Find the point on the curve $y^2 = 4x + 8$ for which the abscissa and ordinate changes at the same rate.
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Applications of Derivatives - ISC Class 12 Mathematics Questions with Answers, Page 4
87 past-paper questions on Applications of Derivatives from ISC Class 12 Mathematics papers (2027-2017), newest first, in full. Questions 61-80 are on this page, 20 to a page. Tap "Show answer" under a question to see its answer.
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Any tangent to the curve $y = 3x^7 + 5x + 3$ :
- (a)is parallel to x – axis
- (b)is parallel to y – axis
- (c)makes an acute angle with x – axis
- (d)makes on obtuse angle with y – axis
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A Norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular window as shown in the figure given below. The total perimeter of the window is 10 m.
Based on the above information answer the following by choosing the correct option:
(i)[2.0]
If the length and breadth of the rectangle portion of the window is $y$ and $x$ respectively (as shown in the figure above) then the relation between the variable is
- (a)$y = \frac{20 + (\pi - 2)x}{4}$
- (b)$y = \frac{20 - (\pi + 2)x}{2}$
- (c)$y = \frac{20 - (\pi + 4)x}{4}$
- (d)$y = \frac{20 - (\pi + 2)x}{4}$
(ii)[2.0]
Let A be the area of the Norman window which admits the sunlight. Then A expressed in terms of $x$ is
- (a)$A = 5x + \frac{\pi}{4}x^2 - 2x^2$
- (b)$A = 5x + \frac{\pi}{8}x^2 - \frac{1}{2}x^2$
- (c)$A = 5x - \frac{\pi}{8}x^2 - \frac{1}{2}x^2$
- (d)$A = 5x - \frac{\pi}{2}x^2 - \frac{1}{4}x^2$
(iii)[2.0]
For the maximum value of A what will be the radius of the semicircle?
- (a)$\frac{10}{2+\pi}$
- (b)$\frac{10}{\pi-2}$
- (c)$\frac{10}{4+\pi}$
- (d)$\frac{20}{4-\pi}$
(iv)[2.0]
For maximum value of A , the length of the rectangle represented by $y$ will be equal to:
- (a)$\frac{10}{4+\pi}$
- (b)$\frac{10}{\pi-2}$
- (c)$\frac{20}{4+\pi}$
- (d)$\frac{20}{4-\pi}$
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Identify from the given options the slope of the normal to the curve $x^2 + 3y + y^2 = 5$ at $(1, 1)$:
- (a)$-\frac{2}{5}$
- (b)$\frac{5}{2}$
- (c)$\frac{2}{5}$
- (d)$-\frac{5}{2}$
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A cone is inscribed in a sphere of radius 12 cm. If the volume of the cone is maximum, find its height.
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The function $f(x) = x^3 - 3x$ is strictly decreasing on:
- (a)$-1 < x < 1$
- (b)$-1 \le x \le 1$
- (c)$1 \le x < \infty$
- (d)$-\infty < x < -1$
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The function $f(x) = \sin x + \cos x$, $x \in (0, \pi/2)$; Maximum value of $f(x)$ is:
- (a)1
- (b)2
- (c)$\frac{1}{\sqrt{2}}$
- (d)$\sqrt{2}$
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Show that the radius of a closed right circular cylinder of given surface area and maximum volume is equal to half of its height.
Given: Closed right circular cylinder of given surface area and maximum volume
To show: Radius of the cylinder is equal to half of its height
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The edge of a variable cube is increasing at the rate of $10\text{ cm/sec}$. How fast is the volume of the cube increasing when the edge is $5\text{ cm}$ long?
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The equation of tangent at $(2, 3)$ on the curve $y^2 = px^3 + q$ is $y = 4x - 7$. Find the values of ‘$p$’ and ‘$q$’.
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Prove that the area of right-angled triangle of given hypotenuse is maximum when the triangle is isosceles.
Given: Right-angled triangle of given hypotenuse
To show: Area of right-angled triangle is maximum when the triangle is isosceles
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A 13 m long ladder is leaning against a wall, touching the wall at a certain height from the ground level. The bottom of the ladder is pulled away from the wall, along the ground, at the rate of 2 m/s. How fast is the height on the wall decreasing when the foot of the ladder is 5 m away from the wall?
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Prove that the function $f(x) = x^3 - 6x^2 + 12x + 5$ is increasing on $\mathbb{R}$.
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Find the point on the straight line $2x + 3y = 6$, which is closest to the origin.
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The volume of a closed rectangular metal box with a square base is $4096\text{ cm}^3$. The cost of polishing the outer surface of the box is ₹ 4 per $\text{cm}^2$. Find the dimensions of the box for the minimum cost of polishing it.
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Find the equations of the normals to the curve $y = x^3 + 2x + 6$ which are parallel to the line $x + 14y + 4 = 0$.
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Find the approximate change in the volume $V$ of a cube of side $x$ metres caused by decreasing the side by $1\%$.
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Find the points on the curve $y = 4x^3 - 3x + 5$ at which the equation of the tangent is parallel to the x-axis.
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A cone is inscribed in a sphere of radius $12\text{ cm}$. If the volume of the cone is maximum, find its height.
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A circular disc of radius 3 cm. is heated. Due to expansion its radius increases at the rate of $0.05\text{ cm/s}$. Find the rate at which its area increases when the radius is $3.2\text{ cm}$.
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