The parabolas: $a x^{2}+2 b x+c y=0$ and $d x^{2}+2 e x+f y=0$ intersect on the line $y=1$. If $a,b,c,d,e,f$ are positive real numbers and $a,b,c$ are in G.P., then:

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If the functions $f(x)=\dfrac{x^{3}}{3}+2bx+\dfrac{a x^{2}}{2}$ and $g(x)=\dfrac{x^{3}}{3}+a x+b x^{2},\ a\ne 2b$ have …

Topic: JEE Main 2023 (30 January Evening Shift)

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Let $q$ be the maximum integral value of $p$ in $[0,10]$ for which the roots of the equation $x^{2}-px+\dfrac{5}{4}p=0$…

Topic: JEE Main 2023 (30 January Evening Shift)

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Let $f,g,h$ be the real valued functions defined on $\mathbb{R}$ as \[ f(x)= \begin{cases} \dfrac{x}{|x|}, & x\neq 0,\\…

Topic: JEE Main 2023 (30 January Evening Shift)

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Let $a,b,c>1$, $a^{3},b^{3}$ and $c^{3}$ be in A.P., and $\log_{a} b,\ \log_{c} a$ and $\log_{b} c$ be in G.P. If the s…

Topic: JEE Main 2023 (30 January Evening Shift)

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Let $S$ be the set of all values of $a_1$ for which the mean deviation about the mean of $100$ consecutive positive int…

Topic: JEE Main 2023 (30 January Evening Shift)

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The number of ways of selecting two numbers $a$ and $b$, $a\in\{2,4,6,\ldots,100\}$ and $b\in\{1,3,5,\ldots,99\}$ such …

Topic: JEE Main 2023 (30 January Evening Shift)

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For $\alpha,\beta\in\mathbb{R}$, suppose the system of linear equations $\begin{aligned} x-y+z&=5,\\ 2x+2y+\alpha z&=…

Topic: JEE Main 2023 (30 January Evening Shift)

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Let $a_1=1,\,a_2,\,a_3,\,a_4,\ldots$ be consecutive natural numbers. Then $\tan^{-1}\!\left(\dfrac{1}{1+a_1a_2}\right…

Topic: JEE Main 2023 (30 January Evening Shift)

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Let $\vec a$ and $\vec b$ be two vectors. Let $|\vec a|=1$, $|\vec b|=4$ and $\vec a\cdot\vec b=2$. If $\vec c=(2\,\vec…

Topic: JEE Main 2023 (30 January Evening Shift)

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The range of the function $f(x)=\sqrt{\,3-x\,}+\sqrt{\,2+x\,}$ is:

Topic: JEE Main 2023 (30 January Evening Shift)

3

The solution of the differential equation $\dfrac{dy}{dx}=-\left(\dfrac{x^{2}+3y^{2}}{3x^{2}+y^{2}}\right),\ y(1)=0$ is:

Topic: JEE Main 2023 (30 January Evening Shift)

4

Let $x=(8\sqrt{3}+13)^{13}$ and $y=(7\sqrt{2}+9)^{9}$. If $[t]$ denotes the greatest integer $\le t$, then:

Topic: JEE Main 2023 (30 January Evening Shift)