The equation $e^{4x} + 8e^{3x} + 13e^{2x} - 8e^x + 1 = 0, x \in R$ has: (1) four solutions two of which are negative (2) two solutions and both are negative (3) no solution (4) two solutions and only one of them is negative
Let $a_1, a_2, a_3, \ldots$ be an A.P. If $a_7 = 3$, the product $(a_1 a_4)$ is minimum and the sum of its first $n$ terms is zero then $n! - 4a_{n(n+2)}$ is equal to (1) $\frac{381}{4}$ (2) 9 (3) $\frac{33}{4}$ (4) 24
If the constant term in the binomial expansion of $\left(\frac{x^{\frac{5}{2}}}{2} - \frac{4}{x^l}\right)^9$ is $-84$ and the coefficient of $x^{-3l}$ is $2^\alpha \beta$ where $\beta < 0$ is an odd number, then $|\alpha l - \beta|$ is equal to $\_\_\_\_$.
The set of all values of $a^2$ for which the line $x + y = 0$ bisects two distinct chords drawn from a point $\mathrm{P}\left(\frac{1+a}{2}, \frac{1-a}{2}\right)$ on the circle $2x^2 + 2y^2 - (1+a)x - (1-a)y = 0$, is equal to: (1) $(8, \infty)$ (2) $(0, 4]$ (3) $(4, \infty)$ (4) $(2, 12]$
Let $S$ be the set of all $a \in N$ such that the area of the triangle formed by the tangent at the point $P(b, c)$, $b, c \in N$, on the parabola $y^2 = 2ax$ and the lines $x = b$, $y = 0$ is 16 unit$^2$, then $\sum_{a \in S} a$ is equal to $\_\_\_\_$.
Let H be the hyperbola, whose foci are $(1 \pm \sqrt{2}, 0)$ and eccentricity is $\sqrt{2}$. Then the length of its latus rectum is: (1) 3 (2) $\frac{5}{2}$ (3) 2 (4) $\frac{3}{2}$
$\lim_{x \rightarrow \infty} \frac{(\sqrt{3x+1} + \sqrt{3x-1})^6 + (\sqrt{3x+1} - \sqrt{3x-1})^6}{\left(x + \sqrt{x^2-1}\right)^6 + \left(x - \sqrt{x^2-1}\right)^6} x^3$ (1) is equal to $\frac{27}{2}$ (2) is equal to 9 (3) does not exist (4) is equal to 27
Let the mean and standard deviation of marks of class A of 100 students be respectively 40 and $\alpha\ (> 0)$, and the mean and standard deviation of marks of class $B$ of $n$ students be respectively 55 and $30 - \alpha$. If the mean and variance of the marks of the combined class of $100 + n$ students are respectively 50 and 350, then the sum of variances of classes $A$ and $B$ is (1) 500 (2) 450 (3) 650 (4) 900
Let $A = [a_{ij}]$, $a_{ij} \in Z \cap [0,4]$, $1 \leq i, j \leq 2$. The number of matrices $A$ such that the sum of all entries is a prime number $p \in (2, 13)$ is $\_\_\_\_$.
Let $A$ be a $n \times n$ matrix such that $|A| = 2$. If the determinant of the matrix $\operatorname{Adj}\left(2 \cdot \operatorname{Adj}\left(2A^{-1}\right)\right)$ is $2^{84}$, then $n$ is equal to $\_\_\_\_$.