Other Maths · JEE Main previous year questions

A body falling from rest under gravity passes a certain point P. It was at a distance of 400 m from P, 4 s prior to passing through P. If $g=10m/{\mathrm{s}}^2$, then the height above the point P from where the body began to fall is

1. A.720 m
2. B.900 m
3. C.320 m
4. D.680 m

For natural numbers $m,n$, if $(1-y{)}^m(1+y{)}^n=1+a_{1}y+a_2{y}^2+\dots$ and $a_1=a_2=10$, then $(m,n)$ is

1. A.$(20,45)$
2. B.$(35,20)$
3. C.$(45,35)$
4. D.$(35,45)$

If $\omega =\frac{z}{z-\frac{1}{3}i}$ and $|\omega |=1$, then $z$ lies on :

1. A.an ellipse
2. B.a circle
3. C.a straight line
4. D.a parabola

The line parallel to the $x$ - axis and passing through the intersection of the lines $ax+2by+3b=0$ and $bx-2ay-3a=0$, where $(a,b)\ne (0,0)$ is :

1. A.below the $x$ - axis at a distance of $\frac{3}{2}$ from it
2. B.below the $x$ - axis at a distance of $\frac{2}{3}$ from it
3. C.above the $x$ - axis at a distance of $\frac{3}{2}$ from it
4. D.above the $x$ - axis at a distance of $\frac{2}{3}$ from it

If $A^2-A+I=0$, then the inverse of $A$ is :

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

If $I_1={\int }_0^1{2}^{x^2}dx$, $I_2={\int }_0^1{2}^{x^3}dx$, $I_3={\int }_1^2{2}^{x^2}dx$ and $I_4={\int }_1^2{2}^{x^3}dx$ then

1. A.$I_2>I_1$
2. B.$I_1>I_2$
3. C.$I_3=I_4$
4. D.$I_3>I_4$

If $C$ is the mid point of $AB$ and $P$ is any point outside $AB$, then :

1. A.$\overrightarrow{PA}+\overrightarrow{PB}=2\overrightarrow{PC}$
2. B.$\overrightarrow{PA}+\overrightarrow{PB}=\overrightarrow{PC}$
3. C.$\overrightarrow{PA}+\overrightarrow{PB}=2\overrightarrow{PC}=\vec{0}$
4. D.$\overrightarrow{PA}+\overrightarrow{PB}=\overrightarrow{PC}=\vec{0}$

The resultant $R$ of two forces acting on a particle is at right angles to one of them and its magnitude is one third of the other force. The ratio of larger force to smaller one is

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

A particle is projected from a point $O$ with velocity $u$ at an angle of $60^{\circ }$ with the horizontal. When it is moving in a direction at right angles to its direction at $O$, its velocity then is given by

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

The value of ${}^{50}{C}_4+{\sum }_{r=1}^6{}^{56-r}{C}_3$ is

1. A.${}^{55}{C}_4$
2. B.${}^{55}{C}_3$
3. C.${}^{56}{C}_3$
4. D.${}^{56}{C}_4$

If the equation $a_n{x}^n+a_{n-1}{x}^{n-1}+\dots \cdots +a_{1}x=0$, $a_1\ne 0,n\ge 2$, has a positive root $x=\alpha$, then the equation $n{a}_n{x}^{n-1}+(n-1)a_{n-1}{x}^{n-2}+\dots \cdots +a_1=0$ has a positive root, which is

1. A.greater than $\alpha$
2. B.smaller than $\alpha$
3. C.greater than or equal to $\alpha$
4. D.equal to $\alpha$

Let $\vec{a}=\hat{i}-\hat{k}$, $\vec{b}=x\hat{i}+\hat{j}+(1-x)\hat{k}$ and $\vec{c}=y\hat{i}+x\hat{j}+(1+x-y)\hat{k}$. Then $\left[\vec{a},\vec{b},\vec{c}\right]$ depends on :

1. A.only $y$
2. B.only $x$
3. C.both $x$ and $y$
4. D.neither $x$ nor $y$

If $|z^2-1|=|z{|}^2+1$, then z lies on :

1. A.an ellipse
2. B.the imaginary axis
3. C.a circle
4. D.the real axis

The value of ${\int }_{-2}^3|1-x^2|dx$ is

1. A.$\frac{1}{3}$
2. B.$\frac{14}{3}$
3. C.$\frac{7}{3}$
4. D.$\frac{28}{3}$

A particle acted on by constant forces $4\hat{i}+\hat{j}-3\hat{k}$ and $3\hat{i}+\hat{j}-\hat{k}$ is displaced from the point $\hat{i}+2\hat{j}+3\hat{k}$ to the point $5\hat{i}+4\hat{j}+\hat{k}$. The total work done by the forces is :

1. A.$50$ units
2. B.$20$ units
3. C.$30$ units
4. D.$40$ units

Let $S(K)=1+3+5\dots +(2K-1)=3+K^2$. Then which of the following is true

1. A.Principle of mathematical induction can be used to prove the formula
2. B.$S(K)\Rightarrow S(K+1)$
3. C.$S(K)\ne S(K+1)$
4. D.$S(1)$ is correct

If $S_n={\sum }_{r=0}^n\frac{1}{{}^n{C}_r}$ and $t_n={\sum }_{r=0}^n\frac{r}{{}^n{C}_r}$, then $\frac{t_n}{S_n}$ is equal to

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

Which one is not periodic?

1. A.$|\sin 3x|+{\sin }^{2}x$
2. B.$\cos \sqrt{x}+{\cos }^{2}x$
3. C.$\cos 4x+{\tan }^{2}x$
4. D.$\cos 2x+\sin x$

If $\alpha \ne \beta$ but ${\alpha }^2=5\alpha -3$ and ${\beta }^2=5\beta -3$ then the equation having $\alpha /\beta$ and $\beta /\alpha$ as its roots is

1. A.$3x^2-19x+3=0$
2. B.$3x^2+19x-3=0$
3. C.$3x^2-19x-3=0$
4. D.$x^2-5x+3=0$

If $a,b,c$ are distinct $+ve$ real numbers and $a^2+b^2+c^2=1$ then $ab+bc+ca$ is

1. A.less than 1
2. B.equal to 1
3. C.greater than 1
4. D.any real no.

The positive integer just greater than $(1+0.0001{)}^{10000}$ is

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

The sum of integers from 1 to 100 that are divisible by 2 or 5 is :

1. A.3000
2. B.3050
3. C.3600
4. D.3250

If $y={\left(x+\sqrt{1+x^2}\right)}^n$, then $(1+x^2)\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}$ is

1. A.$n^{2}y$
2. B.$-n^{2}y$
3. C.$-y$
4. D.$2x^{2}y$

${\int }_0^2[x^2]dx$ is

1. A.$2-\sqrt{2}$
2. B.$2+\sqrt{2}$
3. C.$\sqrt{2}-1$
4. D.$-\sqrt{2}-\sqrt{3}+5$

In a class of 100 students there are 70 boys whose average marks in a subject are 75. If the average marks of the complete class is 72, then what is the average marks of the girls?

1. A.73
2. B.65
3. C.68
4. D.74