Other Maths · JEE Main previous year questions

If $a_n=\frac{-2}{4n^2-16n+15}$, then $a_1+a_2+\dots +a_{25}$ is equal to :

1. A.$\frac{51}{144}$
2. B.$\frac{49}{138}$
3. C.$\frac{50}{141}$
4. D.$\frac{52}{147}$

Let $f^1(x)=\frac{3x+2}{2x+3}$, $x\in \mathbf{R}-\{-\frac{3}{2}\}$ For $n\ge 2$, define $f^n(x)=f^1\circ f^{n-1}(x)$. If $f^5(x)=\frac{ax+b}{bx+a}$, $\gcd (a,b)=1$, then $a+b$ is equal to ____.

Let $A=\left\{(x,y)\in {\mathbb{R}}^2:y\ge 0,2x\le y\le \sqrt{4-(x-1{)}^2}\right\}$ and $B=\left\{(x,y)\in \mathbb{R}\times \mathbb{R}:0\le y\le \min \left\{2x,\sqrt{4-(x-1{)}^2}\right\}\right\}$. Then the ratio of the area of A to the area of B is

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

If the co-efficient of $x^9$ in ${\left(\alpha x^3+\frac{1}{\beta x}\right)}^{11}$ and the co-efficient of $x^{-9}$ in ${\left(\alpha x-\frac{1}{\beta x^3}\right)}^{11}$ are equal, then $(\alpha \beta {)}^2$ is equal to ____.

Let the co-ordinates of one vertex of $\Delta ABC$ be $A(0,2,\alpha )$ and the other two vertices lie on the line $\frac{x+\alpha }{5}=\frac{y-1}{2}=\frac{z+4}{3}$. For $\alpha \in \mathbb{Z}$, if the area of $\Delta ABC$ is 21 sq. units and the line segment $BC$ has length $2\sqrt{21}$ units, then ${\alpha }^2$ is equal to ____.

Let $a_1=b_1=1$ and $a_n=a_{n-1}+(n-1),b_n=b_{n-1}+a_{n-1},\forall n\ge 2$. If $S=\sum _{n=1}^{10}\frac{b_n}{2^n}$ and $T=\sum _{n=1}^8\frac{n}{2^{n-1}}$, then $2^7(2S-T)$ is equal to ____.

The foot of perpendicular of the point (2, 0, 5) on the line $\frac{x+1}{2}=\frac{y-1}{5}=\frac{z+1}{-1}$ is $(\alpha ,\beta ,\gamma )$. Then, which of the following is NOT correct?

1. A.$\frac{\alpha }{\beta }=-8$
2. B.$\frac{\alpha \beta }{\gamma }=\frac{4}{15}$
3. C.$\frac{\beta }{\gamma }=-5$
4. D.$\frac{\gamma }{\alpha }=\frac{5}{8}$

Let $f(x)=2x^n+\lambda ,\lambda \in R,n\in N$, and $f(4)=133,f(5)=255$. Then the sum of all the positive integer divisors of $(f(3)-f(2))$ is

1. A.60
2. B.58
3. C.61
4. D.59

$\sum _{k=0}^6{}^{51-k}{C}_3$ is equal to :

1. A.${}^{51}{\mathrm{C}}_4-{}^{45}{\mathrm{C}}_4$
2. B.${}^{51}{\mathrm{C}}_3-{}^{45}{\mathrm{C}}_3$
3. C.${}^{52}{\mathrm{C}}_3-{}^{45}{\mathrm{C}}_3$
4. D.${}^{52}{\mathrm{C}}_4-{}^{45}{\mathrm{C}}_4$

A triangle is formed by X-axis, Y-axis and the line $3x+4y=60$. Then the number of points P(a, b) which lie strictly inside the triangle, where a is an integer and b is a multiple of a, is ____.

The remainder when $(2023{)}^{2023}$ is divided by 35 is ____.

Let PQR be a triangle. The points A, B and C are on the sides QR, RP and PQ respectively such that $\frac{QA}{AR}=\frac{RB}{BP}=\frac{PC}{CQ}=\frac{1}{2}$. Then $\frac{Area(\Delta PQR)}{Area(\Delta ABC)}$ is equal to :

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

The value of ${\sum }_{r=0}^{22}{}^{22}{C}_r{}^{23}{C}_r$ is

1. A.${}^{44}{C}_{23}$
2. B.${}^{45}{C}_{23}$
3. C.${}^{44}{C}_{22}$
4. D.${}^{45}{C}_{24}$

Let $\vec{u}=\hat{i}-\hat{j}-2\hat{k},\vec{v}=2\hat{i}+\hat{j}-\hat{k},\vec{v}\cdot \vec{w}=2$ and $\vec{v}\times \vec{w}=\vec{u}+\lambda \vec{v}$. Then $\vec{u}\cdot \vec{w}$ is equal to :

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

The value of $12{\int }_0^3|x^2-3x+2|dx$ is ____

The number of real solutions of the equation $3\left(x^2+\frac{1}{x^2}\right)-2\left(x+\frac{1}{x}\right)+5=0$, is

1. A.3
2. B.4
3. C.0
4. D.2

If $f(x)=\frac{2^{2x}}{2^{2x}+2},x\in \mathbb{R}$, then $f\left(\frac{1}{2023}\right)+f\left(\frac{2}{2023}\right)+\dots +f\left(\frac{2022}{2023}\right)$ is equal to

1. A.2011
2. B.2010
3. C.1010
4. D.1011

If $({}^{30}{\mathrm{C}}_1{)}^2+2({}^{30}{\mathrm{C}}_2{)}^2+3({}^{30}{\mathrm{C}}_3{)}^2+\dots +30({}^{30}{\mathrm{C}}_{30}{)}^2=\frac{\alpha 60!}{(30!{)}^2}$ then $\alpha$ is equal to :

1. A.30
2. B.10
3. C.15
4. D.60

If $\frac{1^3+2^3+3^3+\dots uptonterms}{1\cdot 3+2\cdot 5+3\cdot 7+\dots uptonterms}=\frac{9}{5}$, then the value of $n$ is

If $z=2+3i$, then $z^5+(\bar{z}{)}^5$ is equal to :

1. A.244
2. B.224
3. C.245
4. D.265

Let $\{a_n{\}}_{n=0}^{\infty }$ be a sequence such that $a_0=a_1=0$ and $a_{n+2}=3a_{n+1}-2a_n+1,\forall n\ge 0$. Then $a_{25}{a}_{23}-2a_{25}{a}_{22}-2a_{23}{a}_{24}+4a_{22}{a}_{24}$ is equal to

1. A.483
2. B.528
3. C.575
4. D.624

${\sum }_{r=1}^{20}(r^2+1)(r!)$ is equal to

1. A.$22!-21!$
2. B.$22!-2(21!)$
3. C.$21!-2(20!)$
4. D.$21!-20!$

Let $X=\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]$ and $A=\left[\begin{array}{ccc}-1& 2& 3\\ 0& 1& 6\\ 0& 0& -1\end{array}\right]$. For $k\in \mathbb{N}$, if $X^{\prime }{A}^{k}X=33$, then $k$ is equal to ______.

If ${\sum }_{k=1}^{10}{k}^2({}^{10}{C}_k{)}^2=22000L$, then $L$ is equal to ____.

The remainder when $7^{2022}+3^{2022}$ is divided by 5 is :

1. A.0
2. B.2
3. C.3
4. D.4