For the following questions answer them individually
The value of $$\lim_{x \to 0}\frac{\log_e\!\left(\sec(ex)\cdot \sec(e^{2}x)\cdots \sec(e^{10}x)\right)}{e^{2}-e^{2\cos x}}$$ is equal to
For three unit vectors $$\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$$ satisfying $$|\overrightarrow{a}-\overrightarrow{b}|^{2}+|\overrightarrow{b}-\overrightarrow{c}|^{2}+|\overrightarrow{c}-\overrightarrow{a}|^{2}=9$$ and $$|2\overrightarrow{a}+k\overrightarrow{b}+k\overrightarrow{c}|+3$$. the positive value of k is
If $$\alpha, \beta$$, where $$\alpha < \beta$$, the positive value ofk is $$\lambda x^{2}-(\lambda + 3)x+3=0$$ such that $$\frac{1}{\alpha}-\frac{1}{\beta}=\frac{1}{3}$$, then the sum of all possible values of A is
Let y = x be the equation of a chord of the circle $$C_{1}$$ (in the closed half-plane x c $$\geq$$ 0) of diameter 10 passing through the origin. Let $$C_{2}$$ be another circle described on the given chord as its diameter. If the equation of the chord of the circle $$C_{2}$$, which x + ay + b = 0, then a - b is equal to
The common difference ofthe $$A.P.: a_{1},a_{2},.....,a_{m}$$ is 13 more than the common difference of the $$A.P.:b_{1},b_{2},....,b_{n}$$. If $$b_{31}=-277,bb_{43}=-385 \text{ and } a_{78}=327$$ then $$a_{1}$$ is equal to
Let f be a polynomial function such that $$f(x^{2}+1)=x^{4}+5x^{2}+2$$, for all $$x \in R$$. Then $$\int_{0}^{3}f(x)dx$$ is equal to
The mean and variance of 10 observations are 9 and 34.2, respectively. If 8 of these observations are 2, 3, 5, 10, 11 , 13, 15, 21, then the mean deviation about the median of all the 10 observations is
Let A, Band C be three $$2\times 2$$ matrices with real entries such that $$B=(I+A)^{-1}$$ and A+C=1. If $$BC=\begin{bmatrix}1 & -5 \\-1 & 2 \end{bmatrix}$$ and $$CB\begin{bmatrix}x_{1}\\ x_{2} \end{bmatrix}=\begin{bmatrix}12\\-6 \end{bmatrix}$$, then $$x_{1}+x_{2}$$ is
The value of $$\sum_{k=1}^{\infty}(-1)^{k+1}\left(\frac{k(k+1)}{k!}\right)$$ is
Let $$S=\left\{x^{3}+ax^{2}+bx+c:a,b,c, \in N \text{ and }a,b,c \leq 20\right\}$$ be a set of polynomials. Then the number of polynomials in S, which are divisible by $$x^{2}+2$$, is
