Let \(I\) be the identity matrix of order \(3 \times 3\) and for the matrix \(\mathrm{A}=\left[\begin{array}{ccc}\lambda & 2 & 3 \\ 4 & 5 & 6 \\ 7 & -1 & 2\end{array}\right],|\mathrm{A}|=-1\). Let B be the inverse of the matrix \(\operatorname{adj}\left(\mathrm{A} \operatorname{adj}\left(\mathrm{A}^2\right)\right)\). Then \(|(\lambda B+1)|\) is equal to _____

Let \(\vec{a}=\alpha \hat{i}+\hat{j}+\beta \hat{k}\) and \(\vec{b}=3 \hat{i}-5 \hat{j}+4 \hat{k}\) be two vectors, such that \(\vec{a} \times \vec{b}=-\hat{i}+9 \hat{i}+12 k\). Then the projection of \(\vec{b}-2 \vec{a}\) on \(\vec{b}+\vec{a}\) is equal to.

Let \(P\) and \(Q\) be two distinct points on a circle which has center at \(C(2,3)\) and which passes through origin \(O\) , If \(O C\) is perpendicular to both the line segments \(C P\) and \(C Q\), then the set \(\{\mathrm{P}, \mathrm{Q}\}\) is equal to:

If \(f(x)\, = {x^2} - x + 5,\,\,x > \frac{1}{2},\) and \(g(x)\) is its inverse function, then \(g'(7)\) equals

The number of seven digit positive integers formed using the digits \(1,2,3\) and \(4\) only and sum of the digits equal to \(12\) is \(...........\).

Let \([ x ]\) denote the greatest integer function and \(f ( x )=\max \{1+ x +[ x ], 2+ x , x +2[ x ]\}, 0 \leq x \leq 2\). Let \(m\) be the number of points in \([0,2]\), where \(f\) is not continuous and \(n\) be the number of points in \((0,2)\), where \(f\) is not differentiable. Then \((m+n)^2+2\) is equal to:

Among the statements
(S1) : The set \(\left\{\mathrm{z} \in \mathbb{C}-\{-\mathrm{i}\}:|\mathrm{z}|=1\right.\) and \(\frac{\mathrm{z}-\mathrm{i}}{\mathrm{z}+\mathrm{i}}\) is purely real} contains exactly two elements, and (S2) : The set \(\left\{\mathrm{z} \in \mathbb{C}-\{-1\}:|\mathrm{z}|=1\right.\) and \(\frac{\mathrm{z}-1}{\mathrm{z}+1}\) is purely imaginary contains infinitely many elements.

Let \(L_1: \overrightarrow{\mathrm{r}}=(\hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}})+\lambda(\hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}}), \lambda \in \mathrm{R}\) \(L_2: \overrightarrow{\mathrm{r}}=(\hat{\mathrm{j}}-\hat{\mathrm{k}})+\mu(3 \hat{\mathrm{i}}+\hat{\mathrm{j}}+\mathrm{p} \hat{\mathrm{k}}), \mu \in \mathrm{R}\) and  \(L_3: \overrightarrow{\mathrm{r}}=\delta(\ell \hat{\mathrm{i}}+\mathrm{m} \hat{\mathrm{j}}+\mathrm{n} \hat{\mathrm{k}}) \delta \in \mathrm{R}\). Be three lines such that \(\mathrm{L}_1\) is perpendicular to \(\mathrm{L}_2\) and \(L_3\) is perpendicular to both \(L_1\) and \(L_2\). Then the point which lies on \(\mathrm{L}_3\) is

Let a differentiable function \(f\) satisfy \(f ( x )+\int \limits_3^{ x } \frac{ f ( t )}{ t } dt =\sqrt{ x +1}, x \geq 3\). Then \(12 f (8)\) is equal to:

Let \(\vec{v}=\alpha \hat{i}+2 \hat{j}-3 \hat{k}, \vec{w}=2 \alpha \hat{i}+\hat{j}-\hat{k}\), and \(\overrightarrow{ u }\) be a vector such that \(|\vec{u}|=\alpha > 0\). If the minimum value of the scalar triple product \([\vec{u} \vec{v} \vec{w}]\) is \(-\alpha \sqrt{3401}\), and \(|\vec{u} . \hat{i}|^2=\frac{m}{n}\) where \(m\) and \(n\) are coprime natural numbers, then \(m + n\) is equal to \(.........\).

Let \(a_1, a_2, \ldots \ldots, a_n\) be in A.P. If \(a_5=2 a_7\) and \(a_{11}=18\), then \(12\left(\frac{1}{\sqrt{a_{10}}+\sqrt{a_{11}}}+\frac{1}{\sqrt{a_{11}}+\sqrt{a_{12}}}+\ldots . \cdot \frac{1}{\sqrt{a_{17}}+\sqrt{a_{18}}}\right)\) is equal to \(..........\).

Let \(y=y_{1}(x)\) and \(y=y_{2}(x)\) be two distinct solutions of the differential equation \(\frac{d y}{d x}=x+y\), with \(y _{1}(0)=0\) and \(y _{2}(0)=1\) respectively. Then, the number of points of intersection of \(y=y_{1}(x)\) and \(y=y_{2}(x)\) is.

Let \(\alpha, \beta, \gamma\) and \(\delta\) be the coefficients of \(x^7, x^5, x^3\) and \(x\) respectively in the expansion of \(\left(x+\sqrt{x^3-1}\right)^5+\left(x-\sqrt{x^3-1}\right)^5, x\gt1\). If u and v satisfy the equations
\(\begin{aligned}
& \alpha u+\beta v=18 \\
& \gamma u+\delta v=20
\end{aligned}\)
then \(u+v\) equals :