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 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 $${\pi \over 3}$$
3
JEE Advanced 2020 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-2
Let a and b be positive real numbers. Suppose $$PQ = a\widehat i + b\widehat j$$ and $$PS = a\widehat i - b\widehat j$$ are adjacent sides of a parallelogram PQRS. Let u and v be the projection vectors of $$w = \widehat i + \widehat j$$ along PQ and PS, respectively. If |u| + |v| = |w| and if the area of the parallelogram PQRS is 8, then which of the following statements is/are TRUE?
A
a + b = 4
B
a $$-$$ b = 2
C
The length of the diagonal PR of the parallelogram PQRS is 4
D
w is an angle bisector of the vectors PQ and PS
4
JEE Advanced 2016 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-2
Let $$\widehat u = {u_1} \widehat i + {u_2}\widehat j + {u_3}\widehat k$$ be a unit vector in $${{R^3}}$$ and
$$\widehat w = {1 \over {\sqrt 6 }}\left( {\widehat i + \widehat j + 2\widehat k} \right).$$ Given that there exists a vector $${\overrightarrow v }$$ in $${{R^3}}$$ such that $$\left| {\widehat u \times \overrightarrow v } \right| = 1$$ and $$\widehat w.\left( {\widehat u \times \overrightarrow v } \right) = 1.$$ Which of the following statement(s) is (are) correct?
A
There is exactly one choice for such $${\overrightarrow v }$$
B
There are infinitely many choices for such $${\overrightarrow v }$$
C
If $$\widehat u$$ lies in the $$xy$$-plane then $$\left| {{u_1}} \right| = \left| {{u_2}} \right|$$
D
If $$\widehat u$$ lies in the $$xz$$-plane then $$2\left| {{u_1}} \right| = \left| {{u_3}} \right|$$
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