If the length of the latus rectum of an ellipse is \(4\,units\) and the distance between a focus and its nearest vertex on the major axis is \(\frac {3}{2}\,units\) , then its eccentricity is?

The square of the distance of the point of intersection of the lines \(\vec{r} = (\hat{i} + \hat{j} - \hat{k}) + \lambda(a\hat{i} - \hat{j})\), \(a \neq 0\) and \(\vec{r} = (4\hat{i} - \hat{k}) + \mu(2\hat{i} + a\hat{k})\) from the origin is:

Let \(\quad P=\left[\begin{array}{cc}\frac{\sqrt{3}}{2} & \frac{1}{2} \\ -\frac{1}{2} & \frac{\sqrt{3}}{2}\end{array}\right], A=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]\) and \(Q=P Q P^{ T }\). If \(P ^{ T } Q ^{2007} P =\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]\), then \(2 a+b-3 c-4 d\) equal to \(...................\).

If the sum of the first 20 terms of the series
\(\frac{4.1}{4+3.1^2+1^4}+\frac{4.2}{4+3.2^2+2^4}+\frac{4.3}{4+3.3^2+3^4}+\frac{4.4}{4+3.4^2+4^4}+\ldots\)
is \(\frac{m}{n}\), where \(m\) and \(n\) are coprime, then \(m+n\) is equal to :-

Let \(f:[0, \infty) \rightarrow[0, \infty)\) be defined as \(\mathrm{f}(\mathrm{x})= \int_{0}^{x}[y] \,d y\) Where \([x]\) is the greatest integer less than or equal to \(x\). Which of the following is true?

If \(2 \tan ^2 \theta-5 \sec \theta=1\) has exactly \(7\) solutions in the interval \(\left[0, \frac{n \pi}{2}\right]\), for the least value of \(n \in N\) then \(\sum_{\mathrm{k}=1}^{\mathrm{n}} \frac{\mathrm{k}}{2^{\mathrm{k}}}\) is equal to :

Let the tangents drawn from the origin to the circle, \(x^{2}+y^{2}-8 x-4 y+16=0\) touch it at the points \(A\) and \(B .\) The \((A B)^{2}\) is equal to

Suppose the vectors \(x_{1}, x_{2}\) and \(x_{3}\) are the solutions of the system of linear equations, \(Ax = b\) when the vector \(b\) on the right side is equal to \(b _{1}, b _{2}\) and \(b _{3}\) respectively. If \(x =\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right], x _{2}=\left[\begin{array}{l}0 \\ 2 \\ 1\end{array}\right], x _{3}=\left[\begin{array}{l}0 \\ 0 \\ 1\end{array}\right], b _{1}=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]\) \(b _{2}=\left[\begin{array}{l}0 \\ 2 \\ 0\end{array}\right]\) and \(b _{3}=\left[\begin{array}{l}0 \\ 0 \\ 2\end{array}\right],\) then the determinant of \(A\) is equal to

Let \( S=\{(m,n): m, n\in\{1,2,3,.....,50\}\} \). If the number of elements (m, n) in S such that \( 6^{m}+9^{n} \) is a multiple of 5 is p and the number of elements (m, n) in S such that \( m+n \) is a square of a prime number is q, then \( p+q \) is equal to :

Let \(\left\{a_{n}\right\}_{n=0}^{\infty}\) be a sequence such that \(a _{0}= a _{1}=0\) and \(a _{ n +2}=2 a _{ n +1}- a _{ n }+1\) for all \(n \geq 0\). Then, \(\sum\limits_{ n =2}^{\infty} \frac{ a _{ n }}{7^{ n }}\) is equal to

The solution of the differential equation \(\frac{{dy}}{{dx}} = \left( {x - {y}} \right)^2\) when \(y(1) = 1\), is

Let a tangent to the curve \(y^2=24 x\) meet the curve \(xy =2\) at the points \(A\) and \(B\). Then the mid points of such line segments \(A B\) lie on a parabola with the

The eccentricity of an ellipse whose centre is at the origin is \(\frac{1}{2}\) . If one of its directices is \(x = - 4\) then the equation of the normal to it at \(\left( {1,\frac{3}{2}} \right)\) is