If the angle between the pair of lines given by the equation \(a x^2+4 x y+2 y^2=0\) is \(45^{\circ}\) then the possible values of \(a\)

The values of \(p\) and \(q\) so that the function \[ f(x)=\left\{\begin{array}{cl} (1+\mid \sin x)^{\frac{p}{\sin x}}, \frac{-\pi}{6} < x < 0 \ q & , x=0 \ e^{\frac{\sin 2 x}{\sin 3 x}} & , 0 < x < \frac{\pi}{6} \end{array}\right. \] is continuous at \(x=0\), are

The ${13}^{\mathrm{th}}$ term in the expansion of $(1-4x{)}^{-4}$ is

If the points with position vectors \(60 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}, 40 \hat{\mathbf{i}}-8 \hat{\mathbf{j}}\) and \(a \hat{\mathbf{i}}-52 \hat{\mathbf{j}}\) are collinear, then \(a\) is equal to

The value of $\left(1-\mathrm{cos\theta }\right)\left(1+\mathrm{cos\theta }\right)\left(1+{\cot }^2\mathrm{\theta }\right)$, when $\theta =\frac{\pi }{15}$ is

If $\frac{1}{(x-1)(x-2)(x-3)}=\frac{A}{x-1}+\frac{B}{x-2}+\frac{C}{x-3},$ and $\frac{x}{(x-1)(x-2)(x-3)}=\frac{P}{x-1}+\frac{Q}{x-2}+\frac{R}{x-3}$ then $A+2B+3C=$

\(\int \frac{d x}{x\left(x^4+1\right)}=\)

The hybridisation of \(\mathrm{Xe}\) in \(\mathrm{XeO}_3\) is

In the triangle with vertices at \(A(6,3), B(-6,3)\) and \(C(-6,-3)\), the median through \(A\) meets \(B C\) at \(P\), the line \(A C\) meets the \(x\)-axis at \(Q\), while \(R\) and \(S\) respectively denote the orthocentre and centroid of the triangle. Then the correct matching of the coordinates of points in List-I to List-II is \(\begin{array}{llll} & \text { List-I } & & \text { List-II } \ \text { (i) } & P & \text { (A) } & (0,0) \ \text { (ii) } & Q & \text { (B) } & (6,0) \ \text { (iii) } & R & \text { (C) } & (-2,1) \ \text { (iv) } & S & \text { (D) }(-6,0) \ & & \text { (E) }(-6,-3) \ & & \text { (F) }(-6,3)\end{array}\) (i) (ii) (iii) (iv)

The points of intersection of the parabolas \(y^2=5 x\) and \(x^2=5 y\) lie on the line

Solution of \(\frac{d y}{d x}=\frac{x \log x^2+x}{\sin y+y \cos y}\) is

A system is provided with \(50 \mathrm{~J}\) of heat and the work done on the system is \(10 \mathrm{~J}\). What is the change in internal energy of the system in joules?

Let $a, b, c$ be three real numbers. If the function
$f\left(x\right)=\left\{\begin{array}{lll}\cos \left(2x+\pi \right)& \mathrm{if}& x\le 0\\ a{x}^2+b& \mathrm{if}& 0<x<1\\ cx+4& \mathrm{if}& 1\le x\le 2\\ 3a+1& \mathrm{if}& x\ge 2\end{array}\right.$
is continuous everywhere, then $b^2-bc+c^2=$