Other Maths · JEE Main previous year questions

Let $S=2+\frac{6}{7}+\frac{12}{7^2}+\frac{20}{7^3}+\frac{30}{7^4}+\dots$. Then $4S$ is equal to

1. A.${\left(\frac{7}{3}\right)}^2$
2. B.$\frac{7^3}{3^2}$
3. C.${\left(\frac{7}{3}\right)}^3$
4. D.$\frac{7^2}{3^3}$

If $y(x)={\left(x^x\right)}^x$, $x>0$, then $\frac{d^{2}x}{d{y}^2}+20$ at $x=1$ is equal to ________.

Let $f(x)=\frac{x-1}{x+1}$, $x\in R-\{0,-1,1\}$. If $f^{n+1}(x)=f(f^n(x))$ for all $n\in \mathrm{N}$, then $f^6(6)+f^7(7)$ is equal to :

1. A.$\frac{7}{6}$
2. B.$-\frac{3}{2}$
3. C.$\frac{7}{12}$
4. D.$-\frac{11}{12}$

Let $A=\left\{z\in C:\left|\frac{z+1}{z-1}\right|<1\right\}$ and $B=\left\{z\in C:\arg \left(\frac{z-1}{z+1}\right)=\frac{2\pi }{3}\right\}$. Then $A\cap B$ is :

1. A.a portion of a circle centred at $\left(0,-\frac{1}{\sqrt{3}}\right)$ that lies in the second and third quadrants only
2. B.a portion of a circle centred at $\left(0,-\frac{1}{\sqrt{3}}\right)$ that lies in the second quadrant only
3. C.an empty set
4. D.a portion of a circle of radius $\frac{2}{\sqrt{3}}$ that lies in the third quadrant only

The remainder when $(2021{)}^{2023}$ is divided by 7 is :

1. A.1
2. B.2
3. C.5
4. D.6

If $\vec{a}\cdot \vec{b}=1$, $\vec{b}\cdot \vec{c}=2$ and $\vec{c}\cdot \vec{a}=3$, then the value of $\left[\vec{a}\times (\vec{b}\times \vec{c}),\vec{b}\times (\vec{c}\times \vec{a}),\vec{c}\times (\vec{b}\times \vec{a})\right]$ is :

1. A.0
2. B.$-6\vec{a}\cdot (\vec{b}\times \vec{c})$
3. C.$-12\vec{c}\cdot (\vec{a}\times \vec{b})$
4. D.$-12\vec{b}\cdot (\vec{c}\times \vec{a})$

Let $f(x)=\max \{|x+1|,|x+2|,.....,|x+5|\}$. Then ${\int }_{-6}^{0}f(x)dx$ is equal to ____.

Let $A=\{n\in \mathrm{N}:H.C.F.(n,45)=1\}$ and Let $B=\{2k:k\in \{1,2,.......,100\}\}$. Then the sum of all the elements of $A\cap B$ is ____.

Let $A={\sum }_{i=1}^{10}{\sum }_{j=1}^{10}\min \{i,j\}$ and $B={\sum }_{i=1}^{10}{\sum }_{j=1}^{10}\max \{i,j\}$. Then $A+B$ is equal to ____.

If $A={\sum }_{n=1}^{\infty }\frac{1}{(3+(-1{)}^n{)}^n}$ and $B={\sum }_{n=1}^{\infty }\frac{(-1{)}^n}{(3+(-1{)}^n{)}^n}$, then $\frac{A}{B}$ is equal to :

1. A.$\frac{11}{9}$
2. B.1
3. C.$-\frac{11}{9}$
4. D.$-\frac{11}{3}$

Let $f:\mathrm{R}\to \mathrm{R}$ satisfy $f(x+y)=2^{x}f(y)+4^{y}f(x),\forall x,y\in \mathrm{R}$. If $f(2)=3$, then $14.\frac{f^{\prime }(4)}{f^{\prime }(2)}$ is equal to ________.

Let p and q be two real numbers such that $p+q=3$ and $p^4+q^4=369$. Then ${\left(\frac{1}{p}+\frac{1}{q}\right)}^{-2}$ is equal to ________.

If $z^2+z+1=0,z\in C$, then $\left|{\sum }_{n=1}^{15}{\left(z^n+(-1{)}^n\frac{1}{z^n}\right)}^2\right|$ is equal to ________.

If ${(}^{40}{C}_0)+{(}^{41}{C}_1)+{(}^{42}{C}_2)+\cdots +{(}^{60}{C}_{20})=\frac{m}{n}{}^{60}{C}_{20}$ m and n are coprime, then m + n is equal to ________.

If $\frac{1}{2\cdot 3^{10}}+\frac{1}{2^2\cdot 3^9}+\cdots +\frac{1}{2^{10}\cdot 3}=\frac{K}{2^{10}\cdot 3^{10}}$, then the remainder when $K$ is divided by $6$ is :

1. A.$1$
2. B.$2$
3. C.$3$
4. D.$5$

For a natural number n, let ${\alpha }_n=19^n-12^n$. Then, the value of $\frac{31{\alpha }_9-{\alpha }_{10}}{57{\alpha }_8}$ is $\underline{}$.

The sum $1+2\cdot 3+3\cdot 3^2+\cdots +10\cdot 3^9$ is equal to :

1. A.$\frac{2\cdot 3^{12}+10}{4}$
2. B.$\frac{19\cdot 3^{10}+1}{4}$
3. C.$5\cdot 3^{10}-2$
4. D.$\frac{9\cdot 3^{10}+1}{2}$

The value of $b>3$ for which $12\underset{3}{\overset{b}{\int }}\frac{1}{(x^2-1)(x^2-4)}dx={\log }_e\left(\frac{49}{40}\right)$, is equal to $\underline{}$.

Let $x,y>0$. If $x^3{y}^2=2^{15}$, then the least value of $3x+2y$ is

1. A.30
2. B.32
3. C.36
4. D.40

If the shortest distance between the lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{\lambda }$ and $\frac{x-2}{1}=\frac{y-4}{4}=\frac{z-5}{5}$ is $\frac{1}{\sqrt{3}}$, then the sum of all possible value of $\lambda$ is :

1. A.16
2. B.6
3. C.12
4. D.15

Let $S=\left\{\left(\begin{array}{cc}-1& a\\ 0& b\end{array}\right);a,b\in \{1,2,3,\dots ,100\}\right\}$ and let $T_n=\{A\in S:A^{n(n+1)}=I\}$. Then the number of elements in ${\bigcap }_{n=1}^{100}{T}_n$ is ____.

The remainder on dividing $1+3+3^2+3^3+\cdots +3^{2021}$ by 50 is ____.

The distance of line $3y-2z-1=0=3x-z+4$ from the point $(2,-1,6)$ is :

1. A.$\sqrt{26}$
2. B.$2\sqrt{5}$
3. C.$2\sqrt{6}$
4. D.$4\sqrt{2}$

Which of the following is equivalent to the Boolean expression $p\wedge \sim q$ ?

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

Let $S_n=1\cdot (n-1)+2\cdot (n-2)+3\cdot (n-3)+\cdots +(n-1)\cdot 1$, $n\ge 4$. The sum ${\sum }_{n=4}^{\infty }\left(\frac{2S_n}{n!}-\frac{1}{(n-2)!}\right)$ is equal to :

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