Let an ellipse $E:{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$, ${a^2} > {b^2}$, passes through $\left( {\sqrt {{3 \over 2}} ,1} \right)$ and has eccentricity ${1 \over {\sqrt 3 }}$. If a circle, centered at focus F($\alpha$, 0), $\alpha$ > 0, of E and radius ${2 \over {\sqrt 3 }}$, intersects E at two points P and Q, then PQ2 is equal to :
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The number of real roots of the equation ${e^{6x}} - {e^{4x}} - 2{e^{3x}} - 12{e^{2x}} + {e^x} + 1 = 0$ is :
Topic: JEE Main 2021 (25 July Morning Shift)
Let 9 distinct balls be distributed among 4 boxes, B1, B2, B3 and B4. If the probability than B3 contains exactly 3 bal…
Topic: JEE Main 2021 (25 July Morning Shift)
The values of a and b, for which the system of equations 2x + 3y + 6z = 8, x + 2y + az = 5, 3x + 5y + 9z = b, has no so…
Topic: JEE Main 2021 (25 July Morning Shift)
Let $f:[0,\infty ) \to [0,\infty )$ be defined as $f(x) = \int_0^x {[y]dy} $ where [x] is the greatest integer less th…
Topic: JEE Main 2021 (25 July Morning Shift)
Let g : N $\to$ N be defined as g(3n + 1) = 3n + 2, g(3n + 2) = 3n + 3, g(3n + 3) = 3n + 1, for all n $\ge$ 0. Then wh…
Topic: JEE Main 2021 (25 July Morning Shift)
The area (in sq. units) of the region, given by the set $\{ (x,y) \in R \times R|x \ge 0,2{x^2} \le y \le 4 - 2x\} $ is…
Topic: JEE Main 2021 (25 July Morning Shift)
Let y = y(x) be the solution of the differential equation ${{dy} \over {dx}} = 1 + x{e^{y - x}}, - \sqrt 2 < x <…
Topic: JEE Main 2021 (25 July Morning Shift)
If b is very small as compared to the value of a, so that the cube and other higher powers of ${b \over a}$ can be negl…
Topic: JEE Main 2021 (25 July Morning Shift)
The value of the definite integral $\int\limits_{\pi /24}^{5\pi /24} {{{dx} \over {1 + \root 3 \of {\tan 2x} }}} $ is :
Topic: JEE Main 2021 (25 July Morning Shift)
Let f : R $\to$ R be defined as $f(x) = \left\{ {\matrix{ {{{\lambda \left| {{x^2} - 5x + 6} \right|} \over {\mu (5x …
Topic: JEE Main 2021 (25 July Morning Shift)
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The sum of all those terms which are rational numbers in the expansion of (21/3 + 31/4)12 is :
Topic: JEE Main 2021 (25 July Morning Shift)
The total number of functions,$f:\{1,2,3,4\} \rightarrow\{1,2,3,4,5,6\}$ such that $f(1)+f(2)=f(3)$, is equal to :
Topic: JEE Main 2021 (25 July Morning Shift)
If $\alpha, \beta, \gamma, \delta$ are the roots of the equation $x^{4}+x^{3}+x^{2}+x+1=0$, then $\alpha^{2021}+\beta^{…
Topic: JEE Main 2021 (25 July Morning Shift)
For $\mathrm{n} \in \mathbf{N}$, let $\mathrm{S}_{\mathrm{n}}=\left\{z \in \mathbf{C}:|z-3+2 i|=\frac{\mathrm{n}}{4}\ri…
Topic: JEE Main 2021 (25 July Morning Shift)
