From $5$ consonants and $5$ vowels, how many words can be formed using $3$ consonants and $2$ vowels?

The general solution of the differential equation $\tan \left(y\right)dx+{\sec }^2\left(y\right)·\tan \left(x\right)dy=0$ is

\(y=a x+b\) is

If \(\int \log \left(6 \sin ^2 x+17 \sin x+12\right)^{\cos x} d x=f(x)+c\) then, \(f\left(\frac{\pi}{2}\right)=\)

If \(\int f(x) d x=F(x)+C\), then \(\frac{d}{d t} \int_{g(t)}^{h(t)} f(x) d x=\)

let \(a, b, c\) be real numbers such that \(2 a+3 b+6 c=0\) and \(g(x) \equiv a x^2+b x+c=0\) has atleast one root in the interval \((1,2)\). If a function \(f:[1,2] \rightarrow R\) for which Rolle's mean value theorem holds is such that \(f(x)\) is a primitive of \(g(x)\), then \(f(x)=\)

The sum of the coefficients of the last 19 terms in the binomial expansion of \((1+x)^{37}\) is

The image of the point \(P(3,8)\) with respect to the line \(x+3 y=7\), assuming the line to be a plane mirror, is equal to .........

The value of the determinant
\(\left|\begin{array}{ccc}a+b & a+2 b & a+3 b \\ a+2 b & a+3 b & a+4 b \\ a+4 b & a+5 b & a+6 b\end{array}\right|\) is

PQ is a focal chord of the parabola \(\mathrm{y}^2=4 \mathrm{x}\) with focus S. If \(\mathrm{P}=(4,4)\) then \(\mathrm{SQ}=\)

In the circuit, the cells are having negligible resistances. If the galvanometer shows null defection then the value of ' \(\mathrm{V}\) ' is

When ' \(\mathrm{X}\) ' \(\mathrm{g}\) of graphite is completely burnt in a bomb calorimeter in excess of \(\mathrm{O}_2\) at \(298 \mathrm{~K}\) and \(1 \mathrm{~atm}\) pressure as given in the equation
\(\mathrm{C}\) (graphite) \(+\mathrm{O}_2(\mathrm{~g}) \rightarrow \mathrm{CO}_2(\mathrm{~g})\)
The temperature of calorimeter raised from \(298 \mathrm{~K}\) to \(302 \mathrm{~K}\). If the heat capacity of the calorimeter and molar enthalpy change for the reaction at \(1 \mathrm{~atm}\) and \(298 \mathrm{~K}\) are \(20.7 \mathrm{~kJ} \mathrm{~K}^{-1}\) and \(-248.4 \mathrm{~kJ} \mathrm{~mol}^{-1}\), ' \(\mathrm{X}^{\prime}\) in \(\mathrm{g}\) is

Assertion $\left(\mathrm{A}\right):$ If $\sqrt{4{\sin }^4\theta +{\sin }^{2}2\theta }+4{\cos }^2\left(\frac{\pi }{4}-\frac{\theta }{2}\right)=2,$ then $\theta$ lies in $3^{\mathrm{rd}}$ quadrant or $4^{\mathrm{th}}$ quadrant.
Reason $\left(\mathrm{R}\right): \sqrt{{\sin }^2\theta }=\sin \theta$