The area (in sq units) bounded by the curves \(y^2=4 x\) and \(x^2=4 y\) is

If ${\int }_0^{2a}{x}^2\sqrt{2ax-x^2}dx=k{a}^4$, then $k:\pi =$

The product of the perpendicular distances from any point on the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) to its asymptotes is

If \((1+x)^{15}=a_0+a_1 x+\ldots+a_{15} x^{15}, \quad\) then \(\sum_{r=1}^{15} r \frac{a_r}{a_r-1}\) is equal to

If \(\triangle A B C\) is a non isosceles triangle and \(\angle C=90^{\circ}\), then \(\frac{a^2+b^2}{a^2-b^2} \sin (A-B)=\)

A computer program has two modules \(X\) and \(Y\) and errors in them occur independently. \(X\) has an error with probability 0.1 and \(Y\) has an error with probability 0.3 . If an error in \(X\) alone cause the program to crash with probability 0.5 , an error in \(Y\) alone causes the program to crash with probability 0.7 and an error in both \(X\) and \(Y\) cause the program to crash with probability 0.8 , then the probability that the program is crashed is

\(\mathrm{KMnO}_4\) oxidises \(\mathrm{C}_2 \mathrm{H}_2 \mathrm{O}_4\) to form \(\mathrm{CO}_2\). In which of the following, the reaction will be faster?

Statement (I) : The slope of kinetic energy-displacement curve of a body in motion will be directly proportional to its acceleration. Statement (II) : From a height of \(15 \mathrm{~m}\) a ball is projected vertically upwards with a velocity of \(30 \mathrm{~m} / \mathrm{s}\). If the ball rises to the same height after hitting the ground, the loss of its energy on hitting the ground is \(30 \%\). Statement (III) : The velocity acquired by a body of mass ' \(m\) ' after travelling a fixed distance from rest under the action of a constant force is directly proportional to mass ' \(\mathrm{m}\) '. Which of the following is correct?

If \(a, b, c\) are real numbers such that \(a-b=1, b-c=3\), then the number of matrices of the form \(A=\left[\begin{array}{ccc}1 & 1 & 1 \\ a & b & c \\ a^2 & b^2 & c^2\end{array}\right]\) such that \(|A|=-12\), is

\(\int \sqrt{x^2+x+1} d x \times \int \frac{1}{\sqrt{x^2+x+1}} d x\) is equal to

The ratio between kinetic and potential energies of a body executing simple harmonic motion, when it is at a distance of \(\frac{1}{N}\) of its amplitude from the mean position is