The largest \( n\in N \) for which \(7^n\) divides 101!, is :

The angle of elevation of the top of a hill from a point on the horizontal plane passing through the foot of the hill is found to be \(45^{\circ} .\) After walking a distance of \(80\) meters towards the top, up a slope inclined at an angle of \(30^{\circ}\) to the horizontal plane, the angle of elevation of the top of the hill becomes \(75^{\circ} .\) Then the height of the hill (in meters) is

If the standard deviation of the numbers \( 2,3,a \) and \(11\) is \(3.5\)  then which of the following is true ?

Let \(A\) be the point \((1,2)\) and \(B\) be any point on the curve \(x^2+y^2=16\). If the centre of the locus of the point \(P\), which divides the line segment \(A B\) in the ratio \(3: 2\) is the point \(C(\alpha, \beta)\), then the length of the line segment \(AC\) is

If \(\sin ^{-1} \frac{\alpha}{17}+\cos ^{-1} \frac{4}{5}-\tan ^{-1} \frac{77}{36}=0,0 < \alpha < 13\), then \(\sin ^{-1}(\sin \alpha)+\cos ^{-1}(\cos \alpha)\) is equal to \(.........\).

If \( \int_{0}^{1}4~cot^{-1}(1-2x+4x^{2})dx=a~tan^{-1}(2)-b~log_{c}(5), \) where a, b \( \in N \), then \( (2a+b) \) is equal to :

Let the equations of two adjacent sides of a parallelogram \(A B C D\) be \(2 x-3 y=-23\) and \(5 x+4 y\) \(=23\). If the equation of its one diagonal \(AC\) is \(3 x +\) \(7 y=23\) and the distance of A from the other diagonal is \(d\), then \(50 d ^2\) is equal to \(........\).

The number of real solutions of the equation \(\mathrm{x}|\mathrm{x}+5|+2|\mathrm{x}+7|-2=0\) is ...........

An ordinary dice is rolled for a certain number of times. If the probability of getting an odd number \(2\) times is equal to the probability of getting an even number \(3\) times, then the probability of getting an odd number for odd number of times is

Suppose \(\sum \limits_{ r =0}^{2023} r ^{20023} C _{ r }=2023 \times \alpha \times 2^{2022}\). Then the value of \(\alpha\) is \(............\)

If the number of five digit numbers with distinct digits and \(2\) at the \(10^{\text {th }}\) place is \(336 \mathrm{k}\), then \(\mathrm{k}\) is equal to

If \(\lambda_1 < \lambda_2\) are two values of \(\lambda\) such that the angle between the planes \(P_1: \vec{r}(3 \hat{ i }-5 \hat{ j }+\hat{ k })=7\) and \(P_2: \vec{r} \cdot(\lambda \hat{i}+\hat{j}-3 \hat{k})=9\) is \(\sin ^{-1}\left(\frac{2 \sqrt{6}}{5}\right)\), then the square of the length of perpendicular from the point \(\left(38 \lambda_1, 10 \lambda_2, 2\right)\) to the plane \(P _1\) is \(...........\).

All five letter words are made using all the letters \(\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}, \mathrm{E}\) and arranged as in an English dictionary with serial numbers. Let the word at serial number \(n\) be denoted by \(W_n\). Let the probability \(\mathrm{P}\left(\mathrm{W}_{\mathrm{n}}\right)\) of choosing the word \(\mathrm{W}_{\mathrm{n}}\) satisfy \(\mathrm{P}\left(\mathrm{W}_{\mathrm{n}}\right)=2 \mathrm{P}\left(\mathrm{W}_{\mathrm{n}-1}\right), \mathrm{n} \gt 1\).
If \(\mathrm{P}(\mathrm{CDBEA})=\frac{2^\alpha}{2^\beta-1}, \alpha, \beta \in \mathbb{N}\), then \(\alpha+\beta\) is equal to : _______