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:
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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…
Topic: JEE Main 2024 (5 April Morning Shift)
Two straight lines through the origin $O$ intersect the line $3x+4y=12$ at points $P$ and $Q$ such that $\triangle OPQ$…
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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:
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Let a rectangle $ABCD$ of sides $2$ and $4$ be inscribed in another rectangle $PQRS$ such that the vertices of $ABCD$ l…
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Let $A$ and $B$ be two square matrices of order $3$ such that $|A|=3$ and $|B|=2$. Then
$\left|,A^{T}A,( \operatorname{…
Topic: JEE Main 2024 (5 April Morning Shift)
Let a circle $C$ of radius $1$ and closer to the origin be such that the lines passing through the point $(3,2)$ and pa…
Topic: JEE Main 2024 (5 April Morning Shift)
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{…
Topic: JEE Main 2024 (5 April Morning Shift)
The coefficients $a,b,c$ in the quadratic $ax^{2}+bx+c=0$ are chosen from the set ${1,2,3,4,5,6,7,8}$. The probability …
Topic: JEE Main 2024 (5 April Morning Shift)
If the system
$11x+y+\lambda z=-5,\quad 2x+3y+5z=3,\quad 8x-19y-39z=\mu$
has infinitely many solutions, then $\lambda^{…
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The integral $\displaystyle \int_{0}^{\pi/4}\frac{136\sin x}{3\sin x+5\cos x},dx$ is equal to:
Topic: JEE Main 2024 (5 April Morning Shift)
If $\dfrac{1}{\sqrt{1}+\sqrt{2}}+\dfrac{1}{\sqrt{2}+\sqrt{3}}+\cdots+\dfrac{1}{\sqrt{99}+\sqrt{100}}=m$ and $\dfrac{1}{…
Topic: JEE Main 2024 (5 April Morning Shift)
Let $d$ be the distance of the point of intersection of the lines $\dfrac{x+6}{3}=\dfrac{y}{2}=\dfrac{z+1}{1}$ and $\df…
Topic: JEE Main 2024 (5 April Morning Shift)
If the function $f(x)=\dfrac{\sin 3x+\alpha\sin x-\beta\cos 3x}{x^{3}},; x\in\mathbb{R},$ is continuous at $x=0$, then …
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Let the line $2x+3y-k=0,\ k>0$ intersect the $x$-axis and $y$-axis at points $A$ and $B$, respectively. If the circle h…
Topic: JEE Main 2024 (5 April Morning Shift)