Let \(S_1\) and \(S_2\) be respectively the sets of all \(a \in R -\{0\}\) for which the system of linear equations \(a x+2 a y-3 a z=1\) \((2 a+1) x+(2 a+3) y+(a+1) z=2\) \((3 a+5) x+(a+5) y+(a+2) z=3\) has unique solution and infinitely many solutions. Then

Let \(\vec{a}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=2 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}\) and \(\overrightarrow{\mathrm{d}}=\vec{a} \times \overrightarrow{\mathrm{b}}\). If \(\overrightarrow{\mathrm{c}}\) is a vector such that \(\vec{a} \cdot \overrightarrow{\mathrm{c}}=|\overrightarrow{\mathrm{c}}|\), \(|\overrightarrow{\mathrm{c}}-2 \vec{a}|^2=8\) and the angle between \(\overrightarrow{\mathrm{d}}\) and \(\overrightarrow{\mathrm{c}}\) is \(\frac{\pi}{4}\), then \(|10-3 \overrightarrow{\mathrm{~b}} \cdot \overrightarrow{\mathrm{c}}|+|\overrightarrow{\mathrm{d}} \times \overrightarrow{\mathrm{c}}|^2\) is equal to

\(\left|\frac{120}{\pi^3} \int_0^\pi \frac{x^2 \sin x \cos x}{\sin ^4 x+\cos ^4 x} d x\right|\) is equal to ...........

If in the expansion of \((1+x)^{\mathrm{p}}(1-x)^{\mathrm{q}}\), the coefficients of \(x\) and \(x^2\) are 1 and -2 , respectively, then \(\mathrm{p}^2+\mathrm{q}^2\) is equal to :

Let \(\vec{a}, \vec{b}, \vec{c}\) be three mutually perpendicular vectors of the same magnitude and equally inclined at an angle \(\theta\), with the vector \(\vec{a}+\vec{b}+\vec{c}\). Then \(36 \cos ^{2} 2 \theta\) is equal to \(.....\)

The value of \(\sum_{ r =1}^{20}\left(\left|\sqrt{\pi\left(\int_0^{ r } x \mid \sin \pi x dx \right)}\right|\right)\) is ___ .

The area of the region bounded by \(y^{2}=8 x\) and \(y^{2}=16(3-x)\) 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.

The largest value of \(a,\) for which the perpendicular distance of the plane containing the lines \(\vec{r}=(\hat{i}+\hat{j})+\lambda(\hat{i}+a \hat{j}-\hat{k})\) and \(\vec{r}=(\hat{i}+\hat{j})+\mu(-\hat{i}+\hat{j}-a \hat{k})\) from the point \((2,1,4)\) is \(\sqrt{3}\), is\(...\)

The minimum value of the function \(f(x)=\int \limits_0^2 e^{|x-t|} d t\) is

Let \(A\) be a \(3 \times 3\) real matrix such that \(A \left(\begin{array}{l}1 \\ 1 \\ 0\end{array}\right)=\left(\begin{array}{l}1 \\ 1 \\ 0\end{array}\right) ; A \left(\begin{array}{l}1 \\ 0 \\ 1\end{array}\right)=\left(\begin{array}{c}-1 \\ 0 \\ 1\end{array}\right)\) and \(A \left(\begin{array}{l}0 \\ 0 \\ 1\end{array}\right)=\left(\begin{array}{l}1 \\ 1 \\ 2\end{array}\right)\). If \(X =\left( x _{1}, x _{2}, x _{3}\right)^{ T }\) and \(I\) is an identity matrix of order \(3\) , then the system \(( A -2 I ) X =\left(\begin{array}{l}4 \\ 1 \\ 1\end{array}\right)\) has

Let for a differentiable function \(f:(0, \infty) \rightarrow R\), \(f(x)-f(y) \geq \log _e\left(\frac{x}{y}\right)+x-y, \forall x, y \in(0, \infty) \text {. }\) Then \(\sum_{\mathrm{n}=1}^{20} \mathrm{f}^{\prime}\left(\frac{1}{\mathrm{n}^2}\right)\) is equal to

Let \(\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}\) and \(\overrightarrow{\mathrm{b}}=\hat{\mathrm{j}}-\hat{\mathrm{k}} .\) If \(\overrightarrow{\mathrm{c}}\) is a vector such that \(\vec{a} \times \vec{c}=\vec{b}\) and \(\vec{a} \cdot \vec{c}=3\), then \(\vec{a} \cdot(\vec{b} \times \vec{c})\) is equal to :