The value of \(\int_{-1 / \sqrt{2}}^{1 / \sqrt{2}}\left(\left(\frac{x+1}{x-1}\right)^{2}+\left(\frac{x-1}{x+1}\right)^{2}-2\right)^{1 / 2} d x\) is:

The equation of a circle is \(\operatorname{Re}\left(z^{2}\right)+2(\operatorname{Im}(z))^{2}+2 \operatorname{Re}(z)=0\), where \(z=x+\) iy. A line which passes through the center of the given circle and the vertex of the parabola, \(x^{2}-6 x-y+13=0,\) has \(y\)-intercept equal to \(.....\)

For \(\alpha, \beta, z \in C\) and \(\lambda>1\), if \(\sqrt{\lambda-1}\) is the radius of the circle \(|z-\alpha|^2+|z-\beta|^2=2 \lambda\), then \(|\alpha-\beta|\) is equal to \(.............\).

If the system of equations  \(x+2 y+3 z=3\)  ; \(4 x+3 y-4 z=4\) ; \(8 x+4 y-\lambda z=9+\mu\) has infinitely many solutions, then the ordered pair \((\lambda, \mu)\) is equal to

The number of critical points of the function \(f(x)=(x-2)^{2 / 3}(2 x+1)\) is :

Let \(P\) be the plane passing through the points \((5,3\), \(0),(13,3,-2)\) and \((1,6,2)\). For \(\alpha \in N\), if the distances of the points \(A (3,4, \alpha)\) and \(B (2, \alpha\), a) from the plane \(P\) are \(2\) and \(3\) respectively, then the positive value of a is

If \({Z_1} \ne 0\) and \(Z_2\) be two complex numbers such that \(\frac{{{Z_2}}}{{{Z_1}}}\) is a purely imaginary number, then \(\left| {\frac{{2{Z_1} + 3{Z_2}}}{{2{Z_1} - 3{Z_2}}}} \right|\) 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:

The number of natural numbers lying between \(1012\) and \(23421\) that can be formed using the digits \(2,3,4,5,6\) (repetition of digits is not allowed) and divisible by \(55\) is \(....\)

The remainder when \(\left((64)^{(64)}\right)^{(64)}\) is divided by 7 is equal to

Let \(A\) be a \(2 \times 2\) matrix with real entries such that \(A ^{\prime}=\alpha A + I\), where \(\alpha \in R -\{-1,1\}\). If det \(\left(A^2-A\right)=4\), then the sum of all possible values of \(\alpha\) is equal to

Let the plane passing through the point \((-1,0,-2)\) and perpendicular to each of the planes \(2 x+y-\) \(z=2\) and \(x-y-z=3\) be \(a x+b y+c z+8=0\). then the value of \(a+b+c\) is equal to:

Let \(\quad \overrightarrow{\mathrm{a}}=2 \hat{\mathrm{i}}+\alpha \hat{\mathrm{j}}+\hat{\mathrm{k}}, \quad \overrightarrow{\mathrm{b}}=-\hat{\mathrm{i}}+\hat{\mathrm{k}}, \quad \overrightarrow{\mathrm{c}}=\beta \hat{\mathrm{j}}-\hat{\mathrm{k}}\), where \(\alpha\) and \(\beta\) are integers and \(\alpha \beta=-6\). Let the values of the ordered pair \((\alpha, \beta)\) for which the area of the parallelogram of diagonals \(\vec{a}+\vec{b}\) and \(\vec{b}+\vec{c}\) is \(\frac{\sqrt{21}}{2}\), be \(\left(\alpha_1, \beta_1\right)\) and \(\left(\alpha_2, \beta_2\right)\). Then \(\alpha_1^2+\beta_1^2-\alpha_2 \beta_2\) is equal to