For an ellipse with eccentricity \(\frac{1}{2}\) the centre is at the origin. If one directrix is \(x=4\), then the equation of the ellipse is

Let \(A, G, H\) and \(S\) respectively denote the arithmetic mean, geometric mean, harmonic mean and the sum of the numbers \(a_1, a_2, a_3, \ldots, a_n\). Then the value of \(x\) at which the function \(f(x)=\sum_{k=1}^n\left(x-a_k\right)^2\) has minimum is

The maximum value of the variance of Binomial distribution with parameters \(n\) and \(p\) is

If the direction cosines of a line are \(\left(\frac{a}{\sqrt{83}}, \frac{5}{\sqrt{83}}, \frac{c}{\sqrt{83}}\right)\) and \(c-a=4\), then \(c a=\)

A Circle S passes through the points of intersection of the circles \(x^2+y^2-2 x+2 y-2=0\) and \(x^2+y^2+2 x-2 y+1=0\). If the centre of this circle S lies on the line \(x-y+6=0\), then the radius of the circle \(S\) is

Let \(\overrightarrow{\mathbf{a}}=a_1 \hat{\mathbf{i}}+a_2 \hat{\mathbf{j}}+a_3 \hat{\mathbf{k}}\)
Assertion (A) : The identity
\(|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{i}}|^2+|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{j}}|^2+|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{k}}|^2=2|\overrightarrow{\mathbf{a}}|^2\) holds for
\(\vec{a}\).
Reason (R) : \(\overrightarrow{\mathbf{a}} \times \hat{\mathbf{i}}=a_3 \hat{\mathbf{j}}-a_2 \hat{\mathbf{k}}\),
\(\overrightarrow{\mathbf{a}} \times \hat{\mathbf{j}}=a_1 \hat{\mathbf{k}}-a_3 \hat{\mathbf{i}}, \overrightarrow{\mathbf{a}} \times \hat{\mathbf{k}}=a_2 \hat{\mathbf{i}}-a_1 \hat{\mathbf{j}}\)
Which of the following is correct?

A point on the straight line \(3 x+5 y=15\) which is equidistant from the coordinate axes will lie in

A car of mass \(1500 \mathrm{~kg}\) is moving with \(20 \mathrm{~ms}^{-1}\) velocity. If the breaks are applied it comes to rest in 5 seconds, then the retarding force is

If a circle \(S\) passes through the origin and makes an intercept of length 4 units on the line \(x=2\), then the equation of the curve on which the centre of \(S\) lies is

In a system, unit of mass is \(A \mathrm{~kg}\), length is \(B \mathrm{~m}\) and time is \(C \mathrm{~s}\), then the value of \(10 \mathrm{~N}\) in this system is

Identify the correct statements from the following
I. \(\mathrm{B}_2 \mathrm{H}_6\) is an electron deficient hydride.
II. \(\mathrm{NH}_3\) an electron rich hydride.
III. \(\mathrm{NaH}\) is a covalent hydride.
IV. \(\mathrm{YbH}_{255}\) is an interstitial hydride.

Calculate the amount of hydrogen gas required in order to produce \(100 \mathrm{~g}\) of ammonia by the reaction of \(\mathrm{N}_2(g)\) and \(\mathrm{H}_2(g)\) gases.