Question 1: Palette 1/68

JEE Advanced 2026 MCQ

Let $ \vec{a}, \vec{b} $ be two vectors, and let P, Q and R be the points with position vectors $ \vec{a}, \vec{b} $ and $ \vec{a} + \vec{b} $, respectively, with respect to the origin O. If $ |\vec{a} + \vec{b}| = \sqrt{21} $, $ |\vec{a} - \vec{b}| = 3 $, and $ \vec{a} $ and $ (\vec{a} - \vec{b}) $ are perpendicular to each other, then the area of the triangle OPR is :

A

$ \sqrt{3} $

B

$ \frac{\sqrt{3}}{2} $

C

$ \frac{3\sqrt{3}}{2} $

D

$ \frac{3}{2} $

Question 2: Palette 2/68

JEE Advanced 2026 MCQ

For real numbers $\alpha$, $\beta$, $\gamma$, $\delta$ and $\mu$, consider the matrix

$ M = \begin{bmatrix} \alpha & \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \beta & \frac{1}{\sqrt{3}} \\ \gamma & \delta & \mu \end{bmatrix}. $

Suppose that $MM^{T} = I$, where $M^{T}$ is the transpose of the matrix $M$, and $I$ is the $3 \times 3$ identity matrix. Let

$ \vec{u} = \alpha\,\hat{i} + \frac{1}{\sqrt{3}}\hat{j} + \gamma\,\hat{k}, \qquad \vec{v} = \frac{1}{\sqrt{2}}\hat{i} + \beta\hat{j} + \delta\hat{k} \qquad \text{and} \qquad \vec{w} = -\frac{1}{\sqrt{2}}\hat{i} + \frac{1}{\sqrt{3}}\hat{j} + \mu\hat{k}. $

Match each entry in List-I to the correct entry in List-II and choose the correct option.

List-I	List-II
(P) The value of $\gamma^2 + \delta^2$ is	(1) 0
(Q) If $x\vec{u} + y\vec{v} + z\vec{w} = \hat{j}$ for some real numbers $x$, $y$ and $z$, then the value of $x$ is	(2) 1
(R) The value of $\left|\vec{u} \cdot (\vec{v} \times \vec{w})\right|$ is	(3) $\frac{1}{\sqrt{2}}$
(S) The value of $\left|\vec{u} \times (\vec{v} \times \vec{w})\right|$ is	(4) $\frac{1}{\sqrt{3}}$
	(5) $\frac{5}{6}$

A

(P) → (5), (Q) → (4), (R) → (2), (S) → (1)

B

(P) → (4), (Q) → (5), (R) → (1), (S) → (2)

C

(P) → (5), (Q) → (3), (R) → (2), (S) → (1)

D

(P) → (5), (Q) → (4), (R) → (1), (S) → (2)

Question 3: Palette 3/68

JEE Advanced 2025 MCQ

Let $\vec{w} = \hat{i} + \hat{j} - 2\hat{k}$, and $\vec{u}$ and $\vec{v}$ be two vectors, such that $\vec{u} \times \vec{v} = \vec{w}$ and $\vec{v} \times \vec{w} = \vec{u}$. Let $\alpha, \beta, \gamma$, and $t$ be real numbers such that

$\vec{u} = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k},\ \ \ - t \alpha + \beta + \gamma = 0,\ \ \ \alpha - t \beta + \gamma = 0,\ \ \ \alpha + \beta - t \gamma = 0.$

Match each entry in List-I to the correct entry in List-II and choose the correct option.

List – I	List – II
(P) $\lvert \vec{v} \rvert^2$ is equal to	(1) 0
(Q) If $\alpha = \sqrt{3}$, then $\gamma^2$ is equal to	(2) 1
(R) If $\alpha = \sqrt{3}$, then $(\beta + \gamma)^2$ is equal to	(3) 2
(S) If $\alpha = \sqrt{2}$, then $t + 3$ is equal to	(4) 3
	(5) 5

A

(P) $\to$ (2)   (Q) $\to$ (1)   (R) $\to$ (4)   (S) $\to$ (5)

B

(P) $\to$ (2)   (Q) $\to$ (4)   (R) $\to$ (3)   (S) $\to$ (5)

C

(P) $\to$ (2)   (Q) $\to$ (1)   (R) $\to$ (4)   (S) $\to$ (3)

D

(P) $\to$ (5)   (Q) $\to$ (4)   (R) $\to$ (1)   (S) $\to$ (3)

Question 4: Palette 4/68

JEE Advanced 2023 MCQ

Let the position vectors of the points $P, Q, R$ and $S$ be $\vec{a}=\hat{i}+2 \hat{j}-5 \hat{k}, \vec{b}=3 \hat{i}+6 \hat{j}+3 \hat{k}$, $\vec{c}=\frac{17}{5} \hat{i}+\frac{16}{5} \hat{j}+7 \hat{k}$ and $\vec{d}=2 \hat{i}+\hat{j}+\hat{k}$, respectively. Then which of the following statements is true?

A

The points $P, Q, R$ and $S$ are NOT coplanar

B

$\frac{\vec{b}+2 \vec{d}}{3}$ is the position vector of a point which divides $P R$ internally in the ratio $5: 4$

C

$\frac{\vec{b}+2 \vec{d}}{3}$ is the position vector of a point which divides $P R$ externally in the ratio $5: 4$

D

The square of the magnitude of the vector $\vec{b} \times \vec{d}$ is 95

Question 5: Palette 5/68

JEE Advanced 2022 MSQ

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

$|\vec{c}| \leq \sqrt{11}$

Question 6: Palette 6/68

JEE Advanced 2021 MSQ

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 ${\pi \over 3}$

Question 7: Palette 7/68

JEE Advanced 2020 MSQ

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

Question 8: Palette 8/68

JEE Advanced 2017 MCQ

Let O be the origin and let PQR be an arbitrary triangle. The point S is such that

$\overrightarrow{OP}$ . $\overrightarrow{OQ}$ + $\overrightarrow{OR}$ . $\overrightarrow{OS}$ = $\overrightarrow{OR}$ . $\overrightarrow{OP}$ + $\overrightarrow{OQ}$ . $\overrightarrow{OS}$ = $\overrightarrow{OQ}$ . $\overrightarrow{OR}$ + $\overrightarrow{OP}$ . $\overrightarrow{OS}$

Then the triangle PQR has S as its

A

centroid

B

orthocentre

C

incentre

D

circumcentre

Question 9: Palette 9/68

JEE Advanced 2017 MCQ

|$\overrightarrow{OX}$ $ \times $ $\overrightarrow{OY}$| = ?

A

sin(P + Q)

B

sin(P + R)

C

sin(Q + R)

D

sin2R

Question 10: Palette 10/68

JEE Advanced 2016 MSQ

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 $2\left| {{u_1}} \right| = \left| {{u_3}} \right|$

Question 11: Palette 11/68

JEE Advanced 2015 MCQ

Match the following :

Column I	Column II
(A) In $ \mathbb{R}^2 $, if the magnitude of the projection vector of the vector $ \alpha \hat{i} + \beta \hat{j} $ on $ \sqrt{3}\hat{i} + \hat{j} $ is $ \sqrt{3} $ and if $ \alpha = 2 + \sqrt{3}\beta $, then possible value(s) of $ |\alpha| $ is (are)	$(P)\ 1$
(B) Let $ \alpha $ and $ b $ be real numbers such that the function $ f(x)= \begin{cases} -3\alpha x^2-2, & x<1 \\[4pt] bx+\alpha^2, & x\ge 1 \end{cases} $ is differentiable for all $ x \in \mathbb{R} $. Then possible value(s) of $ \alpha $ is (are)	$(Q)\ 2$
(C) Let $ \omega \ne 1 $ be a complex cube root of unity. If $ (3-3\omega+2\omega^2)^{4n+3} +(2+3\omega-3\omega^2)^{4n+3} +(-3+2\omega+3\omega^2)^{4n+3}=0, $ then possible value(s) of $ n $ is (are)	$(R)\ 3$
(D) Let the harmonic mean of two positive real numbers $ a $ and $ b $ be $ 4 $. If $ q $ is a positive real number such that $ a,\ 5,\ q,\ b $ is an arithmetic progression, then the value(s) of $ |q-a| $ is (are)	$(S)\ 4$
	$(T)\ 5$

A

$\left( A \right) \to p, q;\,\,\left( B \right) \to p,q;\,\,\left( C \right) \to p,q,s,t;\,\,\left( D \right) \to q,t$

B

$\left( A \right) \to q;\,\,\left( B \right) \to q;\,\,\left( C \right) \to p,q,s,t;\,\,\left( D \right) \to q,t$

C

$\left( A \right) \to q;\,\,\left( B \right) \to p,q;\,\,\left( C \right) \to p,t;\,\,\left( D \right) \to q,t$

D

$\left( A \right) \to q;\,\,\left( B \right) \to p,q;\,\,\left( C \right) \to p,q,s,t;\,\,\left( D \right) \to q$

Question 12: Palette 12/68

JEE Advanced 2015 MSQ

Let $\Delta PQR$ be a triangle. Let $\vec a = \overrightarrow {QR} ,\vec b = \overrightarrow {RP} $ and $\overrightarrow c = \overrightarrow {PQ} .$ If $\left| {\overrightarrow a } \right| = 12,\,\,\left| {\overrightarrow b } \right| = 4\sqrt 3 ,\,\,\,\overrightarrow b .\overrightarrow c = 24,$ then which of the following is (are) true?

A

${{{{\left| {\overrightarrow c } \right|}^2}} \over 2} - \left| {\overrightarrow a } \right| = 12$

B

${{{{\left| {\overrightarrow c } \right|}^2}} \over 2} + \left| {\overrightarrow a } \right| = 30$

C

$\left| {\overrightarrow a \times \overrightarrow b + \overrightarrow c \times \overrightarrow a } \right| = 48\sqrt 3 $

D

$\overrightarrow a .\overrightarrow b = - 72$

Question 13: Palette 13/68

JEE Advanced 2014 MSQ

Let $\overrightarrow x ,\overrightarrow y $ and $\overrightarrow z $ be three vectors each of magnitude $\sqrt 2 $ and the angle between each pair of them is ${\pi \over 3}$. If $\overrightarrow a $ is a non-zero vector perpendicular to $\overrightarrow x $ and $\overrightarrow y \times \overrightarrow z $ and $\overrightarrow b $ is a non-zero vector perpendicular to $\overrightarrow y $ and $\overrightarrow z \times \overrightarrow x ,$ then

A

$\overrightarrow b = \left( {\overrightarrow b \,.\,\overrightarrow z } \right)\left( {\overrightarrow z - \overrightarrow x } \right)$

B

$\overrightarrow a = \left( {\overrightarrow a \,.\,\overrightarrow y } \right)\left( {\overrightarrow y - \overrightarrow z } \right)$

C

$\overrightarrow a \,.\,\overrightarrow b = - \left( {\overrightarrow a \,.\,\overrightarrow y } \right)\left( {\overrightarrow b \,.\,\overrightarrow z } \right)$

D

$\overrightarrow a = \left( {\overrightarrow a \,.\,\overrightarrow y } \right)\left( {\overrightarrow z - \overrightarrow y } \right)$

Question 14: Palette 14/68

JEE Advanced 2013 MCQ

match List $I$ with List $II$ and select the correct answer using the code given below the lists:

$\,\,\,\,$ $\,\,\,\,$ $\,\,\,\,$ List $I$
(P.)$\,\,\,\,$ Volume of parallelopiped determined by vectors $\overrightarrow a ,\overrightarrow b $ and $\overrightarrow c $ is $2.$ Then the volume of the parallelepiped determined by vectors $2\left( {\overrightarrow a \times \overrightarrow b } \right),3\left( {\overrightarrow b \times \overrightarrow c } \right)$ and $\left( {\overrightarrow c \times \overrightarrow a } \right)$ is
(Q.)$\,\,\,\,$ Volume of parallelopiped determined by vectors $\overrightarrow a ,\overrightarrow b $ and $\overrightarrow c $ is $5.$ Then the volume of the parallelepiped determined by vectors $3\left( {\overrightarrow a + \overrightarrow b } \right),\left( {\overrightarrow b + \overrightarrow c } \right)$ and $2\left( {\overrightarrow c + \overrightarrow a } \right)$ is
(R.)$\,\,\,\,$ Area of a triangle with adjacent sides determined by vectors ${\overrightarrow a }$ and ${\overrightarrow b }$ is $20.$ Then the area of the triangle with adjacent sides determined by vectors $\left( {2\overrightarrow a + 3\overrightarrow b } \right)$ and $\left( {\overrightarrow a - \overrightarrow b } \right)$ is
(S.)$\,\,\,\,$ Area of a parallelogram with adjacent sides determined by vectors ${\overrightarrow a }$ and ${\overrightarrow b }$ is $30.$ Then the area of the parallelogram with adjacent sides determined by vectors $\left( {\overrightarrow a + \overrightarrow b } \right)$ and ${\overrightarrow a }$ is

$\,\,\,\,$ $\,\,\,\,$ $\,\,\,\,$ List $II$
(1.)$\,\,\,\,$ $100$
(2.)$\,\,\,\,$ $30$
(3.)$\,\,\,\,$ $24$
(4.)$\,\,\,\,$ $60$

A

$P = 4,Q = 2,R = 3,S = 1$

B

$P = 2,Q = 3,R = 1,S = 4$

C

$P = 3,Q = 4,R = 1,S = 2$

D

$P = 1,Q = 4,R = 3,S = 2$

Question 15: Palette 15/68

JEE Advanced 2013 MCQ

Let $\overrightarrow{\mathrm{PR}}=3 \hat{i}+\hat{j}-2 \hat{k}$ and $ \overrightarrow{\mathrm{SQ}}=\hat{i}-3 \hat{j}-4 \hat{k}$ determine diagonals of a parallelogram $P Q R S$ and $\overrightarrow{\mathrm{PT}}=\hat{i}+2 \hat{j}+3 \hat{k}$ be another vector. Then the volume of the parallelopiped determined by the vectors $\overrightarrow{\mathrm{PT}}, \overrightarrow{\mathrm{PQ}}$ and $\overrightarrow{\mathrm{PS}}$ is :

A

5 units

B

20 units

C

10 units

D

30 units

Question 16: Palette 16/68

JEE Advanced 2012 MCQ

If $\overrightarrow a $ and $\overrightarrow b $ are vectors such that $\left| {\overrightarrow a + \overrightarrow b } \right| = \sqrt {29} $ and $\,\overrightarrow a \times \left( {2\widehat i + 3\widehat j + 4\widehat k} \right) = \left( {2\widehat i + 3\widehat j + 4\widehat k} \right) \times \widehat b,$ then a possible value of $\left( {\overrightarrow a + \overrightarrow b } \right).\left( { - 7\widehat i + 2\widehat j + 3\widehat k} \right)$ is

A

$0$

B

$3$

C

$4$

D

$8$

Question 17: Palette 17/68

JEE Advanced 2011 MCQ

Let $\overrightarrow a = \widehat i + \widehat j + \widehat k,\,\overrightarrow b = \widehat i - \widehat j + \widehat k$ and $\overrightarrow c = \widehat i - \widehat j - \widehat k$ be three vectors. A vector $\overrightarrow v $ in the plane of $\overrightarrow a $ and $\overrightarrow b ,$ whose projection on $\overrightarrow c $ is ${{1 \over {\sqrt 3 }}}$ , is given by

A

$\widehat i - 3\widehat j + 3\widehat k$

B

$-3\widehat i - 3\widehat j - \widehat k$

C

$3\widehat i - \widehat j + 3\widehat k$

D

$\widehat i + 3\widehat j - 3\widehat k$

Question 18: Palette 18/68

JEE Advanced 2011 MCQ

Match the statements given in Column -$I$ with the values given in Column-$II.$

$\,\,\,\,$ $\,\,\,\,$ $\,\,\,\,$ Column-$I$
(A) $\,\,\,\,$If $\overrightarrow a = \widehat j + \sqrt 3 \widehat k,\overrightarrow b = - \widehat j + \sqrt 3 \widehat k$ and $\overrightarrow c = 2\sqrt 3 \widehat k$ form a triangle, then the internal angle of the triangle between $\overrightarrow a $ and $\overrightarrow b $ is
(B)$\,\,\,\,$ If $\int\limits_a^b {\left( {f\left( x \right) - 3x} \right)dx = {a^2} - {b^2},} $ then the value of $f$ $\left( {{\pi \over 6}} \right)$ is
(C)$\,\,\,\,$ The value of ${{{\pi ^2}} \over {\ell n3}}\int\limits_{7/6}^{5/6} {\sec \left( {\pi x} \right)dx} $ is
(D)$\,\,\,\,$ The maximum value of $\left| {Arg\left( {{1 \over {1 - z}}} \right)} \right|$ for $\left| z \right| = 1,\,z \ne 1$ is given by

$\,\,\,\,$ $\,\,\,\,$ $\,\,\,\,$ Column-$II$
(p)$\,\,\,\,$ ${{\pi \over 6}}$
(q)$\,\,\,\,$ ${{2\pi \over 3}}$
(r)$\,\,\,\,$ ${{\pi \over 3}}$
(s)$\,\,\,\,$ $\pi $
(t) $\,\,\,\,$ ${{\pi \over 2}}$

A

$\left( A \right) \to q;\,\,\left( B \right) \to p;\,\,\left( C \right) \to s;\,\,\left( D \right) \to t$

B

$\left( A \right) \to q;\,\,\left( B \right) \to p;\,\,\left( C \right) \to t;\,\,\left( D \right) \to s$

C

$\left( A \right) \to p;\,\,\left( B \right) \to q;\,\,\left( C \right) \to s;\,\,\left( D \right) \to t$

D

$\left( A \right) \to q;\,\,\left( B \right) \to s;\,\,\left( C \right) \to p;\,\,\left( D \right) \to t$

Question 19: Palette 19/68

JEE Advanced 2011 MSQ

The vector (s) which is/are coplanar with vectors ${\widehat i + \widehat j + 2\widehat k}$ and ${\widehat i + 2\widehat j + \widehat k,}$ and perpendicular to the vector ${\widehat i + \widehat j + \widehat k}$ is/are

A

$\widehat j - \widehat k$

B

$-\widehat i + \widehat j$

C

$\widehat i - \widehat j$

D

$-\widehat j + \widehat k$

Question 20: Palette 20/68

JEE Advanced 2010 MCQ

Two adjacent sides of a parallelogram $ABCD$ are given by
$\overrightarrow {AB} = 2\widehat i + 10\widehat j + 11\widehat k$ and $\,\overrightarrow {AD} = -\widehat i + 2\widehat j + 2\widehat k$
The side $AD$ is rotated by an acute angle $\alpha $ in the plane of the parallelogram so that $AD$ becomes $AD'.$ If $AD'$ makes a right angle with the side $AB,$ then the cosine of the angle $\alpha $ is given by

A

${{8 \over 9}}$

B

${{{\sqrt {17} } \over 9}}$

C

${{1 \over 9}}$

D

${{{4\sqrt 5 } \over 9}}$

Question 21: Palette 21/68

JEE Advanced 2010 MCQ

Let $P,Q,R$ and $S$ be the points on the plane with position vectors ${ - 2\widehat i - \widehat j,4\widehat i,3\widehat i + 3\widehat j}$ and ${ - 3\widehat i + 2\widehat j}$ respectively. The quadrilateral $PQRS$ must be a

A

parallelogram, which is neither a rhombus nor a rectangle

B

square

C

rectangle, but not a square

D

rhombus, but not a square

Question 22: Palette 22/68

JEE Advanced 2009 MCQ

If $\overrightarrow a ,\overrightarrow b ,\overrightarrow c $ and $\overrightarrow d $ are unit vectors such that $(\overrightarrow a \times \overrightarrow b )\,.\,(\overrightarrow c \times \overrightarrow d ) = 1$ and $\overrightarrow a \,.\,\overrightarrow c = {1 \over 2}$, then

A

$\overrightarrow a \,,\,\overrightarrow b ,\overrightarrow c $ are non-coplanar

B

$\overrightarrow b \,,\,\overrightarrow c ,\overrightarrow d $ are non-coplanar

C

$\overrightarrow b \,,\overrightarrow d $ are non-parallel

D

$\overrightarrow a ,\overrightarrow d $ parallel and $\overrightarrow b ,\overrightarrow c $ are parallel

Question 23: Palette 23/68

JEE Advanced 2008 MCQ

The unit vector perpendicular to both ${L_1}$ and ${L_2}$ is :

A

${{ - \widehat i + 7\widehat j + 7\widehat k} \over {\sqrt {99} }}$

B

${{ - \widehat i - 7\widehat j + 5\widehat k} \over {5\sqrt 3 }}$

C

${{ - \widehat i + 7\widehat j + 5\widehat k} \over {5\sqrt 3 }}$

D

${{7\widehat i - 7\widehat j - \widehat k} \over {\sqrt {99} }}$

Question 24: Palette 24/68

JEE Advanced 2008 MCQ

Let two non-collinear unit vectors $\widehat a$ and $\widehat b$ form an acute angle. A point $P$ moves so that at any time $t$ the position vector $\overrightarrow {OP} $ (where $O$ is the origin) is given by $\widehat a\cos t + \widehat b\sin t.$ When $P$ is farthest from origin $O,$ let $M$ be the length of $\overrightarrow {OP} $ and $\widehat u$ be the unit vector along $\overrightarrow {OP} $. Then :

A

$\widehat u = {{\widehat a + \widehat b} \over {\left| {\widehat a + \widehat b} \right|}}\,\,and\,\,M = {\left( {1 + \widehat a.\,\widehat b} \right)^{1/2}}$

B

$\widehat u = {{\widehat a - \widehat b} \over {\left| {\widehat a - \widehat b} \right|}}\,\,and\,\,M = {\left( {1 + \widehat a.\,\widehat b} \right)^{1/2}}$

C

$\widehat u = {{\widehat a + \widehat b} \over {\left| {\widehat a + \widehat b} \right|}}\,\,and\,\,M = {\left( {1 + 2\widehat a.\,\widehat b} \right)^{1/2}}$

D

$\widehat u = {{\widehat a - \widehat b} \over {\left| {\widehat a - \widehat b} \right|}}\,\,and\,\,M = {\left( {1 + 2\widehat a.\,\widehat b} \right)^{1/2}}$

Question 25: Palette 25/68

JEE Advanced 2008 MCQ

The shortest distance between ${L_1}$ and ${L_2}$ is :

A

$0$

B

${17 \over {\sqrt 3 }}$

C

${41 \over {5\sqrt 3 }}$

D

${17 \over {5\sqrt 3 }}$

Question 26: Palette 26/68

JEE Advanced 2008 MCQ

The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vectors $\overrightarrow a \,,\,\overrightarrow b ,\overrightarrow c $ such that $\widehat a\,.\,\widehat b = \widehat b\,.\,\widehat c = \widehat c\,.\,\widehat a = {1 \over 2}.$ Then, the volume of the parallelopiped is :

A

${1 \over {\sqrt 2 }}$

B

${1 \over {2\sqrt 2 }}$

C

${{\sqrt 3 } \over 2}$

D

${1 \over {\sqrt 3 }}$

Question 27: Palette 27/68

JEE Advanced 2007 MCQ

Let $\overrightarrow a \,,\,\overrightarrow b ,\overrightarrow c $ be unit vectors such that ${\overrightarrow a + \overrightarrow b + \overrightarrow c = \overrightarrow 0 .}$ Which one of the following is correct ?

A

$\overrightarrow a \times \overrightarrow b = b \times \overrightarrow c = \overrightarrow c \times \overrightarrow a = \overrightarrow 0 $

B

$\overrightarrow a \times \overrightarrow b = b \times \overrightarrow c = \overrightarrow c \times \overrightarrow a \ne \overrightarrow 0 $

C

$\overrightarrow a \times \overrightarrow b = b \times \overrightarrow c = \overrightarrow a \times \overrightarrow c \ne \overrightarrow 0 $

D

$\overrightarrow a \times \overrightarrow b ,b \times \overrightarrow c ,\overrightarrow c \times \overrightarrow a $ are muturally perpendicular

Question 28: Palette 28/68

JEE Advanced 2007 MCQ

Let the vectors $\overrightarrow {PQ} ,\,\,\overrightarrow {QR} ,\,\,\overrightarrow {RS} ,\,\,\overrightarrow {ST} ,\,\,\overrightarrow {TU} ,$ and $\overrightarrow {UP} ,$ represent the sides of a regular hexagon.

STATEMENT-1: $\overrightarrow {PQ} \times \left( {\overrightarrow {RS} + \overrightarrow {ST} } \right) \ne \overrightarrow 0 .$ because
STATEMENT-2: $\overrightarrow {PQ} \times \overrightarrow {RS} = \overrightarrow 0 $ and $\overrightarrow {PQ} \times \overrightarrow {ST} \ne \overrightarrow 0 \,\,.$

A

Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.

B

Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.

C

Statement-1 is True, Statement-2 is False

D

Statement-1 is False, Statement-2 is True.

Question 29: Palette 29/68

JEE Advanced 2007 MCQ

The minimum of distinct real values of $\lambda ,$ for which the vectors $ - {\lambda ^2}\widehat i + \widehat j + \widehat k,$ $\widehat i - {\lambda ^2}\widehat j + \widehat k$ and $\widehat i + \widehat j - {\lambda ^2}\widehat k$ are coplanar, is

A

zero

B

one

C

two

D

three

Question 30: Palette 30/68

JEE Advanced 2007 MCQ

Let $\vec{a}, \vec{b}, \vec{c}$ be unit vectors such that $\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}$. Which one of the following is correct?

A

$\vec{a} \times \vec{b}=\vec{b} \times \vec{c}=\vec{c} \times \vec{a}=\overrightarrow{0}$

B

$\vec{a} \times \vec{b}=\vec{b} \times \vec{c}=\vec{c} \times \vec{a} \neq \overrightarrow{0}$

C

$\vec{a} \times \vec{b}=\vec{b} \times \vec{c}=\vec{a} \times \vec{c} \neq \overrightarrow{0}$

D

$\vec{a} \times \vec{b}, \vec{b} \times \vec{c}, \vec{c} \times \vec{a}$ are mutually perpendicular

Question 31: Palette 31/68

JEE Advanced 2007 MCQ

The number of distinct real values of $\lambda$, for which the vectors $ - {\lambda ^2}\widehat i + \widehat j + \widehat k,\widehat i - {\lambda ^2}\widehat j + \widehat k$ and $\widehat i + \widehat j - {\lambda ^2}\widehat k$ are coplanar, is :

A

zero

B

one

C

two

D

three

Question 32: Palette 32/68

JEE Advanced 2007 MCQ

Let the vector $\overrightarrow {PQ} ,\overrightarrow {QR} ,\overrightarrow {RS} ,\overrightarrow {ST} ,\overrightarrow {TU} $ and $\overrightarrow {UP} $, represent the sides of a regular hexagon.

Statement 1 : $\overrightarrow {PQ} \times \left( {\overrightarrow {RS} + \overrightarrow {ST} } \right) \ne \overrightarrow 0 $

Statement 2 : $\overrightarrow {PQ} \times \overrightarrow {RS} = \overrightarrow 0 $ and $\overrightarrow {PQ} \times \overrightarrow {ST} \ne \overrightarrow 0 $

A

Statement 1 is True, Statement 2 is True, Statement 2 is a CORRECT explanation for Statement 1

B

Statement 1 is True, Statement 2 is True, Statement 2 is NOT a CORRECT explanation for Statement 1

C

Statement 1 is True, Statement 2 is False

D

Statement 1 is False, Statement 2 is True

Question 33: Palette 33/68

JEE Advanced 2006 MCQ

Let $\overrightarrow a = \widehat i + 2\widehat j + \widehat k,\,\overrightarrow b = \widehat i - \widehat j + \widehat k$ and $\overrightarrow c = \widehat i + \widehat j - \widehat k.$ A vector in the plane of $\overrightarrow a $ and $\overrightarrow b $ whose projection on $\overrightarrow c $ is ${1 \over {\sqrt 3 }},$ is

A

$4\widehat i - \widehat j + 4\widehat k$

B

$3\widehat i + \widehat j - 3\widehat k$

C

$2\widehat i + \widehat j - 2\widehat k$

D

$4\widehat i + \widehat j - 4\widehat k$

Question 34: Palette 34/68

JEE Advanced 2006 MCQ

(i)	Two rays in the first quadrant $x+y=|a|$ and $a x-y=1$ Intersects each other in the interval $a \in\left(a_0, \infty\right)$, the value of $a_0$ is	(A)	2
(ii)	Point $(\alpha, \beta, \gamma)$ lies on the plane $x+y+z=2$. Let $\vec{a}=\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}, \hat{k} \times(\hat{k} \times \vec{a})=0$, then $\gamma=$	(B)	4/3
(iii)	$ \left|\int_0^1\left(1-y^2\right) d y\right|+\left|\int_1^0\left(y^2-1\right) d y\right| $	(C)	$ \left|\int_0^1 \sqrt{1-x} d x\right|+\left|\int_1^0 \sqrt{1+x} d x\right| $
(iv)	If $\sin A \sin B \sin C+\cos A \cos B=1$, then the value of $\sin C=$	(D)	1

A

$ \begin{aligned} & \text { (i)-(D); (ii)-(B); (iii)-(B),(C); } \text { (iv)-(A) } \end{aligned} $

B

$ \begin{aligned} & \text { (i)-(D); (ii)-(A); (iii)-(B); } \text { (iv)-(D) } \end{aligned} $

C

$ \begin{aligned} & \text { (i)-(A); (ii)-(D); (iii)-(B),(C); } \text { (iv)-(D) } \end{aligned} $

D

$ \begin{aligned} & \text { (i)-(D); (ii)-(A); (iii)-(B),(C); } \text { (iv)-(D) } \end{aligned} $

Question 35: Palette 35/68

JEE Advanced 2005 MCQ

If $\overrightarrow a \,,\,\overrightarrow b ,\overrightarrow c $ are three non-zero, non-coplanar vectors and
$\overrightarrow {{b_1}} = \overrightarrow b - {{\overrightarrow b .\,\overrightarrow a } \over {{{\left| {\overrightarrow a } \right|}^2}}}\overrightarrow a ,\overrightarrow {{b_2}} = \overrightarrow b + {{\overrightarrow b .\,\overrightarrow a } \over {{{\left| {\overrightarrow a } \right|}^2}}}\overrightarrow a ,$
$\overrightarrow {{c_1}} = \overrightarrow c - {{\overrightarrow c .\,\overrightarrow a } \over {{{\left| {\overrightarrow a } \right|}^2}}}\overrightarrow a + {{\overrightarrow b .\,\overrightarrow c } \over {{{\left| c \right|}^2}}}{\overrightarrow b _1},\,\,\overrightarrow {{c_2}} = \overrightarrow c - {{\overrightarrow c .\,\overrightarrow a } \over {{{\left| {\overrightarrow a } \right|}^2}}}\overrightarrow a - {{\overrightarrow b \,.\,\overrightarrow c } \over {{{\left| {{{\overrightarrow b }_1}} \right|}^2}}}{\overrightarrow b _1},$
$\overrightarrow {{c_3}} = \overrightarrow c - {{\overrightarrow c .\,\overrightarrow a } \over {{{\left| {\overrightarrow c } \right|}^2}}}\overrightarrow a + {{\overrightarrow b .\,\overrightarrow c } \over {{{\left| c \right|}^2}}}{\overrightarrow b _1},\,\,\overrightarrow {{c_4}} = \overrightarrow c - {{\overrightarrow c .\,\overrightarrow a } \over {{{\left| {\overrightarrow c } \right|}^2}}}\overrightarrow a - {{\overrightarrow b \,.\,\overrightarrow c } \over {{{\left| {{{\overrightarrow b }_1}} \right|}^2}}}{\overrightarrow b _1},$
then the set of orthogonal vectors is

A

$\left( {\overrightarrow a ,\overrightarrow {{b_1}} ,\overrightarrow {{c_3}} } \right)$

B

$\left( {\overrightarrow a ,\overrightarrow {{b_1}} ,\overrightarrow {{c_2}} } \right)$

C

$\left( {\overrightarrow a ,\overrightarrow {{b_1}} ,\overrightarrow {{c_1}} } \right)$

D

$\left( {\overrightarrow a ,\overrightarrow {{b_2}} ,\overrightarrow {{c_2}} } \right)$

Question 36: Palette 36/68

JEE Advanced 2005 MCQ

Incident ray is along the unit vector $\hat{v}$ and the reflected ray is along the unit vector $\widehat{w}$. The normal is along unit vector $\hat{a}$ outwards. Express $\hat{w}$, in terms of $\hat{a}$ and $\hat{v}$.

A

$\widehat{w}=\hat{v}-2(\hat{a} \cdot \hat{v}) \cdot \hat{a}$

B

$\widehat{w}=\hat{v}+2(\hat{a} \cdot \hat{v}) \cdot \hat{a}$

C

$\widehat{w}=\hat{v}-3(\hat{a} \cdot \hat{v}) \cdot \hat{a}$

D

$\widehat{w}=5\hat{v}+3(\hat{a} \cdot \hat{v}) \cdot \hat{a}$

Question 37: Palette 37/68

JEE Advanced 2004 MCQ

If $\overrightarrow a = \left( {\widehat i + \widehat j + \widehat k} \right),\overrightarrow a .\overrightarrow b = 1$ and $\overrightarrow a \times \overrightarrow b = \widehat j - \widehat k,$ then $\overrightarrow b $ is

A

$\widehat i - \widehat j + \widehat k$

B

$2\widehat j - \widehat k$

C

$\widehat i$

D

$2\widehat i$

Question 38: Palette 38/68

JEE Advanced 2004 MCQ

The unit vector which is orthogonal to the vector $3\overrightarrow i + 2\overrightarrow j + 6\overrightarrow k $ and is coplanar with the vectors $\,2\widehat i + \widehat j + \widehat k$ and $\,\widehat i - \widehat j + \widehat k$$\,\,\,$ is

A

${{2\widehat i - 6\widehat j + \widehat k} \over {\sqrt {41} }}$

B

${{2\widehat i - 3\widehat j} \over {\sqrt {13} }}$

C

${{3\widehat i - \widehat k} \over {\sqrt {10} }}$

D

${{4\widehat i + 3\widehat j - 3\widehat k} \over {\sqrt {34} }}$

Question 39: Palette 39/68

JEE Advanced 2003 MCQ

The value of $'a'$ so that the volume of parallelopiped formed by $\widehat i + a\widehat j + \widehat k,\widehat j + a\widehat k$ and $a\widehat i + \widehat k$ becomes minimum is

A

$-3$

B

$3$

C

$1/\sqrt 3 $

D

$\sqrt 3 $

Question 40: Palette 40/68

JEE Advanced 2002 MCQ

Let $\overrightarrow V = 2\overrightarrow i + \overrightarrow j - \overrightarrow k $ and $\overrightarrow W = \overrightarrow i + 3\overrightarrow k .$ If $\overrightarrow U $ is a unit vector, then the maximum value of the scalar triple product $\left| {\overrightarrow U \overrightarrow V \overrightarrow W } \right|$ is

A

$-1$

B

$\sqrt {10} + \sqrt 6 $

C

$\sqrt {59} $

D

$\sqrt {60} $

Question 41: Palette 41/68

JEE Advanced 2002 MCQ

If ${\overrightarrow a }$ and ${\overrightarrow b }$ are two unit vectors such that ${\overrightarrow a + 2\overrightarrow b }$ and ${5\overrightarrow a - 4\overrightarrow b }$ are perpendicular to each other then the angle between $\overrightarrow a $ and $\overrightarrow b $ is

A

${45^ \circ }$

B

${60^ \circ }$

C

${\cos ^{ - 1}}\left( {{1 \over 3}} \right)$

D

${\cos ^{ - 1}}\left( {{2 \over 7}} \right)$

Question 42: Palette 42/68

JEE Advanced 2001 MCQ

If $\overrightarrow a \,,\,\overrightarrow b $ and $\overrightarrow c $ are unit vectors, then ${\left| {\overrightarrow a - \overrightarrow b } \right|^2} + {\left| {\overrightarrow b - \overrightarrow c } \right|^2} + {\left| {\overrightarrow c - \overrightarrow a } \right|^2}$ does NOT exceed

A

$4$

B

$9$

C

$8$

D

$6$

Question 43: Palette 43/68

JEE Advanced 2001 MCQ

Let $\overrightarrow a = \overrightarrow i - \overrightarrow k ,\overrightarrow b = x\overrightarrow i + \overrightarrow j + \left( {1 - x} \right)\overrightarrow k $ and
$\overrightarrow c = y\overrightarrow i - x\overrightarrow j + \left( {1 + x - y} \right)\overrightarrow k .$ Then $\left[ {\overrightarrow a \,\overrightarrow b \,\overrightarrow c } \right]$ depends on

A

only $x$

B

only $y$

C

Neither $x$ Nor $y$

D

both $x$ and $y$

Question 44: Palette 44/68

JEE Advanced 2000 MCQ

If $\overrightarrow a \,,\,\overrightarrow b $ and $\overrightarrow c $ are unit coplanar vectors, then the scalar triple product $\left[ {2\overrightarrow a - \overrightarrow b ,2\overrightarrow b - \overrightarrow c ,2\overrightarrow c - \overrightarrow a } \right] = $

A

$0$

B

$1$

C

$ - \sqrt 3 $

D

$ \sqrt 3 $

Question 45: Palette 45/68

JEE Advanced 2000 MCQ

If the vectors $\overrightarrow a ,\overrightarrow b $ and $\overrightarrow c $ form the sides $BC,$ $CA$ and $AB$ respectively of a triangle $ABC,$ then

A

$\overrightarrow a .\overrightarrow b + \overrightarrow b .\overrightarrow c + \overrightarrow c .\overrightarrow a = 0$

B

$\overrightarrow a \times \overrightarrow b = \overrightarrow b \times \overrightarrow c = \overrightarrow c \times \overrightarrow a $

C

$\overrightarrow a .\overrightarrow b = \overrightarrow b .\overrightarrow c = \overrightarrow c .\overrightarrow a$

D

$\overrightarrow a \times \overrightarrow b + \overrightarrow b \times \overrightarrow c + \overrightarrow c \times \overrightarrow a = \overrightarrow 0 $

Question 46: Palette 46/68

JEE Advanced 2000 MCQ

Let the vectors $\overrightarrow a ,\overrightarrow b ,\overrightarrow c $ and $\overrightarrow d $ be such that
$\left( {\overrightarrow a \times \overrightarrow b } \right) \times \left( {\overrightarrow c \times \overrightarrow d } \right) = 0.$ Let ${P_1}$ and ${P_2}$ be planes determined
by the pairs of vectors $\overrightarrow a .\overrightarrow b $ and $\overrightarrow c .\overrightarrow d $ respectively. Then the angle between ${P_1}$ and ${P_2}$ is

A

$0$

B

${\pi \over 4}$

C

${\pi \over 3}$

D

${\pi \over 2}$

Question 47: Palette 47/68

JEE Advanced 1999 MCQ

Let $a=2i+j-2k$ and $b=i+j.$ If $c$ is a vector such that $a.$ $c = \left| c \right|,\left| {c - a} \right| = 2\sqrt 2 $ and the angle between $\left( {a \times b} \right)$ and $c$ is ${30^ \circ },$ then $\left| {\left( {a \times b} \right) \times c} \right| = $

A

$2/3$

B

$3/2$

C

$2$

D

$3$

Question 48: Palette 48/68

JEE Advanced 1999 MCQ

Let $a=2i+j+k, b=i+2j-k$ and a unit vector $c$ be coplanar. If $c$ is perpendicular to $a,$ then $c =$

A

${1 \over {\sqrt 2 }}\left( { - j + k} \right)$

B

${1 \over {\sqrt 3 }}\left( {- i - j - k} \right)$

C

${1 \over {\sqrt 5 }}\left( {i - 2j} \right)$

D

${1 \over {\sqrt 3 }}\left( {i - j - k} \right)$

Question 49: Palette 49/68

JEE Advanced 1999 MSQ

Let $a$ and $b$ two non-collinear unit vectors. If $u = a - \left( {a\,.\,b} \right)\,b$ and $v = a \times b,$ then $\left| v \right|$ is

A

$\left| u \right|$

B

$\,\left| u \right| + \left| {u\,.\,a} \right|$

C

$\,\left| u \right| + \left| {u\,.\,b} \right|$

D

$\left| u \right| + u.\left( {a + b} \right)$

Question 50: Palette 50/68

JEE Advanced 1998 MCQ

If $a = i + j + k,\overrightarrow b = 4i + 3j + 4k$ and $c = i + \alpha j + \beta k$ are linearly dependent vectors and $\left| c \right| = \sqrt 3 ,$ then

A

$\alpha = 1,\,\,\beta = - 1$

B

$\alpha = 1,\,\,\beta = \pm 1$

C

$\alpha = - 1,\,\,\beta = \pm 1$

D

$\alpha = \pm 1,\,\,\beta = 1$

Question 51: Palette 51/68

JEE Advanced 1998 MCQ

For three vectors $u,v,w$ which of the following expression is not equal to any of the remaining three?

A

$\,u \bullet \left( {v \times w} \right)$

B

$\left( {v \times w} \right) \bullet u$

C

$\,v \bullet \left( {u \times w} \right)$

D

$\left( {u \times v} \right) \bullet w$

Question 52: Palette 52/68

JEE Advanced 1998 MSQ

Which of the following expressions are meaningful?

A

$u\left( {v \times w} \right)$

B

$\left( {u \bullet v} \right) \bullet w$

C

$\left( {u \bullet v} \right)w$

D

$\,u\, \times \left( {v \bullet w} \right)$

Question 53: Palette 53/68

JEE Advanced 1995 MCQ

If $\overrightarrow a ,$ $\overrightarrow b $ and $\overrightarrow c $ are three non coplanar vectors, then
$\left( {\overrightarrow a + \overrightarrow b + \overrightarrow c } \right).\left[ {\left( {\overrightarrow a + \overrightarrow b } \right) \times \left( {\overrightarrow a + \overrightarrow c } \right)} \right]$ equals

A

$0$

B

$\left[ {\overrightarrow a \,\overrightarrow b \,\overrightarrow c } \right]$

C

$2\left[ {\overrightarrow a \,\overrightarrow b \,\overrightarrow c } \right]$

D

$-\left[ {\overrightarrow a \,\overrightarrow b \,\overrightarrow c } \right]$

Question 54: Palette 54/68

JEE Advanced 1995 MCQ

If $\overrightarrow a ,\overrightarrow b ,\overrightarrow c $ are non coplanar unit vectors such that $\overrightarrow a \times \left( {\overrightarrow b \times \overrightarrow c } \right) = {{\left( {\overrightarrow b + \overrightarrow c } \right)} \over {\sqrt 2 }},\,\,$ then the angle between $\overrightarrow a $ and $\overrightarrow b $ is

A

${{3\pi } \over 4}$

B

${{\pi } \over 4}$

C

$\pi /2$

D

$\pi $

Question 55: Palette 55/68

JEE Advanced 1995 MCQ

Let $\overrightarrow a = \widehat i - \widehat j,\overrightarrow b = \widehat j - \widehat k,\overrightarrow c = \widehat k - \widehat i.$ If $\overrightarrow d $ is a unit vector such that $\overrightarrow a .\overrightarrow d = 0 = \left[ {\overrightarrow b \overrightarrow c \overrightarrow d } \right],$ then $\overrightarrow d $ equals

A

$ \pm {{\widehat i + \widehat j - 2k} \over {\sqrt 6 }}$

B

$ \pm {{\widehat i + \widehat j - k} \over {\sqrt 3 }}$

C

$ \pm {{\widehat i + \widehat j + k} \over {\sqrt 3 }}$

D

$ \pm \widehat k$

Question 56: Palette 56/68

JEE Advanced 1995 MCQ

Let $\overrightarrow u ,\overrightarrow v $ and $\overrightarrow w $ be vectors such that $\overrightarrow u + \overrightarrow v + \overrightarrow w = 0.$ If $\left| {\overrightarrow u } \right| = 3,\left| {\overrightarrow v } \right| = 4$ and $\left| {\overrightarrow w } \right| = 5,$ then $\overrightarrow u .\overrightarrow v + \overrightarrow v .\overrightarrow w + \overrightarrow w .\overrightarrow u $ is

A

$47$

B

$-25$

C

$0$

D

$25$

Question 57: Palette 57/68

JEE Advanced 1994 MSQ

The vector $\,{1 \over 3}\left( {2\widehat i - 2\widehat j + \widehat k} \right)$ is

A

a unit vector

B

makes an angle ${\pi \over 3}$ with the vector $\left( {2\widehat i - 4\widehat j + 3\widehat k} \right)$

C

parallel to the vector $\left( { - \widehat i + \widehat j - {1 \over 2}\widehat k} \right)$

D

perpendicular to the vector ${3\widehat i + 2\widehat j - 2\widehat k}$

Question 58: Palette 58/68

JEE Advanced 1993 MCQ

Let $a, b, c$ be distinct non-negative numbers. If the vectors $a\widehat i + a\widehat j + c\widehat k,\widehat i + \widehat k$ and $c\widehat i + c\widehat j + b\widehat k$ lie in a plane, then $c$ is

A

the Arithmetic Mean of $a$ and $b$

B

the Geometric Mean of $a$ and $b$

C

the Harmonic Mean of $a$ and $b$

D

equal to zero

Question 59: Palette 59/68

JEE Advanced 1993 MSQ

Let $\vec a = 2\hat i - \hat j + \hat k,\vec b = \hat i + 2\hat j - \hat k$ and $\overrightarrow c = \widehat i + \widehat j - 2\widehat k - 2\widehat k$ be three vectors. A vector in the plane of ${\overrightarrow b }$ and ${\overrightarrow c }$, whose projection on ${\overrightarrow a }$ is of magnitude $\sqrt {2/3,} $ is :

A

$2\widehat i + 3\widehat j - 3\widehat k$

B

$2\widehat i + 3\widehat j + 3\widehat k$

C

$-2\widehat i - \widehat j + 5\widehat k$

D

$2\widehat i + \widehat j + 5\widehat k$

Question 60: Palette 60/68

JEE Advanced 1989 MCQ

For any three vectors ${\overrightarrow a ,\,\overrightarrow b ,}$ and ${\overrightarrow c ,}$
$\left( {\overrightarrow a - \overrightarrow b } \right)\,.\,\left( {\overrightarrow b - \overrightarrow c } \right)\, \times \,\left( {\overrightarrow c - \overrightarrow a } \right)\, = \,2\overrightarrow {a\,} .\,\overrightarrow {b\,} \times \,\overrightarrow c .$

A

TRUE

B

FALSE

Question 61: Palette 61/68

JEE Advanced 1988 MCQ

Let $\overrightarrow a ,\overrightarrow b ,\overrightarrow c ,$ be three non-coplanar vectors and $\overrightarrow p ,\overrightarrow q ,\overrightarrow r,$ are vectors defined by the relations $\overrightarrow p = {{\overrightarrow b \times \overrightarrow c } \over {\left[ {\overrightarrow a \overrightarrow b \overrightarrow c } \right]}},\,\,\overrightarrow q = {{\overrightarrow c \times \overrightarrow a } \over {\left[ {\overrightarrow a \overrightarrow b \overrightarrow c } \right]}},\,\,\overrightarrow r = {{\overrightarrow a \times \overrightarrow b } \over {\left[ {\overrightarrow a \overrightarrow b \overrightarrow c } \right]}}$ then the value of the expression $\left( {\overrightarrow a + \overrightarrow b } \right).\overrightarrow p + \left( {\overrightarrow b + \overrightarrow c } \right).\overrightarrow q + \left( {\overrightarrow c + \overrightarrow a } \right),\overrightarrow r $ is equal to

A

$0$

B

$1$

C

$2$

D

$3$

Question 62: Palette 62/68

JEE Advanced 1987 MCQ

The number of vectors of unit length perpendicular to vectors $\overrightarrow a = \left( {1,1,0} \right)$ and $\overrightarrow b = \left( {0,1,1} \right)$ is

A

one

B

two

C

three

D

infinite

Question 63: Palette 63/68

JEE Advanced 1986 MCQ

Let $\overrightarrow a = {a_1}i + {a_2}j + {a_3}k,\,\,\,\overrightarrow b = {b_1}i + {b_2}j + {b_3}k$ and $\overrightarrow c = {c_1}i + {c_2}j + {c_3}k$ be three non-zero vectors such that $\overrightarrow c $ is a unit vector perpendicular to both the vectors $\overrightarrow a $ and $\overrightarrow b .$ If the angle between $\overrightarrow a $ and $\overrightarrow b $ is ${\pi \over 6},$ then
${\left| {\matrix{ {{a_1}} & {{a_2}} & {{a_3}} \cr {{b_1}} & {{b_2}} & {{b_3}} \cr {{c_1}} & {{c_2}} & {{c_3}} \cr } } \right|^2}$ is equal to

A

$0$

B

$1$

C

${1 \over 4}\left( {a_1^2 + a_2^2 + a_2^3} \right)\left( {b_1^2 + b_2^2 + b_3^2} \right)$

D

${3 \over 4}\left( {a_1^2 + a_2^2 + a_3^2} \right)\left( {b_1^2 + b_2^2 + b_3^2} \right)\left( {c_1^2 + c_2^2 + c_3^2} \right)$

Question 64: Palette 64/68

JEE Advanced 1984 MCQ

The points with position vectors $a+b,$ $a-b,$ and $a+kb$ are collinear for all real values of $k.$

A

TRUE

B

FALSE

Question 65: Palette 65/68

JEE Advanced 1983 MCQ

If $X.A=0, X.B=0, X.C=0$ for some non-zero vector $X,$ then $\left[ {A\,B\,C} \right] = 0$

A

TRUE

B

FALSE

Question 66: Palette 66/68

JEE Advanced 1982 MCQ

For non-zero vectors ${\overrightarrow a ,\,\overrightarrow b ,\overrightarrow c },$ $\left| {\left( {\overrightarrow a \times \overrightarrow b } \right).\overrightarrow c } \right| = \left| {\overrightarrow a } \right|\left| {\overrightarrow b } \right|\left| {\overrightarrow c } \right|$ holds if and only if

A

$\overrightarrow a \,.\,\overrightarrow b = 0,\overrightarrow b \,.\,\overrightarrow c = 0$

B

$\overrightarrow b \,.\,\overrightarrow c = 0,\overrightarrow c \,.\,\overrightarrow a = 0$

C

$\overrightarrow c \,.\,\overrightarrow a = 0,\overrightarrow a \,.\,\overrightarrow b = 0$

D

$\overrightarrow a \,.\,\overrightarrow b = \overrightarrow b \,.\,\overrightarrow c = \overrightarrow c \,.\,\overrightarrow a = 0$

Question 67: Palette 67/68

JEE Advanced 1981 MCQ

The scalar $\overrightarrow A .\left( {\overrightarrow B + \overrightarrow C } \right) \times \left( {\overrightarrow A + \overrightarrow B + \overrightarrow C } \right)$ equals :

A

$0$

B

$\left[ {\overrightarrow A \,\overrightarrow B \,\overrightarrow C } \right] + \left[ {\overrightarrow B \,\overrightarrow C \,\overrightarrow A } \right]$

C

$\left[ {\overrightarrow A \,\overrightarrow B \,\overrightarrow C } \right]$

D

None of these

Question 68: Palette 68/68

JEE Advanced 1981 MCQ

Let $\overrightarrow A ,\overrightarrow B $ and ${\overrightarrow C }$ be unit vectors suppose that $\overrightarrow A .\overrightarrow B = \overrightarrow A .\overrightarrow C = 0,$ and thatthe angle between ${\overrightarrow B }$ and ${\overrightarrow C }$ is $\pi /6.$ Then $\overrightarrow A = \pm 2\left( {\overrightarrow B \times \overrightarrow C } \right).$

A

TRUE

B

FALSE