1
JEE Advanced 2022 Paper 1 Online
MCQ (More than One Correct Answer)
+4
-2

Let $$|M|$$ denote the determinant of a square matrix $$M$$. Let $$g:\left[0, \frac{\pi}{2}\right] \rightarrow \mathbb{R}$$ be the function defined by

$$g(\theta)=\sqrt{f(\theta)-1}+\sqrt{f\left(\frac{\pi}{2}-\theta\right)-1}$$

where

$$f(\theta)=\frac{1}{2}\left|\begin{array}{ccc} 1 & \sin \theta & 1 \\ -\sin \theta & 1 & \sin \theta \\ -1 & -\sin \theta & 1 \end{array}\right|+\left|\begin{array}{ccc} \sin \pi & \cos \left(\theta+\frac{\pi}{4}\right) & \tan \left(\theta-\frac{\pi}{4}\right) \\ \sin \left(\theta-\frac{\pi}{4}\right) & -\cos \frac{\pi}{2} & \log _{e}\left(\frac{4}{\pi}\right) \\ \cot \left(\theta+\frac{\pi}{4}\right) & \log _{e}\left(\frac{\pi}{4}\right) & \tan \pi \end{array}\right| .$$

Let $$p(x)$$ be a quadratic polynomial whose roots are the maximum and minimum values of the function $$g(\theta)$$, and $$p(2)=2-\sqrt{2}$$. Then, which of the following is/are TRUE ?

A
$$p\left(\frac{3+\sqrt{2}}{4}\right)<0$$
B
$$p\left(\frac{1+3 \sqrt{2}}{4}\right)>0$$
C
$$p\left(\frac{5 \sqrt{2}-1}{4}\right)>0$$
D
$$p\left(\frac{5-\sqrt{2}}{4}\right)<0$$
2
JEE Advanced 2022 Paper 1 Online
+3
-1

Consider the following lists :

List-I List-II
(I) $$\left\{x \in\left[-\frac{2 \pi}{3}, \frac{2 \pi}{3}\right]: \cos x+\sin x=1\right\}$$ (P) has two elements
(II) $$\left\{x \in\left[-\frac{5 \pi}{18}, \frac{5 \pi}{18}\right]: \sqrt{3} \tan 3 x=1\right\}$$ (Q) has three elements
(III) $$\left\{x \in\left[-\frac{6 \pi}{5}, \frac{6 \pi}{5}\right]: 2 \cos (2 x)=\sqrt{3}\right\}$$ (R) has four elements
(IV) $$\left\{x \in\left[-\frac{7 \pi}{4}, \frac{7 \pi}{4}\right]: \sin x-\cos x=1\right\}$$ (S) has five elements
(T) has six elements

The correct option is:

A
(I) $$\rightarrow(\mathrm{P})$$; (II) $$\rightarrow(\mathrm{S})$$; (III) $$\rightarrow(\mathrm{P})$$; (IV) $$\rightarrow(\mathrm{S})$$
B
(I) $$\rightarrow$$ (P); (II) $$\rightarrow$$ (P); (III) $$\rightarrow$$ (T); (IV) $$\rightarrow$$ (R)
C
(I) $$\rightarrow$$ (Q); (II) $$\rightarrow(\mathrm{P})$$; (III) $$\rightarrow$$ (T); (IV) $$\rightarrow$$ (S)
D
(I) $$\rightarrow(\mathrm{Q})$$; (II) $$\rightarrow(\mathrm{S}) ;$$ (III) $$\rightarrow(\mathrm{P})$$; (IV) $$\rightarrow(\mathrm{R})$$
3
JEE Advanced 2022 Paper 1 Online
+3
-1

Two players, $$P_{1}$$ and $$P_{2}$$, play a game against each other. In every round of the game, each player rolls a fair die once, where the six faces of the die have six distinct numbers. Let $$x$$ and $$y$$ denote the readings on the die rolled by $$P_{1}$$ and $$P_{2}$$, respectively. If $$x>y$$, then $$P_{1}$$ scores 5 points and $$P_{2}$$ scores 0 point. If $$x=y$$, then each player scores 2 points. If $$x < y$$, then $$P_{1}$$ scores 0 point and $$P_{2}$$ scores 5 points. Let $$X_{i}$$ and $$Y_{i}$$ be the total scores of $$P_{1}$$ and $$P_{2}$$, respectively, after playing the $$i^{\text {th }}$$ round.

List-I List-II
(I) Probability of $$\left(X_{2} \geq Y_{2}\right)$$ is (P) $$\frac{3}{8}$$
(II) Probability of $$\left(X_{2}>Y_{2}\right)$$ is (Q) $$\frac{11}{16}$$
(III) Probability of $$\left(X_{3}=Y_{3}\right)$$ is (R) $$\frac{5}{16}$$
(IV) Probability of $$\left(X_{3}>Y_{3}\right)$$ is (S) $$\frac{355}{864}$$
(T) $$\frac{77}{432}$$

The correct option is:

A
(I) $$\rightarrow$$ (Q); (II) $$\rightarrow$$ (R); (III) $$\rightarrow$$ (T); (IV) $$\rightarrow(S)$$
B
(I) $$\rightarrow$$ (Q); (II) $$\rightarrow$$ (R); (III) $$\rightarrow$$ (T); (IV) $$\rightarrow$$ (T)
C
(I) $$\rightarrow$$ (P); (II) $$\rightarrow$$ (R); (III) $$\rightarrow(\mathrm{Q}) ;(\mathrm{IV}) \rightarrow(\mathrm{S})$$
D
(I) $$\rightarrow$$ (P); (II) $$\rightarrow$$ (R); (III) $$\rightarrow$$ (Q); (IV) $$\rightarrow$$ (T)
4
JEE Advanced 2022 Paper 1 Online
+3
-1

Let $$p, q, r$$ be nonzero real numbers that are, respectively, the $$10^{\text {th }}, 100^{\text {th }}$$ and $$1000^{\text {th }}$$ terms of a harmonic progression. Consider the system of linear equations

$$\begin{gathered} x+y+z=1 \\ 10 x+100 y+1000 z=0 \\ q r x+p r y+p q z=0 \end{gathered}$$\$

List-I List-II
(I) If $$\frac{q}{r}=10$$, then the system of linear equations has (P) $$x=0, \quad y=\frac{10}{9}, z=-\frac{1}{9}$$ as a solution
(II) If $$\frac{p}{r} \neq 100$$, then the system of linear equations has (Q) $$x=\frac{10}{9}, y=-\frac{1}{9}, z=0$$ as a solution
(III) If $$\frac{p}{q} \neq 10$$, then the system of linear equations has (R) infinitely many solutions
(IV) If $$\frac{p}{q}=10$$, then the system of linear equations has (S) no solution
(T) at least one solution

The correct option is:

A
(I) $$\rightarrow$$ (T); (II) $$\rightarrow$$ (R); (III) $$\rightarrow$$ (S); (IV) $$\rightarrow$$ (T)
B
(I) $$\rightarrow$$ (Q); (II) $$\rightarrow$$ (S); (III) $$\rightarrow$$ (S); (IV) $$\rightarrow$$ (R)
C
(I) $$\rightarrow(\mathrm{Q})$$; (II) $$\rightarrow$$ (R); (III) $$\rightarrow(\mathrm{P})$$; (IV) $$\rightarrow$$ (R)
D
(I) $$\rightarrow$$ (T); (II) $$\rightarrow$$ (S); (III) $$\rightarrow$$ (P); (IV) $$\rightarrow$$ (T)
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