Other Maths · JEE Main previous year questions

A square, of each side 2, lies above the x-axis and has one vertex at the origin. If one of the sides passing through the origin makes an angle $30^{\circ }$ with the positive direction of the x-axis, then the sum of the x-coordinates of the vertices of the square is :

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

Let $$S_n=\frac{1}{1^3}+\frac{1+2}{1^3+2^3}+\frac{1+2+3}{1^3+2^3+3^3}+\dots \dots \cdots +\frac{1+2+\dots \cdots +n}{1^3+2^3+\dots \cdots +n^3}\cdot$$ If $100{\mathrm{S}}_n=n$, then n is equal to :

1. A.$199$
2. B.$99$
3. C.$200$
4. D.$19$

If $(27{)}^{999}$ is divided by $7$, then the remainder is :

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

The sum of all real values of $x$ satisfying the equation $(x^2-5x+5{)}^{x^2+4x-60}=1$ is :

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

Two sides of a rhombus are along the lines, $x-y+1=0$ and $7x-y-5=0$. If its diagonals intersect at $(-1,-2)$, then which one of the following is a vertex of this rhombus?

1. A.$\left(\frac{1}{3},-\frac{8}{3}\right)$
2. B.$\left(-\frac{10}{3},-\frac{7}{3}\right)$
3. C.$(-3,-9)$
4. D.$(-3,-8)$

If $f(x)+2f\left(\frac{1}{x}\right)=3x,x\ne 0$, and $S=\{x\in \mathbf{R}:f(x)=f(-x)\}$; then $S$ :

1. A.is an empty set.
2. B.contains exactly one element.
3. C.contains exactly two elements.
4. D.contains more than two elements.

If $x=-1$ and $x=2$ are extreme points of $f(x)=\alpha \log |x|+\beta x^2+x$ then

1. A.$\alpha =2,\beta =-\frac{1}{2}$
2. B.$\alpha =2,\beta =\frac{1}{2}$
3. C.$\alpha =-6,\beta =\frac{1}{2}$
4. D.$\alpha =-6,\beta =-\frac{1}{2}$

If $\alpha ,\beta \ne 0$, and $f(n)={\alpha }^n+{\beta }^n$ and $$\left|\begin{array}{ccc}3& 1+f(1)& 1+f(2)\\ 1+f(1)& 1+f(2)& 1+f(3)\\ 1+f(2)& 1+f(3)& 1+f(4)\end{array}\right|=K(1-\alpha {)}^2(1-\beta {)}^2(\alpha -\beta {)}^2,$$ then $K$ is equal to :

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

If $(10{)}^9+2(11{)}^1(10{)}^8+3(11{)}^2(10{)}^7+\cdots +10(11{)}^9=k(10{)}^9$, then k is equal to :

1. A.100
2. B.110
3. C.$\frac{121}{10}$
4. D.$\frac{441}{100}$

If $\left[\vec{a}\times \vec{b}\vec{b}\times \vec{c}\vec{c}\times \vec{a}\right]=\lambda {\left[\vec{a}\vec{b}\vec{c}\right]}^2$ then $\lambda$ is equal to :

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

$|z_1+z_2{|}^2+|z_1-z_2{|}^2$ is equal to

1. A.$2(|z_1|+|z_2|)$
2. B.$2(|z_1{|}^2+|z_2{|}^2)$
3. C.$|z_1||z_2|$
4. D.$|z_1{|}^2+|z_2{|}^2$

If $A=\left[\begin{array}{ccc}1& 0& 0\\ 2& 1& 0\\ -3& 2& 1\end{array}\right]$ and $B=\left[\begin{array}{ccc}1& 0& 0\\ -2& 1& 0\\ 7& -2& 1\end{array}\right]$ then $AB$ equals

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

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

1. A.above $x$-axis at a distance $2/3$ from it
2. B.above $x$-axis at a distance $3/2$ from it
3. C.below $x$-axis at a distance $3/2$ from it
4. D.below $x$-axis at a distance $2/3$ from it

If $f(x)=x{e}^{x(1-x)},x\in R$, then $f(x)$ is

1. A.decreasing on $[-1/2,1]$
2. B.decreasing on $R$
3. C.increasing on $[-1/2,1]$
4. D.increasing on $R$

The coordinates of the foot perpendicular from the point $(1,0,0)$ to the line $\frac{x-1}{2}=\frac{y+1}{-3}=\frac{z+10}{8}$ are

1. A.$(2,-3,8)$
2. B.$(1,-1,-10)$
3. C.$(5,-8,-4)$
4. D.$(3,-4,-2)$

The point of intersection of the lines $(a^3+3)x+ay+a-3=0$ and $(a^5+2)x+(a+2)y+2a+3=0$ ($a$ real) lies on the $y$-axis for

1. A.no value of $a$
2. B.more than two values of $a$
3. C.exactly one value of $a$
4. D.exactly two values of $a$

The remainder left out when $8^{2n}-(62{)}^{2n+1}$ is divided by 9 is :

1. A.2
2. B.7
3. C.8
4. D.0

The lines $p\left(p^2+1\right)x-y+q=0$ and ${\left(p^2+1\right)}^{2}x+\left(p^2+1\right)y+2q=0$ are perpendicular to a common line for :

1. A.exactly one values of $p$
2. B.exactly two values of $p$
3. C.more than two values of $p$
4. D.no value of $p$

Given $P(x)=x^4+a{x}^3+b{x}^2+cx+d$ such that $x=0$ is the only real root of $P^{\prime }(x)=0$. If $P(-1)<P(1)$, then in the interval $[-1,1]$ :

1. A.$P(-1)$ is not minimum but $P(1)$ is the maximum of $P$
2. B.$P(-1)$ is the minimum but $P(1)$ is not the maximum of $P$
3. C.Neither $P(-1)$ is the minimum nor $P(1)$ is the maximum of $P$
4. D.$P(-1)$ is the minimum and $P(1)$ is the maximum of $P$

For real x, let $f(x)=x^3+5x+1$, then

1. A.f is one-one but not onto R
2. B.f is onto R but not one-one
3. C.f is one-one and onto R
4. D.f is neither one-one nor onto R

If $(a,a^2)$ falls inside the angle made by the lines $y=\frac{x}{2}$, $x>0$ and $y=3x$, $x>0$, then $a$ belong to :

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

If $x^m\cdot y^n=(x+y{)}^{m+n}$, then $\frac{dy}{dx}$ is

1. A.$\frac{y}{x}$
2. B.$\frac{x+y}{xy}$
3. C.$xy$
4. D.$\frac{x}{y}$

Let $A=\left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right)$ and $B=\left(\begin{array}{cc}a& 0\\ 0& b\end{array}\right),a,b\in N$. Then

1. A.there cannot exist any $B$ such that $AB=BA$
2. B.there exist more then one but finite number of $B$'s such that $AB=BA$
3. C.there exists exactly one $B$ such that $AB=BA$
4. D.there exist infinitely many $B$'s such that $AB=BA$

The two lines $x=ay+b,z=cy+d$; and $x=a^{\prime }y+b^{\prime },z=c^{\prime }y+d^{\prime }$ are perpendicular to each other if :

1. A.$a{a}^{\prime }+c{c}^{\prime }=-1$
2. B.$a{a}^{\prime }+c{c}^{\prime }=1$
3. C.$\frac{a}{a^{\prime }}+\frac{c}{c^{\prime }}=-1$
4. D.$\frac{a}{a^{\prime }}+\frac{c}{c^{\prime }}=1$

A particle has two velocities of equal magnitude inclined to each other at an angle $\theta$. If one of them is halved, the angle between the other and the original resultant velocity is bisected by the new resultant. Then $\theta$ is

1. A.$90^{\circ }$
2. B.$120^{\circ }$
3. C.$45^{\circ }$
4. D.$60^{\circ }$