If the system of equations \(x-2 y+3 z=9\) \(2 x+y+z=b\) \(x-7 y+a z=24\) has infinitely many solutions, then \(a - b\) is equal to

If the domain of the function \( f(x)=\cos^{-1}(\frac{2x-5}{11-3x})+\sin^{-1}(2x^{2}-3x+1) \) is the interval \( [\alpha,\beta] \), then \( \alpha+2\beta \) is equal to:

If the value of the integral \(\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\frac{x^2 \cos x}{1+\pi^x}+\frac{1+\sin ^2 x}{1+e^{\sin x^{323}}}\right) d x=\frac{\pi}{4}(\pi+a)-2\), then the value of \(a\) is

If the area of the region \(\left\{(\mathrm{x}, \mathrm{y}): 1+\mathrm{x}^2 \leq \mathrm{y} \leq \min \{\mathrm{x}+7,11-3 \mathrm{x}\}\right\}\) is A , then 3 A is equal to

The mean and variance of \(7\) observations are \(8\) and \(16\) respectively. If one observation \(14\) is omitted a and \(b\) are respectively mean and variance of remaining \(6\) observation, then \(a+3 b-5\) is equal to \(..........\).

\(\mathop {\lim }\limits_{x \to \infty } \,\left( {\frac{n}{{{n^2}\, + {1^2}}} + \frac{n}{{{n^2} + {2^2}}} + \frac{n}{{{n^2} + {3^2}}} + ...\frac{1}{{5n}}} \right)\) is equal to

Let \(y=y(x)\) be the solution of the differential equation \(x\sqrt{1-x^2}\,dy + \left(y\sqrt{1-x^2} - x\cos^{-1}x\right)dx = 0\), \(x \in (0, 1)\), \(\displaystyle\lim_{x\to 1^-} y(x) = 1\). Then \(y\left(\dfrac{1}{2}\right)\) equals:

Let \(\alpha\) be a root of the equation \(x^{2}+x+1=0\) and the matrix \(A=\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}{1} & {1} & {1} \\ {1} & {\alpha} & {\alpha^{2}} \\ {1} & {\alpha^{2}} & {\alpha^{4}}\end{array}\right],\) then the matrix \(\mathrm{A}^{31}\) is equal to

Let \([t]\) denote the largest integer less than or equal to \(t\). If \(\int_0^3\left(\left[x^2\right]+\left[\frac{x^2}{2}\right]\right) d x=a+b \sqrt{2}-\sqrt{3}-\sqrt{5}+c \sqrt{6}-\sqrt{7},\) where \(a, b, c \in z\), then \(a+b+c\) is equal to ...........

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

The value of the integral \(\int \frac{\sin \theta \cdot \sin 2 \theta\left(\sin ^{6} \theta+\sin ^{4} \theta+\sin ^{2} \theta\right) \sqrt{2 \sin ^{4} \theta+3 \sin ^{2} \theta+6}}{1-\cos 2 \theta} d \theta\) is   (where \(c\) is a constant of integration)

The area enclosed by \(y ^{2}=8 x\) and \(y=\sqrt{2} x\) that lies outside the triangle formed by \(y=\sqrt{2} x, x=\) \(1, y=2 \sqrt{2}\), is equal to

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