Other Maths · JEE Main previous year questions

The number of real solution(s) of the equation $x^2+3x+2=\min \{|x-3|,|x+2|\}$ is :

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

If the square of the shortest distance between the lines $\frac{x-2}{1}=\frac{y-1}{2}=\frac{z+3}{-3}$ and $\frac{x+1}{2}=\frac{y+3}{4}=\frac{z+5}{-5}$ is $\frac{m}{n}$, where $m,n$ are coprime numbers, then $m+n$ is equal to :

1. A.14
2. B.6
3. C.21
4. D.9

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

1. A.$\sqrt{17}$
2. B.$\sqrt{13}$
3. C.$\sqrt{15}$
4. D.$\sqrt{14}$

A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is $1\text{ cm}$ , the ice-cream melts at the rate of $81{\text{ cm}}^3/\min$ and the thickness of the ice-cream layer decreases at the rate of $\frac{1}{4\pi }\text{ cm/min}$. The surface area (in ${\text{cm}}^2$ ) of the chocolate ball (without the ice-cream layer) is :

1. A.$128\pi$
2. B.$196\pi$
3. C.$225\pi$
4. D.$256\pi$

Let $A=\{1,2,3,\dots ,10\}$ and $B=\left\{\frac{m}{n}:m,n\in A,m<n\text{ and }\gcd (m,n)=1\right\}$. Then $n(B)$ is equal to :

1. A.29
2. B.31
3. C.37
4. D.36

The product of all solutions of the equation $e^{5({\log }_{e}x{)}^2+3}=x^8,x>0$, is :

1. A.$e^2$
2. B.$e$
3. C.$e^{6/5}$
4. D.$e^{8/5}$

If ${\sum }_{r=1}^n{T}_r=\frac{(2n-1)(2n+1)(2n+3)(2n+5)}{64}$, then ${\lim }_{n\to \infty }{\sum }_{r=1}^n\left(\frac{1}{T_r}\right)$ is equal to :

1. A.$\frac{2}{3}$
2. B.$\frac{1}{3}$
3. C.1
4. D.0

If ${\sum }_{r=0}^5\frac{{}^{11}{\mathrm{C}}_{2r+1}}{2r+2}=\frac{\mathrm{m}}{\mathrm{n}}$, $\gcd (\mathrm{m},\mathrm{n})=1$, then $\mathrm{m}-\mathrm{n}$ is equal to _________.

Let $L_1:\frac{x-1}{3}=\frac{y-1}{-1}=\frac{z+1}{0}$ and $L_2:\frac{x-2}{2}=\frac{y}{0}=\frac{z+4}{\alpha },\alpha \in \mathbf{R}$, be two lines, which intersect at the point $B$. If $P$ is the foot of perpendicular from the point $A(1,1,-1)$ on $L_2$, then the value of $26\alpha (\mathrm{PB}{)}^2$ is _________.

The shortest distance between the lines $\frac{x-3}{4}=\frac{y+7}{-11}=\frac{z-1}{5}$ and $\frac{x-5}{3}=\frac{y-9}{-6}=\frac{z+2}{1}$ is:

1. A.$\frac{185}{\sqrt{563}}$
2. B.$\frac{187}{\sqrt{563}}$
3. C.$\frac{178}{\sqrt{563}}$
4. D.$\frac{179}{\sqrt{563}}$

A ray of light coming from the point $\mathrm{P}(1,2)$ gets reflected from the point $\mathrm{Q}$ on the $x$-axis and then passes through the point $R(4,3)$. If the point $S(h,k)$ is such that $PQRS$ is a parallelogram, then $h{k}^2$ is equal to:

1. A.60
2. B.70
3. C.80
4. D.90

Let the line $\mathrm{L}$ intersect the lines $x-2=-y=z-1$, $2(x+1)=2(y-1)=z+1$ and be parallel to the line $\frac{x-2}{3}=\frac{y-1}{1}=\frac{z-2}{2}$. Then which of the following points lies on $\mathrm{L}$ ?

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

The remainder when $428^{2024}$ is divided by 21 is ______.

Consider the line $\mathrm{L}$ passing through the points $(1,2,3)$ and $(2,3,5)$. The distance of the point $\left(\frac{11}{3},\frac{11}{3},\frac{19}{3}\right)$ from the line $\mathrm{L}$ along the line $\frac{3x-11}{2}=\frac{3y-11}{1}=\frac{3z-19}{2}$ is equal to

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

Let $\vec{a}=2\hat{i}+\alpha \hat{j}+\hat{k},\vec{b}=-\hat{i}+\hat{k},\vec{c}=\beta \hat{j}-\hat{k}$, where $\alpha$ and $\beta$ are integers and $\alpha \beta =-6$. Let the values of the ordered pair $(\alpha ,\beta )$, for which the area of the parallelogram of diagonals $\vec{a}+\vec{b}$ and $\vec{b}+\vec{c}$ is $\frac{\sqrt{21}}{2}$, be $({\alpha }_1,{\beta }_1)$ and $({\alpha }_2,{\beta }_2)$. Then ${\alpha }_1^2+{\beta }_1^2-{\alpha }_2{\beta }_2$ is equal to

1. A.21
2. B.24
3. C.19
4. D.17

If $\left(\frac{1}{\alpha +1}+\frac{1}{\alpha +2}+\dots \cdots +\frac{1}{\alpha +1012}\right)-\left(\frac{1}{2\cdot 1}+\frac{1}{4\cdot 3}+\frac{1}{6\cdot 5}+\dots \cdots +\frac{1}{2024\cdot 2023}\right)=\frac{1}{2024}$, then $\alpha$ is equal to ____.

The sum of all the solutions of the equation $(8{)}^{2x}-16\cdot (8{)}^x+48=0$ is :

1. A.$1+{\log }_8(6)$
2. B.$1+{\log }_6(8)$
3. C.${\log }_8(6)$
4. D.${\log }_8(4)$

Let the positive integers be written in the form : $$\begin{array}{ccccc}& & 1& & \\ & 2& & 3& \\ 4& & 5& & 6\\ 7& 8& 9& 10& \\ \cdot & & & & \cdot \\ \cdot & & & & \cdot \end{array}$$ If the $k^{\text{th}}$ row contains exactly $k$ numbers for every natural number $k$, then the row in which the number 5310 will be, is ________.

Let $\alpha ={\sum }_{r=0}^n(4r^2+2r+1){}^n{C}_r$ and $\beta =\left({\sum }_{r=0}^n\frac{{}^n{C}_r}{r+1}\right)+\frac{1}{n+1}$. If $140<\frac{2\alpha }{\beta }<281$, then the value of $n$ is ________.

Let $A=\left[\begin{array}{cc}2& -1\\ 1& 1\end{array}\right]$. If the sum of the diagonal elements of $A^{13}$ is $3^n$, then $n$ is equal to ________.

If the line segment joining the points $(5,2)$ and $(2,a)$ subtends an angle $\frac{\pi }{4}$ at the origin, then the absolute value of the product of all possible values of $a$ is :

1. A.4
2. B.8
3. C.6
4. D.2

If $\alpha \ne a,\beta \ne b,\gamma \ne c$ and $\left|\begin{array}{ccc}\alpha & b& c\\ a& \beta & c\\ a& b& \gamma \end{array}\right|=0$, then $\frac{a}{\alpha -a}+\frac{b}{\beta -b}+\frac{\gamma }{\gamma -c}$ is equal to :

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

Let $[t]$ denote the largest integer less than or equal to $t$. If ${\int }_0^3\left(\left[x^2\right]+\left[\frac{x^2}{2}\right]\right)dx=a+b\sqrt{2}-\sqrt{3}-\sqrt{5}+c\sqrt{6}-\sqrt{7}$, where $a,b,c\in \mathbf{Z}$, then $a+b+c$ is equal to ____.

If the shortest distance between the lines $\frac{x-\lambda }{3}=\frac{y-2}{-1}=\frac{z-1}{1}$ and $\frac{x+2}{-3}=\frac{y+5}{2}=\frac{z-4}{4}$ is $\frac{44}{\sqrt{30}}$, then the largest possible value of $|\lambda |$ is equal to ____.

If $1+\frac{\sqrt{3}-\sqrt{2}}{2\sqrt{3}}+\frac{5-2\sqrt{6}}{18}+\frac{9\sqrt{3}-11\sqrt{2}}{36\sqrt{3}}+\frac{49-20\sqrt{6}}{180}+\dots \text{upto }\infty =2+\left(\sqrt{\frac{b}{a}}+1\right){\log }_e\left(\frac{a}{b}\right)$, where a and b are integers with $\gcd (a,b)=1$, then $11a+18b$ is equal to _________.