JEE Main Mathematics, Feb 24 2021, (Shift-1)

Problem 1 of 30

Let $f:\mathbb{R} \longrightarrow \mathbb{R}$ be defined as $f(x) = 2x - 1$ and $g:\mathbb{R - \{1\}} \longrightarrow \mathbb{R}$ be defined as $g(x) = \dfrac{x - \dfrac{1}{2}}{x - 1}$.

Then the composition function $f(g(x))$ is:

Problem 2 of 30

Let $p$ and $q$ be two positive numbers such that $p + q = 2$ and $p^4 + q^4 = 272$. Then $p$ and $q$ are the roots of the equation:

Problem 3 of 30

The system of linear equations:

$3x - 2y - kz = 10$

$2x - 4y - 2z = 6$

$x + 2y - z = 5m$

is inconsistent if:

Problem 4 of 30

The value of

$-\xacomb{15}{1} + 2\cdot \xacomb{15}{2} - 3\cdot \xacomb{15}{3} +... -15\cdot \xacomb{15}{15} + \xacomb{14}{1} + \xacomb{14}{3} + \xacomb{14}{5} + ...$

$+ \xacomb{14}{11}$ is

Problem 5 of 30

If $e^{(\cos^2x + \cos^4x + \cos^6x + ... \infty)log_e2}$ satisfies the equation $t^2 - 9t + 8 = 0$, then the value of

$\dfrac{2\sin x}{\sin x + \sqrt{3}\cos x}\ \left(0 \lt x \lt \dfrac{\pi}{2}\right)$ is:

Problem 6 of 30

$\displaystyle{\lim\limits_{x \rightarrow 0}}\dfrac{\int\limits_0^{x^2}\left(\sin \sqrt{t}\right)dt}{x^3}$ is equal to:

Problem 7 of 30

The function $f(x) = \dfrac{4x^3 - 3x^2}{6} - 2\sin x + (2x - 1)\cos x$:

Problem 8 of 30

A scientific committee is to be formed from $6$ Indians and $8$ foreigners, which includes at least $2$ Indians and double the number of foreigners as Indians. Then the number of ways, the committee can be formed, is :

Problem 9 of 30

If $f : \mathbb{R} \longrightarrow \mathbb{R}$ is defined by $f(x) = \lfloor x - 1\rfloor \cdot \cos\left(\dfrac{2x - 1}{2}\right)\pi$ where $\lfloor \cdot \rfloor$ denotes the greatest integer function, the $f$ is:

Problem 10 of 30

If $\displaystyle{\int} \dfrac{\cos x - \sin x}{\sqrt{8 - \sin 2x}}dx = a\sin^{-1}\left(\dfrac{\sin x + \cos x}{b}\right) + c$, where $c$ is a constant of integration, then the ordered pair $(a, b)$ is equal to:

Problem 11 of 30

The area (in sq. units) of the part of the circle $x^2 + y^2 = 36,$ which is outside the parabola $y^2 = 9x,$ is :

Problem 12 of 30

The population $P = P(t)$ at time $'t'$ of a certain species follows the differential equation $\dfrac{dP}{dt} = 0.5P - 450$. If $P(0) = 850$, then the time at which population becomes zero is:

Problem 13 of 30

A man is walking on a straight line. The arithmetic mean of the reciprocals of the intercepts of this line on the coordinate axes is $\dfrac{1}{4}.$ Three stones $A,\ B$ and $C$ are placed at the points $(1,1),\ (2,2)$ and $(4,4)$ respectively. Then which of these stones is/are on the path of the man?

Problem 14 of 30

The locus of the mid-point of the line segment joining the focus of the parabola $y^2 = 4ax$ to a moving point of the parabola, is another parabola whose directrix is :

Problem 15 of 30

If the tangent to the curve $y = x^3$ at the point $P(t, t^3)$ meets the curve again at $Q$, then the ordinate of the point which divides $PQ$ internally in the ratio $1:2$ is:

Problem 16 of 30

The equation of the plane passing through the point $(1, 2, -3)$ and perpendicular to the planes $3x + y - 2z = 5$ and $2x - 5y - z = 7$, is:

Problem 17 of 30

The distance of the point $(1, 1, 9)$ from the point of intersection of the line

$\dfrac{x-3}{1} = \dfrac{y - 4}{2} = \dfrac{z - 5}{2}$ and the plane $x + y + z = 17$ is:

Problem 18 of 30

An ordinary dice is rolled for a certain number of times. If the probability of getting an odd number $2$ times is equal to the probability of getting an even number $3$ times, then the probability of getting an odd number odd number of times is:

Problem 19 of 30

Two vertical poles are $150$ m apart and the height of one is three times the other. If from the middle point of the line joining their feet, an observer finds the angles of elevation of their tops to be complementary, then the height of the shorter pole (in meters) is:

Problem 20 of 30

The statement among the following that is a tautology is:

Problem 21 of 30

If the least and the largest real values of $\alpha$, for which the equation $z + \alpha|z-1| + 2i = 0$ $z \in \mathbb{C}$ and $i = \sqrt{-1}$ has a solution, are $p$ and $q$ respectively then $4(p^2 + q^2)$ is equal to:

Problem 22 of 30

Let $B_i (i = 1, 2, 3)$ be three independent events in a sample space. The probability that only $B_1$ occurs is $\alpha$, only $B_2$ occurs is $\beta$ and only $B_3$ occurs is $\gamma$. Let $p$ be the probability that none of the events $B_i$ occurs and these $4$ probabilities satisfy the equation $(\alpha - 2\beta)p = \alpha \beta$ and $(\beta - 3\gamma)p = 2\beta \gamma$ (All probabilities are assumed to lie in the interval $(0, 1)$). Then $\dfrac{P(B_1)}{P(B_3)}$ is equal to ___________.

Problem 23 of 30

Let $P = \begin{bmatrix} 3 & -1 & -2 \\ 2 & 0 & \alpha \\ 3 & -5 & 0 \end{bmatrix}$ where $\alpha \in \mathbb{R}$. Suppose $Q = [q_{ij}]$ is a matrix satisfying $PQ = kI_3$ for some $k \in \mathbb{R}$. If $q_{23} = -\dfrac{k}{8}$ and $|Q| = \dfrac{k^2}{2}$ then $\alpha^2 + k^2$ is equal to ____________.

Problem 24 of 30

Let $M$ be any $3 \times 3$ matrix with entries from the set $\{0, 1, 2\}$. The maximum number of such matrices, for which the sum of the diagonal elements of $M^TM$ is seven is ______________.

Problem 25 of 30

Let $A = \{n \in \mathbb{N} : n$ is a $3$-digit number$\}$

$B = \{9k + 2 : k \in \mathbb{N}\}$

$C = \{9k + l : k \in \mathbb{N}\}$ for some $l (0 < l < 9)$

If the sum of all the elements of the set $A \cap (B \cup C)$ is $274 \times 400$, then $l$ is equal to _______.

Problem 26 of 30

The minimum value of $\alpha$ for which the equation $\dfrac{4}{\sin x} + \dfrac{1}{1 - \sin x} = \alpha$ has at least one solution in $\left(0, \dfrac{\pi}{2}\right)$ is ___________.

Problem 27 of 30

If $\displaystyle{\int\limits_{-a}^{a}} (|x| + |x - 2|)dx = 22, (a \gt 22)$ and $\lfloor x \rfloor$ denotes the greatest integer $\le x$ then $\displaystyle{\int\limits_{a}^{-a}} (x + \lfloor x \rfloor)dx$ is equal to ________.

Problem 28 of 30

If one of the diameters of the circle $x^2 + y^2 – 2x – 6y + 6 = 0$ is a chord of another circle $'C'$, whose centre is at $(2, 1),$ then its radius is __________

Problem 29 of 30

Let three vectors $\overrightarrow{a}, \overrightarrow{b}$ and $\overrightarrow{c}$ be such that $\overrightarrow{c}$ is coplanar with $\overrightarrow{a}$ and $\overrightarrow{b}$, $\overrightarrow{a} \cdot \overrightarrow{c} = 7$ and $\overrightarrow{b}$ is perpendicular to $\overrightarrow{c}$, where $\overrightarrow{a} = -\hat{i} + \hat{j} + \hat{k}$ and $\overrightarrow{b} = 2\hat{i} + \hat{k}$, then the value of $2|\overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c}|^2$ is _______.

Problem 30 of 30

$\lim\limits_{n \rightarrow \infty}\tan \left\{\sum\limits_{r=1}^{n}\tan^{-1}\left(\dfrac{1}{1 + r + r^2}\right)\right\}$ is equal to _______.