MCQOPTIONS
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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.
| 1651. |
If \[\mathbf{a},\,\mathbf{b},\,\mathbf{c}\] are non-coplanar vectors and \[\lambda \] is a real number then \[[\lambda (\mathbf{a}+\mathbf{b})\,\,\,\,{{\lambda }^{2}}\mathbf{b}\,\,\,\,\,\lambda \mathbf{c}]=\left[ \mathbf{a}\,\,\mathbf{b}+\mathbf{c}\,\,\mathbf{b} \right]\] for [AIEEE 2005] |
| A. | Exactly three values of \[\lambda \] |
| B. | Exactly two values of \[\lambda \] |
| C. | Exactly one value of \[\lambda \] |
| D. | No value of \[\lambda \] |
| Answer» E. | |
| 1652. |
If \[\mathbf{a}\] is perpendicular to \[\mathbf{b}\]and \[\mathbf{c},|\mathbf{a}|=2,|\mathbf{b}|=3\], \[|\mathbf{c}|=4\] and the angle between \[\mathbf{b}\] and \[\mathbf{c}\]is \[\frac{2\pi }{3}\], then \[[\mathbf{a}\ \mathbf{b}\ \mathbf{c}]\] is equal to [Kerala (Engg.) 2005] |
| A. | \[4\sqrt{3}\] |
| B. | \[6\sqrt{3}\] |
| C. | \[12\sqrt{3}\] |
| D. | \[18\sqrt{3}\] |
| E. | \[8\sqrt{3}\] |
| Answer» D. \[18\sqrt{3}\] | |
| 1653. |
If \[\mathbf{a}=\mathbf{i}+\mathbf{j}-\mathbf{k},\,\,\mathbf{b}=2\mathbf{i}+3\mathbf{j}+\mathbf{k}\] and \[\mathbf{c}=\mathbf{i}+\alpha \mathbf{j}\] are coplanar vectors, the value of \[\alpha \] is [UPSEAT 2004] |
| A. | \[-\frac{4}{3}\] |
| B. | \[\frac{3}{4}\] |
| C. | \[\frac{4}{3}\] |
| D. | 2 |
| Answer» D. 2 | |
| 1654. |
Out of the following which one is not true [Orissa JEE 2004] |
| A. | \[\mathbf{a}\,.\,(\mathbf{b}\times \mathbf{c})\] |
| B. | \[(\mathbf{b}\times \mathbf{c})\,.\,\mathbf{a}\] |
| C. | \[(\mathbf{a}\times \mathbf{b})\,.\,\mathbf{c}\] |
| D. | \[(\mathbf{a}.\mathbf{c})\,\times \,\mathbf{b}\] |
| Answer» E. | |
| 1655. |
. If \[\mathbf{a}\,.\,\mathbf{b}=\mathbf{b}\,.\,\mathbf{c}=\mathbf{c}\,.\,\mathbf{a}=0\] then the value of [a b c] is equal to [Pb. CET 2000] |
| A. | 1 |
| B. | ? 1 |
| C. | \[|\mathbf{a}||\mathbf{b}||\mathbf{c}|\] |
| D. | 0 |
| Answer» D. 0 | |
| 1656. |
Let \[\mathbf{a},\,\mathbf{b}\] and \[\mathbf{c}\] be three vectors. Then scalar triple product x is equal to [UPSEAT 2004] |
| A. | |
| B. | |
| C. | |
| D. | |
| Answer» E. | |
| 1657. |
If \[\mathbf{a},\,\,\mathbf{b},\,\,\mathbf{c}\] are non-coplanar vectors and l is a real number, then the vectors \[\mathbf{a}+2\mathbf{b}+3\mathbf{c},\,\lambda \,\mathbf{b}+4\mathbf{c}\] and \[(2\lambda -1)\mathbf{c}\] are non-coplanar for [AIEEE 2004] |
| A. | No value of l |
| B. | All except one value of l |
| C. | All except two values of l |
| D. | All values of l |
| Answer» E. | |
| 1658. |
Vector coplanar with vectors i + j and j + k and parallel to the vector 2i ? 2j ? 4k, is [Roorkee 2000] |
| A. | i ? k |
| B. | i ? j ? 2k |
| C. | i + j ? k |
| D. | 3i + 3j ? 6k |
| Answer» C. i + j ? k | |
| 1659. |
The value of l for which the four points \[2\mathbf{i}+3\mathbf{j}-\mathbf{k},\] \[\mathbf{i}+2\mathbf{j}+3\mathbf{k}\], \[3\mathbf{i}+4\mathbf{j}-2\mathbf{k},\,\,\mathbf{i}-\lambda \mathbf{j}+6\mathbf{k}\] are coplanar [MP PET 2004] |
| A. | 8 |
| B. | 0 |
| C. | ? 2 |
| D. | 6 |
| Answer» D. 6 | |
| 1660. |
If the vectors \[4i+11j+mk,\,7i+2j+6k\] and \[i+5j+4k\] are coplanar, then m is [Karnataka CET 2003] |
| A. | 38 |
| B. | 0 |
| C. | 10 |
| D. | ? 10 |
| Answer» D. ? 10 | |
| 1661. |
\[a\,.\,[(b+c)\times (a+b+c)]\] is equal to [IIT 1981; UPSEAT 2003; RPET 1988, 2002; MP PET 2004] |
| A. | [a b c] |
| B. | 2[a b c] |
| C. | 3[a b c] |
| D. | 0 |
| Answer» E. | |
| 1662. |
If a and b be parallel vectors, then [a c b] = |
| A. | 0 |
| B. | 1 |
| C. | 2 |
| D. | None of these |
| Answer» B. 1 | |
| 1663. |
If u, v and w are three non-coplanar vectors, then \[(u+v-w)\,.\,[(u-v)\times (v-w)]\] equals [AIEEE 2003; DCE 2005] |
| A. | 0 |
| B. | \[u\,.\,(v\times w)\] |
| C. | \[u\,.\,(w\times v)\] |
| D. | \[3u\,.\,(v\times w)\] |
| Answer» C. \[u\,.\,(w\times v)\] | |
| 1664. |
The volume of the parallelopiped whose conterminous edges are \[i-j+k,\,\,2i-4j+5k\]and \[3i-5j+2k\] is [Kerala (Engg.) 2002] |
| A. | 4 |
| B. | 3 |
| C. | 2 |
| D. | 8 |
| Answer» E. | |
| 1665. |
a.(b × c) is equal to [RPET 2001] |
| A. | b.(a × c) |
| B. | c.(b × a) |
| C. | It is obvious. |
| D. | None of these |
| Answer» D. None of these | |
| 1666. |
If \[a,\,b,\,c\] are vectors such that \[[a\,b\,c\,]=4\], then \[[a\times b\,\,b\times c\,\,c\times a]\] = [AIEEE 2002] |
| A. | 16 |
| B. | 64 |
| C. | 4 |
| D. | 8 |
| Answer» B. 64 | |
| 1667. |
\[(\mathbf{a}+\mathbf{b})\,.\,(\mathbf{b}+\mathbf{c})\times (\mathbf{a}+\mathbf{b}+\mathbf{c})=\] [EAMCET 2002] |
| A. | ? [a b c] |
| B. | [a b c] |
| C. | 0 |
| D. | 2[a b c] |
| Answer» C. 0 | |
| 1668. |
Let \[a=i-k,\,\,\,b=xi+j+(1-x)\,k\],\[c=yi+xj+(1+x-y)k\]. Then \[[a\,\,b\,\,c]\] depends on [IIT Screening 2001; AIEEE 2005] |
| A. | Only x |
| B. | Only y |
| C. | Neither x nor y |
| D. | Both x and y |
| Answer» D. Both x and y | |
| 1669. |
Let \[\overrightarrow{A}=i+j+k\], \[\overrightarrow{B}=i,\,\overrightarrow{C}={{C}_{1}}i+{{C}_{2}}j+{{C}_{3}}k\]. If \[{{C}_{2}}=-1\], and \[{{C}_{3}}=1\], then to make three vectors coplanar [AMU 2000] |
| A. | \[{{C}_{1}}=0\] |
| B. | \[{{C}_{1}}=1\] |
| C. | \[{{C}_{1}}=2\] |
| D. | No value of \[{{C}_{1}}\] can be found |
| Answer» E. | |
| 1670. |
\[a=i+j+k,\,\mathbf{b}=2i-4k,\,c=i+\lambda \,j+3k\] are coplanar, then the value of \[\lambda \] is [MP PET 2000] |
| A. | 5/2 |
| B. | 3/5 |
| C. | 7/3 |
| D. | None of these |
| Answer» E. | |
| 1671. |
If a, b, c are any three coplanar unit vectors, then |
| A. | \[\mathbf{a}\,.\,(\mathbf{b}\times \mathbf{c})=1\] |
| B. | \[\mathbf{a}\,.\,(\mathbf{b}\times \mathbf{c})=3\] |
| C. | \[(\mathbf{a}\times \mathbf{b})\,.\,\mathbf{c}=0\] |
| D. | \[(\mathbf{c}\times \mathbf{a})\,.\,\mathbf{b}=1\] |
| Answer» D. \[(\mathbf{c}\times \mathbf{a})\,.\,\mathbf{b}=1\] | |
| 1672. |
If the vectors \[i+3j-2k\], \[2i-j+4k\] and \[3i+2j+xk\] are coplanar, then the value of x is [Karnataka CET 2000] |
| A. | ? 2 |
| B. | 2 |
| C. | 1 |
| D. | 3 |
| Answer» C. 1 | |
| 1673. |
If \[a,\,b\] and c are unit coplanar vectors then the scalar triple product \[[2a-b\,\,2b-c\,\,2c-a]\] is equal to [IIT Screening 2000; Kerala (Engg.) 2005] |
| A. | 0 |
| B. | 1 |
| C. | \[-\sqrt{3}\] |
| D. | \[\sqrt{3}\] |
| Answer» B. 1 | |
| 1674. |
Given vectors a, b, c such that \[\mathbf{a}\,.(\mathbf{b}\times \mathbf{c})\]\[=\lambda \ne 0,\,\] the value of \[(\mathbf{b}\times \mathbf{c})\,.\,(\mathbf{a}+\mathbf{b}+\mathbf{c})/\lambda \] is [AMU 1999] |
| A. | 3 |
| B. | 1 |
| C. | \[-3\lambda \] |
| D. | \[3/\lambda \] |
| Answer» C. \[-3\lambda \] | |
| 1675. |
If vectors \[\vec{A}=2\mathbf{i}+3\mathbf{j}+4\mathbf{k}\], \[\vec{B}=\mathbf{i}+\mathbf{j}+5\mathbf{k}\], and \[\vec{C}\] form a left handed system, then \[\vec{C}\] is [Roorkee 1999] |
| A. | 11i ? 6j ? k |
| B. | ? 11i + 6j + k |
| C. | 11i ? 6j + k |
| D. | ? 11i + 6j ? k |
| Answer» C. 11i ? 6j + k | |
| 1676. |
If \[\mathbf{a,}\,\mathbf{b,}\,\mathbf{c}\] are non-coplanar vectors and \[\mathbf{d}=\lambda \mathbf{a}+\mu \,\mathbf{b}+\nu \mathbf{c},\] then \[\lambda \] is equal to [Roorkee 1999] |
| A. | \[\frac{[\mathbf{d}\,\mathbf{b}\,\mathbf{c}]}{[\mathbf{b}\,\mathbf{a}\,\mathbf{c}]}\] |
| B. | \[\frac{[\mathbf{b}\,\mathbf{c}\,\mathbf{d}]}{[\mathbf{b}\,\mathbf{c}\,\mathbf{a}]}\] |
| C. | \[\frac{[\mathbf{b}\,\mathbf{d}\,\mathbf{c}]}{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]}\] |
| D. | \[\frac{[\mathbf{c}\,\mathbf{b}\,\mathbf{d}\,]}{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]}\] |
| Answer» C. \[\frac{[\mathbf{b}\,\mathbf{d}\,\mathbf{c}]}{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]}\] | |
| 1677. |
Which of the following expressions are meaningful [IIT 1998; RPET 2001] |
| A. | \[\mathbf{u}\,.\,(\mathbf{v}\times \mathbf{w})\] |
| B. | \[(\mathbf{u}\,.\,\mathbf{v})\,.\,\mathbf{w}\] |
| C. | \[(\mathbf{u}\,.\,\mathbf{v})\,\mathbf{w}\] |
| D. | \[\mathbf{u}\times (\mathbf{v}\,.\,\mathbf{w})\] |
| Answer» B. \[(\mathbf{u}\,.\,\mathbf{v})\,.\,\mathbf{w}\] | |
| 1678. |
For three vectors u, v, w which of the following expressions is not equal to any of the remaining three [IIT 1998] |
| A. | \[\mathbf{u}\,.\,(\mathbf{v}\times \mathbf{w})\] |
| B. | \[\,(\mathbf{v}\times \mathbf{w})\,.\,\mathbf{u}\,\] |
| C. | \[\mathbf{v}\,.\,(\mathbf{u}\times \mathbf{w})\] |
| D. | \[(\mathbf{u}\times \mathbf{v})\,.\,\mathbf{w}\] |
| Answer» D. \[(\mathbf{u}\times \mathbf{v})\,.\,\mathbf{w}\] | |
| 1679. |
If three conterminous edges of a parallelopiped are represented by \[\mathbf{a}-\mathbf{b},\,\,\mathbf{b}-\mathbf{c}\] and \[\mathbf{c}-\mathbf{a}\], then its volume is [MP PET 1999; Pb. CET 2003] |
| A. | [a b c] |
| B. | 2 [a b c] |
| C. | \[\,{{[\mathbf{a}\,\,\mathbf{b}\,\,\mathbf{c}]}^{2}}\] |
| D. | 0 |
| Answer» E. | |
| 1680. |
If the vectors \[2\mathbf{i}-3\mathbf{j},\,\,\mathbf{i}+\mathbf{j}-\mathbf{k}\] and \[3\mathbf{i}-\mathbf{k}\] form three concurrent edges of a parallelopiped, then the volume of the parallelopiped is [IIT 1983; RPET 1995; DCE 2001; Kurukshetra CEE 1998; MP PET 2001] |
| A. | 8 |
| B. | 10 |
| C. | 4 |
| D. | 14 |
| Answer» D. 14 | |
| 1681. |
\[\mathbf{a}\,.\,(\mathbf{a}\times \mathbf{b})=\] [MP PET 1996] |
| A. | b . b |
| B. | \[{{a}^{2}}b\] |
| C. | 0 |
| D. | \[{{a}^{2}}+ab\] |
| Answer» D. \[{{a}^{2}}+ab\] | |
| 1682. |
If \[\mathbf{a}=-3\mathbf{i}+7\mathbf{j}+5\mathbf{k},\] \[\mathbf{b}=-3\mathbf{i}+7\mathbf{j}-3\mathbf{k}\], \[\mathbf{c}=7\mathbf{i}-5\mathbf{j}-3\mathbf{k}\] are the three coterminous edges of a parallelopiped, then its volume is [MP PET 1996] |
| A. | 108 |
| B. | 210 |
| C. | 272 |
| D. | 308 |
| Answer» D. 308 | |
| 1683. |
If \[\mathbf{a}=3\mathbf{i}-\mathbf{j}+2\mathbf{k},\] \[\mathbf{b}=2\mathbf{i}+\mathbf{j}-\mathbf{k},\] then \[\mathbf{a}\times (\mathbf{a}\,.\,\mathbf{b})=\] [Karnataka CET 1994] |
| A. | 3a |
| B. | \[3\sqrt{14}\] |
| C. | 0 |
| D. | None of these |
| Answer» E. | |
| 1684. |
Volume of the parallelopiped whose coterminous edges are \[2\mathbf{i}-3\mathbf{j}+4\mathbf{k},\,\,\mathbf{i}+2\mathbf{j}-2\mathbf{k},\,\,3\mathbf{i}-\mathbf{j}+\mathbf{k},\] is [EAMCET 1993] |
| A. | 5 cubic unit |
| B. | 6 cubic unit |
| C. | 7 cubic unit |
| D. | 8 cubic unit |
| Answer» D. 8 cubic unit | |
| 1685. |
If the vectors \[2\mathbf{i}-3\mathbf{j}+4\mathbf{k},\,\,\mathbf{i}+2\mathbf{j}-\mathbf{k}\] and \[x\mathbf{i}-\mathbf{j}+2\mathbf{k}\] are coplanar, then \[x=\] [EAMCET 1994] |
| A. | \[\frac{8}{5}\] |
| B. | \[\frac{5}{8}\] |
| C. | 0 |
| D. | 1 |
| Answer» B. \[\frac{5}{8}\] | |
| 1686. |
If a,b,c are three coplanar vectors, then \[[\mathbf{a}+\mathbf{b}\,\,\mathbf{b}+\mathbf{c}\,\,\mathbf{c}+\mathbf{a}]=\] [MP PET 1995] |
| A. | [a b c] |
| B. | 2 [a b c] |
| C. | 3 [a b c] |
| D. | 0 |
| Answer» E. | |
| 1687. |
If a, b, c be any three non-coplanar vectors, then \[[\mathbf{a}+\mathbf{b}\,\,\,\mathbf{b}+\mathbf{c}\,\,\,\mathbf{c}+\mathbf{a}]=\] [RPET 1988; MP PET 1990, 02; Kerala (Engg.) 2002] |
| A. | \[|\mathbf{a}\,\mathbf{b}\,\mathbf{c}|\] |
| B. | 2\[[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]\] |
| C. | \[{{[\,\mathbf{a}\,\mathbf{b}\,\mathbf{c}\,]}^{2}}\] |
| D. | \[2\,{{[\,\mathbf{a}\,\mathbf{b}\,\mathbf{c}\,]}^{2}}\] |
| Answer» C. \[{{[\,\mathbf{a}\,\mathbf{b}\,\mathbf{c}\,]}^{2}}\] | |
| 1688. |
If the given vectors \[(-bc,\,{{b}^{2}}+bc,\,{{c}^{2}}+bc),\] \[({{a}^{2}}+ac,\,-ac,\,{{c}^{2}}+ac)\] and \[({{a}^{2}}+ab,\,{{b}^{2}}+ab,\,-ab)\] are coplanar, where none of a, b and c is zero, then |
| A. | \[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}=1\] |
| B. | \[bc+ca+ab=0\] |
| C. | \[a+b+c=0\] |
| D. | \[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}=bc+ca+ab\] |
| Answer» C. \[a+b+c=0\] | |
| 1689. |
Three concurrent edges OA, OB, OC of a parallelopiped are represented by three vectors \[2\mathbf{i}+\mathbf{j}-\mathbf{k},\,\,\mathbf{i}+2\mathbf{j}+3\mathbf{k}\] and \[-3\mathbf{i}-\mathbf{j}+\mathbf{k},\] the volume of the solid so formed in cubic unit is [Kurukshetra CEE 1998] |
| A. | 5 |
| B. | 6 |
| C. | 7 |
| D. | 8 |
| Answer» B. 6 | |
| 1690. |
If \[\mathbf{a}=2\mathbf{i}+\mathbf{j}-\mathbf{k},\,\,\mathbf{b}=\mathbf{i}+2\mathbf{j}+\mathbf{k}\] and \[\mathbf{c}=\mathbf{i}-\mathbf{j}+2\mathbf{k},\] then \[\mathbf{a}\,.\,(\mathbf{b}\times \mathbf{c})=\] [RPET 1989, 2001] |
| A. | 6 |
| B. | 10 |
| C. | 12 |
| D. | 24 |
| Answer» D. 24 | |
| 1691. |
If three vectors \[\mathbf{a}=12\mathbf{i}+4\mathbf{j}+3\mathbf{k},\,\,\]\[\mathbf{b}=8\mathbf{i}-12\mathbf{j}-9\mathbf{k}\] and \[\mathbf{c}=33\mathbf{i}-4\mathbf{j}-24\mathbf{k}\] represents a cube, then its volume will be [Roorkee 1988] |
| A. | 616 |
| B. | 308 |
| C. | 154 |
| D. | None of these |
| Answer» E. | |
| 1692. |
If \[\mathbf{i},\,\mathbf{j},\,\mathbf{k}\] are the unit vectors and mutually perpendicular, then \[[\mathbf{i}\,\mathbf{k}\,\mathbf{j}]\] is equal to [RPET 1986] |
| A. | 0 |
| B. | ? 1 |
| C. | 1 |
| D. | None of these |
| Answer» C. 1 | |
| 1693. |
If a, b, c are any three vectors and their inverse are \[{{\mathbf{a}}^{-1}},\,{{\mathbf{b}}^{-1}},\,{{\mathbf{c}}^{-1}}\]and \[[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]\ne 0,\] then \[[{{\mathbf{a}}^{-1}}\,{{\mathbf{b}}^{-1}}\,{{\mathbf{c}}^{-1}}]\] will be [Roorkee 1989] |
| A. | Zero |
| B. | One |
| C. | Non-zero |
| D. | [a b c] |
| Answer» D. [a b c] | |
| 1694. |
Let a, b, c be distinct non-negative numbers. If the vectors \[a\mathbf{i}+a\mathbf{j}+c\mathbf{k},\,\,\mathbf{i}+\mathbf{k}\] and \[c\mathbf{i}+c\mathbf{j}+b\mathbf{k}\] lie in a plane, then c is [IIT 1993; AIEEE 2005] |
| 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 |
| Answer» C. The harmonic mean of a and b | |
| 1695. |
The volume of the parallelopiped whose edges are represented by \[-12\mathbf{i}+\alpha \mathbf{k},\,\,3\mathbf{j}-\mathbf{k}\] and \[2\mathbf{i}+\mathbf{j}-15\mathbf{k}\] is 546. Then \[\alpha =\] [IIT Screening 1989; MNR 1987] |
| A. | 3 |
| B. | 2 |
| C. | ? 3 |
| D. | ? 2 |
| Answer» D. ? 2 | |
| 1696. |
If a, b, c are three non-coplanar vector, then \[\frac{\mathbf{a}\,.\,\mathbf{b}\times \mathbf{c}}{\mathbf{c}\times \mathbf{a}\,.\,\mathbf{b}}+\frac{\mathbf{b}\,.\,\mathbf{a}\times \mathbf{c}}{\mathbf{c}\,.\,\mathbf{a}\times \mathbf{b}}\]= [IIT 1985, 86; UPSEAT 2003] |
| A. | 0 |
| B. | 2 |
| C. | ? 2 |
| D. | None of these |
| Answer» B. 2 | |
| 1697. |
The function \[f(x)=x(x+3){{e}^{-(1/2)x}}\] satisfies all the conditions of Rolle's theorem in [?3, 0]. The value of c is |
| A. | 0 |
| B. | ?1 |
| C. | ? 2 |
| D. | ? 3 |
| Answer» D. ? 3 | |
| 1698. |
If from mean value theorem, \[f'({{x}_{1}})=\frac{f(b)-f(a)}{b-a}\], then [MP PET 1999] |
| A. | \[a<{{x}_{1}}\le b\] |
| B. | \[a\le {{x}_{1}}<b\] |
| C. | \[a<{{x}_{1}}<b\] |
| D. | \[a\le {{x}_{1}}\le b\] |
| Answer» D. \[a\le {{x}_{1}}\le b\] | |
| 1699. |
Rolle's theorem is true for the function \[f(x)={{x}^{2}}-4\]in the interval |
| A. | [?2, 0] |
| B. | [?2, 2] |
| C. | \[\left[ 0,\,\frac{1}{2} \right]\] |
| D. | \[[0,\,\,2]\] |
| Answer» C. \[\left[ 0,\,\frac{1}{2} \right]\] | |
| 1700. |
From mean value theorem \[f(b)-f(a)=\] \[(b-a)f'({{x}_{1}});\] \[a |
| A. | \[\sqrt{ab}\] |
| B. | \[\frac{a+b}{2}\] |
| C. | \[\frac{2ab}{a+b}\] |
| D. | \[\frac{b-a}{b+a}\] |
| Answer» B. \[\frac{a+b}{2}\] | |