Consider the curves $C_1:y^2=4x$ and $C_2:x^2+y^2-6x+1=0$  Assertion $\left(\mathrm{A}\right)$: The common tangents to the curves $C_1$ and $C_2$ are orthogonal. Reason $\left(\mathrm{R}\right)$: $x-y+1=0$ and $x+y+1=0$ are the common tangents to the curves $C_1$ and $C_2$

If a function \(f\) is defined by \(f(x)=\frac{\cot ^3 x-\tan x}{\cos (x+\pi / 4)}(x \neq \pi / 4)\), then \(\lim _{x \rightarrow \frac{\pi}{4}} f(x)=\)

The value of ${\int }_0^{\pi /2}\frac{dx}{1+{\left(\tan x\right)}^{\sqrt{2018}}}$ is equal to

If \(S_n\) is the sum of the first \(n\) terms of the series \(1^2+2 \times 2^2+3^2+2 \times 4^2+5^2+2 \times 6^2+\ldots \infty,\) then, when \(n\) is even \(S_n=\)

If \(\alpha\) and \(\beta\) are the roots of \(x^2+2 x-3=0\) and \(\gamma, \delta\) are the roots of \(y^2-y-4=0\), then the equation of the circle having \((\alpha, \gamma)\) and \((\beta, \delta)\) as ends of a diameter is

\((\overrightarrow{\mathbf{a}}+2 \overrightarrow{\mathbf{b}}-\overrightarrow{\mathbf{c}}) \cdot(\overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}}) \times(\overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}}-\overrightarrow{\mathbf{c}})\) is equal to

\(f:[-2,2] \rightarrow[-2,2]\) and \(g:[-2,2] \rightarrow[0,4]\) are two functions defined as \(f(x)=\left\{\begin{array}{l}-2,-2 \leq x \leq 0 \ x^2-2,0 \leq x \leq 2\end{array}\right.\) and \(g(x)=|f(x)|+f(|x|)\), then

Let \(\overrightarrow{\mathbf{a}}=a_1 \hat{\mathbf{i}}+a_2 \hat{\mathbf{j}}+a_3 \hat{\mathbf{k}}\) Assertion (A) : The identity \(|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{i}}|^2+|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{j}}|^2+|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{k}}|^2=2|\overrightarrow{\mathbf{a}}|^2\) holds for \(\vec{a}\). Reason (R) : \(\overrightarrow{\mathbf{a}} \times \hat{\mathbf{i}}=a_3 \hat{\mathbf{j}}-a_2 \hat{\mathbf{k}}\), \(\overrightarrow{\mathbf{a}} \times \hat{\mathbf{j}}=a_1 \hat{\mathbf{k}}-a_3 \hat{\mathbf{i}}, \overrightarrow{\mathbf{a}} \times \hat{\mathbf{k}}=a_2 \hat{\mathbf{i}}-a_1 \hat{\mathbf{j}}\) Which of the following is correct?

Let $A\left({\theta }_1\right)$ and $B\left({\theta }_2\right)$ be two points on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and $S$ be the focus of the hyperbola. If $A,S,B$ are collinear and $a\cos \left(\frac{{\theta }_1+{\theta }_2}{2}\right)=k\cos \left(\frac{{\theta }_1-{\theta }_2}{2}\right)$ then $k=$

In a right angled triangle, if the difference between the two acute angles is \(60^{\circ}\), then the ratio of the lengths of the hypotenuse and the perpendicular drawn to the hypotenuse from its opposite vertex is

If \(f(x)=\left|\begin{array}{ccc}1+\sin x+\sin 2 x+\sin 3 x & \frac{3+\sin 2 x}{2} & \frac{-2+\sin 3 x}{3} \3+4 \sin x & \frac{3}{2} & \frac{4}{3} \sin x \1+\sin x & \frac{1}{2} \sin x & \frac{1}{3}\end{array}\right|\) then \(\int_0^{\pi / 2}\left(f(x)+f^{\prime}(x)\right) d x=\)

A random variable \(\mathrm{X}\) has the following probability distribution \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline \mathrm{X}=\mathrm{x} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \ \hline \mathrm{P}(\mathrm{X}=\mathrm{x}) & 0.15 & 0.23 & \mathrm{k} & 0.10 & 0.20 & 0.08 & 0.07 & 0.05 \ \hline \end{array} For the events \(E=\{x / x\) is a prime number \(\}\) and \(F=\{x / x < 4\}\) then \(P(E \cup F)=\)

If \(\alpha \in R, n \in N\) and \(n+2(n-1)+3(n-2)+\ldots+\) \((n-1) 2+n .1=\alpha n(n+1)(n+2)\), then \(\alpha=\)