The function $f : \mathbb{R} \to \mathbb{R}$ defined by $$ f(x) = \lim_{n \to \infty} \frac{\cos(2 \pi x) - x^{2n} \sin(x-1)}{1 + x^{2n+1} - x^{2n}} $$ is continuous for all $x$ in :

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$ \text{Let } f(x)=ax^{2}+bx+c \text{ be such that } f(1)=3,\ f(-2)=\lambda \text{ and } f(3)=4. $ $ \text{If } f(0)+f(…

Topic: JEE Main 2022 (28 July Evening Shift)

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$ \text{Let } A \text{ and } B \text{ be any two } 3\times 3 \text{ symmetric and skew-symmetric matrices respectively.…

Topic: JEE Main 2022 (28 July Evening Shift)

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$S = \{\, x \in [-6,3] \setminus \{-2,2\} \;:\; \dfrac{|x+3|-1}{|x|-2} \geq 0 \,\}$ $T = \{\, x \in \mathbb{Z} \;:\;…

Topic: JEE Main 2022 (28 July Evening Shift)

4

$ \text{Let } \alpha, \beta \text{ be the roots of the equation } x^{2} - \sqrt{2}x + \sqrt{6} = 0 \text{ and } \dfrac{…

Topic: JEE Main 2022 (28 July Evening Shift)

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The function $f(x) = x e^{\,x(1-x)}, \, x \in \mathbb{R}$ is :

Topic: JEE Main 2022 (28 July Evening Shift)

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The sum of the absolute maximum and absolute minimum values of the function $$f(x) = \tan^{-1}(\sin x - \cos x)$$ i…

Topic: JEE Main 2022 (28 July Evening Shift)

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Let $x(t) = 2\sqrt{2}\cos t \sqrt{\sin 2t}$ and $y(t) = 2\sqrt{2}\sin t \sqrt{\sin 2t}, \; t \in (0,\tfrac{\pi}{2…

Topic: JEE Main 2022 (28 July Evening Shift)

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Let $I_n(x) = \int_0^x \dfrac{1}{(t^2+5)^n} \, dt, \; n = 1, 2, 3, \dots$ Then :

Topic: JEE Main 2022 (28 July Evening Shift)

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$ \text{The area enclosed by the curves } y=\log_{e}(x+e^{2}),; x=\log_{e}!\left(\dfrac{2}{y}\right) \text{ and } x=\lo…

Topic: JEE Main 2022 (28 July Evening Shift)

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$ \text{Let } y=y(x) \text{ be the solution curve of the differential equation } \dfrac{dy}{dx}+\dfrac{1}{x^{2}-1},y=\l…

Topic: JEE Main 2022 (28 July Evening Shift)

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Let the hyperbola $H : \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ pass through the point $(2\sqrt{2}, -2\sqrt{2})$. A pa…

Topic: JEE Main 2022 (28 July Evening Shift)

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$ \text{Let } S \text{ be the set of all } a \in \mathbb{R} \text{ for which the angle between the vectors } \vec{u}=a(…

Topic: JEE Main 2022 (28 July Evening Shift)

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$ \text{Let A and B be two events such that } P(B|A)=\frac{2}{5}, P(A|B)=\frac{1}{7},; \text{and } P(A\cap B)=\frac{1}{…

Topic: JEE Main 2022 (28 July Evening Shift)