1
JEE Advanced 2022 Paper 2 Online
MCQ (More than One Correct Answer)
+4
-2 Let $\hat{\imath}, \hat{\jmath}$ and $\hat{k}$ be the unit vectors along the three positive coordinate axes. Let

\begin{aligned} & \vec{a}=3 \hat{\imath}+\hat{\jmath}-\hat{k} \text {, } \\ & \vec{b}=\hat{\imath}+b_{2} \hat{\jmath}+b_{3} \hat{k}, \quad b_{2}, b_{3} \in \mathbb{R} \text {, } \\ & \vec{c}=c_{1} \hat{\imath}+c_{2} \hat{\jmath}+c_{3} \hat{k}, \quad c_{1}, c_{2}, c_{3} \in \mathbb{R} \end{aligned}

be three vectors such that $b_{2} b_{3}>0, \vec{a} \cdot \vec{b}=0$ and

$$\left(\begin{array}{ccc} 0 & -c_{3} & c_{2} \\ c_{3} & 0 & -c_{1} \\ -c_{2} & c_{1} & 0 \end{array}\right)\left(\begin{array}{l} 1 \\ b_{2} \\ b_{3} \end{array}\right)=\left(\begin{array}{r} 3-c_{1} \\ 1-c_{2} \\ -1-c_{3} \end{array}\right) .$$

Then, which of the following is/are TRUE?
A
$\vec{a} \cdot \vec{c}=0$
B
$\vec{b} \cdot \vec{c}=0$
C
$|\vec{b}|>\sqrt{10}$
D
$|\vec{c}| \leq \sqrt{11}$
2
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
$$\frac{x}{1}=\frac{y-4}{-2}=\frac{z}{3}$$
3
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
$$3(\alpha+\beta+\gamma)=-121$$
4
JEE Advanced 2021 Paper 2 Online
MCQ (More than One Correct Answer)
+4
-2 Let O be the origin and $$\overrightarrow {OA} = 2\widehat i + 2\widehat j + \widehat k$$ and $$\overrightarrow {OB} = \widehat i - 2\widehat j + 2\widehat k$$ and $$\overrightarrow {OC} = {1 \over 2}\left( {\overrightarrow {OB} - \lambda \overrightarrow {OA} } \right)$$ for some $$\lambda$$ > 0. If $$\left| {\overrightarrow {OB} \times \overrightarrow {OC} } \right| = {9 \over 2}$$, then which of the following statements is (are) TRUE?
A
Projection of $$\overrightarrow {OC}$$ on $$\overrightarrow {OA}$$ is $$- {3 \over 2}$$
B
Area of the triangle OAB is $${9 \over 2}$$
C
Area of the triangle ABC is $${9 \over 2}$$
D
The acute angle between the diagonals of the parallelogram with adjacent sides $${\overrightarrow {OA} }$$ and $${\overrightarrow {OC} }$$ is $${\pi \over 3}$$
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