For all x > 0, let y₁(x), y₂(x), and y₃(x) be the functions satisfying
$ \frac{dy_1}{dx} - (\sin x)^2 y_1 = 0, \quad y_1(1) = 5, $
$ \frac{dy_2}{dx} - (\cos x)^2 y_2 = 0, \quad y_2(1) = \frac{1}{3}, $
$ \frac{dy_3}{dx} - \frac{(2-x^3)}{x^3} y_3 = 0, \quad y_3(1) = \frac{3}{5e}, $
respectively. Then
$ \lim\limits_{x \to 0^+} \frac{y_1(x)y_2(x)y_3(x) + 2x}{e^{3x} \sin x} $
is equal to __________________.
Consider the following frequency distribution:
| Value | 4 | 5 | 8 | 9 | 6 | 12 | 11 |
|---|---|---|---|---|---|---|---|
| Frequency | 5 | $f_1$ | $f_2$ | 2 | 1 | 1 | 3 |
Suppose that the sum of the frequencies is 19 and the median of this frequency distribution is 6.
For the given frequency distribution, let $\alpha$ denote the mean deviation about the mean, $\beta$ denote the mean deviation about the median, and $\sigma^2$ denote the variance.
Match each entry in List-I to the correct entry in List-II and choose the correct option.
| List – I | List – II |
|---|---|
| (P) 7 f1 + 9 f2 is equal to | (1) 146 |
| (Q) 19 α is equal to | (2) 47 |
| (R) 19 β is equal to | (3) 48 |
| (S) 19 σ2 is equal to | (4) 145 |
| (5) 55 |
Let $\mathbb{R}$ denote the set of all real numbers. For a real number $x$, let [ x ] denote the greatest integer less than or equal to $x$. Let $n$ denote a natural number.
Match each entry in List-I to the correct entry in List-II and choose the correct option.
| List–I | List–II |
|---|---|
| (P) The minimum value of $n$ for which the function $$ f(x)=\left[\frac{10 x^3-45 x^2+60 x+35}{n}\right] $$ is continuous on the interval $[1,2]$, is | (1) 8 |
| (Q) The minimum value of $n$ for which $g(x)=\left(2 n^2-13 n-15\right)\left(x^3+3 x\right)$, $x \in \mathbb{R}$, is an increasing function on $\mathbb{R}$, is | (2) 9 |
| (R) The smallest natural number $n$ which is greater than 5 , such that $x=3$ is a point of local minima of $$ h(x)=\left(x^2-9\right)^n\left(x^2+2 x+3\right) $$ is | (3) 5 |
| (S) Number of $x_0 \in \mathbb{R}$ such that
$$ l(x)=\sum\limits_{k=0}^4\left(\sin |x-k|+\cos \left|x-k+\frac{1}{2}\right|\right) $$ $x \in \mathbb{R}$, is NOT differentiable at $x_0$, is |
(4) 6 |
| (5) 10 |
Let $\vec{w} = \hat{i} + \hat{j} - 2\hat{k}$, and $\vec{u}$ and $\vec{v}$ be two vectors, such that $\vec{u} \times \vec{v} = \vec{w}$ and $\vec{v} \times \vec{w} = \vec{u}$. Let $\alpha, \beta, \gamma$, and $t$ be real numbers such that
$\vec{u} = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k},\ \ \ - t \alpha + \beta + \gamma = 0,\ \ \ \alpha - t \beta + \gamma = 0,\ \ \ \alpha + \beta - t \gamma = 0.$
Match each entry in List-I to the correct entry in List-II and choose the correct option.
| List – I | List – II |
|---|---|
| (P) $\lvert \vec{v} \rvert^2$ is equal to | (1) 0 |
| (Q) If $\alpha = \sqrt{3}$, then $\gamma^2$ is equal to | (2) 1 |
| (R) If $\alpha = \sqrt{3}$, then $(\beta + \gamma)^2$ is equal to | (3) 2 |
| (S) If $\alpha = \sqrt{2}$, then $t + 3$ is equal to | (4) 3 |
| (5) 5 |