1
JEE Main 2023 (Online) 11th April Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Consider ellipses $$\mathrm{E}_{k}: k x^{2}+k^{2} y^{2}=1, k=1,2, \ldots, 20$$. Let $$\mathrm{C}_{k}$$ be the circle which touches the four chords joining the end points (one on minor axis and another on major axis) of the ellipse $$\mathrm{E}_{k}$$. If $$r_{k}$$ is the radius of the circle $$\mathrm{C}_{k}$$, then the value of $$\sum_\limits{k=1}^{20} \frac{1}{r_{k}^{2}}$$ is :

A
2870
B
3210
C
3320
D
3080
2
JEE Main 2023 (Online) 11th April Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Area of the region $$\left\{(x, y): x^{2}+(y-2)^{2} \leq 4, x^{2} \geq 2 y\right\}$$ is

A
$$2 \pi+\frac{16}{3}$$
B
$$\pi-\frac{8}{3}$$
C
$$\pi+\frac{8}{3}$$
D
$$2 \pi-\frac{16}{3}$$
3
JEE Main 2023 (Online) 11th April Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let $$\vec{a}$$ be a non-zero vector parallel to the line of intersection of the two planes described by $$\hat{i}+\hat{j}, \hat{i}+\hat{k}$$ and $$\hat{i}-\hat{j}, \hat{j}-\hat{k}$$. If $$\theta$$ is the angle between the vector $$\vec{a}$$ and the vector $$\vec{b}=2 \hat{i}-2 \hat{j}+\hat{k}$$ and $$\vec{a} \cdot \vec{b}=6$$, then the ordered pair $$(\theta,|\vec{a} \times \vec{b}|)$$ is equal to :

A
$$\left(\frac{\pi}{3}, 3 \sqrt{6}\right)$$
B
$$\left(\frac{\pi}{3}, 6\right)$$
C
$$\left(\frac{\pi}{4}, 3 \sqrt{6}\right)$$
D
$$\left(\frac{\pi}{4}, 6\right)$$
4
JEE Main 2023 (Online) 11th April Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

The number of triplets $$(x, \mathrm{y}, \mathrm{z})$$, where $$x, \mathrm{y}, \mathrm{z}$$ are distinct non negative integers satisfying $$x+y+z=15$$, is :

A
136
B
80
C
92
D
114
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