Chapter 21

For a scalar function \(\phi\), possessing continuous second order partial derivatives \(\nabla \times(\nabla \phi)=\) (A) \(\phi\) (B) 0 (C) \(\nabla \phi\) (D) none of these

For a vector function \(A\) possessing continuous second order partial derivatives, \(\nabla \cdot(\nabla \times A)=\) (A) \(A\) (B) \(\nabla \times A\) (C) 0 (D) none of these

Assertion: If \(a, b, c\) are three non-coplanar, non-zero vectors, then \((a \cdot a) b \times c+(a \cdot b) c \times a+(a \cdot c) a \times b=[b c a] a\) Reason: If the vectors \(a, b, c\) are non-coplanar, then so are \(b \times c, c \times a, a \times b\)

Given two vectors are \(\hat{i}-\hat{j}\) and \(\hat{i}+2 \hat{j}\) the unit vector coplanar with the two vectors and perpendicular to first is: (B) \(\frac{1}{\sqrt{5}}(2 \hat{i}+\hat{j})\) (C) \(\pm \frac{1}{\sqrt{2}}(\hat{i}+\hat{k})\) (D) none of these

The vector \(\hat{i}+x \hat{j}+3 \hat{k}\) is rotated through an angle \(\theta\) and doubled in magnitude, then it becomes \(4 \hat{i}+(4 x-2) \hat{j}+2 \hat{k} .\) The value of \(x\) are: \(\quad\) [2002] (A) \(\left\\{-\frac{2}{3}, 2\right\\}\) (B) \(\left\\{\frac{1}{3}, 2\right\\}\) (C) \(\left\\{\frac{2}{3}, 0\right\\}\) (D) \(\\{2,7\\}\)

If \(\left|\begin{array}{lll}a & a^{2} & 1+a^{3} \\ b & b^{2} & 1+b^{3} \\ c & c^{2} & 1+c^{3}\end{array}\right|=0\) and vectors \(\left(1, a, a^{2}\right)\left(1, b, b^{2}\right)\) and \(\left(1, c, c^{2}\right)\) are non-coplanar, then the product \(a b c\) equals [2003] (A) 2 (B) \(-1\) (C) 1 (D) 0

Let \(\vec{a}, \vec{b}\) and \(\vec{c}\) be three non-zero vectors such that no two of these are collinear. If the vector \(\vec{a}+2 \vec{b}\) is collinear with \(\vec{c}\) and \(\vec{b}+3 \vec{c}\) is collinear with \(\vec{a}(\lambda\) being some non-zero scalar) then \(\vec{a}+2 \vec{b}+6 \vec{c}\) equals [2004](A) \(\lambda \vec{a}\) (B) \(\lambda \vec{b}\) (C) \(\lambda \vec{c}\) (D) 0

A particle is acted upon by constant forces \(4 \hat{i}+\hat{j}-3 \hat{k}\) and \(3 \hat{i}+\hat{j}-\hat{k}\) which displace it from a point \(\hat{i}+2 \hat{j}+3 \hat{k}\) to the point \(5 \hat{i}+4 \hat{j}+\hat{k}\). The work done in standard units by the forces is given by \([\mathbf{2 0 0 4}]\) (A) 40 (B) 30 (C) 25 (D) 15

If \(\bar{a}, \bar{b}, \bar{c}\) are non-coplanar vectors and \(\lambda\) is a real number, then the vectors \(\bar{a}+2 \bar{b}+3 \bar{c}, \lambda \bar{b}+4 \bar{c}\) and \((2 \lambda-1) \bar{c}\) are non-coplanar for (A) all values of \(\lambda\) (B) all except one value of \(\lambda\) (C) all except two values of \(\lambda\) (D) no value or \(\lambda\)

Let \(\bar{u}, \bar{v}, \bar{w}\) be such that \(|\bar{u}|=1,|\bar{v}|=2,|\bar{w}|=3\). If the projection \(\bar{v}\) along \(\bar{u}\) is equal to that of \(\bar{w}\) along \(\bar{u}\) and \(\bar{v}, \bar{w}\) are perpendicular to each other then \(|\bar{u}-\bar{v}+\bar{w}|\) equals [2004] (A) 2 (B) \(\sqrt{7}\) (C) \(\sqrt{14}\) (D) 14

Let \(\bar{a}, \bar{b}\) and \(\bar{c}\) be non-zero vectors such that \((\bar{a} \times \bar{b}) \times \bar{c}=-|\bar{b}| \bar{c} \mid \bar{a} . \quad\) If \(\theta\) is the acute angle between the vectors \(\bar{b}\) and \(\bar{c}\) then \(\sin \theta\) equals [2004] (A) \(\frac{1}{3}\) (B) \(\frac{\sqrt{2}}{3}\) (C) \(\frac{2}{3}\) (D) \(\frac{2 \sqrt{2}}{3}\)

If \(C\) is the mid point of \(A B\) and \(P\) is any point outside \(A B\), then [2005] (A) \(\overline{P A}+\overline{P B}=2 \overrightarrow{P C}\) (B) \(\overline{P A}+\overrightarrow{P B}=\overrightarrow{P C}\) (C) \(\overrightarrow{P A}+\overrightarrow{P B}+2 \overrightarrow{P C}=0\) (D) \(\overrightarrow{P A}+\overrightarrow{P B}+\overrightarrow{P C}=0\)

The distance between the line \(\vec{r}=2 \hat{i}-2 \hat{j}+3 \hat{k}\) \(+\lambda(\hat{i}+\hat{j}+4 \hat{k})\) and the plane \(\vec{r} \cdot(\hat{i}+5 \hat{j}+\hat{k})=5\) is \([\mathbf{2 0 0 5}]\) (A) \(\frac{10}{9}\) (B) \(\frac{10}{3 \sqrt{3}}\) (C) \(\frac{3}{10}\) (D) \(\frac{10}{3}\)

For any vector \(\vec{a}\), the value of \((\vec{a} \times \hat{i})^{2}+(\vec{a} \times \hat{j})^{2}+(\vec{a} \times \hat{k})^{2}\) is equal to \([2005]\)(A) \(3 \vec{a}^{2}\) (B) \(\vec{a}^{2}\) (C) \(2 \vec{a}^{2}\) (D) \(4 \vec{a}^{2}\)

If non-zero numbers \(a, b, c\) are in H.P., then the straight line \(\frac{x}{a}+\frac{y}{b}+\frac{1}{c}=0\) always passes through a fixed point. That point is (A) \((-1,2)\) (B) \((-1,-2)\) (C) \((1,-2)\) (D) \(\left(1,-\frac{1}{2}\right)\)

Let \(a, b\) and \(c\) be distinct non-negative numbers. If the vectors \(a \hat{i}+a \hat{j}+c \hat{k}, \hat{i}+\hat{k}\) and \(c \hat{i}+c \hat{j}+b \hat{k}\) lie in a plane, then \(c\) is \([2005]\) (A) the Geometric Mean of \(a\) and \(b\) (B) the Arithmetic Mean of \(a\) and \(b\) (C) equal to zero (D) the Harmonic Mean of \(a\) and \(b\)

If \(\vec{a}, \vec{b}, \vec{c}\) are non-coplanar vectors and \(\lambda\), is a real number then \(\left[\lambda(\vec{a}+\vec{b}) \lambda^{2} \vec{b} \lambda \vec{c}\right]=[\vec{a} \vec{b}+\vec{c} \vec{b}]\) for \([\mathbf{2 0 0 5}]\) (A) exactly one value of \(\lambda\) (B) no value of \(\lambda\) (C) exactly three values of \(\lambda\) (D) exactly two values of \(\lambda\)

Let \(\vec{a}=\hat{i}-\hat{k}, \vec{b}=x \hat{i}+\hat{j}+(1-x) \hat{k} \quad\) and \(\vec{c}=y \hat{i}+x \hat{j}\) \(+(1+x-y) \hat{k}\). Then \([\vec{a}, \vec{b}, \vec{c}]\) depends on \(\quad[\mathbf{2 0 0 5}]\) (A) only \(y\) (B) only \(x\) (C) both \(x\) and \(y\) (D) neither \(x\) nor \(y\)

Let \(\vec{a}=\hat{i}-\hat{k}, \vec{b}=x \hat{i}+\hat{j}+(1-x) \hat{k} \quad\) and \(\vec{c}=y \hat{i}+x \hat{j}\) \(+(1+x-y) \hat{k}\). Then \([\vec{a}, \vec{b}, \vec{c}]\) depends on \(\quad[\mathbf{2 0 0 5}]\) (A) only \(y\) (B) only \(x\) (C) both \(x\) and \(y\) (D) neither \(x\) nor \(y\)

The values of \(\mathrm{a}\), for which the points \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) with position vectors \(2 \hat{i}-\hat{j}+\hat{k}, \hat{i}-3 \hat{j}-5 \hat{k}\) and \(a \hat{i}-3 \hat{j}+\hat{k}\) respectively are the vertices of a right-angled triangle with \(C=\frac{\pi}{2}\) are [2006] (A) 2 and 1 (B) \(-2\) and \(-1\) (C) \(-2\) and 1 (D) 2 and \(-1\)

If \(\hat{u}\) and \(\hat{v}\) are unit vectors and \(\theta\) is the acute angle between them, then \(2 \hat{u} \times 3 \hat{v}\) is a unit vector for \([\mathbf{2 0 0 7}]\)(A) exactly two values of \(\theta\) (B) more than two values of \(\theta\) (C) no value of \(\theta\) (D) exactly one value of \(\theta\)

Let \(\bar{a}=\hat{i}+\hat{j}+\hat{k}, \bar{b}=\hat{i}-\hat{j}+2 \hat{k}\) and \(\bar{c}=x \hat{i}+(x-2) \hat{j}-\hat{k}\). If the vector \(\bar{c}\) lies in the plane of \(\bar{a}\) and \(\bar{b}\), then \(x\) equals [2007] (A) 0 (B) 1 (C) \(-4\) (D) \(-2\)

The vector \(\vec{a}=\alpha \hat{i}+2 \hat{j}+\beta \hat{k}\) lies in the plane of the vectors \(\vec{b}=\hat{i}+\hat{j}\) and \(\vec{c}=\hat{j}+\hat{k}\) and bisects the angle between \(\vec{b}\) and \(\vec{c}\). Then which one of the following gives possible values of \(\alpha\) and \(\beta ?\) [2008] (A) \(\alpha=2, \beta=2\) (B) \(\alpha=1, \beta=2\) (C) \(\alpha=2, \beta=1\) (D) \(\alpha=1, \beta=1\)

The non-zero vectors \(\vec{a}, \vec{b}\) and \(\vec{c}\) are related by \(\vec{a}=8 \vec{b}\) and \(\vec{c}=7 \vec{b}\). Then the angle between \(\vec{a}\) and \(\vec{c}\) is [2008] (A) 0 (B) \(\pi / 4\) (C) \(\pi / 2\) (D) \(\pi\)

If \(\vec{u}, \vec{v}, \vec{w}\) are non-coplanar vectors and \(p, q\) are real numbers, then the equality \([3 \vec{u} p \vec{v} p \vec{w}]-[p \vec{v} \vec{w} q \vec{u}]\) \(-\left[\begin{array}{lll}2 \vec{w} & q \vec{v} & q \vec{u}\end{array}\right]=0\) holds for [2009] (A) exactly one value of \((p, q)\) (B) exactly two values of \((p, q)\) (C) more than two but not all values of \((p, q)\) (D) all values of \((p, q)\)

The projections of a vector on the three coordinate axis are \(6,-3,2\) respectively. The direction cosines of the vector are (A) \(6,-3,2\) (B) \(\frac{6}{5},-\frac{3}{5}, \frac{2}{5}\) (C) \(\frac{6}{7},-\frac{3}{7}, \frac{2}{7}\) (D) \(-\frac{6}{7},-\frac{3}{7}, \frac{2}{7}\)

Let \(\vec{a}=\vec{j}-\vec{k}\) and \(\vec{c}=\vec{i}-\vec{j}-\vec{k}\). Then, the vector \(\vec{b}\) satisfying \(\vec{a} \times \vec{b}+\vec{c}=\overrightarrow{0}\) and \(a \cdot \vec{b}=3\) is (A) \(2 \hat{i}-\hat{j}+2 \hat{k}\) (B) \(\hat{i}-\hat{j}-2 \hat{k}\) (C) \(\hat{i}+\hat{j}-2 \hat{k}\) (D) \(-\hat{i}+\hat{j}-2 \hat{k}\)

If the vectors \(\vec{a}=\hat{i}-\hat{j}+2 \hat{k}, \hat{b}=2 \hat{i}+4 \hat{j}+\hat{k}\) and \(\vec{c}=\lambda \hat{i}+\hat{j}\) \(+\mu \hat{k}\) are mutually orthogonal, then the tuple \((\lambda, \mu)=\) [2010] (A) \((2,-3)\) (B) \((-2,3)\) (C) \((3,-2)\) (D) \((-3,2)\)

The vectors \(\vec{a}\) and \(\vec{b}\) are not perpendicular and \(\vec{c}\) and \(\vec{d}\) are two vectors satisfying: \(\vec{b} \times \vec{c}=\vec{b} \times \vec{d}\) and \(\vec{a} \cdot \vec{d}=0 .\) Then, the vector \(\vec{d}\) is equal to (A) \(c+\left(\frac{a . c}{a \cdot b}\right) b\) (B) \(b+\left(\frac{b . c}{a \cdot b}\right) c\) (C) \(c-\left(\frac{a . c}{a \cdot b}\right) b\) (D) \(b-\left(\frac{b \cdot c}{a \cdot b}\right) c\)

Let \(\hat{a}\) and \(\hat{b}\) be two unit vectors. If the vectors \(\vec{c}=\hat{a}+2 \hat{b}\) and \(d=5 \hat{a}-4 \hat{b}\) are perpendicular to each other, then the angle between \(\hat{a}\) and \(\hat{b}\) is [2012] (A) \(\frac{\pi}{6}\) (B) \(\frac{\pi}{2}\) (C) \(\frac{\pi}{3}\) (D) \(\frac{\pi}{4}\)

Let \(A B C D\) be a parallelogram such that \(\overrightarrow{A B}=\vec{q}, \overrightarrow{A D}=\bar{p}\) and \(\square B A D\) be an acute angle. If \(\vec{r}\) is the vector which coincides with the altitude directed from the vertex \(B\) to the side \(A D\), then \(\vec{r}\) is given by \([2012]\) (A) \(\vec{r}=3 \vec{q}-\frac{3(\vec{p} \cdot \vec{q})}{(\vec{p} \cdot \vec{p})} \vec{p}\) (B) \(\vec{r}=-\vec{q}+\left(\frac{\vec{p} \cdot \vec{q}}{\vec{p} \cdot \vec{p}}\right) \vec{p}\)

If the vectors \(\overrightarrow{A B}=3 \hat{i}+4 \hat{k}\) and \(\overrightarrow{A C}=5 \hat{i}+2 \hat{j}+4 \hat{k}\) represent the sides of a triangle \(A B C\), then the length of the median through \(A\) is (A) \(\sqrt{72}\) (B) \(\sqrt{33}\) (C) \(\sqrt{45}\) (D) \(\sqrt{18}\)

If \([\vec{a} \times \vec{b} \vec{b} \times \vec{c} \vec{c} \times \vec{a}]=\lambda[\vec{a} \vec{b} \vec{c}]^{2}\), then the value of \(\lambda\) is equal to [2014] (A) 2 (B) 3 (C) 0 (D) 1

Let \(\vec{a}, \vec{b}\) and \(\vec{c}\) be three non-zero vectors such that no two of them are collinear and \((\vec{a} \times \vec{b}) \times \vec{c}=\frac{1}{3}|\vec{b} \| \vec{c}| \vec{a}\). If \(\theta\) is the angle between vectors \(\vec{b}\) and \(\vec{c}\), then a value of \(\sin \theta\) (A) \(\frac{-\sqrt{2}}{3}\) (B) \(\frac{2}{3}\) (C) \(\frac{-2 \sqrt{3}}{3}\) (D) \(\frac{2 \sqrt{2}}{3}\)

Let \(\vec{a}, \vec{b}\) and \(\vec{c}\) be three unit vectors such that \(\vec{a} \times(\vec{b} \times \vec{c})=\frac{\sqrt{3}}{2}(\vec{b}+\vec{c})\). if \(\vec{b}\) is not parallel to \(\vec{c}\), then the angle between \(\vec{a}\) and \(\vec{b}\) is (A) \(\frac{5 \pi}{6}\) (B) \(\frac{3 \pi}{4}\) (C) \(\frac{\pi}{2}\) (D) \(\frac{2 \pi}{3}\)