Fог \(\theta \in\left(0, \frac{\pi}{2}\right)\), \(\operatorname{sech}^{-1}(\cos \theta)\) is equal to

If the normal drawn at \(\mathrm{P}(8,16)\) to the parabola \(y^2=32 x\) meets the parabola again at Q, then the equation of the tangent drawn at Q to the parabola is

If \(\alpha\) is a root of \(z^2-z+1=0\), then \(\left(\alpha^{2014}+\frac{1}{\alpha^{2014}}\right)+\left(\alpha^{2015}+\frac{1}{\alpha^{2015}}\right)^2\) \[ \begin{aligned} & +\left(\alpha^{2016}+\frac{1}{\alpha^{2016}}\right)^3+\left(\alpha^{2017}+\frac{1}{\alpha^{2017}}\right)^4+ \ & \left(\alpha^{2018}+\frac{1}{\alpha^{2018}}\right)^5= \end{aligned} \]

If \(\int \frac{d x}{\sqrt{\sin ^3 x \cos x}}=g(x)+c\), then \(g(x)\) is equal to

The period of \(\tan k y+\sin k y\), where \(k=1+4+9+\ldots 20\) terms, is

The centroid of the triangle formed by the pair of straight lines \(12 x^2-20 x y+7 y^2=0\) and the line \(2 x-3 y+4=0\) is :

Let $\overrightarrow{a}$ be a vector in the plane containing vectors $\overrightarrow{b}=\hat{i}+2\hat{j}+\hat{k}$ and $\overrightarrow{c}=2\hat{i}-\hat{j}+\hat{k}$. If $\overrightarrow{a}$ is perpendicular to $\hat{i}+\hat{j}+3\hat{k}$ and its projection on $\overrightarrow{b}$ is $3\sqrt{6}$, then ${\left|\overrightarrow{a}\right|}^2=$

If \(\int \frac{(2 x+3)}{x(x+1)(x+2)(x+3)+1} d x\) \(=-\frac{1}{p x^2+q x+r}+c\), then \(\frac{3 p-q}{r}=\)

For the reaction \(\mathrm{N}_2 \mathrm{O}_4(g) \rightleftharpoons 2 \mathrm{NO}_2(g)\), the correct relation between degree of dissociation \((\alpha)\) of \(\mathrm{N}_2 \mathrm{O}_4(g)\) and equilibrium constant, \(\mathrm{K}_{\mathrm{p}}\) is \((\mathrm{P}=\) total pressure of mixture \()\)

If $\left(\frac{9}{4}, \frac{5}{4}, \frac{15}{4}\right)$ is the centroid of a tetrahedron whose vertices are $(a, 2, 1), (1, b, 4),$ $(4, 0, c)$ and $(1, 1, 7),$ then

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)

A charge \(q\) enters a region having electric field \(\mathbf{E}\) and magnetic field \(\mathbf{B}\) with velocity \(\mathbf{v}\). If it continues to move with the same velocity, then which of the following statement is not true

Acid rain is mainly caused by the emissions of which of the following gases? I. Sulphur dioxide II. Carbon dioxide III. Nitrogen dioxide IV. Methane