Let the domains of the functions $f(x)=\log_{4}\big(\log_{3}\big(\log_{7}\big(8-\log_{2}(x^{2}+4x+5)\big)\big)\big)$ and $g(x)=\sin^{-1}\left(\dfrac{7x+10}{x-2}\right)$ be $(\alpha,\beta)$ and $[\gamma,\delta]$, respectively. Then $\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2}$ is equal to

1

Let $A={-3,-2,-1,0,1,2,3}$ and $R$ be a relation on $A$ defined by $xRy$ iff $2x-y\in{0,1}$. Let $l$ be the number of e…

Topic: JEE Main 2025 (4 April Evening Shift)

1

Let the mean and the standard deviation of the observation $2,3,3,3,4,5,7,a,b$ be $4$ and $\sqrt{2}$ respectively. Then…

Topic: JEE Main 2025 (4 April Evening Shift)

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Let the sum of the focal distances of the point $P(4,3)$ on the hyperbola $H:\ \dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}…

Topic: JEE Main 2025 (4 April Evening Shift)

3

Let the matrix $A=\left[\begin{array}{lll}1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0\end{array}\right]$ satisfy $A^n=A^{n-2}+A…

Topic: JEE Main 2025 (4 April Evening Shift)

4

Let $a>0$. If the function $f(x)=6x^3-45ax^2+108a^2x+1$ attains its local maximum and minimum values at the points $x_1…

Topic: JEE Main 2025 (4 April Evening Shift)

5

If a curve $y=y(x)$ passes through the point $\left(1,\dfrac{\pi}{2}\right)$ and satisfies the differential equation $(…

Topic: JEE Main 2025 (4 April Evening Shift)

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If the sum of the first $20$ terms of the series $\dfrac{4\cdot1}{4+3\cdot1^{2}+1^{4}}+\dfrac{4\cdot2}{4+3\cdot2^{2}+2^…

Topic: JEE Main 2025 (4 April Evening Shift)

7

Let for two distinct values of $p$ the lines $y=x+p$ touch the ellipse $E:\ \dfrac{x^{2}}{4^{2}}+\dfrac{y^{2}}{3^{2}}=1…

Topic: JEE Main 2025 (4 April Evening Shift)

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The centre of a circle $C$ is at the centre of the ellipse $E:\ \dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1,\ a>b$. Let…

Topic: JEE Main 2025 (4 April Evening Shift)

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A line passing through the point $A(-2,0)$ touches the parabola $P: y^2=x-2$ at the point $B$ in the first quadrant. Th…

Topic: JEE Main 2025 (4 April Evening Shift)

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Let the product of $\omega_1=(8+i)\sin\theta+(7+4i)\cos\theta$ and $\omega_2=(1+8i)\sin\theta+(4+7i)\cos\theta$ be $\al…

Topic: JEE Main 2025 (4 April Evening Shift)