JEE Main Mathematics — Algebra previous year questions with solutions.
Let $S={1, 2, 3,\ldots .,100}$, then number of non-empty subsets $A$ of $S$ such that the product of elements in $A$ is even is :
The Number of ways of choosing $10$ objects out of $31$ objects of which $10$ are identical and the remaining $21$ are distinct, is:
If the fractional part of the number $\frac{{2}^{403}}{15}$ is $\frac{k}{15},$ then $k$ is equal to
A committee of $11$ member is to be formed from $8$ males and $5$ females. If $m$ is the number of ways the committee is formed with at least $6$ males and $n$ is the number of ways the committee is formed with at least $3$ females, then:
The domain of the definition of the function $f(x)=\frac{1}{4-{x}^{2}}+{log}_{10}({x}^{3}-x)$ is:
Let ${a}_{1},{a}_{2},{a}_{3}...$ be an $A.P.$ with ${a}_{6}=2.$ Then, the common difference of this $A.P.,$ which maximise the product ${a}_{1}\cdot {a}_{4}\cdot {a}_{5},$is :
If $A=[\begin{matrix}1 & \mathrm{sin}\theta & 1 \\ -\mathrm{sin}\theta & 1 & \mathrm{sin}\theta \\ -1 & -\mathrm{sin}\theta & 1\end{matrix}]$, then for all $\theta \in (\frac{3\pi }{4},\frac{5\pi }{4})$, $\mathrm{det}(A)$ lies in the interval :
Let $\left(-2-\frac{1}{3} i\right)^{3}=\frac{x+i y}{27}(i=\sqrt{-1}),$ where $x$ and $y$ are real numbers then $\mathrm{y}-\mathrm{x}$ equals
If the function $f:R-{1, -1}\rightarrow A$ defined by $f(x)=\frac{{x}^{2}}{1-{x}^{2}},$ is surjective, then $A$ is equal to
If the sum of the first $15$ terms of the series ${(\frac{3}{4})}^{3}+{(1\frac{1}{2})}^{3}+{(2\frac{1}{4})}^{3}+{3}^{3}+{(3\frac{3}{4})}_{ }^{3}+\ldots$ is equal to $225K$, then $K$ is equal to :
All possible numbers are formed using the digits $1, 1, 2, 2, 2, 2, 3, 4, 4$ taken all at a time. The number of such numbers in which the odd digits occupy even places is
If both the roots of the quadratic equation ${x}^{2}-mx+4=0$ are real and distinct and they lie in the interval $(1, 5),$ then $m$ lies in the interval: Note: In the actual JEE paper interval was $[1, 5]$
A group of students comprises of $5$ boys and n girls. If the number of ways, in which a team of $3$ students can randomly be selected from this group such that there is at least one boy and at least one girl in each team, is $1750,$ then n is equal to
If ${\Delta }_{1}=|\begin{matrix}x & sin\theta & cos\theta \\ -sin\theta & -x & 1 \\ cos\theta & 1 & x\end{matrix}|$ and ${\Delta }_{2}=|\begin{matrix}x & sin2\theta & cos2\theta \\ -sin2\theta & -x & 1 \\ cos2\theta & 1 & x\end{matrix}|,$ $x\neq 0;$ then for all $\theta \in (0,\frac{\pi }{2})$ :
A ratio of the ${5}^{th}$ term from the beginning to the ${5}^{th}$ term from the end in the binomial expansion of ${({2}^{\frac{1}{3}}+\frac{1}{2{(3)}^{\frac{1}{3}}})}^{10}$ is
The smallest natural number $n$ , such that the coefficient of $x$ in the expansion of ${({x}^{2}+\frac{1}{{x}^{3}})}^{n}$ is $C23 n$ , is
Let $A, B$ and $C$ be sets such that $\phi \neq A\cap B\subseteq C.$ Then which of the following statements is not true?
Consider the quadratic equation $(c-5){x}^{2}-2cx+(c-4)=0,c\neq 5.$ Let $S$ be the set of all integral values of $c$ for which one root of the equation lies in the interval $(0, 2)$ and its other root lies in the interval $(2, 3).$ Then the number of elements in $S$ is
Let the sum of the first $n$ terms of a non-constant $A.P.,{a}_{1},{a}_{2},{a}_{3},....,{a}_{n}$ be $50n+\frac{n(n-7)}{2}A,$ where $A$ is a constant. If $d$ is the common difference of this $A.P.$, then the ordered pair $(d,{a}_{50})$ is equal to
Let $A$ and $B$ be two invertible matrices of order $3 \times 3$. If $\operatorname{det}\left(\mathrm{ABA}^{\mathrm{T}}\right)=8$ and det $\left(\mathrm{AB}^{-1}\right)=8$, then det $\left(\mathrm{BA}^{-1} \mathrm{~B}^{\mathrm{T}}\right)$ is equal to
If ${a}_{1},{a}_{2},{a}_{3}.........,{a}_{n}$ are in $A.P.$ and ${a}_{1}+{a}_{4}+{a}_{7}.........+{a}_{16}=114$ , then ${a}_{1}+{a}_{6}+{a}_{11}+{a}_{16}$ is equal to :
The sum of the co-efficient of all even degree terms in $x$ in the expansion of ${(x+\sqrt{{x}^{3}-1})}^{6}{+(x-\sqrt{{x}^{3}-1})}^{6},(x>1)$ is equal to
Let $z\in C$ with $Im(z)=10$ and it satisfies $\frac{2 z-n}{2 z+n}=2i-1$ for some natural number $n.$ Then
If $|\begin{matrix}x-4 & 2x & 2x \\ 2x & x-4 & 2x \\ 2x & 2x & x-4\end{matrix}|=(A+Bx) {(x-A)}^{2},$ then the ordered pair $(A,B)$ is equal to