List of practice Questions

Evaluate \[ \int \frac{5^x}{\sqrt{5^{-2x}} - 5^{2x}}\,dx \]

Evaluate $\displaystyle \int_{0}^{1} x(1-x)^5 \, dx$

If the radius of a circular blot of oil is increasing at the rate of $2$ cm/min, then the rate of change of its area when its radius is $3$ cm is

If $B$ is the end point of minor axis of the ellipse $b^2x^2+a^2y^2=a^2b^2\ (a>b)$ and $S$ and $S'$ are the foci of the ellipse such that $\triangle BSS'$ is an equilateral triangle, then the eccentricity $e$ is

If $x^2+y^2=t+\dfrac{1}{t}$ and $x^4+y^4=t^2+\dfrac{1}{t^2}$, then $\dfrac{dy}{dx}=$

The coordinates of the point where the line $\dfrac{x-1}{2}=\dfrac{y-2}{-3}=\dfrac{z+3}{4}$ meets the plane $2x+4y-z=1$ are

The particular solution of the differential equation \[ \left( y + x \frac{dy}{dx} \right) \sin y = \cos x \quad \text{at} \quad x = 0 \, \text{is:} \]

The population \( P(t) \) of a certain mouse species at time \( t \) satisfies the differential equation \[ \frac{dP(t)}{dt} = 0.5P(t) - 450. \quad \text{If} \, P(0) = 850, \, \text{then the time at which the population becomes zero is} \]

If the points \( (1, 1, \lambda) \) and \( (-3, 0, 1) \) are equidistant from the plane \( 3x + 4y - 12z + 13 = 0 \), then the integer value of \( \lambda \) is

The cumulative distribution function of a continuous random variable \( X \) is given by \( F(X = x) = \dfrac{\sqrt{x}}{2} \). Then \( P(X>1) \) is

The logical expression \( [p \wedge (q \vee r)] \vee [\neg r \wedge \neg q \wedge p] \) is equivalent to

The length of the perpendicular from the point \( P(a,b) \) to the line \( \dfrac{x}{a} + \dfrac{y}{b} = 1 \) is

If the elements of matrix \( A \) are the reciprocals of the elements of the matrix \( \begin{pmatrix} 1 & \omega & \omega^2 \\ \omega & \omega^2 & 1 \\ \omega^2 & 1 & \omega \end{pmatrix} \), where \( \omega \) is a complex cube root of unity, then

Evaluate the integral \[ \int \frac{dx}{\cos 2x - \cos^2 x} \]

The particular solution of the differential equation \[ y \frac{dx}{dy} = x \log x \quad \text{at} \quad x = e \text{ and } y = 1 \] is

Evaluate \[ \int \frac{\log x - 1}{1 + (\log x)^2} \, dx \]

The integrating factor of the differential equation \[ \frac{dy}{dx} + \frac{1}{x}y = x^3 - 3 \] is

If \[ \mathbf{a} = \hat{i} + \hat{j} + \hat{k}, \quad \mathbf{b} = \hat{i} - \hat{j} + 2\hat{k}, \quad \mathbf{c} = x\hat{i} + \hat{j} + (x - 1)\hat{k} \] If the vector \( \mathbf{c} \) lies in the plane of \( \mathbf{a} \) and \( \mathbf{b} \), then \( x = \)

Degree of the differential equation \[ \frac{dy}{dx} e^x + \left( \frac{dy}{dx} \right)^3 = x \]

If \[ O = (0, 0, 0), \quad P = (1, \sqrt{2}, 1), \] then the acute angles made by the line OP with the \( XOY, \, YOZ, \, ZOX \text{ planes are, respectively,} \)

Find the value of the following expression:
\[ 5^2 + 6^2 + 7^2 + \cdots + 20^2 = \]

The perimeter of a triangle is \(10\) cm. If one of its sides is \(4\) cm, then the remaining sides of the triangle, when the area of the triangle is maximum, are

If $\displaystyle \int_{1}^{k} (3x^2 + 2x + 1)\,dx = 11$, then $k =$

A body is heated to $110^\circ$C and placed in air at $10^\circ$C. After $1$ hour its temperature is $60^\circ$C. The additional time required for it to cool to $30^\circ$C is

The derivative of $\sin^{-1}\!\left(\dfrac{\sqrt{1+x}+\sqrt{1-x}}{2}\right)$ with respect to $\cos^{-1}x$ is