Chapter 25

If sdaa \(\vec{a}, \vec{b}, \vec{c}\) are vectors such that \(\vec{a}+\vec{b}+\vec{c}=0\) and \(|\vec{a}|=7,|\vec{b}|=5,|\vec{c}|=3\) then angle between vector \(\vec{b}\) and \(\rightarrow\) \(c\) is (a) \(60^{\circ}\) (b) \(30^{\circ}\) (c) \(45^{\circ}\) (d) \(90^{\circ}\)

If the volume of a parallelopiped, whose coterminus edges are given by the vectors \(\vec{a}=\hat{i}+\hat{j}+n \hat{k}, \vec{b}=2 \hat{i}+4 \hat{j}-n \hat{k}\) and \(\vec{c}=\hat{i}+n \hat{j}+3 \hat{k}(n \geq 0)\), is 158 cu.units, then: \([\) Sep. \(05,2020(\mathrm{I})]\) (a) \(\vec{a} \cdot \vec{c}=17\) (b) \(\vec{b} \cdot \vec{c}=10\) (c) \(n=7\) (d) \(n=9\)

Let \(x_{0}\) be the point of local maxima of \(f(x)=\vec{a} \cdot(\vec{b} \times \vec{c})\), where \(\vec{a}=x \hat{i}-2 \hat{j}+3 \hat{k}, \vec{b}=-2 \hat{i}+x \hat{j}-\hat{k}\) and \(\vec{c}=7 \hat{i}-2 \hat{j}+x \hat{k}\) Then the value of \(\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}\) at \(x=x_{0}\) is : \(\\{\) Sep. \(04,2020(\mathrm{I})]\) (a) \(-4\) (b) \(-30\) (c) 14 (d) \(-22\)

If \(\vec{a}=2 \hat{i}+\hat{j}+2 \hat{k}\), then the value of \(|\hat{i} \times(\vec{a} \times \hat{i})|^{2}+|\hat{j} \times(\vec{a} \times \hat{j})|^{2}+|\hat{k} \times(\vec{a} \times \hat{k})|^{2}\) is equal to

Let \(\vec{a}=\hat{i}-2 \hat{j}+\hat{k}\) and \(\vec{b}=\hat{i}-\hat{j}+\hat{k}\) be two vectors. If \(\vec{c}\) is a vector such that \(\vec{b} \times \vec{c}=\vec{b} \times \vec{a}\) and \(\vec{c} \cdot \vec{a}=0\), then \(\begin{array}{ll}\vec{c} \cdot \vec{b} \text { is equal to: } & \text { [Jan. 8, 2020 (II)] }\end{array}\) (a) \(-\frac{3}{2}\) (b) \(\frac{1}{2}\) (c) \(-\frac{1}{2}\) (d) \(-1\)

Let \(\vec{a}, \vec{b}\) and \(\vec{c}\) be three unit vectors such that \(\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}\). if \(\lambda=\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}\) and \(\vec{d}=\vec{a} \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}\), then the ordered pair, \((\lambda, \vec{d})\) is equal to: \(\quad\) [Jan. 7, 2020 (II)] (a) \(\left(\frac{3}{2}, 3 \vec{a} \times \vec{c}\right)\) (b) \(\left(-\frac{3}{2}, 3 \vec{c} \times \vec{b}\right)\) (c) \(\left(\frac{3}{2}, 3 \vec{b} \times \vec{c}\right)\) (d) \(\left(-\frac{3}{2}, 3 \vec{a} \times \vec{b}\right)\)

Let \(\alpha=3 \hat{\mathrm{i}}+\hat{\mathrm{j}}\) and \(\vec{\beta}=2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+3 \hat{\mathrm{k}}\). If \(\vec{\beta}=\vec{\beta}_{1}-\vec{\beta}_{2}\), where \(\vec{\beta}_{1}\) is parallel to \(\bar{\alpha}\) and \(\vec{\beta}_{2}\) is perpendicular to \(\bar{\alpha}\), then \(\vec{\beta}_{1} \times \vec{\beta}_{2}\) is equal to: \(\quad\) [April 09, 2019 (I)] (a) \(-3 \hat{\mathrm{i}}+9 \hat{\mathrm{j}}+5 \hat{\mathrm{k}}\) (b) \(3 \hat{\mathrm{i}}-9 \hat{\mathrm{j}}-5 \hat{\mathrm{k}}\) (c) \(\frac{1}{2}(-3 \hat{\mathrm{i}}+9 \hat{\mathrm{j}}+5 \hat{\mathrm{k}})\) (d) \(\frac{1}{2}(3 \hat{\mathrm{i}}-9 \hat{\mathrm{j}}+5 \hat{\mathrm{k}})\)

The magnitude of the projection of the vector \(2 \hat{i}+3 \hat{j}+\hat{k}\) on the vector perpendicular to the plane containing the vectors \(\hat{i}+\hat{j}+\hat{k}\) and \(\hat{i}+2 \hat{j}+3 \hat{k}\), is : \([\) April 08,2019 (I)] (a) \(\frac{\sqrt{3}}{2}\) (b) \(\sqrt{6}\) (c) \(3 \sqrt{6}\) (d) \(\sqrt{\frac{3}{2}}\)

Let \(\vec{a}=3 \hat{i}+2 \hat{j}+x \hat{k}\) and \(\vec{b}=\hat{i}-\hat{j}+\hat{k}\), for some real \(x\). Then \(|\vec{a} \times \vec{b}|=\mathrm{r}\) is possible if : \(\quad\) [April08, 2019 (II)] (a) \(\sqrt{\frac{3}{2}}

Let \(\vec{a}, \vec{b}\) and \(\vec{c}\) be three unit vectors, out of which vectors \(\vec{b}\) and \(\vec{c}\) are non-parallel. If \(\alpha\) and \(\beta\) are the angles which vector \(\vec{a}\) makes with vectors \(\vec{b}\) and \(\vec{c}\) respectively and \(\vec{a} \times(\vec{b} \times \vec{c})=\frac{1}{2} \vec{b}\), then \(|\alpha-\beta|\) is equal to : [Jan. 12, 2019 (II)\\} (a) \(30^{\circ}\) (b) \(90^{\circ}\) (c) \(60^{\circ}\) (d) \(45^{\circ}\)

Let \(\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}-\hat{\mathrm{j}}, \overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}\) and \(\overrightarrow{\mathrm{c}}\) be a vector such that \(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{b}}=\overrightarrow{0}\) and \(\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{c}}=4\), then \(|\overrightarrow{\mathrm{c}}|^{2}\) is equal to: [Jan 09, 2019] (a) \(\frac{19}{2}\) (b) 9 (c) 8 (d) \(\frac{17}{2}\)

If the position vectors of the vertices \(A, B\) and \(C\) of a \(\triangle \mathrm{ABC}\) are respectively \(4 \hat{i}+7 \hat{j}+8 \hat{k}, 2 \hat{i}+3 \hat{j}+4 \hat{k}\) and \(2 \hat{i}+5 \hat{j}+7 \hat{k}\), then the position vector of the point, where 73 the bisector of \(\angle A\) meets \(B C\) is \([\) Online April \(\mathbf{1 5}\), 2018] (a) \(\frac{1}{2}(4 \hat{i}+8 \hat{j}+11 \hat{k})\) (b) \(\frac{1}{3}(6 \hat{i}+13 \hat{j}+18 \hat{k})\) (c) \(\frac{1}{4}(8 \hat{i}+14 \hat{j}+9 \hat{k})\) (d) \(\frac{1}{3}(6 \hat{i}+11 \hat{j}+15 \hat{k})\)

Let \(\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{c}=\hat{j}-\hat{k}\) and a vector \(\vec{b}\) be such that \(\vec{a} \times \vec{b}=\vec{c}\) and \(\vec{a} \cdot \vec{b}=3 .\) Then \(|\vec{b}|\) equals? [Online April 16, 2018] (a) \(\sqrt{\frac{11}{3}}\) (b) \(\frac{\sqrt{11}}{3}\) (c) \(\frac{11}{\sqrt{3}}\) (d) \(\frac{11}{3}\)

If \(\vec{a}, \vec{b}\), and \(\overrightarrow{\mathrm{c}}\) are unit vectors such that \(\vec{a}+2 \vec{b}+2 \overrightarrow{\mathbf{c}}=\overrightarrow{0}\), then \(|\vec{a} \times \overrightarrow{\mathrm{c}}|\) is equal to [Online April 15, 2018] (a) \(\frac{1}{4}\) (b) \(\frac{\sqrt{15}}{4}\) (c) \(\frac{15}{16}\) (d) \(\frac{\sqrt{15}}{16}\)

Let \(\overrightarrow{\mathrm{a}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}}\) and \(\overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}\). Let \(\overrightarrow{\mathrm{c}}\) be a vector such that \(|\overrightarrow{\mathrm{c}}-\overrightarrow{\mathrm{a}}|=3,|(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}) \times \overrightarrow{\mathrm{c}}|=3\) and the angle between \(\overrightarrow{\mathrm{c}}\) and \(\vec{a} \times \vec{b}\) be \(30^{\circ}\). Then \(\vec{a} \cdot \vec{c}\) is equal to : (a) \(\frac{1}{8}\) (b) \(\frac{25}{8}\) (c) 2 (d) 5

If the vector \(\vec{b}=3 \hat{j}+4 \hat{k}\) is written as the sum of a vec- tor \(\vec{b}_{1}\), parallel to \(\vec{a}=\hat{i}+\hat{j}\) and a vector \(\overrightarrow{b_{2}}\), perpendicu- lar to \(\vec{a}\), then \(\overrightarrow{b_{1}} \times \overline{b_{2}}\) is equal to: \([\) Online April 9, 2017] (a) \(-3 \hat{i}+3 \hat{j}-9 \hat{k}\) (b) \(6 \hat{i}-6 \hat{j}+\frac{9}{2} \hat{k}\) (c) \(-6 \hat{i}+6 \hat{j}-\frac{9}{2} \hat{k}\) (d) \(3 \hat{i}-3 \hat{j}+9 \hat{k}\)

The area (in sq. units) of the parallelogram whose diagonals are along the vectors \(8 \hat{i}-6 \hat{j}\) and \(3 \hat{i}+4 \hat{j}-12 \hat{k}\), is: [Online April 8, 2017] (a) 26 (b) 65 (c) 20 (d) 52

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 \(\Rightarrow\) the angle between a and \(b\) is: (a) \(\frac{2 \pi}{3}\) (b) \(\frac{5 \pi}{6}\) (c) \(\frac{3 \pi}{4}\) (d) \(\frac{\pi}{2}\)

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 \(\sin \theta\) is : [2015] (a) \(\frac{2}{3}\) (b) \(\frac{-2 \sqrt{3}}{3}\) (c) \(\frac{2 \sqrt{2}}{3}\) (d) \(\frac{-\sqrt{2}}{3}\)

Let a and \(b\) be two unit vectors such that \(|\vec{a}+\vec{b}|=\sqrt{3}\). If \(\overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{a}}+2 \overrightarrow{\mathrm{b}}+3(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}})\), then \(2|\overrightarrow{\mathrm{c}}|\) is equal to [Online April 10, 2015] (a) \(\sqrt{55}\) (b) \(\sqrt{37}\) (c) \(\sqrt{51}\) (d) \(\sqrt{43}\)

If \([\vec{a} \times \vec{b} \quad \vec{b} \times \vec{c} \quad \vec{c} \times \vec{a}]=\lambda[\vec{a} \vec{b} \vec{c}]^{2}\) then \(\lambda\) is equal to \([2014]\) (a) 0 (b) 1 (c) 2 (d) 3

If \(\vec{x}=3 \hat{i}-6 \hat{j}-\hat{k}, \vec{y}=\hat{i}+4 \hat{j}-3 \hat{k}\) and \(\vec{z}=3 \hat{i}-4 \hat{j}-12 \hat{k}\) then the magnitude of the projection of \(\vec{x} \times \vec{y}\) on \(\vec{z}\) is: [Online April 19, 2014] (a) 12 (b) 15 (c) 14 (d) 13

If \(\overrightarrow{\left.\mathrm{c}\right|^{2}}=60\) and \(\overrightarrow{\mathrm{c}} \times(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+5 \hat{\mathrm{k}})=\overrightarrow{0}\), then a value of \(\overrightarrow{\text { c }} \cdot(-7 \hat{i}+2 \hat{j}+3 \hat{k})\) is: \(\quad\) [Online April 11, 2014] (a) \(4 \sqrt{2}\) (b) 12 (c) 24 (d) \(12 \sqrt{2}\)

Let \(\vec{a}=2 \hat{i}+\hat{j}-2 \hat{k}, \vec{b}=\hat{i}+\hat{j}\). If \(\vec{c}\) is a vector such that \(\vec{a} \bullet \vec{c}=|\vec{c}|,|\vec{c}-\vec{a}|=2 \sqrt{2}\) and the angle between \(\vec{a} \times \vec{b}\) and \(\vec{c}\) is \(30^{\circ}\), then \(|(\vec{a} \times \vec{b}) \times \vec{c}|\) equals: [Online April 25, 2013] (a) \(\frac{1}{2}\) (b) \(\frac{3 \sqrt{3}}{2}\) (c) 3 (d) \(\frac{3}{2}\)

Statement 1: The vectors \(\vec{a}, \vec{b}\) and \(\vec{c}\) lie in the same plane if and only if \(\vec{a} \cdot(\vec{b} \times \vec{c})=0\) \(\Rightarrow\) Statement \(2:\) The vectors \(u\) and \(v\) are perpendicular if and only if \(\vec{u} \cdot \vec{v}=0\) where \(\vec{u} \times \vec{v}\) is a vector perpendicular to the plane of \(\vec{u}\) and \(\vec{v}\). [Online May 26, 2012] (a) Statement 1 is false, Statement 2 is true. (b) Statement 1 is true, Statement 2 is true, Statement 2 is correct explanation for Statement 1 . (c) Statement 1 is true, Statement 2 is false. (d) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1 .

If \(\vec{u}=\hat{j}+4 \hat{k}, \vec{v}=\hat{i}+3 \hat{k}\) and \(\vec{w}=\cos \theta \hat{i}+\sin \theta \hat{j}\) are vectors in 3 -dimensional space, then the maximum possible value of \(|\vec{u} \times \vec{v} \cdot \vec{w}|\) is \(\quad\) [Online May 12, 2012] (a) \(\sqrt{3}\) (b) 5 (c) \(\sqrt{14}\) (d) 7

Statement 1: If the points \((1,2,2),(2,1,2)\) and \((2,2, z)\) and \((1,1,1)\) are coplanar, then \(z=2\). Statement 2: If the 4 points \(P, Q, R\) and \(S\) are coplanar, then the volume of the tetrahedron \(P Q R S\) is 0 . [Online May 12, 2012] (a) Statement 1 is false,, Statement 2 is true. (b) Statement 1 is true, Statement 2 is false. (c) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation of Statement 1 . (d) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation of Statement 1 .

If \(\vec{a}=\hat{i}-2 \hat{j}+3 \hat{k}, \vec{b}=2 \hat{i}+3 \hat{j}-\hat{k}\) and \(\vec{c}=\lambda \hat{i}+\hat{j}+(2 \lambda-1) \hat{k}\) are coplanar vectors, then \(\lambda\) is equal to [Online May 7,2012] (a) 0 (b) \(-1\) (c) 2 (d) 1

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) \(\vec{c}+\left(\frac{\vec{a} \vec{c}}{\vec{a} \cdot \vec{b}}\right) \vec{b}\) (b) \(\vec{b}+\left(\frac{\vec{b} \cdot \vec{c}}{\vec{a} \vec{b}}\right) \vec{c}\) (c) \(\vec{c}-\left(\frac{\vec{a} \cdot \vec{c}}{\vec{a} \cdot \vec{b}}\right) \vec{b}\) (d) \(\vec{b}-\left(\frac{\vec{b} \cdot \vec{c}}{\vec{a} \vec{b}}\right) \vec{c}\)

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

Let \(\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}=\hat{i}-\hat{j}+2 \hat{k}\) and \(\vec{c}=x \hat{i}+(x-2) \hat{j}-\hat{k}\). If the vector \(\vec{c}\) lies in the plane of \(\vec{a}\) and \(\vec{b}\), then \(x\) equals (a) \(-4\) (b) \(-2\) (c) 0 (d) 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 \(\quad[2007]\) (a) no value of \(\theta\) (b) exactly one value of \(\theta\) (c) exactly two values of \(\theta\) (d) more than two values of \(\theta\)

If \((\bar{a} \times \bar{b}) \times \bar{c}=\bar{a} \times \bar{b} \times \bar{c})\) where \(\bar{a}, \bar{b}\) and \(\bar{c}\) are any three vectors such that \(\bar{a} \bar{b} \neq 0, \bar{b} \cdot \bar{c} \neq 0\) then \(\bar{a}\) and \(\bar{c}\) are [2006] (a) inclined at an angle of \(\frac{\pi}{3}\) between them (b) inclined at an angle of \(\frac{\pi}{6}\) between them (c) perpendicular (d) parallel

Let \(\vec{a}=\hat{i}-\hat{k}, \bar{b}=x \hat{i}+\hat{j}+(1-x) \hat{k}\) and \(\vec{c}=y \hat{i}+x \hat{j}+(1+x-y) \hat{k}\). Then \([\vec{a}, \bar{b}, \vec{c}]\) depends on \(\\{2005]\) (a) only y (b) only \(\mathrm{x}\) (c) both \(\mathrm{x}\) and \(\mathrm{y}\) (d) neither \(\mathrm{x}\) nor \(\mathrm{y}\)

If \(\vec{a}, \vec{b}, \vec{c}\) arenon coplanar vectors and \(\lambda\) is a real number then \(\quad[2005]\) \(\left[\lambda(\vec{a}+\vec{b}) \lambda^{2} \bar{b} \lambda \vec{c}\right]=\left[\begin{array}{lll}\vec{a} & \vec{b}+\vec{c} & \vec{b}\end{array}\right]\) for (a) exactly one value of \(\lambda\) (b) no value of \(\lambda\) (c) exactly three values of \(\lambda\) (d) exactly two values of \(\lambda\)

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 \([\mathbf{2 0 0 5}]\) (a) \(3 \vec{a}^{2}\) (b) \(\vec{a}^{2}\) (c) \(2 \vec{a}^{2}\) (d) \(4 \vec{a}^{2}\)

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

If \(\vec{u}, \vec{v}\) and \(\vec{w}\) are three non- coplanar vectors, then \((\vec{u}+\vec{v}-\vec{w}) \cdot(\vec{u}-\vec{v}) \times(\vec{v}-\vec{w})\) equals (a) \(3 \vec{u} \cdot \vec{v} \times \vec{w}\) (b) 0 (c) \(\vec{u} \cdot(\vec{v} \times \vec{w})\) (d) \(\vec{u} \cdot \vec{w} \times \vec{v}\).

A tetrahedron has vertices at \(\mathrm{O}(0,0,0), \mathrm{A}(1,2,1) \mathrm{B}(2,1,3)\) and \(\mathrm{C}(-1,1,2)\). Then the angle between the faces \(\mathrm{OAB}\) and ABC will be (a) \(90^{\circ}\) (b) \(\cos ^{-1}\left(\frac{19}{35}\right)\) (c) \(\cos ^{-1}\left(\frac{17}{31}\right)\) (d) \(30^{\circ}\)

Let \(\vec{u}=\hat{i}+\hat{j}, \vec{v}=\hat{i}-\hat{j}\) and \(\vec{w}=\hat{i}+2 \hat{j}+3 \hat{k}\). If \(\hat{n}\) is a unit vector such that \(\vec{u}, \hat{n}=0\) and \(\vec{v} \cdot \hat{n}=0\), then \(|\vec{w} \cdot \hat{n}|\) is equal to \(\quad[2003]\) (a) 3 (b) 0 (c) 1 (d) 2

If \(a \times b=b \times c=c \times a\) then \(a+b+c=\) (a) abc (b) \(-1\) (c) 0 (d) 2

\(\vec{a}=3 \hat{i}-5 \hat{j}\) and \(\vec{b}=6 \hat{i}+3 \hat{j}\) are two vectors and \(\vec{c}\) is a vector such that \(\vec{c}=\vec{a} \times \vec{b}\) then \(|\vec{a}|:|\vec{b}|:|\vec{c}| \quad[2002]\) (a) \(\sqrt{34}: \sqrt{45}: \sqrt{39}\) (b) \(\sqrt{34}: \sqrt{45}: 39\) (c) \(34: 39: 45\) (d) \(39: 35: 34\)

If the vectors \(\vec{c}, \vec{a}=x \hat{i}+y \hat{j}+z \hat{k}\) and \(\hat{b}=\hat{j}\) are such that \(\vec{a}, \vec{c}\) and \(\vec{b}\) form a right handed system then \(\vec{c}\) is: [2002] (a) \(z \hat{i}-x \hat{k}\) (b) \(\overrightarrow{0}\) (c) \(\hat{y j}\) (d) \(-z \hat{i}+x \hat{k}\)

If \(\vec{a}, \vec{b}, \vec{c}\) are vectors such that \([\vec{a} \vec{b} \vec{c}]=4\) then \([\vec{a} \times \vec{b} \vec{b} \times \vec{c} \vec{c} \times \vec{a}]=\) (a) 16 (b) 64 (c) 4 (d) 8

A particle just clears a wall of height \(\mathrm{b}\) at a distance a and strikes the ground at a distance \(c\) from the point of projection. The angle of projection is [2007] (a) \(\tan ^{-1} \frac{b c}{a(c-a)}\) (b) \(\tan ^{-1} \frac{b c}{a}\) (c) \(\tan ^{-1} \frac{b}{a c}\) (d) \(45^{\circ}\).

A body weighing \(13 \mathrm{~kg}\) is suspended by two strings \(5 \mathrm{~m}\) and \(12 \mathrm{~m}\) long, their other ends being fastened to the extremities of a rod \(13 \mathrm{~m}\) long. If the rod be so held that the body hangs immediately below the middle point, then tensions in the strings are (a) \(5 \mathrm{~kg}\) and \(12 \mathrm{~kg}\) (b) \(5 \mathrm{~kg}\) and \(13 \mathrm{~kg}\) (c) \(12 \mathrm{~kg}\) and \(13 \mathrm{~kg}\) (d) \(5 \mathrm{~kg}\) and \(5 \mathrm{~kg}\)

The resultant of two forces \(\mathrm{P} n\) and \(3 n\) is a force of \(7 n\). If the direction of \(3 n\) force were reversed, the resultant would be \(\sqrt{19} n\). The value of \(P\) is (a) \(3 n\) (b) \(4 n\) (c) \(5 n\) (d) \(6 n\).

A body falling from rest under gravity passes a certain point \(P\). It was at a distance of \(400 \mathrm{~m}\) from \(P, 4\) s prior to passing through \(P\). If \(g=10 \mathrm{~m} / \mathrm{s}^{2}\), then the height above the point \(P\) from where the body began to fall is [2006] (a) \(720 \mathrm{~m}\) (b) \(900 \mathrm{~m}\) (c) \(320 \mathrm{~m}\) (d) \(680 \mathrm{~m}\)

A particle has two velocities of equal magnitude inclined to each other at an angle \(\theta\). If one of them is halved, the angle between the other and the original resultant velocity is bisected by the new resultant. Then \(\theta\) is \(\\{2006]\) (a) \(90^{\circ}\) (b) \(120^{\circ}\) (c) \(45^{\circ}\) (d) \(60^{\circ}\)

The resultant \(R\) of two forces acting on a particle is at right angles to one of them and its magnitude is one third of the other force. The ratio of larger force to smaller one is: \([2005]\) (a) \(2: 1\) (b) \(3: \sqrt{2}\) (c) \(3: 2\) (d) \(3: 2 \sqrt{2}\)