JEE Main Mathematics, Feb 25, 2021 (Shift-2)

Problem 1 of 30

If for the matrix, $A = \begin{bmatrix} 1 & - \alpha \\ \alpha & \beta \end{bmatrix}$, $AA^T=I_2,$ then the value of $\alpha^4 + \beta^4$ is :

Problem 2 of 30

Let $A$ be a $3 \times 3$ matrix with det$(A)=4.$ Let $R_i$ denote the $i^{th}$ row of $A.$ If a matrix $B$ is obtained by performing the operation $R_2 \rightarrow 2R_2 + 5 R_3$ on $2A,$ then det$(B)$ is equal to

Problem 3 of 30

The following system of linear equations

$\phantom{0000} 2x + 3y + 2z = 9$

$\phantom{0000} 3x + 2y + 2z = 9$

$\phantom{0000} x - y + 4z = 8$

Problem 4 of 30

If $I_n = \displaystyle{\int\limits_{\dfrac{\pi}{4}}^{\dfrac{\pi}{2}}} \cot^n x dx$, then :

Problem 5 of 30

A function $f(x)$ is given by $f(x) = \dfrac{5^x}{5^x + 5}$, then the sum of the series

$f\left( \dfrac{1}{20} \right) + f\left( \dfrac{2}{20} \right) + f\left( \dfrac{3}{20} \right) + ... + f\left( \dfrac{39}{20} \right)$

is equal to:

Problem 6 of 30

Let $\alpha$ and $\beta$ be the roots of $x^2 - 6x - 2 = 0$. If $a_n = \alpha^n - \beta^n$ for $n \geqslant 1$, then the value of $\dfrac{a_{10} - 2a_8}{3a_9}$

Problem 7 of 30

The minimum value of $f(x) = a^{a^x} + a^{1 - a^x}$, where $a, x \in \mathbb{R}$ and $a \gt 0$, is equal to:

Problem 8 of 30

The integral $\displaystyle{\int} \dfrac{e^{3\log_{e}2x} + 5e^{2\log_{e}2x}}{e^{4\log_{e}x} + 5e^{3\log_{e}x} - 7e^{2\log_{e}x}} dx$, $x \gt 0$, is equal to:

(where $c$ is a constant of integration)

Problem 9 of 30

If $\alpha, \beta \in \mathbb{R}$ are such that $1 - 2i$ (here $i^2 = -1$) is a root of the equation $z^2 + \alpha z + \beta = 0$, then $(\alpha - \beta)$ is equal to:

Problem 10 of 30

If the curve $x^2 + 2y^2 = 2$ intersects the line $x + y = 1$ at two points $P$ and $Q$, then the angle subtended by the line segment $PQ$ at the origin is:

Problem 11 of 30

The shortest distance between the line $x - y = 1$ and the curve $x^2 = 2y$ is:

Problem 12 of 30

A hyperbola passes through the foci of the ellipse $\dfrac{x^2}{25} + \dfrac{y^2}{16} = 1$ and its transverse and conjugate axes coincide with the major and minor axes of the ellipse, respectively. If the product of their eccentricities is one, then the equation of the hyperbola is:

Problem 13 of 30

A plane passes through the points $A(1, 2, 3), B(2, 3, 1)$ and $C(2, 4, 2)$. If $O$ is the origin and $P$ is $(2, -1, 1)$, then the projection of $\overrightarrow{OP}$ on this plane is of length:

Problem 14 of 30

$\lim\limits_{n \rightarrow \infty} \left[ \dfrac{1}{n} + \dfrac{n}{(n + 1)^2} + \dfrac{n}{(n + 2)^2} + ... + \dfrac{n}{(2n - 1)^2}\right]$ is equal to:

Problem 15 of 30

In a group of $400$ people, $160$ are smokers and non-vegetarian; $100$ are smokers and vegetarian and the remaining $140$ are non-smokers and vegetarian. Their chances of getting a particular chest disorder are $35\%$, $20\%$ and $10\%$ respectively. A person is chosen from the group at random and is found to be suffering from the chest disorder. The probability that the selected person is a smoker and non-vegetarian is:

Problem 16 of 30

Let $A$ be a set of all $4$-digit natural numbers whose exactly one digit is $7$. Then the probability that a randomly chosen element of $A$ leaves a remainder of $2$ when divided by $5$ is:

Problem 17 of 30

If $0 \lt x, y \lt \pi$ and $\cos x + \cos y - \cos{(x + y)} = \dfrac{3}{2}$, then $\sin x + \cos y$ is equal to:

Problem 18 of 30

Let $x$ denote the total number of one-one functions from a set $A$ with $3$ elements to a set $B$ with $5$ elements and $y$ denote the total number of one-one functions from set $A$ to the set $A \times B$. Then:

Problem 19 of 30

$\text{cosec} \left[2\cot^{-1}(5) + \cos^{-1}\left( \dfrac{4}{5} \right)\right]$ is equal to:

Problem 20 of 30

The contrapositive of the statement $\unicode{0x201C} \text{If you will work, you will earn money} \unicode{0x201D}$ is:

Problem 21 of 30

A function $f$ is defined on $[-3, 3]$ as

$f(x) = \left\{\begin{array}{cc} \min\{|x|, 2-x^2\}, & -2 \le x \le 2 \\ \lfloor |x| \rfloor, & 2 \lt |x| \le 3 \end{array} \right.$

where $\lfloor x \rfloor$ denotes the greatest integer $\le x$. The number of points, where $f$ is not differentiable in $(-3, 3)$ is ______.

Problem 22 of 30

If the curve, $y = y(x)$ represented by the solution of the differential equation $(2xy^2 - y)dx + xdy = 0$, passes through the intersection of the lines, $2x - 3y = 1$ and $3x + 2y = 8$, then $|y(1)|$ is equal to ______.

Problem 23 of 30

The total number of two digit numbers $'n'$, such that $3^n + 7^n$ is a multiple of $10$ is ______.

Problem 24 of 30

If $\lim\limits_{x \rightarrow 0} \dfrac{ax - (e^{4x} - 1)}{ax(e^{4x} - 1)}$ exists and is equal to $b$, then the value of $a - 2b$ is ______.

Problem 25 of 30

If the curves $x = y^4$ and $xy = k$ cut at right angles, then $(4k)^6$ is equal to ______.

Problem 26 of 30

The value of $\displaystyle{\int\limits_{-2}^{2}} |3x^2 - 3x - 6| dx$ is ______.

Problem 27 of 30

If the remainder when $x$ is divided by $4$ is $3$, then the remainder when $(2020 + x)^{2022}$ is divided by $8$ is ________.

Problem 28 of 30

A line $'l'$ passing through the origin is perpendicular to the lines

$l_1 : \overrightarrow{r} = (3 + t)\hat{i} + (-1 + 2t)\hat{j} + (4 + 2t)\hat{k}$

$l_2 : \overrightarrow{r} = (3 + 2s)\hat{i} + (3 + 2s)\hat{j} + (2 + s)\hat{k}$

If the co-ordinates of the point in the first octant on $'l_2'$ at a distance of $\sqrt{17}$ from the point of intersection of $'l'$ and $'l_1'$ are $(a, b, c)$ then $18(a + b + c)$ is equal to ______.

Problem 29 of 30

A line is a common tangent to the circle $(x-3)^2 + y^2 = 9$ and the parabola $y^2 = 4x.$ If the two points of contact $(a,b)$ and $(c,d)$ are distinct and lie in the first quadrant, then $2(a+c)$ is equal to ________

Problem 30 of 30

Let $\overrightarrow{a} = \hat{i} + \alpha \hat{j} + 3\hat{k}$ and $\overrightarrow{b} = 3\hat{i} - \alpha \hat{j} + \hat{k}$. If the area of the parallelogram whose adjacent sides are represented by the vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $8\sqrt{3}$ square units, then $\overrightarrow{a} \cdot \overrightarrow{b}$ is equal to ______.