Mathematics Tes...

Let \(R\) be a relation defined as \(x R y\) if and only if \(2 x+3 y=20\), where \(x, y \in N\). How many elements of the form \((x, y)\) are there in \(R\) ?

Let \(\mathrm{P}\) be a square matrix such that \(\mathrm{P}^2=\mathrm{I}-\mathrm{P}\). For \(\alpha, \beta, \gamma, \delta \in \mathrm{N}\), if \(\mathrm{P}^\alpha+\mathrm{P}^\beta=\gamma \mathrm{I}-29 \mathrm{P}\) and \(\mathrm{P}^\alpha-\mathrm{P}^\beta=\delta \mathrm{I}-13 \mathrm{P}\), then \(\alpha+\beta+\gamma-\delta\) is equal to :

If \(\underset{{{{x} \rightarrow \infty}}}{\lim}\left(\frac{{x}^{2}+{x}+1}{{x}+1}-{px}-{q}\right)=-3\), then \({p}\) and \({q}\) is:

Consider the following statements:

1. The null set is a subset of every set.

2. Every set is a subset of itself.

3. If a set has \(10\) elements, then its power set will have \(1024\) elements.

Which of the above statements are correct?

If \((a, b)\) be the orthocentre of the triangle whose vertices are \((1,2),(2,3)\) and \((3,1)\), and \(I_1=\int_a^b x \sin \left(4 x-x^2\right) d x, I_2=\int_a^b \sin \left(4 x-x^2\right) d x\), then \(36 \frac{I_1}{I_2}\) is equal to :

If \({ }^{2 n} P_{3}=100^{n} P_{2}\), then the value of \(n\) is:

If \(\mathrm{A}=\left[\begin{array}{ll}x & 2 \\ 4 & 3\end{array}\right]\) and \(\mathrm{A}^{-1}=\left[\begin{array}{cc}\frac{1}{8} & \frac{-1}{12} \\ \frac{-1}{6} & \frac{4}{9}\end{array}\right]\), then find the value of \({x}\) ?

The Cartesian equations of a line are \(5 x-3=3 y+4=2 z-3\). Direction ratios of the line are:

Evaluate the following expression: \(\sin x+\sin (x-\pi)+\sin (x+\pi)\)

A water tank has the shape of an inverted right circular cone, whose semi-vertical angle is tan^–1(1/2) . Water is poured into it at a constant rate of 5 cubic meter per minute. Then the rate (in m/min.), at which the level of water is rising at the instant when the depth of water in the tank is 10m; is: