If a random variable \(X\) takes the values \(x_1, x_2, x_3, \ldots, x_{100}\) with probability \(P\left(X=x_i\right)=\) \(K i(i+1)\), then \(200 K=\)

Find the eccentricity of an ellipse, if the length of its latus rectum is 4 units and distance between its vertex and the nearest focus is \(3 / 2\) units.

If all the letters of the word COMBINATION are arranged in all possible ways to form 11 letter words (with or without meaning), then the number of words among them in which C and N occupy the end positions and no vowel appears exactly in the middle position is

Let \(\overline{\mathrm{i}}-2 \overline{\mathrm{j}}+\overline{\mathrm{k}}, \overline{\mathrm{i}}+\overline{\mathrm{j}}-2 \overline{\mathrm{k}}, 2 \overline{\mathrm{i}}-\overline{\mathrm{j}}-\overline{\mathrm{k}}\) and \(\overline{\mathrm{i}}+\overline{\mathrm{j}}+\overline{\mathrm{k}}\) be the position vectors of four points \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) and D respectively. If a point P divides AB in the ratio \(2: 1\) internally and a point \(Q\) divides \(C D\) in the ratio \(1: 2\) externally, then the ratio in which the point with position vector \(5 \overline{\mathrm{i}}-6 \overline{\mathrm{j}}-5 \overline{\mathrm{k}}\) divides PQ is

If $f:R\to R$ is defined as $f\left(x\right)=x-\left[x\right]+3,\forall x\in R,$ then $f$ is

If the centroid of the triangle formed by the lines $2y^2+5xy-3x^2=0$ and $x+y=k$ is $\left(\frac{1}{18},\frac{11}{18}\right)$, then the value of $\text{'}k\text{'}$ equals_____

\(\int e^{3 \log x}\left(x^4+1\right)^{-1} d x=\)

If \(f(x)=\int \frac{5 x^8+7 x^6}{\left(x^2+2 x^7+1\right)^2} d x(x \geq 0)\) and \(f(0)=0\), then the value of \(f(1)=\)

In planetary motion the areal velocity of position vector of a planet depends on angular velocity \(\omega\) and the distance of the planet from sun \(r\). The correct relation for areal velocity is

Which among the following pairs is not an acidic buffer?

In which of the following sets the sum of bond orders of three species is maximum?

The ratio of the speed of sound in a monatomic gas at \(27^{\circ} \mathrm{C}\) and rms speed of the molecules of the same gas at a temperature of \(127^{\circ} \mathrm{C}\) is

The centripetal acceleration of a particle in uniform circular motion is \(18 \mathrm{~ms}^{-2}\). If the radius of the circular path is 50 cm , the change in velocity of the particle in a time of \(\frac{\pi}{18} \mathrm{~s}\) is