The general solution of \(\frac{d y}{d x}=\frac{x^3\left(y^4+1\right)}{[}\) is \[ \left[2 y^{-\frac{2}{3}}+3\left(\frac{x}{\sqrt[3]{y}}\right)^2\right]^{\frac{3}{2}} \]

The value \(C\) of the Lagrange's mean value theorem for the function \(f(x)=x(x-1)(x-2)\) in the interval \(\left[0, \frac{1}{2}\right]\) is

\(\int_0^{\pi / 4} x^2 \sin 2 x d x=\)

The value of \(\cos \frac{2 \pi}{15} \cos \frac{4 \pi}{15} \cos \frac{8 \pi}{15} \cos \frac{14 \pi}{15}\) is :

There are $10$ red and $5$ yellow roses of different sizes. If $x$ is the number of garlands that can be formed with all these flowers so that no two yellow roses come together and $y$ is the number of garlands formed with all these flowers so that all the red roses coming together, then $\frac{2\left(x-y\right)}{10!}=$

If $f\left(x\right)=\left\{\begin{array}{cl}\frac{x^2\ln \cos x}{\ln \left(1+x^2\right)}& , x\ne 0\\ 0& , x=0\end{array}\right.$, then $f\left(x\right)$ is

If \(\alpha, \beta, \gamma\) are the roots of the equation \(x^3+3 x^2-x-3=0\), then \(\left(1+\alpha^2\right)\left(1+\beta^2\right)\left(1+\gamma^2\right)=\)

If \(f: R \rightarrow R\) is defined by \(f(x)=\left\{\begin{array}{ccc} \frac{x+2}{x^2+3 x+2} & \text { if } & x \in R-\{-1,-2\} \ -1 & \text { if } & x=-2 \ 0 & \text { if } & x=-1 \end{array}\right.\) then \(f\) is continuous on the set

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?

When the right gap of a meter bridge consists of two equal resistors in series, the balancing point is at 50 cm. When one of the resistors in the right gap is removed and is connected in parallel to the resistor in the left gap, the balancing point is at

The \(C_p\) of an ideal gas is \(10.314 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\). One mole of this gas is expanded against a constant pressure of p atm. The change in temperature during expansion is 1.0 K. The values of q (in J ) and \(\Delta \mathrm{H}\) (in \(\mathrm{J} \mathrm{mol}^{-1}\) ) are respectively

All the letters of the word 'COLLEGE' are arranged in all possible ways and all the seven letter words (with or without meaning) thus formed are arranged in the dictionary order. Then the rank of the word 'COLLEGE' is

The coefficient of $x^4$ in the expansion of ${\left(1-x-x^2+x^3\right)}^6$ is