Maths

Let a and b be the roots of the equation x² + (2i – 1) = 0. Then, the value of   $\left|{\alpha }^8+{\beta }^8\right|$ is equal to:

If the system of linear equations

2x + y – z = 7

x – 3y + 2z = 1

x + 4y + $\delta z=k,$ where $\delta ,k\in R$

has infinitely many solutions, then $\delta +k$ is equal to:

The area enclosed by y2 = 8x 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:

Let $\overrightarrow{a}=\alpha \widehat{i}a+3\widehat{j}-\widehat{k},\overrightarrow{b}=3\widehat{i}-\beta \widehat{j}+4\widehat{k}and\overrightarrow{c}=\widehat{i}+2\widehat{j}-2\widehat{k}where\alpha ,\beta \in R,$ be three vectors. If the projection of $\overrightarrow{a}on\overrightarrow{c}is\frac{10}{3}and\overrightarrow{b}*\overrightarrow{c}=-6\widehat{i}+10\widehat{j}+7\widehat{k},$ then the value of + is equal to:

Let f : R R be a function defined by:

$f\left(x\right)=[\begin{array}{cccc}\hfill max\left(t^3-3t\right)\hfill & \hfill ;\hfill & \hfill t\le x;x\le 2\hfill & \hfill x^2+2x-6\hfill \\ \hfill ;\hfill & \hfill 2<x<3\hfill & \hfill \left[x-3\right]+9\hfill & \hfill ;\hfill \\ \hfill 3\le x\le 5\hfill & \hfill 2x+1\hfill & \hfill ;\hfill & \hfill x>5\hfill \end{array}$

where [t] is the greatest integer less than or equal to t. Let me be the number of points where f is not differentiable and $I={\displaystyle \underset{-2}{\overset{2}{\int }}f\left(x\right)dx.}$ Then the ordered pair (m, I) is equal to:

If the mirror image of the point (2, 4,7) in the plane 3x – y + 4z = 2 is (a, b, c), then 2a + b + 2c is equal to:

Let the solution curve of the differential equation

$x\frac{dy}{dx}-y=\sqrt[]{y^2+16x^2},y\left(1\right)=3bey=y\left(x\right).$ Then y(2) is equal to:

The probability that a randomly chosen 2 * 2 matrix with all the entries from the set of first 10 primes, is singular, is equal to:

Let Q and R be two points on the line $\frac{x+1}{2}=\frac{y+2}{3}=\frac{z-1}{2}$  at a distance $\sqrt[]{26}$  from the point P(4, 2, 7). Then the surface of the area of the triangle PQR is……………

Let the function f(x) = 2x² – log_e x, x  > 0, be decreasing in (0, a) and increasing in (a, 4). A tangent to the parabola y² = 4ax at a point P on it passes through the point (8a, 8a – 1) but does not pass through the point $\left(-\frac{1}{a},0\right).$  If the equation of the normal at P is $\frac{x}{\alpha }+\frac{\gamma }{\beta }=1,$  then a + b is equal to…………….