Other Maths · JEE Main previous year questions

If $\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 2\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 3\\ 0& 1\end{array}\right]\cdots \left[\begin{array}{cc}1& n-1\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}1& 78\\ 0& 1\end{array}\right]$, then the inverse of $\left[\begin{array}{cc}1& n\\ 0& 1\end{array}\right]$ is

1. A.$\left[\begin{array}{cc}1& 0\\ 12& 1\end{array}\right]$
2. B.$\left[\begin{array}{cc}1& 0\\ 13& 1\end{array}\right]$
3. C.$\left[\begin{array}{cc}1& -13\\ 0& 1\end{array}\right]$
4. D.$\left[\begin{array}{cc}1& -12\\ 0& 1\end{array}\right]$

Let $S=\{\theta \in [-2\pi ,2\pi ]:2{\cos }^2\theta +3\sin \theta =0\}$. Then the sum of the elements of S is

1. A.$\pi$
2. B.$2\pi$
3. C.$\frac{13\pi }{6}$
4. D.$\frac{5\pi }{3}$

If the two lines $x+(a-1)y=1$ and $2x+a^{2}y=1$ $(a\in \mathbb{R}-\{0,1\})$ are perpendicular, then the distance of their point of intersection from the origin is :

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

The vertices B and C of a $\Delta \mathrm{ABC}$ lie on the line, $\frac{x+2}{3}=\frac{y-1}{0}=\frac{z}{4}$ such that $\mathrm{BC}=5\mathrm{units}$. Then the area (in sq. units) of this triangle, given that the point $\mathrm{A}(1,-1,2)$, is :

1. A.6
2. B.$5\sqrt{17}$
3. C.$\sqrt{34}$
4. D.$2\sqrt{34}$

Two newspapers A and B are published in a city. It is known that 25% of the city populations reads A and 20% reads B while 8% reads both A and B. Further, 30% of those who read A but not B look into advertisements and 40% of those who read B but not A also look into advertisements, while 50% of those who read both A and B look into advertisements. Then the percentage of the population who look into advertisement is :-

1. A.13.5
2. B.13
3. C.12.8
4. D.13.9

Two poles standing on a horizontal ground are of heights 5m and 10 m respectively. The line joining their tops makes an angle of $15^{\circ }$ with ground. Then the distance (in m) between the poles, is :-

1. A.$5\left(2+\sqrt{3}\right)$
2. B.$\frac{5}{2}\left(2+\sqrt{3}\right)$
3. C.$10\left(\sqrt{3}-1\right)$
4. D.$5\left(\sqrt{3}+1\right)$

If a point R(4, y, z) lies on the line segment joining the points P(2, –3, 4) and Q(8, 0, 10), then the distance of R from the origin is :

1. A.$2\sqrt{14}$
2. B.$\sqrt{53}$
3. C.$2\sqrt{21}$
4. D.6

If $z=\frac{\sqrt{3}}{2}+\frac{i}{2}\left(i=\sqrt{-1}\right)$, then $(1+iz+z^5+i{z}^8{)}^9$ is equal to :

1. A.1
2. B.−1
3. C.0
4. D.$(-1+2i{)}^9$

Suppose that the points (h,k), (1,2) and (−3,4) lie on the line $L_1$. If a line $L_2$ passing through the points (h,k) and (4,3) is perpendicular to $L_1$, then $\frac{k}{h}$ equals :

1. A.$\frac{1}{3}$
2. B.3
3. C.0
4. D.$-\frac{1}{7}$

The sum $\sum _{k=1}^{20}k\frac{1}{2^k}$ is equal to

1. A.$2-\frac{11}{2^{19}}$
2. B.$2-\frac{3}{2^{17}}$
3. C.$1-\frac{11}{2^{20}}$
4. D.$2-\frac{21}{2^{20}}$

If $\frac{z-\alpha }{z+\alpha }$ ($\alpha \in \mathbb{R}$) is a purely imaginary number and $|z|=2$, then a value of $\alpha$ is :

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

For $x>1$, if $(2x{)}^{2y}=4e^{2x-2y}$, then $(1+{\log }_{e}2x{)}^2\frac{dy}{dx}$ is equal to :

1. A.$\frac{x{\log }_{e}2x-{\log }_{e}2}{x}$
2. B.${\log }_{e}2x$
3. C.$x{\log }_{e}2x$
4. D.$\frac{x{\log }_{e}2x+{\log }_{e}2}{x}$

Let $S_k=\frac{1+2+3+\dots +k}{k}$. If $S_1^2+S_2^2+\dots +S_{10}^2=\frac{5}{12}A$, then A is equal to :

1. A.283
2. B.156
3. C.301
4. D.303

Let $f(x)=\frac{x}{\sqrt{a^2+x^2}}-\frac{d-x}{\sqrt{b^2+(d-x{)}^2}}$, $x\in \mathbb{R}$, where a, b and d are non-zero real constants. Then :

1. A.f is an increasing function of x
2. B.f is neither increasing nor decreasing function of x
3. C.f ' is not a continuous function of x
4. D.f is a decreasing function of x

If in a parallelogram ABDC, the coordinates of A, B and C are respectively (1, 2), (3, 4) and (2, 5), then the equation of the diagonal AD is :

1. A.5x + 3y – 11 = 0
2. B.5x – 3y + 1 = 0
3. C.3x – 5y + 7 = 0
4. D.3x + 5y – 13 = 0

Let $S_n=1+q+q^2+\dots +q^n$ and $T_n=1+\left(\frac{q+1}{2}\right)+{\left(\frac{q+1}{2}\right)}^2+\dots +{\left(\frac{q+1}{2}\right)}^n$ where q is a real number and $q\ne 1$. If ${}^{101}{\mathrm{C}}_1+{}^{101}{\mathrm{C}}_2\cdot S_1+\dots +{}^{101}{\mathrm{C}}_{101}\cdot S_{100}=\alpha T_{100}$ then $\alpha$ is equal to

1. A.202
2. B.200
3. C.$2^{100}$
4. D.$2^{99}$

Consider a class of 5 girls and 7 boys. The number of different teams consisting of 2 girls and 3 boys that can be formed from this class, if there are two specific boys A and B, who refuse to be the members of the same team, is :

1. A.500
2. B.350
3. C.200
4. D.300

If the fractional part of the number $\left\{\frac{2^{403}}{15}\right\}$ is $\frac{k}{15}$, then k is equal to :

1. A.8
2. B.14
3. C.6
4. D.1

Let $A=\left\{\theta \in \left(-\frac{\pi }{2},\pi \right):\frac{3+2i\sin \theta }{1-2i\sin \theta }\text{ is purely imaginary}\right\}$. Then the sum of the elements in A is :

1. A.$\frac{5\pi }{6}$
2. B.$\pi$
3. C.$\frac{3\pi }{4}$
4. D.$\frac{2\pi }{3}$

The least positive integer n for which ${\left(\frac{1+i\sqrt{3}}{1-i\sqrt{3}}\right)}^n=1$, is :

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

Let $\text{A}=\left[\begin{array}{ccc}1& 0& 0\\ 1& 1& 0\\ 1& 1& 1\end{array}\right]$ and $\text{B}={\text{A}}^{20}$. Then the sum of the elements of the first column of B is :

1. A.210
2. B.211
3. C.231
4. D.251

If $x=\sqrt{2^{{\text{cosec}}^{-1}t}}$ and $y=\sqrt{2^{{\sec }^{-1}t}}$ $(|t|\ge 1)$, then $\frac{dy}{dx}$ is equal to :

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

If $\left|\begin{array}{ccc}x-4& 2x& 2x\\ 2x& x-4& 2x\\ 2x& 2x& x-4\end{array}\right|=(A+Bx)(x-A{)}^2$ then the ordered pair (A, B) is equal to :

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

Let $\mathrm{S}=\{x\in \mathrm{R}:x\ge 0$ and $2|\sqrt{x}-3|+\sqrt{x}(\sqrt{x}-6)+6=0\}$. Then S

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

If ${\int }_1^2\frac{dx}{(x^2-2x+4{)}^{\frac{3}{2}}}=\frac{k}{k+5}$, then k is equal to :

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