Calculus PYQ — Page 22

$\int \left(\frac{1}{\log_e x} - \frac{1}{(\log_e x)^2}\right)dx$ is equal to

Match List-I with List-II | List-I | List-II | |---|---| | Differential equation | Integrating factor | | (A) $x\frac{dy}{dx} - y = 2x^2$ | (I) $e^{-y}$ | | (B) $\frac{dy}{dx} + \frac{y}{x} = 2x$ | (II) $\frac{1}{x}$ | | (C) $x\frac{dy}{dx} + 2y = x^2logx$ | (III) $x$ | | (D) $\frac{dx}{dy} - x = y$ | (IV) $x^2$ | Choose the correct answer from the options given below:

The area bounded by $y = 3x + 1$, $x = 0$, $y = 0$ and $x = a$ is 8 Sq.units. Then value of $a$ (where $a > 0$) is

Match List-I with List-II | List-I | List-II | |---|---| | (A) Marginal average cost if cost function $C(x) = \frac{50}{\sqrt{x}}$ | (I) $50\sqrt{x}$ | | (B) Marginal average cost if cost function $C(x) = 50\sqrt{x}$ | (II) $-\frac{75}{x^2\sqrt{x}}$ | | (C) Revenue function if demand function $P=\frac{50}{\sqrt{x}}$ | (III) $\frac{-25}{x\sqrt{x}}$ | | (D) Marginal revenue if demand function $P=50\sqrt{x}$ | (IV) $75\sqrt{x}$ | Choose the **correct** answer from the options given below:

The particular solution of the differential equation $\left[x \sin^2\left(\frac{y}{x}\right) - y\right]dx + xdy = 0$, $y = \frac{\pi}{4}$ when $x = 1$ is

The area of the smaller region bounded by the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$ and the straight line $3x + 4y = 12$ is:

In the following differential equation $\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2 = 2x^2 \log\left(\frac{d^2y}{dx^2}\right)$ order and degree is:

$\int_{1}^{2} \frac{1}{x(x+1)} dx, x > 0$ equals

The function $f(x) = x + \frac{1}{x}$ has

A square board of side 36cm is made into a box without top by cutting a square from each corner and folding up the flaps to form a box then maximum volume of the box is

The edge of a cube is increasing at a rate of 7 cm/s. The rate of change of area of the cube when its side is 3 cm is:

If $e^x + e^y = e^{x+y}$, then $\frac{dy}{dx}$ equals

Area (in sq. units) of the region bounded by the curves $y = -1$, $y = 2$, $x = y^3$ and $x = 0$ is

If $f(x)$ and $g(x)$ are continuous functions in [0, a] such that $f(x) = f(a - x)$ and $g(x) + g(a - x) = a$ then $\int_{0}^{a} f(x)g(x)dx =$

The marginal cost of production of x units of a commodity is $56 + \frac{3}{2}x$. It is known that fixed costs are Rs.115. Then the total cost of producing 50 units is:-

The largest open interval in which the function $f(x) = 4x^3 - 5x^2 - 8x + 12$ increases, is:

A cylindrical drum of radius 7 cm and height 2 m is being kept in a vertical position filled with milk. If the milk is leaking at 14 cm³/sec from its lower base, then the rate of decrease in the level of milk is: [Take $\pi = \frac{22}{7}$]

If $x = a\sec^3 \theta$, $y = a \tan^3 \theta$, then $\frac{d^2y}{dx^2}$ equals.

Which of the following functions $f(x)$ are differentiable at $x = 0$? (A) $|x|$ (B) $|x - 1|$ (C) $[x]$, where $[t]$ denotes the greatest integer $\leq t$ (D) $|x + 1|$ (E) $x^2$ Choose the correct answer from the options given below:

The area of the region bounded by the curves $y = x$ and $y = x^3$ is:

The value of lim (x→0) (sin x)/x is

The area of the region bounded by the lines $\frac{x}{7 \sqrt{3} a}+\frac{y}{b}=4, x=0$ and $y=0$ is :

If $f(x)$, defined by $f(x)=\left\{\begin{array}{lll}k x+1 & \text { if } & x \leq \pi \\ \cos x & \text { if } & x>\pi\end{array}\right.$ is continuous at $x=\pi$, then the value of $k$ is :

The value of $\int_{0}^{1} \frac{a-b x^{2}}{\left(a+b x^{2}\right)^{2}} d x$ is :