The sum of the first three terms of G.P is $S$ and their products is 27. Then all such $S$ lie in (1) $(-\infty, -9] \cup [3, \infty)$ (2) $[-3, \infty)$ (3) $(-\infty, -3] \cup [9, \infty)$ (4) $(-\infty, 9]$
Let $\alpha > 0, \beta > 0$ be such that $\alpha^{3} + \beta^{2} = 4$. If the maximum value of the term independent of $x$ in the binomial expansion of $\left(\alpha x^{\frac{1}{9}} + \beta x^{-\frac{1}{6}}\right)^{10}$ is $10k$, then $k$ is equal to (1) 336 (2) 352 (3) 84 (4) 176
A line parallel to the straight line $2x - y = 0$ is tangent to the hyperbola $\frac{x^{2}}{4} - \frac{y^{2}}{2} = 1$ at the point $(x_{1}, y_{1})$. Then $x_{1}^{2} + 5y_{1}^{2}$ is equal to (1) 6 (2) 8 (3) 10 (4) 5
Let $X = \{x \in N : 1 \leq x \leq 17\}$ and $Y = \{ax + b : x \in X$ and $a, b \in R, a > 0\}$. If mean and variance of elements of $Y$ are 17 and 216 respectively then $a + b$ is equal to (1) 7 (2) $-7$ (3) $-27$ (4) 9
If $R = \{(x, y) : x, y \in Z, x^{2} + 3y^{2} \leq 8\}$ is a relation on the set of integers $Z$, then the domain of $R^{-1}$ is (1) $\{-2, -1, 1, 2\}$ (2) $\{0, 1\}$ (3) $\{-2, -1, 0, 1, 2\}$ (4) $\{-1, 0, 1\}$
Let $A$ be a $2 \times 2$ real matrix with entries from $\{0, 1\}$ and $|A| \neq 0$. Consider the following two statements: $(P)$ If $A \neq I_{2}$, then $|A| = -1$ $(Q)$ If $|A| = 1$, then $\operatorname{tr}(A) = 2$ Where $I_{2}$ denotes $2 \times 2$ identity matrix and $\operatorname{tr}(A)$ denotes the sum of the diagonal entries of $A$. Then (1) $(P)$ is false and $(Q)$ is true (2) Both $(P)$ and $(Q)$ are false (3) $(P)$ is true and $(Q)$ is false (4) Both $(P)$ and $(Q)$ are true
Let $S$ be the set of all $\lambda \in R$ for which the system of linear equations $$2x - y + 2z = 2$$ $$x - 2y + \lambda z = -4$$ $$x + \lambda y + z = 4$$ has no solution. Then the set $S$ (1) Contains more than two elements (2) Is an empty set (3) Is a singleton (4) Contains exactly two elements
The domain of the function $f(x) = \sin^{-1}\left(\frac{|x| + 5}{x^{2} + 1}\right)$ is $(-\infty, -a] \cup [a, \infty)$, then $a$ is equal to (1) $\frac{\sqrt{17}}{2}$ (2) $\frac{\sqrt{17} - 1}{2}$ (3) $\frac{1 + \sqrt{17}}{2}$ (4) $\frac{\sqrt{17}}{2} + 1$
If a function $f(x)$ defined by $$f(x) = \begin{cases} ae^{x} + be^{-x}, & -1 \leq x < 1 \\ cx^{2}, & 1 \leq x \leq 3 \\ ax^{2} + 2cx, & 3 < x \leq 4 \end{cases}$$ be continuous for some $a, b, c \in R$ and $f'(0) + f'(2) = e$, then the value of $a$ is (1) $\frac{1}{e^{2} - 3e + 13}$ (2) $\frac{e}{e^{2} - 3e - 13}$ (3) $\frac{e}{e^{2} + 3e + 13}$ (4) $\frac{e}{e^{2} - 3e + 13}$
If the tangent to the curve $y = x + \sin y$ at a point $(a, b)$ is parallel to the line joining $\left(0, \frac{3}{2}\right)$ and $\left(\frac{1}{2}, 2\right)$, then (1) $b = a$ (2) $|b - a| = 1$ (3) $|a + b| = 1$ (4) $b = \frac{\pi}{2} + a$
If $p(x)$ be a polynomial of degree three that has a local maximum value 8 at $x = 1$ and a local minimum value 4 at $x = 2$ then $p(0)$ is equal to (1) 6 (2) $-12$ (3) 24 (4) 12
Let $P(h, k)$ be a point on the curve $y = x^{2} + 7x + 2$, nearest to the line, $y = 3x - 3$. Then the equation of the normal to the curve at $P$ is (1) $x + 3y + 26 = 0$ (2) $x + 3y - 62 = 0$ (3) $x - 3y - 11 = 0$ (4) $x - 3y + 22 = 0$
Area (in sq. units) of the region outside $\frac{|x|}{2} + \frac{|y|}{3} = 1$ and inside the ellipse $\frac{x^{2}}{4} + \frac{y^{2}}{9} = 1$ is (1) $6(\pi - 2)$ (2) $3(\pi - 2)$ (3) $3(4 - \pi)$ (4) $6(4 - \pi)$
Let $y = y(x)$ be the solution of the differential equation, $\frac{2 + \sin x}{y + 1} \cdot \frac{dy}{dx} = -\cos x, y > 0, y(0) = 1$. If $y(\pi) = a$ and $\frac{dy}{dx}$ at $x = \pi$ is $b$, then the ordered pair $(a, b)$ is equal to (1) $\left(2, \frac{3}{2}\right)$ (2) $(1, -1)$ (3) $(1, 1)$ (4) $(2, 1)$
The plane passing through the points $(1, 2, 1)$, $(2, 1, 2)$ and parallel to the line, $2x = 3y, z = 1$ also passes through the point (1) $(0, 6, -2)$ (2) $(-2, 0, 1)$ (3) $(0, -6, 2)$ (4) $(2, 0, -1)$