1
JEE Advanced 2022 Paper 1 Online
MCQ (More than One Correct Answer)
+4
-2
Let $$S$$ be the reflection of a point $$Q$$ with respect to the plane given by

$$\vec{r}=-(t+p) \hat{\imath}+t \hat{\jmath}+(1+p) \hat{k}$$

where $$t, p$$ are real parameters and $$\hat{\imath}, \hat{\jmath}, \hat{k}$$ are the unit vectors along the three positive coordinate axes. If the position vectors of $$Q$$ and $$S$$ are $$10 \hat{\imath}+15 \hat{\jmath}+20 \hat{k}$$ and $$\alpha \hat{\imath}+\beta \hat{\jmath}+\gamma \hat{k}$$ respectively, then which of the following is/are TRUE ?
A
$$3(\alpha+\beta)=-101$$
B
$$3(\beta+\gamma)=-71$$
C
$$3(\gamma+\alpha)=-86$$
D
$$3(\alpha+\beta+\gamma)=-121$$
2
JEE Advanced 2022 Paper 1 Online
MCQ (More than One Correct Answer)
+4
-2

Consider the parabola $$y^{2}=4 x$$. Let $$S$$ be the focus of the parabola. A pair of tangents drawn to the parabola from the point $$P=(-2,1)$$ meet the parabola at $$P_{1}$$ and $$P_{2}$$. Let $$Q_{1}$$ and $$Q_{2}$$ be points on the lines $$S P_{1}$$ and $$S P_{2}$$ respectively such that $$P Q_{1}$$ is perpendicular to $$S P_{1}$$ and $$P Q_{2}$$ is perpendicular to $$S P_{2}$$. Then, which of the following is/are TRUE?

A
$$S Q_{1}=2$$
B
$$Q_{1} Q_{2}=\frac{3 \sqrt{10}}{5}$$
C
$$P Q_{1}=3$$
D
$$S Q_{2}=1$$
3
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
$$p\left(\frac{5-\sqrt{2}}{4}\right)<0$$
4
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) $$\rightarrow(\mathrm{R})$$
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