JEE MAIN 2023 Previous Year Questions (PYQs) – Page 23 of 34
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Let $\lambda\ne 0$ be a real number. Let $\alpha,\beta$ be the roots of the equation $14x^{2}-31x+3\lambda=0$ and $\alpha,\gamma$ be the roots of the equation $35x^{2}-53x+4\lambda=0$.
Then $\dfrac{3\alpha}{\beta}$ and $\dfrac{4\alpha}{\gamma}$ are the roots of the equation
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All words, with or without meaning, are made using all the letters of the word MONDAY. These words are written as in a dictionary with serial numbers. The serial number of the word MONDAY is :
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Let $B$ and $C$ be the two points on the line $y+x=0$ such that $B$ and $C$ are symmetric with respect to the origin.
Suppose $A$ is a point on $y-2x=2$ such that $\triangle ABC$ is an equilateral triangle.
Then, the area of the $\triangle ABC$ is:
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Let $|\vec a|=2$, $|\vec b|=3$ and the angle between the vectors $\vec a$ and $\vec b$ be $\dfrac{\pi}{4}$. Then $|(\vec a+2\vec b)\times(2\vec a-3\vec b)|^2$ is equal to:
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Three rotten apples are mixed accidentally with seven good apples and four apples are drawn one by one without replacement.
Let the random variable $X$ denote the number of rotten apples. If $\mu$ and $\sigma^{2}$ represent the mean and variance of $X$, respectively, then $10(\mu^{2}+\sigma^{2})$ is equal to:
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Let $S=\{\,z\in\mathbb{C}:\ \overline{z}=i\big(z^2+\operatorname{Re}(\overline{z})\big)\,\}$. Then $\displaystyle \sum_{z\in S}|z|^2$ is equal to:
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Let $f(\theta)=3\big(\sin^{4}\!\left(\tfrac{3\pi}{2}-\theta\right)+\sin^{4}\!(3\pi+\theta)\big)-2\big(1-\sin^{2}2\theta\big)$ and
$S=\left\{\theta\in[0,\pi]:\, f'(\theta)=-\dfrac{\sqrt{3}}{2}\right\}$.
If $4\beta=\displaystyle\sum_{\theta\in S}\theta$, then $f(\beta)$ is equal to:
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If $\displaystyle \lim_{x\to0}\frac{e^{ax}-\cos(bx)-\dfrac{e^{x}-e^{-x}}{2}}{1-\cos(2x)}=17$, then $5a^2+b^2$ is equal to:
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Fifteen football players of a club-team are given 15 T-shirts with their names written on the backside.
If the players pick up the T-shirts randomly, then the probability that at least 3 players pick the correct T-shirt is:
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The line that is coplanar to the line $\dfrac{x+3}{-3}=\dfrac{y-1}{1}=\dfrac{z-5}{5}$ is:
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