Two sets $A$ and $B$ are as under : $A = {(a,b) \in \mathbb{R} \times \mathbb{R} : |a-5| < 1 \text{ and } |b-5| < 1}$ $B = {(a,b) \in \mathbb{R} \times \mathbb{R} : 4(a-6)^{2} + 9(b-5)^{2} \le 36}$ Then

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If the system of linear equations$x + ky + 3z = 0$$3x + ky - 2z = 0$$2x + 4y - 3z = 0$has a non-zero solution $(x, y, z…

Topic: JEE Main 2018 (Offline)

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If $\alpha, \beta \in \mathbb{C}$ are the distinct roots of the equation $x^{2} - x + 1 = 0$, then $\alpha^{101} + \bet…

Topic: JEE Main 2018 (Offline)

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Let $S = {x \in \mathbb{R} : x \ge 0 \text{ and } 2\lvert\sqrt{x}-3\rvert + \sqrt{x}(\sqrt{x}-6)+6=0}$. Then $S$ :

Topic: JEE Main 2018 (Offline)

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From $6$ different novels and $3$ different dictionaries, $4$ novels and $1$ dictionary are to be selected and arranged…

Topic: JEE Main 2018 (Offline)

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If $\left|\begin{bmatrix} x-4 & 2x & 2x \\ 2x & x-4 & 2x \\ 2x & 2x & x-4 \end{bmatrix}\right| = (A+Bx)(x-A)^{2}$ then …

Topic: JEE Main 2018 (Offline)

2

Let $a_1, a_2, a_3, \ldots, a_{49}$ be in A.P. such that $\displaystyle \sum_{k=0}^{12} a_{4k+1} = 416$ and $a_9 + a_{4…

Topic: JEE Main 2018 (Offline)

3

A bag contains $4$ red and $6$ black balls. A ball is drawn at random from the bag, its colour is observed and this bal…

Topic: JEE Main 2018 (Offline)

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If $\displaystyle \sum_{i=1}^{9}(x_i-5)=9$ and $\displaystyle \sum_{i=1}^{9}(x_i-5)^{2}=45$, then the standard deviatio…

Topic: JEE Main 2018 (Offline)

5

Let $\vec{u}$ be a vector coplanar with the vectors $\vec{a}=2\hat{i}+3\hat{j}-\hat{k}$ and $\vec{b}=\hat{j}+\hat{k}$. …

Topic: JEE Main 2018 (Offline)

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A straight line through a fixed point $(2,3)$ intersects the coordinate axes at distinct points $P$ and $Q$. If $O$ is …

Topic: JEE Main 2018 (Offline)

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Let $y = y(x)$ be the solution of the differential equation $\sin x \dfrac{dy}{dx} + y \cos x = 4x,\ x \in (0,\pi).$ If…

Topic: JEE Main 2018 (Offline)

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The integral $\displaystyle \int \frac{\sin^{2}x \cos^{2}x}{\left(\sin^{5}x + \cos^{3}x \sin^{2}x + \sin^{3}x \cos^{2}x…

Topic: JEE Main 2018 (Offline)

9

The value of $\displaystyle \int_{-\pi/2}^{\pi/2} \frac{\sin^{2}x}{1+2^{x}},dx$ is :

Topic: JEE Main 2018 (Offline)

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Let $g(x)=\cos x^{2}$, $f(x)=\sqrt{x}$ and $\alpha,\beta\ (\alpha

Topic: JEE Main 2018 (Offline)