Other Maths · JEE Main previous year questions

The imaginary part of ${\left(3+2\sqrt{-54}\right)}^{\frac{1}{2}}-{\left(3-2\sqrt{-54}\right)}^{\frac{1}{2}}$ can be :

1. A.$-2\sqrt{6}$
2. B.6
3. C.$\sqrt{6}$
4. D.$-\sqrt{6}$

Let $[t]$ denote the greatest integer less than or equal to $t$. Then the value of ${\int }_1^2|2x-[3x]|dx$ is _______.

If for all real triplets $(a,b,c)$, $f(x)=a+bx+c{x}^2$; then ${\int }_0^{1}f(x)dx$ is equal to :

1. A.$\frac{1}{6}\left\{f(0)+f(1)+4f\left(\frac{1}{2}\right)\right\}$
2. B.$2\left\{3f(1)+2f\left(\frac{1}{2}\right)\right\}$
3. C.$\frac{1}{3}\left\{f(0)+f\left(\frac{1}{2}\right)\right\}$
4. D.$\frac{1}{2}\left\{f(1)+3f\left(\frac{1}{2}\right)\right\}$

The product $2^{\frac{1}{4}}\cdot 4^{\frac{1}{16}}\cdot 8^{\frac{1}{48}}\cdot 16^{\frac{1}{128}}\dots \text{ to }\infty$ is equal to :

1. A.$2^{\frac{1}{4}}$
2. B.$2^{\frac{1}{2}}$
3. C.1
4. D.2

If $\mathrm{A}=\{x\in \mathbb{R}:|x|<2\}$ and $\mathrm{B}=\{x\in \mathbb{R}:|x-2|\ge 3\}$; then :

1. A.$\mathrm{A}-\mathrm{B}=[-1,2)$
2. B.$\mathrm{A}\cup \mathrm{B}=\mathbb{R}-(2,5)$
3. C.$\mathrm{A}\cap \mathrm{B}=(-2,-1)$
4. D.$\mathrm{B}-\mathrm{A}=\mathbb{R}-(-2,5)$

If $\frac{dy}{dx}=\frac{xy}{x^2+y^2}$; $y(1)=1$; then a value of x satisfying $y(x)=e$ is :

1. A.$\sqrt{2}e$
2. B.$\frac{1}{2}\sqrt{3}e$
3. C.$\frac{e}{\sqrt{2}}$
4. D.$\sqrt{3}e$

If ${\mathrm{C}}_r\equiv {}^{25}{\mathrm{C}}_r$ and ${\mathrm{C}}_0+5.{\mathrm{C}}_1+9.{\mathrm{C}}_2+\dots +(101).{\mathrm{C}}_{25}=2^{25}.k$, then k is equal to _______.

If a, b and c are the greatest value of ${}^{19}{\mathrm{C}}_p$, ${}^{20}{\mathrm{C}}_q$ and ${}^{21}{\mathrm{C}}_r$ respectively, then :

1. A.$\frac{a}{11}=\frac{b}{22}=\frac{c}{21}$
2. B.$\frac{a}{10}=\frac{b}{22}=\frac{c}{21}$
3. C.$\frac{a}{10}=\frac{b}{11}=\frac{c}{42}$
4. D.$\frac{a}{11}=\frac{b}{22}=\frac{c}{42}$

Let two points be A(1, -1) and B(0, 2). If a point P(x', y') be such that the area of $\Delta \mathrm{PAB}=5\text{ sq. units}$ and it lies on the line, $3x+y-4\lambda =0$, then a value of $\lambda$ is :

1. A.4
2. B.1
3. C.-3
4. D.3

The sum ${\sum }_{k=1}^{20}(1+2+3+\dots +k)$ is :

If $A=\left(\begin{array}{cc}2& 2\\ 9& 4\end{array}\right)$ and $I=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$ then $10A^{-1}$ is equal to :

1. A.$6I-A$
2. B.$4I-A$
3. C.$A-6I$
4. D.$A-4I$

The sum, ${\sum }_{n=1}^7\frac{n(n+1)(2n+1)}{4}$ is equal to _______.

Let $x^k+y^k=a^k$, $(a,k>0)$ and $\frac{dy}{dx}+{\left(\frac{y}{x}\right)}^{\frac{1}{3}}=0$, then k is:

1. A.$\frac{1}{3}$
2. B.$\frac{2}{3}$
3. C.$\frac{4}{3}$
4. D.$\frac{3}{2}$

The greatest positive integer k, for which $49^k+1$ is a factor of the sum $49^{125}+49^{124}+\cdots +49^2+49+1$, is:

1. A.32
2. B.60
3. C.63
4. D.65

The number of ordered pairs $(r,k)$ for which $6\cdot {}^{35}{\mathrm{C}}_r=(k^2-3)\cdot {}^{36}{\mathrm{C}}_{r+1}$, where $k$ is an integer, is :

1. A.6
2. B.3
3. C.2
4. D.4

If $e^y+xy=e$, the ordered pair $\left(\frac{dy}{dx},\frac{d^{2}y}{d{x}^2}\right)$ at $x=0$ is equal to :

1. A.$\left(\frac{1}{e},-\frac{1}{e^2}\right)$
2. B.$\left(-\frac{1}{e},\frac{1}{e^2}\right)$
3. C.$\left(-\frac{1}{e},-\frac{1}{e^2}\right)$
4. D.$\left(\frac{1}{e},\frac{1}{e^2}\right)$

A circle touching the x-axis at (3, 0) and making an intercept of length 8 on the y-axis passes through the point :

1. A.(1, 5)
2. B.( 2, 3)
3. C.(3, 5)
4. D.(3, 10)

A value of $\alpha$ such that ${\int }_{\alpha }^{\alpha +1}\frac{dx}{(x+\alpha )(x+\alpha +1)}={\log }_e\left(\frac{9}{8}\right)$ is :

1. A.2
2. B.-2
3. C.$\frac{1}{2}$
4. D.$-\frac{1}{2}$

The Boolean expression $\sim (p\Rightarrow (\sim q))$ is equivalent to :

1. A.$p\wedge q$
2. B.$q\Rightarrow \sim p$
3. C.$p\vee q$
4. D.$(\sim p)\Rightarrow q$

If ${}^{20}{\text{C}}_1+(2^2){}^{20}{\text{C}}_2+(3^2){}^{20}{\text{C}}_3+\cdots +(20^2){}^{20}{\text{C}}_{20}=A(2^{\beta })$, then the ordered pair $(A,\beta )$ is equal to :

1. A.$(420,19)$
2. B.$(420,18)$
3. C.$(380,18)$
4. D.$(380,19)$

All the pairs (x, y) that satisfy the inequality $2^{\sqrt{{\sin }^{2}x-2\sin x+5}}\cdot \frac{1}{4^{{\sin }^{2}y}}\le 1$ also satisfy the equation

1. A.$\sin x=|\sin y|$
2. B.$\sin x=2\sin y$
3. C.$2\sin x=\sin y$
4. D.$2|\sin x|=3\sin y$

If the length of the perpendicular from the point $(\beta ,0,\beta )$ $(\beta \ne 0)$ to the line, $\frac{x}{1}=\frac{y-1}{0}=\frac{z+1}{-1}$ is $\sqrt{\frac{3}{2}}$, then $\beta$ is equal to :

1. A.2
2. B.1
3. C.-2
4. D.-1

If ${\lim }_{x\to 1}\frac{x^4-1}{x-1}={\lim }_{x\to k}\frac{x^3-k^3}{x^2-k^2}$, then k is :

1. A.$\frac{3}{2}$
2. B.$\frac{8}{3}$
3. C.$\frac{4}{3}$
4. D.$\frac{3}{8}$

Lines are drawn parallel to the line $4x-3y+2=0$, at a distance $\frac{3}{5}$ from the origin. Then which one of the following points lies on any of these lines ?

1. A.$\left(\frac{1}{4},-\frac{1}{3}\right)$
2. B.$\left(-\frac{1}{4},\frac{2}{3}\right)$
3. C.$\left(-\frac{1}{4},-\frac{2}{3}\right)$
4. D.$\left(\frac{1}{4},\frac{1}{3}\right)$

The sum $1+\frac{1^3+2^3}{1+2}+\frac{1^3+2^3+3^3}{1+2+3}+\cdots +\frac{1^3+2^3+3^3+\cdots +15^3}{1+2+3+\cdots +15}-\frac{1}{2}(1+2+3+\cdots +15)$ is equal to :

1. A.620
2. B.1240
3. C.1860
4. D.660