JEE Main Mathematics, Feb 26, 2021 (Shift-2)

Problem 1 of 30

If the mirror image of the point $(1, 3, 5)$ with respect to the plane $4x - 5y + 2z = 8$ is $(\alpha, \beta, \gamma)$, then $5(\alpha, \beta, \gamma)$ equals:

Problem 2 of 30

Let $A = \{1, 2, 3, ..., 10\}$ and $f:A \rightarrow A$ be defined as

$f(k) = \left\{\begin{array}{cl} k + 1 & \text{if k is odd} \\ k & \text{if k is even} \end{array}\right.$

Then the number of possible functions $g:A \rightarrow A$ such that $g \circ f = f$ is:

Problem 3 of 30

Let $A_1$ be the area of the region bounded by the curves $y = \sin x$, $y = \cos x$ and the $y$-axis in the first quadrant. Also, let $A_2$ be the area of the region bounded by the curves $y = \sin x$, $y = \cos x$, $x$-axis and $x = \dfrac{\pi}{2}$ in the first quadrant. Then,

Problem 4 of 30

If $0 \lt a, b \lt 1$, and $\tan^{-1}a + \tan^{-1}b = \dfrac{\pi}{4}$, then the value of

$a + b - \dfrac{a^2 + b^2}{4} + \dfrac{a^3 + b^3}{3} - \dfrac{a^4 + b^4}{4} + ...$ is:

Problem 5 of 30

Let slope of the tangent line to a curve at any point $P(x, y)$ be given by $\dfrac{xy^2 + y}{x}$. If the curve intersects the line $x + 2y = 4$ at $x = -2$, then the value of $y$, for which the point $(3, y)$ lies on the curve, is:

Problem 6 of 30

The sum of the series $\displaystyle{\sum\limits_{n = 1}^{\infty}} \dfrac{n^2 + 6n + 10}{(2n + 1)!}$ is equal to:

Problem 7 of 30

Let $f(x) = \displaystyle{\int\limits_{0}^{x}} e^t f(t) dt + e^x$ be a differentiable function for all $x \in \mathbb{R}$. Then $f(x)$ equals:

Problem 8 of 30

Let $f(x)$ be a differentiable function at $x = a$ with $f'(a) = 2$ and $f(a) = 4$. Then $\lim\limits_{x \rightarrow a} \dfrac{xf(a) - af(x)}{x - a}$ equals:

Problem 9 of 30

Let $f(x) = \sin^{-1}x$ and $g(x) = \dfrac{x^2 - x - 2}{2x^2 - x - 6}$. If $g(2) = \lim\limits_{x \rightarrow 2} g(x)$, then the domain of the function $f \circ g$ is:

Problem 10 of 30

Let $A(1, 4)$ and $B(1, -5)$ be two points. Let $P$ be a point on the circle $(x - 1)^2 + (y - 1)^2 = 1$ such that $(PA)^2 + (PB)^2$ have maximum value, then the points $P, A$ and $B$ lie on:

Problem 11 of 30

If vectors $\overrightarrow{a_1} = x\hat{i} - \hat{j} + \hat{k}$ and $\overrightarrow{a_2} = \hat{i} + y\hat{j} + z\hat{k}$ are collinear, then a possible unit vector parallel to the vector $x\hat{i} + y\hat{j} + z\hat{k}$ is:

Problem 12 of 30

Let $F_1(A, B, C) = (A \land \lnot B) \lor [\lnot C \land (A \lor B)] \lor \lnot A$ and $F_2(A, B) = (A \lor B) \lor (B \rightarrow \lnot A)$ be two logical expressions. Then:

Problem 13 of 30

A seven digit number is formed using digits $\{3, 3, 4, 4, 4, 5, 5\}$. The probability, that number so formed is divisible by $2$, is:

Problem 14 of 30

Consider the system of equations:

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

$\phantom{0000} 2x + 6y - 11z = b$

$\phantom{0000} x - 2y + 7z = c$

where $a, b$ and $c$ are real constants. Then the system of equations:

Problem 15 of 30

The triangle of maximum area that can be inscribed in a given circle of radius $'r'$ is:

Problem 16 of 30

Let $L$ be a line obtained from the intersection of two planes $x + 2y + z = 6$ and $y + 2z = 4$. If point $P(\alpha, \beta, \gamma)$ is the foot of the perpendicular from $(3, 2, 1)$ on $L$, then the value of $21(\alpha + \beta + \gamma)$ equals:

Problem 17 of 30

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

$f(x) = \left\{\begin{array}{ll}2\sin\left(-\dfrac{\pi x}{2}\right), & \text{ if }x \lt -1 \\ |ax^2 + x + b|, & \text{ if }-1 \le x \le 1 \\ \sin\left(\pi x \right), & \text{ if }x \gt 1\end{array}\right.$

If $f(x)$ is continuous on $\mathbb{R}$, then $a + b$ equals:

Problem 18 of 30

If the locus of the mid-point of the line segment from the point $(3, 2)$ to a point on the circle, $x^2 + y^2 = 1$ is a circle of radius $r$, then $r$ is equal to:

Problem 19 of 30

A natural number has prime factorization given by $n = 2^x3^y5^z$, where $y$ and $z$ are such that $y + z = 5$ and $y^{-1} + z^{-1} = \dfrac{5}{6}$, $y \gt z$. Then the number of odd divisors of $n$, including $1$, is:

Problem 20 of 30

For $x \gt 0$, $f(x) = \displaystyle \int\limits_{1}^{x}\dfrac{\log_{e}t}{(1+t)}dt$, then $f(e) + f\left(\dfrac{1}{e}\right)$ is equal to:

Problem 21 of 30

If $I_{m, n} = \displaystyle{\int\limits_{0}^{1}} x^{m-1}(1-x)^{n-1} dx,$ for $m,n \geqslant 1,$ and $\displaystyle{\int\limits_{0}^{1}} \dfrac{x^{m-1} + x^{n-1}}{(1+x)^{m+n}} dx = \alpha I_{m,n}, \alpha \in R,$ then $\alpha$ equals _________.

Problem 22 of 30

Let $z$ be those complex numbers which satisfy $|z+5| \leq 4$ and $z(1+i) + \overline{z}(1-i) \geqslant -10, \ i = \sqrt{-1}.$

If the maximum value of $|z+1|^2$ is $\alpha + \beta \sqrt{2}$ then the value of $(\alpha + \beta)$ is ________.

Problem 23 of 30

Let the normals at all points on a given curve pass through a fixed point $(a,b).$ If the curve passes through $(3,-3)$ and $(4, -2 \sqrt{2})$, and given that $a - 2 \sqrt{2} b = 3,$ then $(a^2+b^2+ab)$ is equal to ________.

Problem 24 of 30

Let $a$ be an integer such that all real roots of the polynomial $2x^5+5x^4+10x^3+10x^2+10x+10$ lie in the interval $(a,a+1)$ Then $|a|$ is equal to _________.

Problem 25 of 30

If $X_1, X_2, ..., X_{18}$ be eighteen observations such that $\displaystyle{\sum\limits_{i = 1}^{18}}(X_i - \alpha) = 36$ and $\displaystyle{\sum\limits_{i = 1}^{18}}(X_i - \beta)^2 = 90$ where $\alpha$ and $\beta$ are distinct real numbers. If the standard deviation of these observations is $1$, then the value of $|\alpha - \beta|$ is ______.

Problem 26 of 30

If the matrix $A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 3 & 0 & -1\end{bmatrix}$ satisfies the equation $A^{20} + \alpha A^{19} + \beta A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 1\end{bmatrix}$ for some real numbers $\alpha$ and $\beta$, then $\beta - \alpha$ is equal to ______.

Problem 27 of 30

Let $\alpha$ and $\beta$ be two real numbers such that $\alpha + \beta = 1$ and $\alpha \beta = -1$. Let $p_n = (\alpha)^n + (\beta)^n$, $p_{n-1} = 11$ and $p_{n + 1} = 29$ for some integer $n \geqslant 1$. Then the value of $p_n^2$ is ______.

Problem 28 of 30

The total number of $4$-digit numbers whose greatest common divisor with $18$ is $3$, is ______.

Problem 29 of 30

If the arithmetic mean and geometric mean of the $p\xasuper{th}$ and $q\xasuper{th}$ terms of the sequence $-16, 8, -4, 2, ...$ satisfy the equation $4x^2 - 9x + 5 = 0$, the $p + q$ is equal to ______.

Problem 30 of 30

Let $L$ be a common tangent line to the curves $4x^2 + 9y^2 = 36$ and $(2x)^2 + (2y)^2 = 31$. Then the square of the slope of the line $L$ is ______.