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

1

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. …

Topic: JEE Main 2023 (29 January Morning Shift)

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Three rotten apples are mixed accidentally with seven good apples and four apples are drawn one by one without replacem…

Topic: JEE Main 2023 (29 January Morning Shift)

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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…

Topic: JEE Main 2023 (29 January Morning Shift)

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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 player…

Topic: JEE Main 2023 (29 January Morning Shift)

5

Let the tangents at the points $A(4,-11)$ and $B(8,-5)$ on the circle $x^{2}+y^{2}-3x+10y-15=0$, intersect at the point…

Topic: JEE Main 2023 (29 January Morning Shift)

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Let $f(x)=x+\dfrac{a}{\pi^{2}-4}\sin x+\dfrac{b}{\pi^{2}-4}\cos x,\ x\in\mathbb{R}$ be a function which satisfies $\d…

Topic: JEE Main 2023 (29 January Morning Shift)

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Let $\alpha$ and $\beta$ be real numbers. Consider a $3\times 3$ matrix $A$ such that $A^{2}=3A+\alpha I$. If $A^{4}=…

Topic: JEE Main 2023 (29 January Morning Shift)

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A light ray emits from the origin making an angle $30^\circ$ with the positive $x$-axis. After getting reflected by t…

Topic: JEE Main 2023 (29 January Morning Shift)

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For two non-zero complex numbers $z_{1}$ and $z_{2}$, if $\operatorname{Re}(z_{1}z_{2})=0$ and $\operatorname{Re}(z_{1}…

Topic: JEE Main 2023 (29 January Morning Shift)

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Let $\Delta$ be the area of the region $\{(x,y)\in\mathbb{R}^{2}:\ x^{2}+y^{2}\le 21,\ y^{2}\le 4x,\ x\ge 1\}$. Then …

Topic: JEE Main 2023 (29 January Morning Shift)