List of practice Questions

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

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

Consider graph of function \(f(x)=2\cos\left(\frac{x}{2}\right)+3\), \(g(x)=4\). The number of points of intersection of two graphs in interval \([0,4\pi]\) is:

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?

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

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 \(AB\).

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:

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

Find Mode

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

Which of the following is not same as \((A\Delta B)\cap C\)?

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.

In a tuition batch of 2 students, the probability that \(X\) will pass the examination is \(\frac25\) and that of \(Y\) is \(\frac34\). Assuming independence, what is the probability that neither \(X\) nor \(Y\) will pass?

Find the acute angle at which curves \[ y=(x-2)^2 \] and \[ y=-4+6x-x^2 \] intersect.

Find the value of \(n\) if \[ \left(\sum_{k=1}^{n}(-1)^{k-1}k\right)^2 - \sum_{k=1}^{n}(-1)^{k-1}k^2 +2450=0. \]

Let \(A_1,A_2,A_3\) be events in a sample space with \(A_1\cap A_2\neq \phi\). Then always:

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:

If \(x,y,z\) satisfy \[ x+y+z=1, \] \[ 4x+9y+16z=25, \] \[ 16x+18y+256z=625, \] find the value of \(x\):

The number of ordered tuples \((p,q,r)\) in truth table for which the statement \((\neg p \lor q)\Rightarrow r\) is true is:

If \[ \tan^{-1}(3x)+\tan^{-1}(2x)=\frac{\pi}{4}, \] then find the value of \(x\):

Let \(R\subset \mathbb N\times \mathbb N\). Which of the following statements is necessarily true?

Let \(f:\mathbb{R}\to\mathbb{R}\), \(f(x)=|x+1|e^{-x^2}\), then which of the following option is true?

If \[ f(x)= \begin{cases} 1,& |x|\le 1\\ 0,& |x|>1 \end{cases} \] \[ g(x)= \begin{cases} 2-x^2,& |x|\le 2\\ 2,& |x|>2 \end{cases} \] and \(h(x)=f(g(x))\), then the interval in which \(h(x)=1\) for all values of \(x\) is:

The number of values of \(\theta\) in \([0,2\pi]\) for which the following homogeneous system has a non-trivial solution is: \[ x+(\sin\theta)y+(\cos\theta)z=0 \] \[ x+(\cos\theta)y+(\sin\theta)z=0 \] \[ x-(\sin\theta)y-(\cos\theta)z=0 \]

Find the value of determinant at \(x=2026\): \[ \begin{vmatrix} x & x+1 & x+3\\ x+1 & x+3 & x+6\\ x+3 & x+6 & x+10 \end{vmatrix} \]