Other Maths · JEE Main previous year questions

${}^{n-1}{\mathrm{C}}_r=\left(k^2-8\right){}^n{\mathrm{C}}_{r+1}$ if and only if :

1. A.$2\sqrt{2}<k<2\sqrt{3}$
2. B.$2\sqrt{2}<k\le 3$
3. C.$2\sqrt{3}<k<3\sqrt{3}$
4. D.$2\sqrt{3}<k\le 3\sqrt{2}$

If $8=3+\frac{1}{4}(3+p)+\frac{1}{4^2}(3+2p)+\frac{1}{4^3}(3+3p)+\cdots \cdots \infty$, then the value of $p$ is ____.

Let $\alpha =\frac{(4!)!}{(4!{)}^{3!}}$ and $\beta =\frac{(5!)!}{(5!{)}^{4!}}$. Then :

1. A.$\alpha \in \mathbf{N}$ and $\beta \in \mathbf{N}$
2. B.$\alpha \in \mathbf{N}$ and $\beta \notin \mathbf{N}$
3. C.$\alpha \notin \mathbf{N}$ and $\beta \in \mathbf{N}$
4. D.$\alpha \notin \mathbf{N}$ and $\beta \notin \mathbf{N}$

The values of $\alpha$, for which $\left|\begin{array}{ccc}1& \frac{3}{2}& \alpha +\frac{3}{2}\\ 1& \frac{1}{3}& \alpha +\frac{1}{3}\\ 2\alpha +3& 3\alpha +1& 0\end{array}\right|=0$, lie in the interval

1. A.$(-2,1)$
2. B.$(-\frac{3}{2},\frac{3}{2})$
3. C.$(-3,0)$
4. D.$(0,3)$

The lines $\frac{x-2}{2}=\frac{y}{-2}=\frac{z-7}{16}$ and $\frac{x+3}{4}=\frac{y+2}{3}=\frac{z+2}{1}$ intersect at the point $P$. If the distance of P from the line $\frac{x+1}{2}=\frac{y-1}{3}=\frac{z-1}{1}$ is $l$, then $14l^2$ is equal to _______.

If the sum of squares of all real values of $\alpha$, for which the lines $2x-y+3=0,6x+3y+1=0$ and $\alpha x+2y-2=0$ do not form a triangle is $p$, then the greatest integer less than or equal to $p$ is _______.

$$\underset{0}{\overset{\infty }{\int }}\frac{6}{e^{3x}+6e^{2x}+11e^x+6}dx=$$

1. A.${\log }_e\left(\frac{256}{81}\right)$
2. B.${\log }_e\left(\frac{64}{27}\right)$
3. C.${\log }_e\left(\frac{32}{27}\right)$
4. D.${\log }_e\left(\frac{512}{81}\right)$

Fractional part of the number $\frac{4^{2022}}{15}$ is equal to

1. A.$\frac{8}{15}$
2. B.$\frac{4}{15}$
3. C.$\frac{1}{15}$
4. D.$\frac{14}{15}$

Let $S=\left\{z\in \mathbb{C}:\bar{z}=i\left(z^2+\mathrm{Re}(\bar{z})\right)\right\}$. Then $\sum _{z\in S}|z{|}^2$ is equal to :

1. A.$\frac{7}{2}$
2. B.$4$
3. C.$3$
4. D.$\frac{5}{2}$

The remainder, when $7^{103}$ is divided by 17, is _________

Let $[\alpha ]$ denote the greatest integer $\le \alpha$. Then $[\sqrt{1}]+[\sqrt{2}]+[\sqrt{3}]+\cdots +[\sqrt{120}]$ is equal to __________

If $\frac{1}{n+1}{}^n{\mathrm{C}}_n+\frac{1}{n}{}^n{\mathrm{C}}_{n-1}+\cdots +\frac{1}{2}{}^n{\mathrm{C}}_1+{}^n{\mathrm{C}}_0=\frac{1023}{10}$ then $n$ is equal to :

1. A.$9$
2. B.$6$
3. C.$7$
4. D.$8$

Let $<a_n>$ be a sequence such that $a_1+a_2+\cdots +a_n=\frac{n^2+3n}{(n+1)(n+2)}$. If $28{\sum }_{k=1}^{10}\frac{1}{a_k}=p_1{p}_2{p}_3\dots p_m$, where $p_1,p_2,\dots ,p_m$ are the first $m$ prime numbers, then $m$ is equal to

1. A.$5$
2. B.$7$
3. C.$6$
4. D.$8$

Let R be a rectangle given by the lines $x=0,x=2,y=0$ and $y=5$. Let $A(\alpha ,0)$ and $B(0,\beta ),\alpha \in [0,2]$ and $\beta \in [0,5]$, be such that the line segment AB divides the area of the rectangle R in the ratio $4:1$. Then, the mid-point of AB lies on a :

1. A.hyperbola
2. B.straight line
3. C.parabola
4. D.circle

Let $f(x)=[x^2-x]+|-x+[x]|$, where $x\in \mathbb{R}$ and $[t]$ denotes the greatest integer less than or equal to $t$. Then, $f$ is :

1. A.continuous at $x=0$, but not continuous at $x=1$
2. B.continuous at $x=0$ and $x=1$
3. C.continuous at $x=1$, but not continuous at $x=0$
4. D.not continuous at $x=0$ and $x=1$

Let $A=\left[\begin{array}{ccc}0& 1& 2\\ a& 0& 3\\ 1& c& 0\end{array}\right]$, where $a,c\in \mathbb{R}$. If $A^3=A$ and the positive value of $a$ belongs to the interval $(n-1,n]$, where $n\in \mathbb{N}$, then $n$ is equal to _________.

For $m,n>0$, let $\alpha (m,n)={\int }_0^2{t}^m(1+3t{)}^{n}dt$. If $11\alpha (10,6)+18\alpha (11,5)=p(14{)}^6$, then $p$ is equal to _________.

Let $S=109+\frac{108}{5}+\frac{107}{5^2}+\dots +\frac{2}{5^{107}}+\frac{1}{5^{108}}$. Then the value of $(16S-(25{)}^{-54})$ is equal to _________.

Let $\mathrm{S}=\left\{z\in \mathbb{C}-\{i,2i\}:\frac{z^2+8iz-15}{z^2-3iz-2}\in \mathbb{R}\right\}$. If $\alpha -\frac{13}{11}i\in \mathrm{S}$, $\alpha \in \mathbb{R}-\{0\}$, then $242{\alpha }^2$ is equal to _________.

The number of points, where the curve $f(x)=e^{8x}-e^{6x}-3e^{4x}-e^{2x}+1$, $x\in \mathbb{R}$ cuts $x$-axis, is equal to _________.

Let a, b, c be three distinct positive real numbers such that $(2a{)}^{{\log }_{e}a}=(bc{)}^{{\log }_{e}b}$ and $b^{{\log }_{e}2}=a^{{\log }_{e}c}$. Then, $6a+5bc$ is equal to _________.

Let $S=\left\{z=x+iy:\frac{2z-3i}{4z+2i}\text{ is a real number}\right\}$. Then which of the following is NOT correct?

1. A.$y+x^2+y^2\ne -\frac{1}{4}$
2. B.$(x,y)=\left(0,-\frac{1}{2}\right)$
3. C.$x=0$
4. D.$y\in \left(-\infty ,-\frac{1}{2}\right)\cup \left(-\frac{1}{2},\infty \right)$

Let the number $(22{)}^{2022}+(2022{)}^{22}$ leave the remainder $\alpha$ when divided by 3 and $\beta$ when divided by 7. Then $({\alpha }^2+{\beta }^2)$ is equal to :

1. A.13
2. B.10
3. C.20
4. D.5

If $S_n=4+11+21+34+50+\dots$ to $n$ terms, then $\frac{1}{60}(S_{29}-S_9)$ is equal to :

1. A.227
2. B.226
3. C.220
4. D.223