JEE Main Mathematics, Feb 26, 2021 (Shift-1)

Problem 1 of 30

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are perpendicular then $\overrightarrow{a} \times \left(\overrightarrow{a} \times \left(\overrightarrow{a} \times \left(\overrightarrow{a} \times \overrightarrow{b}\right)\right)\right)$ is equal to:

Problem 2 of 30

If $(1, 5, 35)$, $(7, 5, 5)$, $(1, \lambda, 7)$ and $(2\lambda, 1, 2)$ are coplanar, then the sum of all possible values of $\lambda$ is:

Problem 3 of 30

The intersection of three lines $x - y = 0$, $x + 2y = 3$ and $2x + y = 6$ is a :

Problem 4 of 30

The value of $\begin{vmatrix} (a + 1)(a + 2) & a + 2 & 1 \\ (a + 2)(a + 3) & a + 3 & 1 \\ (a + 3)(a + 4) & a + 4 & 1 \end{vmatrix}$ is:

Problem 5 of 30

The rate of growth of bacteria in a culture is proportional to the number of bacteria present and the bacteria count is $1000$ at initial time $t = 0$. The number of bacteria is increased by $20\%$ in $2$ hours. If the population of bacteria is $2000$ after $\dfrac{k}{\log_e\left(\dfrac{6}{5}\right)}$ hours, then $\left(\dfrac{k}{\log_e2}\right)^2$ is equal to:

Problem 6 of 30

In the circle given below, let $OA = 1$ unit, $OB = 13$ units and $PQ \perp OB$. Then, the area of the area of the triangle $PQB$ (in square units) is:

Problem 7 of 30

The value of $\lim\limits_{h \rightarrow 0} 2\left\{\dfrac{\sqrt{3}\sin\left(\dfrac{\pi}{6} + h\right) - \cos\left(\dfrac{\pi}{6} + h\right)}{\sqrt{3}h\left(\sqrt{3}\cos h - \sin h\right)}\right\}$

Problem 8 of 30

The maximum slope of the curve $y = \dfrac{1}{2}x^4 - 5x^3 + 18x^2 - 19x$ occurs at the point:

Problem 9 of 30

The value of $\displaystyle{\int\limits_{-\dfrac{\pi}{2}}^{\dfrac{\pi}{2}}} \dfrac{\cos^2 x}{1 + 3^x} dx$ is:

Problem 10 of 30

The number of seven digit integers with sum of the digits equal to $10$ and formed by using the digits $1, 2$ and $3$ only is:

Problem 11 of 30

The maximum value of the term independent of $'t'$ in the expansion of $\left(\large{tx}^{\frac{1}{5}} + \dfrac{(1-x)^{\frac{1}{10}}}{t}\right)^{10}$ where $x \in (0, 1)$ is:

Problem 12 of 30

Let $R = \{(P, Q) | P$ and $Q$ are at the same distance from the origin$\}$ be a relation, then the equivalence class of $(1, -1)$ is the set:

Problem 13 of 30

Let $A$ be a symmetric matrix of order $2$ with integer entries. If the sum of the diagonal elements of $A^2$ is $1$, then the possible number of such matrices is:

Problem 14 of 30

Let $f$ be any function defined on $\mathbb{R}$ and let it satisfy the condition:

$\phantom{00} |f(x) - f(y)| \le |(x - y)^2|, \forall (x, y) \in \mathbb{R}$

If $f(0) = 1$, then:

Problem 15 of 30

A fair coin is tossed a fixed number of times. If the probability of getting $7$ heads is equal to probability of getting $9$ heads, then the probability of getting $2$ heads is:

Problem 16 of 30

If $\dfrac{\sin^{-1}x}{a} = \dfrac{\cos^{-1}x}{b} = \dfrac{\tan^{-1}y}{c}; 0 \lt x \lt 1$, then the value of $\cos\left(\dfrac{\pi c}{a + b}\right)$ is:

Problem 17 of 30

In an increasing geometric series, the sum of the second and the sixth term is $\dfrac{25}{2}$ and the product of the third and the fifth term is $25$, then the sum of $4\xasuper{th}$, $6\xasuper{th}$ and $8\xasuper{th}$ terms is equal to:

Problem 18 of 30

The sum of the infinite series $1 + \dfrac{2}{3} + \dfrac{7}{3^2} + \dfrac{12}{3^3} + \dfrac{17}{3^4} + \dfrac{22}{3^5} + ...$ is equal to:

Problem 19 of 30

Consider the three planes

$\phantom{0000} 3x + 15y + 21z = 9$

$\phantom{0000} x - 3y - z = 5$

$\phantom{0000} 2x + 10y + 14z = 5$

Then, which of the following is true?

Problem 20 of 30

The value of $\displaystyle{\sum\limits_{n = 1}^{100} \int\limits_{n-1}^{n}} \large{e^{x - \lfloor x \rfloor}}dx$, where $\lfloor x \rfloor$ is the greatest integer $\le x$, is:

Problem 21 of 30

The difference between degree and order of a differential equation that represents the family of curves $y^2 = a\left(x + \dfrac{\sqrt{a}}{2}\right), a \gt 0$ is ______.

Problem 22 of 30

The sum of $162\xasuper{th}$ power of the roots of the equation $x^3 - 2x^2 + 2x - 1 = 0$ is ______.

Problem 23 of 30

The area bounded by the lines $y = ||x - 1| - 2|$ is ______.

Problem 24 of 30

If $y = y(x)$ is the solution of the equation $e^{\sin y}\cos y \dfrac{dy}{dx} + e^{\sin y}\cos x = \cos x$, $y(0) = 0;$ then $1 + y\left( \dfrac{\pi}{6} \right) + \dfrac{\sqrt{3}}{2}y\left( \dfrac{\pi}{3} \right) + \dfrac{1}{\sqrt{2}}y\left( \dfrac{\pi}{4} \right)$ is equal to ______.

Problem 25 of 30

The number of solutions of the equation $\log_4(x - 1) = \log_2(x - 3)$ is ______.

Problem 26 of 30

The number of integral values of $'k'$ for which the equation $3\sin x + 4\cos x = k + 1$ has a solution, $k \in \mathbb{R}$ is ______.

Problem 27 of 30

Let $(\lambda, 2, 1)$ be a point on the plane which passes through the point $(4, -2, 2)$. If the plane is perpendicular to the line joining the points $(-2, -21, 29)$ and $(-1, -16, 23)$, then

$\left(\dfrac{\lambda}{11}\right)^2 - \dfrac{4\lambda}{11} - 4$ is equal to ______.

Problem 28 of 30

The value of the integral $\displaystyle{\int\limits_{0}^{\pi}}|\sin 2x|dx$ is ______.

Problem 29 of 30

If $\sqrt{3} (\cos^{2}x) = (\sqrt{3} - 1)\cos x + 1$, the number of solutions of the given equation when $x \in \left[0, \dfrac{\pi}{2}\right]$ is ______.

Problem 30 of 30

Let $m, n \in \mathbb{N}$ and $gcd(2, n) = 1$. If $\displaystyle{30{30\choose 0} + 29{30\choose 1}} + ... + 2{30\choose 28} + 1{30\choose 29} = n \cdot 2^m$, then $n + m$ is equal to ______.

$\left(\text{Here} \displaystyle{n \choose k} = \xacomb{n}{k}\right)$