Class 12th

Let $\frac{x-2}{3}=\frac{y+1}{-2}=\frac{z+3}{-1}$ lie on the plane px – qy + z = 5, for some $p,q\in R.$ The shortest distance of the plane form the origin is

If y = y(x) is the solution of the differential equation

$\left(1+e^{2x}\right)\frac{dy}{dx}+2\left(1+y^2\right)e^x=0andy\left(0\right)=0,$ then $6\left(y\text{'}\left(0\right)+{\left(y\left(lo{g}_e\sqrt[]{3}\right)\right)}^2\right)$ is equal to

If ${\displaystyle \underset{0}{\overset{2}{\int }}\left(\sqrt[]{2x}-\sqrt[]{2x-x^2}\right)dx}={\displaystyle \underset{0}{\overset{1}{\int }}\left(1-\sqrt[]{1-y^2}-\frac{y^2}{2}\right)dy}+{\displaystyle \underset{1}{\overset{2}{\int }}\left(2-\frac{y^2}{2}\right)dy+IthenIequals}$

Let f be a real valued continuous function on [0, 1] and f(x) = $x+{\displaystyle \underset{0}{\overset{1}{\int }}\left(x-t\right)f\left(t\right)dt.}$ Then, which of the following points (x, y) lies on the curve y = f(x)?

Let f : R → R be a function defined by f(x) = ${\left(x-3\right)}^{n_1}{\left(x-5\right)}^{n_2},n_1,n_2\in N.$  Then which of the following is NOT true?

The value of $\underset{x\to 1}{lim}\frac{\left(x^2-1\right)si{n}^2\left(\pi x\right)}{x^4-2x^3+2x-1}$ is equal to

Let $A=\left(\begin{array}{cc}\hfill 2\hfill & \hfill -1\hfill \\ \hfill 0\hfill & \hfill 2\hfill \end{array}\right).$ If $B=I{-}^5\text{?}{C}_1\left(adjA\right){+}^5\text{?}{C}_2{\left(adjA\right)}^2-..{.}^5\text{?}{C}_5{\left(adjA\right)}^5,$ then the sum of all elements of the matrix B is

An inductor of 0.5 mH, a capacitor of 200 $\mu F$  and a resistor of 2 $\Omega$  are connected in series with a 220V ac source. If the current is in phase with the emf, the frequency of ac source will be__________ * 10² Hz.

In a double slit experiment with monochromatic light, fringes are obtained on a screen placed at some distance from the plane of slits. If the screen is moved by 5 * 10^-2m towards the slits, the change in fringe width is 3 * 10^-3cm. If the distance between the slits is 1mm, then the wavelength of the light will be_____________ nm.

The half life of a radioactive substance is 5 years. After x years a given sample of the radioactive substance gets reduced to 6/25% of its initial value. The value of x is___________.