List of practice Questions

If \[ 3P(A)=P(B)=\frac{3}{5} \] and \[ P(A|B)=\frac{2}{3}, \] then \(P(A\cup B)\) is:

For a given sample the computed values of variance and fourth central moment are \(3\) and \(63\) respectively. Then underlying frequency distribution is classified as:

Assertion (A): A relation \(R\) on the set \(\{1,2,3\}\) defined as \[ R=\{(1,1),(1,2),(2,1),(2,2),(3,3)\} \] is an equivalence relation.

Reason (R): A relation that is reflexive, symmetric and transitive is an equivalence relation.

Determine the values of \(x\) for which \[ f(x)=\frac{x-3}{x+1}, \quad x\neq -1 \] is an increasing function.

Solve the differential equation \[ (x+2y^3)\frac{dy}{dx}=y. \]

An engineer standing at point \(P\) wishes to determine the width of a rectangular pond. She finds distance to westernmost point \(A\) to be \(60\) m and distance to northernmost point \(B\) to be \(80\) m. If angle \(APB=60^\circ\), find \(AB\).

If \[ y\sqrt{x^2+1}=\log\left(\sqrt{x^2+1}-x\right), \] show that \[ (x^2+1)\frac{dy}{dx}+xy+1=0. \]

If \[ 3P(A)=P(B)=\frac{3}{5} \] and \[ P(A|B)=\frac{2}{3}, \] then \(P(A\cup B)\) is:

The order and degree of the differential equation \[ y=\left(\frac{d^2y}{dx^2}\right)^2-\lambda\frac{dy}{dx} \] is:

Determine the interval(s) in which \[ f(x)=\frac{x}{3}+\frac{3}{x}, \quad x\neq 0 \] is increasing.

\(x,y,z\) are positive real numbers such that \[ \sqrt{x+y}-3\sqrt{y+z}=2 \] and \[ 4x-5y-9z=8. \] Find \[ \sqrt{\frac{20x+38y+18z+1}{9z+9y+2}}. \]

Let \(x-y\tan35^\circ=\tan25^\circ(y+x\tan35^\circ)\) for some \(x,y\in\mathbb{R}\). Then which of the following is true?

If \[ y\sqrt{x^2+1}=\log\left(\sqrt{x^2+1}-x\right), \] show that \[ (x^2+1)\frac{dy}{dx}+xy+1=0. \]

\[ \lim_{x\to0} \frac{|x|\log_e(1+|\sin2x|)} {x^2(|x|+3)} \]

Sketch the graph defined by \[ \left\{(x,y): \frac{x^2}{25}+\frac{y^2}{25}=1\right\}. \] Find the area of the region of the minor segment cut off by the line \[ x=\frac{5}{2}, \] using integration.

Which of the following equation represents common tangent to parabola \(y=-x^2\) and \(y=(x-2)^2\)?

The distance (in meter) for 7 throws of a shotputter are \(14.5,15.2,16.8,17.1,15.9,16.3,14.7\). Calculate sample mean and sample standard deviation.

\(BC=a,\; AC=b,\; AB=c\) are sides of \(\triangle ABC\) and \(\angle C\neq \frac{\pi}{2}\). Which of the following is not correct?

Given observations \[ 10,4,11,6,17,15,9,8,x \] and mean = mode = median. Find the value of \(x\).

Let \(X\) and \(Y\) be two independent identically distributed Bernoulli random variables with \[ P(X=1)=\frac12, \qquad P(X=0)=\frac12. \] If \(Z=XY\), then distribution of \(Z\) is:

Segments of lines \[ 2x+3y=1 \] and \[ 4x-3y=11 \] are diameters of a circle of area \(153.94\) square units. Then the equation of circle with integer radius is:

Let \(f:[0,\infty)\to\mathbb{R}\) be defined by \[ f(x)=\frac{3x^2+4x+1}{x^2+3x+2}. \] Then the value of \((f^{-1})'(2)\) is:

If \( S_n = 3n^2 + 5n \) represents the sum of \( n \) terms of an A.P., find the \(10^{th}\) term \( (a_{10}) \).

A racing track is build around an elliptical ground whose equation is given by \( 9x^2 + 16y^2 = 144 \).The width of the track is 3 m as shown below :