JEE MAIN 2024 Previous Year Questions (PYQs) – Page 3 of 39

Let $\vec a=\hat i+\hat j+\hat k,;\vec b=2\hat i+4\hat j-5\hat k$ and $\vec c=x\hat i+2\hat j+3\hat k,;x\in\mathbb R$. If $\vec d$ is the unit vector in the direction of $(\vec b+\vec c)$ such that $\vec a\cdot\vec d=1$, then $(\vec a\times\vec b)\cdot\vec c$ equals:

Let $A=\left[\begin{array}{ll}1 & 2 \\ 0 & 1\end{array}\right]$ and $B=I+\operatorname{adj}(A)+(\operatorname{adj} A)^2+\ldots+(\operatorname{adj} A)^{10}$. Then, the sum of all the elements of the matrix

Let a rectangle $ABCD$ of sides $2$ and $4$ be inscribed in another rectangle $PQRS$ such that the vertices of $ABCD$ lie on the sides of $PQRS$. Let $a$ and $b$ be the sides of $PQRS$ when its area is maximum. Then $(a+b)^2$ is equal to:

Let $A={1,3,7,9,11}$ and $B={2,4,5,7,8,10,12}$. The total number of one–one maps $f:A\to B$ such that $f(1)+f(3)=14$ is:

Two straight lines through the origin $O$ intersect the line $3x+4y=12$ at points $P$ and $Q$ such that $\triangle OPQ$ is isosceles and $\angle POQ=90^\circ$. If $I=OP^2+PQ^2+QO^2$, then the greatest integer $\le I$ is:

If the line $\dfrac{2-x}{3}=\dfrac{3y-2}{4\lambda+1}=4-z$ makes a right angle with the line $\dfrac{x+3}{3\mu}=\dfrac{1-2y}{6}=\dfrac{5-z}{7}$, then $4\lambda+9\mu$ equals:

Consider the following two statements: Statement I: For any two non-zero complex numbers $z_1,z_2$, $(|z_1|+|z_2|)\left|\dfrac{z_1}{|z_1|}+\dfrac{z_2}{|z_2|}\right|\le 2(|z_1|+|z_2|)$. Statement II: If $x,y,z$ are three distinct complex numbers and $a,b,c$ are positive real numbers such that $\dfrac{a}{|,y-z,|}=\dfrac{b}{|,z-x,|}=\dfrac{c}{|,x-y,|}$, then $\dfrac{a^{2}}{,y-z,}+\dfrac{b^{2}}{,z-x,}+\dfrac{c^{2}}{,x-y,}=1$. Between the above two statements:

Let $A$ and $B$ be two square matrices of order $3$ such that $|A|=3$ and $|B|=2$. Then $\left|,A^{T}A,( \operatorname{adj}(2A))^{-1},(\operatorname{adj}(4B)),(\operatorname{adj}(AB))^{-1},A A^{T}\right|$ is equal to:

Let a circle $C$ of radius $1$ and closer to the origin be such that the lines passing through the point $(3,2)$ and parallel to the coordinate axes touch it. Then the shortest distance of the circle $C$ from the point $(5,5)$ is:

Let $f(x)=x^{5}+2x^{3}+3x+1,; x\in\mathbb{R}$, and let $g(x)$ be a function such that $g(f(x))=x$ for all $x\in\mathbb{R}$. Then $\dfrac{g'(7)}{g'(7)}$ is equal to: