MCQOPTIONS
Saved Bookmarks
MCQOPTIONS
This section includes 8666 Mcqs, each offering curated multiple-choice questions to sharpen your Joint Entrance Exam - Main (JEE Main) knowledge and support exam preparation. Choose a topic below to get started.
| 1601. |
The angle between the vectors \[\mathbf{i}-\mathbf{j}+\mathbf{k}\] and \[\mathbf{i}+2\mathbf{j}+\mathbf{k}\] is [BIT Ranchi 1991] |
| A. | \[{{\cos }^{-1}}\left( \frac{1}{\sqrt{15}} \right)\] |
| B. | \[{{\cos }^{-1}}\left( \frac{4}{\sqrt{15}} \right)\] |
| C. | \[{{\cos }^{-1}}\left( \frac{4}{15} \right)\] |
| D. | \[\frac{\pi }{2}\] |
| Answer» E. | |
| 1602. |
If the angle between a and b be \[{{30}^{o}}\], then the angle between 3 a and ? 4 b will be |
| A. | \[{{150}^{o}}\] |
| B. | \[{{90}^{o}}\] |
| C. | \[{{120}^{o}}\] |
| D. | \[{{30}^{o}}\] |
| Answer» B. \[{{90}^{o}}\] | |
| 1603. |
If q be the angle between the unit vectors a and b, then \[\mathbf{a}-\sqrt{2}\,\mathbf{b}\] will be a unit vector if \[\theta =\] |
| A. | \[\frac{\pi }{6}\] |
| B. | \[\frac{\pi }{4}\] |
| C. | \[p\mathbf{i}+q\mathbf{j}+r\mathbf{k}\] |
| D. | \[\frac{2\pi }{3}\] |
| Answer» C. \[p\mathbf{i}+q\mathbf{j}+r\mathbf{k}\] | |
| 1604. |
The angle between the vectors \[3\,\mathbf{i}+\mathbf{j}+2\,\mathbf{k}\] and \[2\,\mathbf{i}-2\,\mathbf{j}+4\,\mathbf{k}\] is [MP PET 1990] |
| A. | \[{{\cos }^{-1}}\frac{2}{\sqrt{7}}\] |
| B. | \[{{\sin }^{-1}}\frac{2}{\sqrt{7}}\] |
| C. | \[{{\cos }^{-1}}\frac{2}{\sqrt{5}}\] |
| D. | \[{{\sin }^{-1}}\frac{2}{\sqrt{5}}\] |
| Answer» C. \[{{\cos }^{-1}}\frac{2}{\sqrt{5}}\] | |
| 1605. |
If a, b, c are unit vectors such that \[\mathbf{a}+\mathbf{b}+\mathbf{c}=\mathbf{0},\] then \[\mathbf{a}\,\,.\,\,\mathbf{b}+\mathbf{b}\,\,.\,\,\mathbf{c}+\mathbf{c}\,\,.\,\,\mathbf{a}=\] [MP PET 1988; Karnataka CET 2000; UPSEAT 2003, 04] |
| A. | 1 |
| B. | 3 |
| C. | ? 3/2 |
| D. | 3/2 |
| Answer» D. 3/2 | |
| 1606. |
\[\mathbf{a},\,\mathbf{b}\] and c are three vectors with magnitude \[|\mathbf{a}|\,=4,\] \[|\mathbf{b}|\,=4,\] \[|\mathbf{c}|\,=2\] and such that \[\mathbf{a}\] is perpendicular to \[(\mathbf{b}+\mathbf{c}),\,\mathbf{b}\] is perpendicular to \[(\mathbf{c}+\mathbf{a})\] and \[\mathbf{c}\] is perpendicular to \[(\mathbf{a}+\mathbf{b}).\] It follows that \[|\mathbf{a}+\mathbf{b}+\mathbf{c}|\] is equal to [UPSEAT 2004] |
| A. | 9 |
| B. | 6 |
| C. | 5 |
| D. | 4 |
| Answer» C. 5 | |
| 1607. |
\[a,\,b,\,c\] are three vectors, such that \[a+b+c=0\], \[|a|\,=1,\,|b|\,=2,\,|c|\,=3\], then \[a.b+b.c+c.a\] is equal to [AIEEE 2003] |
| A. | 0 |
| B. | ? 7 |
| C. | 7 |
| D. | 1 |
| Answer» C. 7 | |
| 1608. |
If \[|\mathbf{a}|\,=3,\,\,|\mathbf{b}|\,=4\] then a value of l for which \[\mathbf{a}+\lambda \mathbf{b}\] is perpendicular to \[\mathbf{a}-\lambda \mathbf{b}\] is [Karnataka CET 2004] |
| A. | \[\mathbf{a}=2\,\mathbf{i}+2\,\mathbf{j}+3\,\mathbf{k},\,\,\mathbf{b}=-\mathbf{i}+2\,\mathbf{j}+\mathbf{k}\] |
| B. | \[\frac{3}{4}\] |
| C. | \[\frac{3}{2}\] |
| D. | \[\frac{4}{3}\] |
| Answer» C. \[\frac{3}{2}\] | |
| 1609. |
If \[|a|\,=\,|\mathbf{b}|,\] then \[(a+b)\,.\,(a-b)\] is [MP PET 2002] |
| A. | Positive |
| B. | Negative |
| C. | Zero |
| D. | None of these |
| Answer» D. None of these | |
| 1610. |
If i, j, k are unit vectors, then [MP PET 2001] |
| A. | i . j \[=\]1 |
| B. | i . i \[=\]1 |
| C. | \[\mathbf{i}\times \mathbf{j}=1\] |
| D. | \[\mathbf{i}\times (\mathbf{j}\times \mathbf{k})=1\] |
| Answer» C. \[\mathbf{i}\times \mathbf{j}=1\] | |
| 1611. |
If x and y are two unit vectors and \[\pi \] is the angle between them, then \[\frac{1}{2}|x-y|\] is equal to [UPSEAT 2001] |
| A. | 0 |
| B. | \[\pi /2\] |
| C. | 1 |
| D. | \[\pi /4\] |
| Answer» D. \[\pi /4\] | |
| 1612. |
If \[a\,.\,i=a\,.\,(i+j)=a\,.\,(i+j+k)\], then a = [EAMCET 2002] |
| A. | i |
| B. | k |
| C. | j |
| D. | i + j + k |
| Answer» B. k | |
| 1613. |
If a and b are adjacent sides of a rhombus, then [RPET 2001] |
| A. | a.b = 0 |
| B. | a × b = 0 |
| C. | a.a = b.b |
| D. | None of these |
| Answer» D. None of these | |
| 1614. |
If a and b be unlike vectors, then a . b = |
| A. | | a | | b | |
| B. | ? | a | | b | |
| C. | 0 |
| D. | None of these |
| Answer» C. 0 | |
| 1615. |
Let a, b and c be vectors with magnitudes 3, 4 and 5 respectively and a + b + c = 0, then the values of a.b + b.c + c.a is [IIT 1995; DCE 2001; AIEEE 2002; UPSEAT 2002; Kerala (Engg.) 2005] |
| A. | 47 |
| B. | 25 |
| C. | 50 |
| D. | ? 25 |
| Answer» E. | |
| 1616. |
If \[a=(1,\,-1,\,2),\ b=(-2,\,3,\,5)\], \[\mathbf{c}=(2\,,\,-2,\,4)\] and i is the unit vector in the x-direction, then \[(a-2b+3c)\,.\,i=\] [Karnataka CET 2001] |
| A. | 11 |
| B. | 15 |
| C. | 18 |
| D. | 36 |
| Answer» B. 15 | |
| 1617. |
For any three non-zero vectors \[{{r}_{1}},\,{{r}_{2}}\] and \[{{r}_{3}}\], \[\left| \,\begin{matrix} {{r}_{1}}\,.\,{{r}_{1}} & {{r}_{1}}\,.\,{{r}_{2}} & {{r}_{1}}\,.\,{{r}_{3}} \\ {{r}_{2}}\,.\,{{r}_{1}} & {{r}_{2}}\,.\,{{r}_{2}} & {{r}_{2}}\,.\,{{r}_{3}} \\ {{r}_{3}}\,.\,{{r}_{1}} & {{r}_{3}}\,.\,{{r}_{2}} & {{r}_{3}}\,.\,{{r}_{3}} \\ \end{matrix} \right|=0\]. Then which of the following is false [AMU 2000] |
| A. | All the three vectors are parallel to one and the same plane |
| B. | All the three vectors are linearly dependent |
| C. | This system of equation has a non-trivial solution |
| D. | All the three vectors are perpendicular to each other |
| Answer» B. All the three vectors are linearly dependent | |
| 1618. |
(a .b) c and (a.c) b are [RPET 2000] |
| A. | Two like vectors |
| B. | Two equal vectors |
| C. | Two vectors in direction of a |
| D. | None of these |
| Answer» E. | |
| 1619. |
If vectors \[\mathbf{a},\,b,\,\mathbf{c}\] satisfy the condition \[|\mathbf{a}-\mathbf{c}|=|\mathbf{b}-\mathbf{c}|\], then \[(\mathbf{b}-\mathbf{a})\,.\,\left( \mathbf{c}-\frac{\mathbf{a}+\mathbf{b}}{\mathbf{2}} \right)\]is equal to [AMU 1999] |
| A. | 0 |
| B. | ?1 |
| C. | 1 |
| D. | 2 |
| Answer» B. ?1 | |
| 1620. |
If a is any vector in space, then [MP PET 1997] |
| A. | \[\mathbf{a}=(\mathbf{a}\,.\,\mathbf{i})\,\mathbf{i}+\,(\mathbf{a}\,.\,\mathbf{j})\,\mathbf{j}+\,(\mathbf{a}\,.\,\mathbf{k})\,\mathbf{k}\] |
| B. | \[\mathbf{a}=(\mathbf{a}\,\times \,\mathbf{i})+\,(\mathbf{a}\,\times \,\mathbf{j})\,+\,(\mathbf{a}\,\times \,\mathbf{k})\,\] |
| C. | \[\mathbf{a}=\mathbf{j}\,(\mathbf{a}\,.\,\mathbf{i})\,+\mathbf{k}\,(\mathbf{a}\,.\,\mathbf{j})\,+\,\mathbf{i}\,(\mathbf{a}\,.\,\mathbf{k})\,\] |
| D. | \[\mathbf{a}=(\mathbf{a}\,\times \,\mathbf{i})\times \mathbf{i}+\,(\mathbf{a}\,\times \,\mathbf{j})\times \mathbf{j}\,+\,(\mathbf{a}\,\times \,\mathbf{k})\times \mathbf{k}\,\] |
| Answer» B. \[\mathbf{a}=(\mathbf{a}\,\times \,\mathbf{i})+\,(\mathbf{a}\,\times \,\mathbf{j})\,+\,(\mathbf{a}\,\times \,\mathbf{k})\,\] | |
| 1621. |
A, B, C, D are any four points, then \[\overrightarrow{AB}\,\,.\,\,\overrightarrow{CD}\,\,+\,\overrightarrow{\,BC}\,\,.\,\,\overrightarrow{AD}\,\,+\overrightarrow{CA}\,\,.\,\,\overrightarrow{BD}\,\,=\] [MNR 1986] |
| A. | \[2\,\,\overrightarrow{AB}\,\,.\,\,\overrightarrow{BC}\,\,.\,\,\overrightarrow{CD}\] |
| B. | \[\overrightarrow{AB}\,\,+\,\,\overrightarrow{BC}\,\,+\,\,\overrightarrow{CD}\] |
| C. | \[5\sqrt{3}\] |
| D. | 0 |
| Answer» E. | |
| 1622. |
If in a right angled triangle ABC, the hypotenuse \[AB=p,\] then \[\overrightarrow{AB}\,\,.\,\,\overrightarrow{AC}+\overrightarrow{BC}\,.\,\,\overrightarrow{BA}+\overrightarrow{CA}\,\,.\,\,\overrightarrow{CB}\] is equal to |
| A. | \[2{{p}^{2}}\] |
| B. | \[\frac{{{p}^{2}}}{2}\] |
| C. | \[{{p}^{2}}\] |
| D. | None of these |
| Answer» D. None of these | |
| 1623. |
If ABCDEF is regular hexagon, the length of whose side is a, then \[\overrightarrow{AB}\,\,.\,\overrightarrow{AF}+\frac{1}{2}\,{{\overrightarrow{BC}}^{2}}=\] |
| A. | a |
| B. | \[{{a}^{2}}\] |
| C. | \[2\,{{a}^{2}}\] |
| D. | 0 |
| Answer» E. | |
| 1624. |
If a, b, c are non-zero vectors such that \[\mathbf{a}\,\,.\,\,\mathbf{b}=\mathbf{a}\,\,.\,\,\mathbf{c},\] then which statement is true [RPET 2001] |
| A. | b = c |
| B. | \[\mathbf{a}\,\bot \,(\mathbf{b}-\mathbf{c})\] |
| C. | \[\mathbf{b}=\mathbf{c}\] or \[\mathbf{a}\,\bot \,(\mathbf{b}-\mathbf{c})\] |
| D. | None of these |
| Answer» D. None of these | |
| 1625. |
\[\mathbf{a}\,.\,\mathbf{b}=0,\] then [RPET 1995] |
| A. | \[\mathbf{a}\,\bot \,\mathbf{b}\] |
| B. | a || b |
| C. | Angle between a and b is \[{{60}^{o}}\] |
| D. | None of these |
| Answer» B. a || b | |
| 1626. |
If a and b are mutually perpendicular vectors, then \[{{(\mathbf{a}+\mathbf{b})}^{2}}=\] [MP PET 1994; Pb. CET 2002] |
| A. | \[\mathbf{a}+\mathbf{b}\] |
| B. | \[\mathbf{a}-\mathbf{b}\] |
| C. | \[{{a}^{2}}-{{b}^{2}}\] |
| D. | \[{{(\mathbf{a}-\mathbf{b})}^{2}}\] |
| Answer» E. | |
| 1627. |
The horizontal force and the force inclined at an angle \[{{60}^{o}}\] with the vertical, whose resultant is in vertical direction of P kg, are [IIT 1983] |
| A. | \[P,2P\] |
| B. | \[P,\,\,P\sqrt{3}\] |
| C. | \[2P,\,\,P\sqrt{3}\] |
| D. | None of these |
| Answer» D. None of these | |
| 1628. |
If \[\mathbf{d}=\lambda \,(\mathbf{a}\times \mathbf{b})+\mu \,(\mathbf{b}\times \mathbf{c})+\nu \,(\mathbf{c}\times \mathbf{a})\]and \[[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]=\frac{1}{8},\] then \[\lambda +\mu +\nu \] is equal to |
| A. | \[8\mathbf{d}\,.\,(\mathbf{a}+\mathbf{b}+\mathbf{c})\] |
| B. | \[8\mathbf{d}\,\times \,(\mathbf{a}+\mathbf{b}+\mathbf{c})\] |
| C. | \[\frac{\mathbf{d}\,}{8}.\,(\mathbf{a}+\mathbf{b}+\mathbf{c})\] |
| D. | \[\frac{\mathbf{d}\,}{8}\times \,(\mathbf{a}+\mathbf{b}+\mathbf{c})\] |
| Answer» B. \[8\mathbf{d}\,\times \,(\mathbf{a}+\mathbf{b}+\mathbf{c})\] | |
| 1629. |
If \[\mathbf{p}=\mathbf{i}-2\mathbf{j}+3\mathbf{k}\] and \[\mathbf{q}=3\mathbf{i}+\mathbf{j}+2\mathbf{k},\] then a vector along r which is linear combination of p and q and also perpendicular to q is [MNR 1986] |
| A. | \[\mathbf{i}+5\mathbf{j}-4\mathbf{k}\] |
| B. | \[\mathbf{i}-5\mathbf{j}+4\mathbf{k}\] |
| C. | \[-\frac{1}{2}\,(\mathbf{i}+5\mathbf{j}-4\mathbf{k})\] |
| D. | None of these |
| Answer» D. None of these | |
| 1630. |
If \[\vec{\lambda }\] is a unit vector perpendicular to plane of vector a and b and angle between them is q, then a . b will be [RPET 1985] |
| A. | \[|\mathbf{a}|\,\,|\mathbf{b}|\,\sin \theta \,\vec{\lambda }\] |
| B. | \[|\mathbf{a}|\,\,|\mathbf{b}|\,\cos \,\,\theta \,\vec{\lambda }\] |
| C. | \[|\mathbf{a}|\,\,|\mathbf{b}|\,\cos \,\,\theta \,\] |
| D. | \[|\mathbf{a}|\,\,|\mathbf{b}|\,\sin \theta \,\] |
| Answer» D. \[|\mathbf{a}|\,\,|\mathbf{b}|\,\sin \theta \,\] | |
| 1631. |
If a, b, c are coplanar vectors, then [IIT 1989] |
| A. | \[\left| \,\begin{matrix} \mathbf{a} & \mathbf{b} & \mathbf{c} \\ \mathbf{b} & \mathbf{c} & \mathbf{a} \\ \mathbf{c} & \mathbf{a} & \mathbf{b} \\ \end{matrix}\, \right|=\mathbf{0}\] |
| B. | \[\left| \,\begin{matrix} \mathbf{a} & \mathbf{b} & \mathbf{c} \\ \mathbf{a}\,.\,\mathbf{a} & \mathbf{a}\,.\,\mathbf{b} & \mathbf{a}\,.\,\mathbf{c} \\ \mathbf{b}\,.\,\mathbf{a} & \mathbf{b}\,.\,\mathbf{b} & \mathbf{b}\,.\,\mathbf{c} \\ \end{matrix}\, \right|=\mathbf{0}\] |
| C. | \[\left| \,\begin{matrix} \mathbf{a} & \mathbf{b} & \mathbf{c} \\ \mathbf{c}\,.\,\mathbf{a} & \mathbf{c}\,.\,\mathbf{b} & \mathbf{c}\,.\,\mathbf{c} \\ \mathbf{b}\,.\,\mathbf{a} & \mathbf{b}\,.\,\mathbf{c} & \mathbf{b}\,.\,\mathbf{b} \\ \end{matrix}\, \right|=\mathbf{0}\] |
| D. | \[\left| \,\begin{matrix} \mathbf{a} & \mathbf{b} & \mathbf{c} \\ \mathbf{a}\,.\,\mathbf{b} & \mathbf{a}\,.\,\mathbf{a} & \mathbf{a}\,.\,\mathbf{c} \\ \mathbf{c}\,.\,\mathbf{a} & \mathbf{c}\,.\,\mathbf{c} & \mathbf{c}\,.\,\mathbf{b} \\ \end{matrix}\, \right|=\mathbf{0}\] |
| Answer» C. \[\left| \,\begin{matrix} \mathbf{a} & \mathbf{b} & \mathbf{c} \\ \mathbf{c}\,.\,\mathbf{a} & \mathbf{c}\,.\,\mathbf{b} & \mathbf{c}\,.\,\mathbf{c} \\ \mathbf{b}\,.\,\mathbf{a} & \mathbf{b}\,.\,\mathbf{c} & \mathbf{b}\,.\,\mathbf{b} \\ \end{matrix}\, \right|=\mathbf{0}\] | |
| 1632. |
A vector whose modulus is \[\sqrt{51}\] and makes the same angle with \[\mathbf{a}=\frac{\mathbf{i}-2\mathbf{j}+2\mathbf{k}}{3},\,\,\mathbf{b}=\frac{-\,4\mathbf{i}-3\mathbf{k}}{5}\] and \[\mathbf{c}=\mathbf{j},\] will be [Roorkee 1987] |
| A. | \[5\mathbf{i}+5\mathbf{j}+\mathbf{k}\] |
| B. | \[5\mathbf{i}+\mathbf{j}-5\mathbf{k}\] |
| C. | \[5\mathbf{i}+\mathbf{j}+5\mathbf{k}\] |
| D. | \[\pm \,(5\mathbf{i}-\mathbf{j}-5\mathbf{k})\] |
| Answer» E. | |
| 1633. |
If \[\mathbf{r}\,.\,\mathbf{i}=\mathbf{r}\,.\,\mathbf{j}=\mathbf{r}\,.\,\mathbf{k}\] and \[|\mathbf{r}|\,\,=3,\] then \[\mathbf{r}=\] |
| A. | \[\pm \,3\,(\mathbf{i}+\mathbf{j}+\mathbf{k})\] |
| B. | \[\pm \,\frac{1}{3}\,(\mathbf{i}+\mathbf{j}+\mathbf{k})\] |
| C. | \[\pm \,\frac{1}{\sqrt{3}}\,(\mathbf{i}+\mathbf{j}+\mathbf{k})\] |
| D. | \[\pm \,\sqrt{3}\,(\mathbf{i}+\mathbf{j}+\mathbf{k})\] |
| Answer» E. | |
| 1634. |
If the moduli of a and b are equal and angle between them is \[{{120}^{o}}\] and \[\mathbf{a}\,.\,\mathbf{b}=-\,8,\] then | a | is equal to [RPET 1986] |
| A. | ? 5 |
| B. | ? 4 |
| C. | 4 |
| D. | 5 |
| Answer» D. 5 | |
| 1635. |
If \[\overrightarrow{{{F}_{1}}}=\mathbf{i}-\mathbf{j}+\mathbf{k},\] \[\overrightarrow{{{F}_{2}}}=-\mathbf{i}+2\mathbf{j}-\mathbf{k},\] \[\overrightarrow{{{F}_{3}}}=\mathbf{j}-\mathbf{k},\] \[\vec{A}=4\mathbf{i}-3\mathbf{j}-2\mathbf{k}\] and \[\vec{B}=6\mathbf{i}+\mathbf{j}-3\mathbf{k},\] then the scalar product of \[\overrightarrow{{{F}_{1}}}+\overrightarrow{{{F}_{2}}}+\overrightarrow{{{F}_{3}}}\]and \[\overrightarrow{AB}\] will be [Roorkee 1980] |
| A. | 3 |
| B. | 6 |
| C. | 9 |
| D. | 12 |
| Answer» D. 12 | |
| 1636. |
The value of b such that scalar product of the vectors \[(\mathbf{i}+\mathbf{j}+\mathbf{k})\] with the unit vector parallel to the sum of the vectors \[(2\mathbf{i}+4\mathbf{j}-5\mathbf{k})\] and \[(b\mathbf{i}+2\mathbf{j}+3\mathbf{k})\] is 1, is [MNR 1992; Roorkee 1985, 95; Kurukshetra CEE 1998; UPSEAT 2000] |
| A. | ? 2 |
| B. | ? 1 |
| C. | 0 |
| D. | 1 |
| Answer» E. | |
| 1637. |
If the vector \[\mathbf{i}+\mathbf{j}+\mathbf{k}\] makes angles \[\alpha ,\,\beta ,\,\gamma \]with vectors \[\mathbf{i},\,\mathbf{j},\mathbf{k}\]respectively, then |
| A. | \[\alpha =\beta \ne \gamma \] |
| B. | \[\alpha =\gamma \ne \beta \] |
| C. | \[\beta =\gamma \ne \alpha \] |
| D. | \[\alpha =\beta =\gamma \] |
| Answer» E. | |
| 1638. |
\[{{(\mathbf{r}\,.\,\mathbf{i})}^{2}}+{{(\mathbf{r}\,.\,\mathbf{j})}^{2}}+{{(\mathbf{r}\,.\,\mathbf{k})}^{2}}=\] |
| A. | \[3{{r}^{2}}\] |
| B. | \[{{r}^{2}}\] |
| C. | 0 |
| D. | None of these |
| Answer» C. 0 | |
| 1639. |
If the angle between the vectors a and b be q and \[\mathbf{a}\,.\,\mathbf{b}=\cos \theta ,\] then the true statement is |
| A. | a and b are equal vectors |
| B. | a and b are like vectors |
| C. | a and b are unlike vectors |
| D. | a and b are unit vectors |
| Answer» E. | |
| 1640. |
If a, b, c are three vectors such that \[\mathbf{a}=\mathbf{b}+\mathbf{c}\] and the angle between b and c is \[\pi /2,\] then [EAMCET 2003] |
| A. | \[{{a}^{2}}={{b}^{2}}+{{c}^{2}}\] |
| B. | \[{{b}^{2}}={{c}^{2}}+{{a}^{2}}\] |
| C. | \[{{c}^{2}}={{a}^{2}}+{{b}^{2}}\] |
| D. | \[2{{a}^{2}}-{{b}^{2}}={{c}^{2}}\] (Note : Here \[a=\,\,|\mathbf{a}|,\,\,b=\,|\,\mathbf{b}|,\,\,c=\,|\mathbf{c}|)\] |
| Answer» B. \[{{b}^{2}}={{c}^{2}}+{{a}^{2}}\] | |
| 1641. |
If \[|\mathbf{a}+\mathbf{b}|\,\,>\,\,|\mathbf{a}-\mathbf{b}|,\] then the angle between a and b is |
| A. | Acute |
| B. | Obtuse |
| C. | \[\frac{\pi }{2}\] |
| D. | \[\pi \] |
| Answer» B. Obtuse | |
| 1642. |
If \[|\mathbf{a}|\,=3,\,\,|\mathbf{b}|\,=4,\,\,|\mathbf{c}|\,=\,\,5\] and \[\mathbf{a}+\mathbf{b}+\mathbf{c}=0,\] then the angle between a and b is [MP PET 1989; Bihar CEE 1994] |
| A. | 0 |
| B. | \[\frac{\pi }{6}\] |
| C. | \[\frac{\pi }{3}\] |
| D. | \[\frac{\pi }{2}\] |
| Answer» E. | |
| 1643. |
If q be the angle between the unit vectors a and b, then \[\cos \frac{\theta }{2}=\] [MP PET 1998; Pb. CET 2002] |
| A. | \[\frac{1}{2}\,|\mathbf{a}-\mathbf{b}|\] |
| B. | \[\frac{1}{2}\,|\mathbf{a}+\mathbf{b}|\] |
| C. | \[\frac{|\mathbf{a}-\mathbf{b}|}{|\mathbf{a}+\mathbf{b}|}\] |
| D. | \[\frac{|\mathbf{a}+\mathbf{b}|}{|\mathbf{a}-\mathbf{b}|}\] |
| Answer» C. \[\frac{|\mathbf{a}-\mathbf{b}|}{|\mathbf{a}+\mathbf{b}|}\] | |
| 1644. |
\[(\mathbf{a}\,.\,\mathbf{i})\,\mathbf{i}+(\mathbf{a}\,.\,\mathbf{j})\mathbf{j}+(\mathbf{a}\,.\,\mathbf{k})\,\mathbf{k}=\] [Karnataka CET 2004] |
| A. | a |
| B. | 2 a |
| C. | 0 |
| D. | None of these |
| Answer» B. 2 a | |
| 1645. |
If \[\mathbf{p}=\frac{\mathbf{b}\times \mathbf{c}}{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]},\,\,\mathbf{q}=\frac{\mathbf{c}\times \mathbf{a}}{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]},\,\,\mathbf{r}=\frac{\mathbf{a}\times \mathbf{b}}{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]},\,\,\]where a, b, c are three non-coplanar vectors, then the value of \[(\mathbf{a}+\mathbf{b}+\mathbf{c})\,.\,(\mathbf{p}+\mathbf{q}+\mathbf{r})\] is given by [MNR 1992; UPSEAT 2000] |
| A. | 3 |
| B. | 2 |
| C. | 1 |
| D. | 0 |
| Answer» B. 2 | |
| 1646. |
If a, b, c are the three non-coplanar vectors and p, q, r are defined by the relations \[\mathbf{p}=\frac{\mathbf{b}\times \mathbf{c}}{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]},\,\,\mathbf{q}=\frac{\mathbf{c}\times \mathbf{a}}{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]},\,\,\mathbf{r}=\frac{\mathbf{a}\times \mathbf{b}}{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]}\] then (a+b) . p +(b+c) . q +(c+a) . r = [IIT 1988; BIT Mesra 1996; AMU 2002] |
| A. | 0 |
| B. | 1 |
| C. | 2 |
| D. | 3 |
| Answer» E. | |
| 1647. |
If the points whose position, vectors are \[3\mathbf{i}-2\mathbf{j}-\mathbf{k},\] \[2\mathbf{i}+3\mathbf{j}-4\mathbf{k},\] \[-\mathbf{i}+\mathbf{j}+2\mathbf{k}\]and \[4\mathbf{i}+5\mathbf{j}+\lambda \mathbf{k}\] lie on a plane, then \[\lambda =\] [IIT 1986; Pb. CET 2003] |
| A. | \[-\frac{146}{17}\] |
| B. | \[\frac{146}{17}\] |
| C. | \[-\frac{17}{146}\] |
| D. | \[\frac{17}{146}\] |
| Answer» B. \[\frac{146}{17}\] | |
| 1648. |
If a vector a lie in the plane b and g then which is correct [Orissa JEE 2005] |
| A. | \[[\alpha \,\,\beta \,\,\gamma ]=0\] |
| B. | \[[\alpha \,\,\beta \,\,\gamma ]=1\] |
| C. | \[[\alpha \,\,\beta \,\,\gamma ]=3\] |
| D. | \[[\beta \,\,\gamma \,\,\alpha ]=1\] |
| Answer» B. \[[\alpha \,\,\beta \,\,\gamma ]=1\] | |
| 1649. |
If a,b,c are three non-zero, non-coplanar vectors and \[{{b}_{1}}=b-\frac{b.a}{|a{{|}^{2}}}a,\,{{b}_{2}}=b+\frac{b.a}{|a{{|}^{2}}}a\],\[{{c}_{1}}=c-\frac{c.a}{|a{{|}^{2}}}a-\frac{c.b}{|b{{|}^{2}}}b\], \[{{c}_{2}}=c-\frac{c.a}{|a{{|}^{2}}}a\frac{c.{{b}_{1}}}{|{{b}_{1}}{{|}^{2}}}{{b}_{1}}\], \[{{c}_{3}}=c-\frac{c.a}{|a{{|}^{2}}}a\frac{c.{{b}_{2}}}{|{{b}_{2}}{{|}^{2}}}{{b}_{2}}\], \[{{c}_{4}}=a-\frac{c.a}{|a{{|}^{2}}}a\]. Then which of the following is a set of mutually orthogonal vectors is [IIT Screening 2005] |
| A. | \[\{\mathbf{a},\,{{\mathbf{b}}_{1}},\,{{c}_{1}}\}\] |
| B. | \[\{\mathbf{a},\,{{\mathbf{b}}_{1}},\,{{c}_{2}}\}\] |
| C. | \[\{\mathbf{a},\,{{\mathbf{b}}_{2}},\,{{c}_{3}}\}\] |
| D. | \[\{\mathbf{a},\,{{\mathbf{b}}_{2}},\,{{c}_{4}}\}\] |
| Answer» C. \[\{\mathbf{a},\,{{\mathbf{b}}_{2}},\,{{c}_{3}}\}\] | |
| 1650. |
If the vectors \[2\mathbf{i}+\mathbf{j}-\mathbf{k},\,-\mathbf{i}+2\mathbf{j}+\lambda \mathbf{k}\] and \[-5\mathbf{i}+2\mathbf{j}-\mathbf{k}\] are coplanar, then the value of \[\lambda \] is equal [J & K 2005] |
| A. | ? 13 |
| B. | 13/9 |
| C. | ? 13/9 |
| D. | ? 9/13 |
| Answer» D. ? 9/13 | |