The mean and variance of five observations are $\dfrac{24}{5}$ and $\dfrac{194}{25}$ respectively. If the mean of the first four observations is $\dfrac{7}{2}$, then the variance of the first four observations is:

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If $R$ is the smallest equivalence relation on the set ${1,2,3,4}$ such that ${(1,2),(1,3)}\subset R$, then the number …

Topic: JEE Main 2025 (29 January Evening Shift)

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The function $f(x)=\frac{x}{x^2-6 x-16}, x \in \mathbb{R}-\{-2,8\}$

Topic: JEE Main 2025 (29 January Evening Shift)

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Let a unit vector $\hat{\mathbf{u}}=x\mathbf{i}+y\mathbf{j}+z\mathbf{k}$ make angles $\dfrac{\pi}{2},\ \dfrac{\pi}{3}$ …

Topic: JEE Main 2025 (29 January Evening Shift)

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The distance of the point $(2,3)$ from the line $2x-3y+28=0$, measured parallel to the line $\sqrt{3},x-y+1=0$, is equa…

Topic: JEE Main 2025 (29 January Evening Shift)

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Let $A$ be the point of intersection of the lines $3x+2y=14$ and $5x-y=6$, and $B$ be the point of intersection of the …

Topic: JEE Main 2025 (29 January Evening Shift)

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If $\displaystyle \int \frac{\sin^{3/2}x+\cos^{3/2}x}{\sqrt{\sin^2 x,\cos^2 x},\sin(x-\theta)},dx = A\sqrt{\cos\theta,\…

Topic: JEE Main 2025 (29 January Evening Shift)

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Number of ways of arranging 8 identical books into 4 identical shelves where any number of shelves may remain empty is …

Topic: JEE Main 2025 (29 January Evening Shift)

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Let $P(3,2,3)$, $Q(4,6,2)$ and $R(7,3,2)$ be the vertices of $\triangle PQR$. The angle $\angle QPR$ is:

Topic: JEE Main 2025 (29 January Evening Shift)

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Let $y=\log_e!\left(\dfrac{1-x^2}{1+x^2}\right)$, with $-1

Topic: JEE Main 2025 (29 January Evening Shift)

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If $\log_e a,;\log_e b,;\log_e c$ are in an A.P. and $\log_e a-\log_e(2b),;\log_e(2b)-\log_e(3c),;\log_e(3c)-\log_e a$ …

Topic: JEE Main 2025 (29 January Evening Shift)

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If $\sin\left(\dfrac{y}{x}\right)=\log_e|x|+\dfrac{\alpha}{x}$ is a solution of the differential equation $x\cos\left(\…

Topic: JEE Main 2025 (29 January Evening Shift)

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An integer is chosen at random from the integers $1,2,3,\dots,50$. The probability that the chosen integer is a multipl…

Topic: JEE Main 2025 (29 January Evening Shift)

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Let $\overrightarrow{OA}=\vec a,\ \overrightarrow{OB}=12\vec a+4\vec b$ and $\overrightarrow{OC}=\vec b$, where $O$ is …

Topic: JEE Main 2025 (29 January Evening Shift)

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The function $f(x)=2x+3x^{\frac{1}{3}},; x\in\mathbb{R}$ has:

Topic: JEE Main 2025 (29 January Evening Shift)

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If each term of a geometric progression $a_1,a_2,a_3,\dots$ with $a_1=\dfrac{1}{8}$ and $a_2\neq a_1$ is the arithmetic…

Topic: JEE Main 2025 (29 January Evening Shift)

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Let $r$ and $\theta$ respectively be the modulus and amplitude of the complex number $z = 2 - i\left(2\tan\dfrac{5\pi}{…

Topic: JEE Main 2025 (29 January Evening Shift)

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The sum of the solutions $x \in \mathbb{R}$ of the equation $\dfrac{3\cos 2x + \cos^3 2x}{\cos^6 x - \sin^6 x} = x^3 - …

Topic: JEE Main 2025 (29 January Evening Shift)