1
JEE Advanced 2022 Paper 1 Online
Numerical
+3
-0
Let $$A B C$$ be the triangle with $$A B=1, A C=3$$ and $$\angle B A C=\frac{\pi}{2}$$. If a circle of radius $$r>0$$ touches the sides $$A B, A C$$ and also touches internally the circumcircle of the triangle $$A B C$$, then the value of $$r$$ is __________ .
2
JEE Advanced 2022 Paper 1 Online
MCQ (More than One Correct Answer)
+4
-2

Consider the equation

$$\int_{1}^{e} \frac{\left(\log _{\mathrm{e}} x\right)^{1 / 2}}{x\left(a-\left(\log _{\mathrm{e}} x\right)^{3 / 2}\right)^{2}} d x=1, \quad a \in(-\infty, 0) \cup(1, \infty)$$

Which of the following statements is/are TRUE?

A
No $$a$$ satisfies the above equation
B
An integer $$a$$ satisfies the above equation
C
An irrational number $$a$$ satisfies the above equation
D
More than one $$a$$ satisfy the above equation
3
JEE Advanced 2022 Paper 1 Online
MCQ (More than One Correct Answer)
+4
-2

Let $$a_{1}, a_{2}, a_{3}, \ldots$$ be an arithmetic progression with $$a_{1}=7$$ and common difference 8. Let $$T_{1}, T_{2}, T_{3}, \ldots$$ be such that $$T_{1}=3$$ and $$T_{n+1}-T_{n}=a_{n}$$ for $$n \geq 1$$. Then, which of the following is/are TRUE ?

A
$$T_{20}=1604$$
B
$$\sum\limits_{k=1}^{20} T_{k}=10510$$
C
$$T_{30}=3454$$
D
$$\sum\limits_{k=1}^{30} T_{k}=35610$$
4
JEE Advanced 2022 Paper 1 Online
MCQ (More than One Correct Answer)
+4
-2
Let $$P_{1}$$ and $$P_{2}$$ be two planes given by

\begin{aligned} &P_{1}: 10 x+15 y+12 z-60=0 \\\\ &P_{2}:-2 x+5 y+4 z-20=0 \end{aligned}

Which of the following straight lines can be an edge of some tetrahedron whose two faces lie on $$P_{1}$$ and $$P_{2}$$ ?
A
$$\frac{x-1}{0}=\frac{y-1}{0}=\frac{z-1}{5}$$
B
$$\frac{x-6}{-5}=\frac{y}{2}=\frac{z}{3}$$
C
$$\frac{x}{-2}=\frac{y-4}{5}=\frac{z}{4}$$
D
$$\frac{x}{1}=\frac{y-4}{-2}=\frac{z}{3}$$
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