NUMERICAL

If $a_{1}=1$ and for all $n\ge1$, $a_{n+1}=\frac{1}{2}a_{n}+\frac{n^{2}-2n-1}{n^{2}(n+1)^{2}}$, then the value of $\sum_{n=1}^{\infty}(a_{n}-\frac{2}{n^{2}})$ is equal to:

MCQ_SINGLE

If $L_{1}$ and $L_{2}$ are two parallel lines and $\triangle ABC$ is an equilateral triangle as shown in the figure, then the area of triangle ABC is:

NUMERICAL

Let $A$ satisfy the equation $\lambda\begin{bmatrix}1\\2\\3\end{bmatrix}=\begin{bmatrix}-1\\0\\3\end{bmatrix}$, and $|2\text{adj}(A+I)|=2^{\alpha}3^{\beta}11^{\gamma}$ then $\alpha+\beta+\gamma$ is equal to .

MCQ_SINGLE

If $a_{1}, a_{2}, a_3,...$ are in increasing geometric progression such that $a_{1}+a_{3}+a_{5}=21$ and $a_{1}a_{3}a_{5}=64$, then $a_{1}+a_{2}+a_{3}$ is:

NUMERICAL

Chord joining at two points $(x_{1},y_{1})$ and $(x_{2},y_{2})$ on the parabola $y^{2}=12x$ intersect at right angle at the vertex, then $x_{1}x_{2}-y_{1}y_{2}=$ .

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