ISC Class 12 Application of Derivatives — practice questions
48 free MCQs with worked solutions. Tap any question for the answer + explanation, or practice them all in the app.
Practice ISC Class 12 Application of Derivatives in the app →The area of a circle is $A=\pi r^2$. The rate of change of the area with respect to the radius $r$, at $r=5$ cA stone dropped into a lake makes circular waves whose radius grows at $4$ cm/s. When the radius is $10$ cm, tThe volume of a cube is increasing at $9$ cm$^3$/s. When the edge length is $10$ cm, the surface area is increThe length $x$ of a rectangle decreases at $3$ cm/min and the width $y$ increases at $2$ cm/min. The rate of cThe length $x$ of a rectangle decreases at $3$ cm/min and the width $y$ increases at $2$ cm/min. When $x=10$ cThe total cost (in Rupees) of producing $x$ units is $C(x)=0.005x^3-0.02x^2+30x+5000$. The marginal cost when The total revenue (in Rupees) from selling $x$ units is $R(x)=3x^2+36x+5$. The marginal revenue when $x=15$ isA particle moves along the curve $6y=x^3+2$. The points at which the $y$-coordinate changes $8$ times as fast Sand pours from a pipe at $12$ cm$^3$/s, forming a cone whose height is always one-sixth of the base radius. WA balloon remains spherical while inflated by $900$ cm$^3$/s of gas. When the radius is $15$ cm, the radius isA ladder $5$ m long leans against a wall. The foot is pulled away along the ground at $2$ cm/s. When the foot A balloon remains spherical with diameter $\dfrac{3}{2}(2x+1)$. The rate of change of its volume with respect The function $f(x)=x^2-4x+6$ is decreasing on the intervalFor $f(x)=4x^3-6x^2-72x+30$, the function is decreasing on the intervalWhich statement about $f(x)=x^3-3x^2+4x$ on $\mathbf{R}$ is correct?The function $f(x)=\cos x$ is decreasing on the intervalThe function $f(x)= in 3x$ on $\left[0,\tfrac{\pi}{2}\right]$ is increasing on the intervalThe function $f(x)= in x+\cos x$ on $[0,2\pi]$ is decreasing on the intervalOn which interval is the function $f(x)=x^{100}+ in x-1$ decreasing?The interval in which $y=x^2 e^{-x}$ is increasing isWhich of the following functions is decreasing on $\left(0,\tfrac{\pi}{2}\right)$?For the function $f(x)=x^2+ax+1$ to be increasing on $[1,2]$, the values of $a$ must satisfyFor $f(x)=x^3-3x+3$, the point of local maxima and the local maximum value areFor the function $f(x)=2x^3-6x^2+6x+5$, the point $x=1$ isFor $f(x)=4x^3+12x^2-24x+10$ (so $f'(x)=12x(x-1)(x+2)$ after dividing constants), the second-derivative test aThe local minimum value of $f(x)=3+|x|$ on $\mathbf{R}$ isTwo positive numbers have a sum of $15$. The sum of their squares is minimum when the numbers arePoles AP$=16$ m and BQ$=22$ m stand at A and B with AB$=20$ m. A point R on AB at distance $x$ from A minimiseA trapezium has its three sides other than the base each equal to $10$ cm. Its area is maximum when the slant The maximum area of the trapezium whose three non-base sides are each $10$ cm (maximum at $x=5$) isA cylinder of greatest curved surface area is inscribed in a cone of base radius $r$. The radius of this cylinFor $f(x)=2x^3-15x^2+36x+1$ on $[1,5]$, the absolute maximum value isFor $f(x)=2x^3-15x^2+36x+1$ on $[1,5]$, the absolute minimum value isFor $f(x)=12x^{4/3}-6x^{1/3}$ on $[-1,1]$, the absolute maximum value isA helicopter flies along $y=x^2+7$. A soldier at $(3,7)$ wants the nearest distance. The nearest distance isThe function $f(x)=2x^3-6x^2+6x+5$ has critical point(s) atA car's distance is $x=t^2\left(2-\dfrac{t}{3}\right)$ metres. It starts at P ($t=0$) and stops at Q. The timeFor the car with $x=t^2\left(2-\dfrac{t}{3}\right)$ reaching Q at $t=4$ s, the distance PQ isA man $2$ m tall walks at $5$ km/h away from a lamp post $6$ m high. The length of his shadow increases at theAn open box is made from a $3$ m by $8$ m sheet by cutting squares of side $x$ from each corner. The volume isA manufacturer sells $x$ items at price $\left(5-\dfrac{x}{100}\right)$ each; cost of $x$ items is $\left(\dfrA water tank is an inverted cone with semi-vertical angle $\tan^{-1}(0.5)$. Water is poured at $5$ m$^3$/h. WhA circular disc of radius $3$ cm is heated; its radius grows at $0.05$ cm/s. The approximate rate of increase The point on the curve $x^2=2y$ which is nearest to the point $(0,5)$ isThe maximum value of $[x(x-1)+1]^{1/3}$ for $0\le x\le 1$ isA cylindrical tank of radius $10$ m is filled with wheat at $314$ m$^3$/h. The depth of the wheat is increasinFor $f(x)=3x^4-8x^3+12x^2-48x+25$ on $[0,3]$, the critical point inside the interval isTwo numbers have a sum of $24$. Their product is largest when the numbers are