x)=\sin 3x$ on $\left[0,\tfrac{\pi}{2}\right]$ is increasing on the intervalRead the answer and worked solution →The function $f(x)=\sin x+\cos x$ on $[0,2\pi]$ is decreasing on the intervalRead the answer and worked solution →The interval in which $y=x^2 e^{-x}$ is increasing isRead the answer and worked solution →Which of the following functions is decreasing on $\left(0,\tfrac{\pi}{2}\right)$?Read the answer and worked solution →For the function $f(x)=x^2+ax+1$ to be increasing on $[1,2]$, the values of $a$ must satisfyRead the answer and worked solution →For $f(x)=x^3-3x+3$, the point of local maxima and the local maximum value areRead the answer and worked solution →For the function $f(x)=2x^3-6x^2+6x+5$, the point $x=1$ isRead the answer and worked solution →The local minimum value of $f(x)=3+|x|$ on $\mathbf{R}$ isRead the answer and worked solution →Two positive numbers have a sum of $15$. The sum of their squares is minimum when the numbers areRead the answer and worked solution →A trapezium has its three sides other than the base each equal to $10$ cm. Its area is maximum when the slant foot length $x$ isRead the answer and worked solution →The maximum area of the trapezium whose three non-base sides are each $10$ cm (maximum at $x=5$) isRead the answer and worked solution →A cylinder of greatest curved surface area is inscribed in a cone of base radius $r$. The radius of this cylinder isRead the answer and worked solution →For $f(x)=2x^3-15x^2+36x+1$ on $[1,5]$, the absolute maximum value isRead the answer and worked solution →For $f(x)=2x^3-15x^2+36x+1$ on $[1,5]$, the absolute minimum value isRead the answer and worked solution →For $f(x)=12x^{4/3}-6x^{1/3}$ on $[-1,1]$, the absolute maximum value isRead the answer and worked solution →A helicopter flies along $y=x^2+7$. A soldier at $(3,7)$ wants the nearest distance. The nearest distance isRead the answer and worked solution →The function $f(x)=2x^3-6x^2+6x+5$ has critical point(s) atRead the answer and worked solution →The point on the curve $x^2=2y$ which is nearest to the point $(0,5)$ isRead the answer and worked solution →The maximum value of $[x(x-1)+1]^{1/3}$ for $0\le x\le 1$ isRead the answer and worked solution →A cylindrical tank of radius $10$ m is filled with wheat at $314$ m$^3$/h. The depth of the wheat is increasing at the rate of (take $\pi=3.14$)Read the answer and worked solution →For $f(x)=3x^4-8x^3+12x^2-48x+25$ on $[0,3]$, the critical point inside the interval isRead the answer and worked solution →Two numbers have a sum of $24$. Their product is largest when the numbers areRead the answer and worked solution →

Worked practice questions

The area of a circle is $A=\pi r^2$. The rate of change of the area with respect to the radius $r$, at $r=5$ cm, is

  1. $5\pi$ cm$^2$/cm
  2. $\pi$ cm$^2$/cm
  3. $25\pi$ cm$^2$/cm
  4. $10\pi$ cm$^2$/cm
Show answer and solution

Correct answer: D. $10\pi$ cm$^2$/cm

1. Area: $A=\pi r^2$. 2. Differentiate w.r.t. $r$: $\dfrac{dA}{dr}=2\pi r$. 3. Put $r=5$: $\dfrac{dA}{dr}=2\pi(5)=10\pi$. _Source: syllabus Mathematics Ch 6 "Application of Derivatives", p.1_

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The volume of a cube is increasing at $9$ cm$^3$/s. When the edge length is $10$ cm, the surface area is increasing at the rate of

  1. $1.8$ cm$^2$/s
  2. $2.4$ cm$^2$/s
  3. $3.6$ cm$^2$/s
  4. $7.2$ cm$^2$/s
Show answer and solution

Correct answer: C. $3.6$ cm$^2$/s

1. $V=x^3$, $S=6x^2$. 2. $\dfrac{dV}{dt}=3x^2\dfrac{dx}{dt}=9 \Rightarrow \dfrac{dx}{dt}=\dfrac{3}{x^2}$. 3. $\dfrac{dS}{dt}=12x\dfrac{dx}{dt}=12x\cdot\dfrac{3}{x^2}=\dfrac{36}{x}$. 4. At $x=10$: $\dfrac{dS}{dt}=\dfrac{36}{10}=3.6$. 5. The trap $7.2$ wrongly uses $S=3x^2$. _Source: syllabus Mathematics Ch 6 "Application of Derivatives", p.1_

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The length $x$ of a rectangle decreases at $3$ cm/min and the width $y$ increases at $2$ cm/min. The rate of change of the perimeter is

  1. $+2$ cm/min
  2. $-2$ cm/min
  3. $+5$ cm/min
  4. $-5$ cm/min
Show answer and solution

Correct answer: B. $-2$ cm/min

1. $\dfrac{dx}{dt}=-3$, $\dfrac{dy}{dt}=2$. 2. Perimeter $P=2(x+y)$. 3. $\dfrac{dP}{dt}=2\left(\dfrac{dx}{dt}+\dfrac{dy}{dt}\right)=2(-3+2)=-2$. _Source: syllabus Mathematics Ch 6 "Application of Derivatives", p.2_

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