Consider ellipses $\mathbf{E_k} : kx^2 + k^2y^2 = 1, \; k = 1, 2, \ldots, 20$. Let $\mathbf{C_k}$ be the circle which touches the four chords joining the end points (one on minor axis and another on major axis) of the ellipse $\mathbf{E_k}$. If $r_k$ is the radius of the circle $\mathbf{C_k}$, then the value of \[ \sum_{k=1}^{20} \dfrac{1}{r_k^2} \] is:
Previous 10 Questions — JEE Main 2023 (11 April Morning Shift)
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Let $f(x) = \lfloor x^2 - x \rfloor + | -x + \lfloor x \rfloor |$, where $x \in \mathbb{R}$
and $\lfloor t \rfloor$ d…
Topic: JEE Main 2023 (11 April Morning Shift)
Let $\mathbf{A}$ be a $2 \times 2$ matrix with real entries such that
$\mathbf{A}' = \alpha \mathbf{A} + \mathbf{I}$,…
Topic: JEE Main 2023 (11 April Morning Shift)
For any vector $\vec{a} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}$, with $10|a_i| < 1, \; i = 1, 2, 3$,
consider the …
Topic: JEE Main 2023 (11 April Morning Shift)
Let sets A and B have 5 elements each. Let the mean of the elements in sets A and B be 5 and 8 respectively and the var…
Topic: JEE Main 2023 (11 April Morning Shift)
The value of the integral
\[
\int_{-\log_e 2}^{\log_e 2} e^x \left( \log_e\!\left( e^x + \sqrt{1 + e^{2x}} \right) \r…
Topic: JEE Main 2023 (11 April Morning Shift)
Let $y = y(x)$ be a solution curve of the differential equation
\[
(1 - x^2 y^2)\,dx = y\,dx + x\,dy.
\]
If the line …
Topic: JEE Main 2023 (11 April Morning Shift)
Let $w_1$ be the point obtained by the rotation of $z_1 = 5 + 4i$ about the origin through a right angle in the anticlo…
Topic: JEE Main 2023 (11 April Morning Shift)
Let $x_1, x_2, \ldots, x_{100}$ be in an arithmetic progression, with $x_1 = 2$ and their mean equal to $200$.
If $y_…
Topic: JEE Main 2023 (11 April Morning Shift)
Let $S = \{ M = [a_{ij}], \; a_{ij} \in \{0, 1, 2\}, \; 1 \le i, j \le 2 \}$ be a sample space
and $A = \{ M \in S : …
Topic: JEE Main 2023 (11 April Morning Shift)
Let $R$ be a rectangle given by the lines $x = 0$, $x = 2$, $y = 0$ and $y = 5$.
Let $A(\alpha, 0)$ and $B(0, \beta)$…
Topic: JEE Main 2023 (11 April Morning Shift)
Next 10 Questions — JEE Main 2023 (11 April Morning Shift)
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Area of the region
\[
\{(x, y) : x^2 + (y - 2)^2 \le 4, \; x^2 \ge 2y\}
\]
is:
Topic: JEE Main 2023 (11 April Morning Shift)
Let $\vec{a}$ be a non-zero vector parallel to the line of intersection of the two planes described by
$\hat{i} + \ha…
Topic: JEE Main 2023 (11 April Morning Shift)
The number of triplets , where are distinct non negative integers satisfying , is :
Topic: JEE Main 2023 (11 April Morning Shift)
An organization awarded 48 medals in event 'A', 25 in event 'B' and 18 in event 'C'. If these medals went to total 60 m…
Topic: JEE Main 2023 (11 April Morning Shift)
The number of integral solutions $x$ of $\log _{\left(x+\frac{7}{2}\right)}\left(\frac{x-7}{2 x-3}\right)^{2} \geq 0$ i…
Topic: JEE Main 2023 (11 April Morning Shift)
