JEE Main Mathematics, Feb 24 2021, Shift-2

Problem 1 of 30

Let $a, b \in \mathbb{R}$. If the mirror image of the point $P(a, 6, 9)$ with respect to the line $\dfrac{x - 3}{7} = \dfrac{y - 2}{5} = \dfrac{z - 1}{-9}$ is $(20, b, -a - 9)$ then $|a + b|$ is equal to:

Problem 2 of 30

Let $a, b, c$ be in arithmetic progression. Let the centroid of the triangle with vertices $(a, c), (2, b)$ and $(a, b)$ be $\left(\dfrac{10}{3}, \dfrac{7}{3}\right)$. If $\alpha , \beta$ are the roots of the equation $ax^2 + bx + 1 = 0$, then the value of $\alpha^2 + \beta^2 - \alpha \beta$ is:

Problem 3 of 30

If $P$ is a point on the parabola $y = x^2 + 4$ which is closest to the straight line $y = 4x - 1$, then the coordinates of $P$ are:

Problem 4 of 30

For which of the following curves, the line $x + \sqrt{3}y = 2\sqrt{3}$ is the tangent at the point $\dfrac{3\sqrt{3}}{2}, \dfrac{1}{2}$?

Problem 5 of 30

The value of the integral $\displaystyle{\int\limits_{1}^{3}}\lfloor x^2 - 2x - 2 \rfloor dx$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$ is:

Problem 6 of 30

The negation of the statement

$\lnot p \land (p \lor q)$ is:

Problem 7 of 30

Let $A$ and $B$ be $3 \times 3$ real matrices such that $A$ is a symmetric matrix and $B$ is a skew-symmetric matrix. Then the system of linear equations $(A^2B^2 - B^2A^2)X = O$, where $X$ is a $3 \times 1$ column matrix of unknown variables and $O$ is a $3 \times 1$ null matrix, has:

Problem 8 of 30

Let $f(x)$ be a differentiable function defined on $[0, 2]$ such that $f'(x) = f'(2 - x)$ for all $x \in (0, 2), f(0) = 1$ and $f(2) = e^2$.

Then the value of $\displaystyle{\int\limits_{0}^{2}}f(x)dx$ is:

Problem 9 of 30

Let $f$ be a twice differentiable function defined on $\mathbb{R}$ such that $f(0) = 1, f'(0) = 2$ and $f'(x) \ne 0$ for all $x \in \mathbb{R}$. If $\begin{vmatrix} f(x) & f'(x) \\ f'(x) & f''(x) \end{vmatrix} = 0$, for all $x \in \mathbb{R}$, then the value of $f(1)$ lies in the interval:

Problem 10 of 30

If the curve $y = ax^2 + bx + c, x \in \mathbb{R}$, passes through the point $(1, 2)$ and the tangent line to this curve at the origin is $y = x$, then the possible values of $a, b, c$ are:

Problem 11 of 30

If $n \ge 2$ is a positive integer, then the sum of the series

$\xacomb{n+1}{2} + 2\left(\xacomb{2}{2} + \xacomb{3}{2} + \xacomb{4}{2} + ... + \xacomb{n}{2}\right)$ is:

Problem 12 of 30

For the system of linear equations:

$x - 2y = 1$, $x - y + kz = -2$, $ky + 4z = 6$, $k \in \mathbb{R}$ consider the following statements:

$(A)$ The system has unique solution if $k \ne 2, k \ne -2$

$(B)$ The system has unique solution if $k = -2$

$(C)$ The system has unique solution if $k = 2$

$(D)$ The system has no-solution if $k = 2$

$(E)$ The system has infinite number of solutions if $k \ne 2$

Which of the following statements are correct?

Problem 13 of 30

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be defined as

$f(x) = \left\{\begin{array}{ll} -55x, & \text{ if } x < -5 \\ 2x^3 - 3x^2 -120x, & \text{ if } -5 \le x \le 4 \\ 2x^3 - 3x^2 - 36x -336, & \text{ if } x \gt 4\end{array}\right.$

Let $A = \{x \in R:f$ is increasing$\}$. Then $A$ is equal to:

Problem 14 of 30

A possible value of $\tan\left(\dfrac{1}{4}\sin^{-1}\dfrac{\sqrt{63}}{8}\right)$ is:

Problem 15 of 30

The angle of elevation of a jet plane from a point $A$ on the ground is $60^\circ$. After a flight of $20$ seconds at the speed of $432$ km/hour, the angle of elevation changes to $30^\circ$. If the jet plane is flying at a constant height of, then the height is:

Problem 16 of 30

The vector equation of the plane passing through the intersection of the planes

$\overrightarrow{r} \cdot \left(\hat{i} + \hat{j} + \hat{k}\right) = 1$ and $\overrightarrow{r} \cdot \left(\hat{i} - 2\hat{j}\right) = -2$, and the point $(1, 0, 2)$ is:

Problem 17 of 30

For the statements $p$ and $q$, consider the following compound statements:

$(a) \phantom{000} (\lnot q \land (p \rightarrow q)) \rightarrow \lnot p$

$(b) \phantom{000} ((p \lor q) \land \lnot p) \rightarrow q$

Problem 18 of 30

If a curve $y = f(x)$ passes through the point $(1, 2)$ and satisfies the $x\dfrac{dy}{dx} + y = bx^4$, then for what value of $b$, $\displaystyle{\int\limits_{1}^{2}}f(x)dx = \dfrac{62}{5}$?

Problem 19 of 30

The probability that two randomly selected subsets of the set $\{1, 2, 3, 4, 5\}$ have exactly two elements in their intersection, is:

Problem 20 of 30

The area of the region $R = \{(x, y) : 5x^2 \le y \le 2x^2 + 9\}$ is:

Problem 21 of 30

If the variance of $10$ natural number $1, 1, 1, ..., 1, k$ is less than $10$, then the maximum possible value of $k$ is ______.

Problem 22 of 30

Let a point $P$ be such that its distance from the point $(5, 0)$ is thrice the distance of $P$ from the point $(-5, 0)$. If the locus of the point $P$ is a circle of radius $r$, then $4r^2$ is closest to the integer ______.

Problem 23 of 30

If the area of the triangle formed by the $x$-axis, and the normal and tangent to the circle $(x - 2)^2 + (y - 3)^2 = 25$ at the point $(5, 7)$ is $A$, then $24A$ is equal to ______.

Problem 24 of 30

Let $i = \sqrt{-1}$. If $\dfrac{(-1 + i\sqrt{3})^{21}}{(1 - i)^{24}} + \dfrac{(1 + i\sqrt{3})^{21}}{(1 + i)^{24}} = k$ and $n = \lfloor |k| \rfloor$ be the greatest integral part of $|k|$. Then $\sum\limits_{j = 0}^{n+5}(j + 5)^2 - \sum\limits_{j = 0}^{n+5}(j + 5)$ is equal to ______.

Problem 25 of 30

The students $S_1, S_2, S_3, ..., S_{10}$ are to be divided into $3$ groups $A$, $B$ and $C$ such that each group has at least one student and the group $C$ has at most $3$ students. Then the total number of possibilities of forming such groups is ______.

Problem 26 of 30

For integers $n$ and $r$ let $\left(\begin{matrix} n \\ r \end{matrix}\right) = \left\{\begin{array}{ll} \xacomb{n}{r}, & \text{ if } n \ge r \ge 0 \\ 0, & \text{ otherwise} \end{array}\right.$

The maximum value of $k$ for which the sum

$\displaystyle{\sum\limits_{i = 0}^{k} {10 \choose i}{15 \choose {k-i}}} + \displaystyle{\sum\limits_{i = 0}^{k+1} {12 \choose i}{13 \choose {k+1-i}}}$ exists, is equal to ______.

This question, as published, was incorrect and was declared a bonus question.

Do not attempt this question, just enter $0$ as answer.

Problem 27 of 30

If $a + \alpha = 1, b + \beta = 2$ and $a f(x) + \alpha f\left(\dfrac{1}{x}\right) = bx + \dfrac{\beta}{x}, x \ne 0$, then the value of the expression

$\dfrac{f(x) + \left(\dfrac{1}{x}\right)}{x + \dfrac{1}{x}}$ is ______.

Problem 28 of 30

Let $\lambda$ be an integer. If the shortest distance between the lines $x - \lambda = 2y - 1 = -2z$ and $x = y + 2\lambda = z - \lambda$ is $\dfrac{\sqrt{7}}{2\sqrt{2}}$, then the value of $| \lambda |$ is ______.

Problem 29 of 30

The sum of the first four terms of a geometric progression (G.P.) is $\dfrac{65}{12}$ and the sum of their respective reciprocals is $\dfrac{65}{18}$. If the product of first three terms of the G.P. is $1$, and the third term is $\alpha$, then $2\alpha$ is ______.

Problem 30 of 30

The number of the real roots of the equation $(x + 1)^2 + |x - 5| = \dfrac{27}{4}$ is ______.