Group A consists of 7 boys and 3 girls, while group $B$ consists of 6 boys and 5 girls. The number of ways, 4 boys and 4 girls can be invited for a picnic if 5 of them must be from group $A$ and the remaining 3 from group $B$, is equal to : (1) 8750 (2) 9100 (3) 8925 (4) 8575
If the system of equations $$x + 2y - 3z = 2$$ $$2x + \lambda y + 5z = 5$$ $$14x + 3y + \mu z = 33$$ has infinitely many solutions, then $\lambda + \mu$ is equal to : (1) 13 (2) 10 (3) 12 (4) 11
The area of the region enclosed by the curves $y = \mathrm{e}^{x}$, $y = \left| \mathrm{e}^{x} - 1 \right|$ and $y$-axis is: (1) $1 - \log_{e} 2$ (2) $\log_{e} 2$ (3) $1 + \log_{e} 2$ (4) $2 \log_{e} 2 - 1$
Let the points $\left( \frac{11}{2}, \alpha \right)$ lie on or inside the triangle with sides $x + y = 11$, $x + 2y = 16$ and $2x + 3y = 29$. Then the product of the smallest and the largest values of $\alpha$ is equal to: (1) 44 (2) 22 (3) 33 (4) 55
Let $f : (0, \infty) \rightarrow \mathbf{R}$ be a function which is differentiable at all points of its domain and satisfies the condition $x^{2} f'(x) = 2x f(x) + 3$, with $f(1) = 4$. Then $2f(2)$ is equal to: (1) 39 (2) 19 (3) 29 (4) 23
Let $[x]$ denote the greatest integer function, and let m and n respectively be the numbers of the points, where the function $f(x) = [x] + |x - 2|$, $-2 < x < 3$, is not continuous and not differentiable. Then $\mathrm{m} + \mathrm{n}$ is equal to: (1) 6 (2) 8 (3) 9 (4) 7
Let $A = \left[ a_{ij} \right]$ be a square matrix of order 2 with entries either 0 or 1. Let $E$ be the event that $A$ is an invertible matrix. Then the probability $\mathrm{P}(\mathrm{E})$ is: (1) $\frac{3}{16}$ (2) $\frac{5}{8}$ (3) $\frac{3}{8}$ (4) $\frac{1}{8}$
Let the position vectors of three vertices of a triangle be $4\vec{p} + \vec{q} - 3\vec{r}$, $-5\vec{p} + \vec{q} + 2\vec{r}$ and $2\vec{p} - \vec{q} + 2\vec{r}$. If the position vectors of the orthocenter and the circumcenter of the triangle are $\frac{\vec{p} + \vec{q} + \vec{r}}{4}$ and $\alpha\vec{p} + \beta\vec{q} + \gamma\vec{r}$ respectively, then $\alpha + 2\beta + 5\gamma$ is equal to: (1) 3 (2) 4 (3) 1 (4) 6
The function $f : (-\infty, \infty) \rightarrow (-\infty, 1)$, defined by $f(x) = \frac{2^{x} - 2^{-x}}{2^{x} + 2^{-x}}$ is: (1) Neither one-one nor onto (2) Onto but not one-one (3) Both one-one and onto (4) One-one but not onto
Suppose A and B are the coefficients of $30^{\text{th}}$ and $12^{\text{th}}$ terms respectively in the binomial expansion of $(1 + x)^{2\mathrm{n}-1}$. If $2\mathrm{A} = 5\mathrm{B}$, then n is equal to: (1) 22 (2) 20 (3) 21 (4) 19
Let $(2, 3)$ be the largest open interval in which the function $f(x) = 2\log_{\mathrm{e}}(x - 2) - x^{2} + ax + 1$ is strictly increasing and $(\mathrm{b}, \mathrm{c})$ be the largest open interval, in which the function $\mathrm{g}(x) = (x - 1)^{3}(x + 2 - \mathrm{a})^{2}$ is strictly decreasing. Then $100(a + b - c)$ is equal to: (1) 420 (2) 360 (3) 160 (4) 280
If the equation of the parabola with vertex $\mathrm{V}\left(\frac{3}{2}, 3\right)$ and the directrix $x + 2y = 0$ is $\alpha x^{2} + \beta y^{2} - \gamma xy - 30x - 60y + 225 = 0$, then $\alpha + \beta + \gamma$ is equal to: (1) 7 (2) 9 (3) 8 (4) 6
Let P be the image of the point $\mathrm{Q}(7, -2, 5)$ in the line $\mathrm{L} : \frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z}{4}$ and $\mathrm{R}(5, \mathrm{p}, \mathrm{q})$ be a point on $L$. Then the square of the area of $\triangle PQR$ is $\_\_\_\_$.
If $\int \frac{2x^{2} + 5x + 9}{\sqrt{x^{2} + x + 1}} \, \mathrm{d}x = x\sqrt{x^{2} + x + 1} + \alpha\sqrt{x^{2} + x + 1} + \beta \log_{e}\left| x + \frac{1}{2} + \sqrt{x^{2} + x + 1} \right| + \mathrm{C}$, where $C$ is the constant of integration, then $\alpha + 2\beta$ is equal to $\_\_\_\_$.
Q23
First order differential equations (integrating factor)View
Let $y = y(x)$ be the solution of the differential equation $2\cos x \frac{\mathrm{d}y}{\mathrm{d}x} = \sin 2x - 4y \sin x$, $x \in \left(0, \frac{\pi}{2}\right)$. If $y\left(\frac{\pi}{3}\right) = 0$, then $y'\left(\frac{\pi}{4}\right) + y\left(\frac{\pi}{4}\right)$ is equal to $\_\_\_\_$.
Number of functions $f : \{1, 2, \ldots, 100\} \rightarrow \{0, 1\}$, that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to $\_\_\_\_$.
Let $\mathrm{H}_{1} : \frac{x^{2}}{\mathrm{a}^{2}} - \frac{y^{2}}{\mathrm{b}^{2}} = 1$ and $\mathrm{H}_{2} : -\frac{x^{2}}{\mathrm{A}^{2}} + \frac{y^{2}}{\mathrm{B}^{2}} = 1$ be two hyperbolas having length of latus rectums $15\sqrt{2}$ and $12\sqrt{5}$ respectively. Let their eccentricities be $e_{1} = \sqrt{\frac{5}{2}}$ and $e_{2}$ respectively. If the product of the lengths of their transverse axes is $100\sqrt{10}$, then $25\mathrm{e}_{2}^{2}$ is equal to $\_\_\_\_$.