Let \(g(t)=\int \limits_{-\pi / 2}^{\pi / 2} \cos \left(\frac{\pi}{4} t+f(x)\right) \,d x\), where \(f(x)=\log _{e}\left(x+\sqrt{x^{2}+1}\right), x \in R\). Then which one of the following is correct?

Let \({ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}-1}=28,{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}=56\) and \({ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}+1}=70\). Let \(\mathrm{A}(4 \cos t, 4 \sin t), \mathrm{B}(2 \sin t,-2 \cos \mathrm{t})\) and \(C\left(3 r-n, r^2-n-1\right)\) be the vertices of a triangle \(A B C\), where \(t\) is a parameter. If \((3 x-1)^2+(3 y)^2\) \(=\alpha\), is the locus of the centroid of triangle ABC , then \(\alpha\) equals

Let \(A\) be a \(n \times n\) matrix such that \(| A |=2\). If the determinant of the matrix \(\operatorname{Adj}\left(2 . \operatorname{Adj}\left(2 A ^{-1}\right)\right.\) ). is \(2^{84}\), then \(n\) is equal to \(................\)

Consider the following two statements : Statement \(I\) : For any two non-zero complex numbers \(\mathrm{z}_1, \mathrm{z}_2\) \(\left(\left|z_1\right|+\left|z_2\right|\right)\left|\frac{z_1}{\left|z_1\right|}+\frac{z_2}{\left|z_2\right|}\right| \leq 2\left(\left|z_1\right|+\left|z_2\right|\right)\) and Statement \(II\) : If \(\mathrm{x}, \mathrm{y}, \mathrm{z}\) are three distinct complex numbers and a, b, c are three positive real numbers such that \(\frac{a}{|y-z|}=\frac{b}{|z-x|}=\frac{c}{|x-y|}\), then \(\frac{\mathrm{a}^2}{\mathrm{y}-\mathrm{z}}+\frac{\mathrm{b}^2}{\mathrm{z}-\mathrm{x}}+\frac{\mathrm{c}^2}{\mathrm{x}-\mathrm{y}}=1\) Between the above two statements,

Let \({A_n} = \left( {\frac{3}{4}} \right) - {\left( {\frac{3}{4}} \right)^2} + {\left( {\frac{3}{4}} \right)^3} - ..... + {\left( { - 1} \right)^{n - 1}}{\left( {\frac{3}{4}} \right)^n}\) and \(B_n \,= 1 - A_n\) . Then, the least odd natural number \(p\) , so that \({B_n} > {A_n}\), for all \(n \geq p\) is

Let \(A=\left[\begin{array}{ll}2 & 3 \\ a & 0\end{array}\right], a \in R\) be written as \(P+Q\) where \(P\) is a symmetric matrix and \(Q\) is skew symmetric matrix. If \(\operatorname{det}(Q)=9\), then the modulus of the sum of all possible values of determinant of \(P\) is equal to:

Let a vector \(\vec{a}\) has a magnitude \(9\) . Let a vector \(\vec{b}\) be such that for every \((x, y) \in R \times R-\{(0,0)\}\), the vector \((x \vec{a}+y \vec{b})\) is perpendicular to the vector (6y \(\vec{a}-18 \times \vec{b}\) ). Then the value of \(|\vec{a} \times \vec{b}|\) is equal to.

Consider the system of linear equations \(x+y+z=5, x+2 y+\lambda^2 z=9\) \(x+3 y+\lambda z=\mu\), where \(\lambda, \mu \in R\). Then, which of the following statement is NOT correct?

If the domain of the function \(f(x)=\log _e\) \(\left(\frac{2 x+3}{4 x^2+x-3}\right)+\cos ^{-1}\left(\frac{2 x-1}{x+2}\right)\) is \((\alpha, \beta]\), then the value of \(5 \beta-4 \alpha\) is equal to

Let \(A=\left\{a_{i}\right\}\) be a \(3 \times 3\) matrix, where \(a_{i j}=\left\{\begin{aligned}(-1)^{j-i} & \text { if } i < j \\ 2 & \text { if } i=j \\(-1)^{i+j} & \text { if } i > j \end{aligned}\right.\)  then \(\operatorname{det}\left(3 \operatorname{Adj}\left(2 \mathrm{~A}^{-1}\right)\right)\) is equal to \(.....\)

If \( \overrightarrow{ a }=2 \hat{ i }+\hat{ j }+3 \hat{ k },  \overrightarrow{ b }=3 \hat{ i }+3 \hat{ j }+\hat{ k } \) and \(\overrightarrow{ c }= c _{1} \hat{ i }+ c _{2} \hat{ j }+ c _{3} \hat{ k }\) are coplanar vectors and \(\overrightarrow{ a } \cdot \overrightarrow{ c }=5, \overrightarrow{ b } \perp \overrightarrow{ c }\), then \(122\left( c _{1}+ c _{2}+ c _{3}\right)\) is equal to.......

Let the function \(f(x)=\left(x^2+1\right)\left|x^2-a x+2\right|+\cos |x|\) be not differentiable at the two points \(x=\alpha=2\) and \(x=\beta\). Then the distance of the point \((\alpha, \beta)\) from the line \(12 x+5 y+10=0\) is equal to :

Let a line \(L\) be perpendicular to both the lines \(L_1: \dfrac{x+1}{3} = \dfrac{y+3}{5} = \dfrac{z+5}{7}\) and \(L_2: \dfrac{x-2}{1} = \dfrac{y-4}{4} = \dfrac{z-6}{7}\). If \(\theta\) is the acute angle between the lines \(L\) and \(L_3: \dfrac{x - \dfrac{8}{7}}{2} = \dfrac{y - \dfrac{4}{7}}{1} = \dfrac{z}{2}\), then \(\tan\theta\) is equal to: