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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.
| 7351. |
If the sides of a triangle are in ratio 3 : 7 : 8, then R : r is equal to |
| A. | 0.0881944444444445 |
| B. | 0.293055555555556 |
| C. | 0.129861111111111 |
| D. | 0.29375 |
| Answer» C. 0.129861111111111 | |
| 7352. |
Radius of a circle is increasing uniformly at the rate of \[3cm/\sec .\]The rate of increasing of area when radius is \[10cm\], will be |
| A. | \[\pi \,c{{m}^{2}}/s\] |
| B. | \[2\pi \,c{{m}^{2}}/s\] |
| C. | \[10\pi \,c{{m}^{2}}/s\] |
| D. | None of these |
| Answer» E. | |
| 7353. |
The intercept on the line \[y=x\] by the circle \[{{x}^{2}}+{{y}^{2}}-2x=0\] is AB, equation of the circle on AB as a diameter is [AIEEE 2004] |
| A. | \[{{x}^{2}}+{{y}^{2}}+x-y=0\] |
| B. | \[{{x}^{2}}+{{y}^{2}}-x+y=0\] |
| C. | \[{{x}^{2}}+{{y}^{2}}+x+y=\]0 |
| D. | \[{{x}^{2}}+{{y}^{2}}-x-y=0\] |
| Answer» E. | |
| 7354. |
A letter is known to have come either from LONDON or CLIFTON; on the postmark only the two consecutive letters ON are legible. The probability that it came from LONDON is |
| A. | \[\frac{5}{17}\] |
| B. | \[\frac{12}{17}\] |
| C. | \[\frac{17}{30}\] |
| D. | \[\frac{3}{5}\] |
| Answer» C. \[\frac{17}{30}\] | |
| 7355. |
If \[{{x}^{2}}+ax+10=0\] and \[{{x}^{2}}+bx-10=0\] have a common root, then \[{{a}^{2}}-{{b}^{2}}\] is equal to [Kerala (Engg.) 2002] |
| A. | 10 |
| B. | 20 |
| C. | 30 |
| D. | 40 |
| Answer» E. | |
| 7356. |
If the lines \[y=(2+\sqrt{3})x+4\] and \[y=kx+6\]are inclined at an angle \[{{60}^{o}}\]to each other, then the value of k will be |
| A. | 1 |
| B. | 2 |
| C. | 1 |
| D. | 2 |
| Answer» D. 2 | |
| 7357. |
The smallest value of \[{{x}^{2}}-3x+3\] in the interval \[(-3,\,3/2)\] is [EAMCET 1991; 93] |
| A. | 44289 |
| B. | 5 |
| C. | -15 |
| D. | -20 |
| Answer» B. 5 | |
| 7358. |
The sum of \[1+n\left( 1-\frac{1}{x} \right)+\frac{n(n+1)}{2!}\text{ }{{\left( 1-\frac{1}{x} \right)}^{2}}+.....\infty ,\] will be[Roorkee 1975] |
| A. | \[{{x}^{n}}\] |
| B. | \[{{x}^{-n}}\] |
| C. | \[{{\left( 1-\frac{1}{x} \right)}^{n}}\] |
| D. | None of these |
| Answer» B. \[{{x}^{-n}}\] | |
| 7359. |
A ladder 5 m in length is resting against vertical wall. The bottom of the ladder is pulled along the ground away from the wall at the rate of \[1.5\,m/\sec \]. The length of the highest point of the ladder when the foot of the ladder \[4.0\,m\] away from the wall decreases at the rate of [Kurukshetra CEE 1996] |
| A. | 2 m/sec |
| B. | 3 m/sec |
| C. | 2.5 m/sec |
| D. | 1.5 m/sec |
| Answer» B. 3 m/sec | |
| 7360. |
In a box containing 100 eggs, 10 eggs are rotten. The probability that out of a sample of 5 eggs none is rotten if the sampling is with replacement is [MP PET 1991; MNR 1986; RPET 1995; UPSEAT 2000] |
| A. | \[{{\left( \frac{1}{10} \right)}^{5}}\] |
| B. | \[{{\left( \frac{1}{5} \right)}^{5}}\] |
| C. | \[{{\left( \frac{9}{5} \right)}^{5}}\] |
| D. | \[{{\left( \frac{9}{10} \right)}^{5}}\] |
| Answer» E. | |
| 7361. |
\[1+3+7+15+31+..........\]to \[n\] terms = [IIT 1963] |
| A. | \[{{2}^{n+1}}-n\] |
| B. | \[{{2}^{n+1}}-n-2\] |
| C. | \[{{2}^{n}}-n-2\] |
| D. | None of these |
| Answer» C. \[{{2}^{n}}-n-2\] | |
| 7362. |
In how many ways can 15 members of a council sit along a circular table, when the Secretary is to sit on one side of the Chairman and the Deputy Secretary on the other side |
| A. | \[2\times 12\,!\] |
| B. | 24 |
| C. | \[2\times 15\,!\] |
| D. | None of these |
| Answer» B. 24 | |
| 7363. |
A spherical iron ball 10 cm in radius is coated with a layer of ice of uniform thickness that melts at a rate of 50 cm3/min. When the thickness of ice is 5 cm, then the rate at which the thickness of ice decreases, is [AIEEE 2005] |
| A. | \[\frac{1}{54\pi }\]cm/min |
| B. | \[\frac{5}{6\pi }\] cm/min |
| C. | \[\frac{1}{36\pi }\] cm/min |
| D. | \[\frac{1}{18\pi }\] cm/min |
| Answer» E. | |
| 7364. |
If \[\mathbf{a}=\mathbf{i}+2\mathbf{j}-2\mathbf{k},\,\,\mathbf{b}=2\mathbf{i}-\mathbf{j}+\mathbf{k}\]and \[\mathbf{c}=\mathbf{i}+3\mathbf{j}-\mathbf{k},\] then \[\mathbf{a}\times (\mathbf{b}\times \mathbf{c})\] is equal to [RPET 1989] |
| A. | \[20\mathbf{i}-3\mathbf{j}+7\mathbf{k}\] |
| B. | \[20\mathbf{i}-3\mathbf{j}-7\mathbf{k}\] |
| C. | \[20\mathbf{i}+3\mathbf{j}-7\mathbf{k}\] |
| D. | None of these |
| Answer» B. \[20\mathbf{i}-3\mathbf{j}-7\mathbf{k}\] | |
| 7365. |
If \[\mathbf{a}=3\mathbf{i}-\mathbf{j}+2\mathbf{k},\] \[\mathbf{b}=2\mathbf{i}+\mathbf{j}-\mathbf{k}\] and \[\mathbf{c}=\mathbf{i}-2\mathbf{j}+2\mathbf{k},\]then \[(\mathbf{a}\times \mathbf{b})\times \mathbf{c}\] is equal to |
| A. | \[24\mathbf{i}+7\mathbf{j}-5\mathbf{k}\] |
| B. | \[7\mathbf{i}-24\mathbf{j}+5\mathbf{k}\] |
| C. | \[12\mathbf{i}+3\mathbf{j}-5\mathbf{k}\] |
| D. | \[\mathbf{i}+\mathbf{j}-7\mathbf{k}\] |
| Answer» B. \[7\mathbf{i}-24\mathbf{j}+5\mathbf{k}\] | |
| 7366. |
If a, b, c, d are coplanar vectors, then \[(\mathbf{a}\times \mathbf{b})\times (\mathbf{c}\times \mathbf{d})=\] [MP PET 1998] |
| A. | \[|\,\mathbf{a}\,\times \,\mathbf{c}{{|}^{2}}\] |
| B. | \[|\mathbf{a}\times \mathbf{d}{{|}^{2}}\] |
| C. | \[|\mathbf{b}\times \mathbf{c}{{|}^{2}}\] |
| D. | 0 |
| Answer» E. | |
| 7367. |
Let \[a,\,b,\,c\] be three vectors from \[a\times (b\times c)=(a\times b)\times c\], if [Orissa JEE 2003] |
| A. | \[b\times (a\times c)=0\] |
| B. | \[a(b\times c)=0\] |
| C. | \[c\times a=a\times b\] |
| D. | \[c\times b=b\times a\] |
| Answer» B. \[a(b\times c)=0\] | |
| 7368. |
\[[\mathbf{b}\times \mathbf{c}\,\,\mathbf{c}\times \mathbf{a}\,\,\mathbf{a}\times \mathbf{b}]\] is equal to [MP PET 2004] |
| A. | \[\mathbf{a}\times (\mathbf{b}\times \mathbf{c})\] |
| B. | \[2\,[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]\] |
| C. | \[{{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]}^{2}}\] |
| D. | \[[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]\] |
| Answer» D. \[[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]\] | |
| 7369. |
If \[\mathbf{a}=\mathbf{i}-\mathbf{j},\mathbf{b}=\mathbf{i}+\mathbf{j},\,\,\,\mathbf{c}=\mathbf{i}+3\mathbf{j}+5\mathbf{k}\]and\[\mathbf{n}\]is a unit vector such that \[\mathbf{b}.\mathbf{n}=0,\mathbf{a}.\mathbf{n}=0\]then the value of \[|\mathbf{c}\ .\ \mathbf{n}|\] is equal to [DCE 2005] |
| A. | 1 |
| B. | 3 |
| C. | 5 |
| D. | 2 |
| Answer» D. 2 | |
| 7370. |
Three forces \[\mathbf{i}+2\,\mathbf{j}-3\,\mathbf{k},\,\,2\,\mathbf{i}+3\,\mathbf{j}+4\,\mathbf{k}\] and \[\mathbf{i}-\mathbf{j}+\mathbf{k}\] are acting on a particle at the point (0, 1, 2). The magnitude of the moment of the forces about the point \[(1,\,-2,\,0)\] is [MNR 1983] |
| A. | \[2\sqrt{35}\] |
| B. | \[6\sqrt{10}\] |
| C. | \[4\sqrt{17}\] |
| D. | None of these |
| Answer» C. \[4\sqrt{17}\] | |
| 7371. |
If a and b are two vectors such that a . b = 0 and \[\mathbf{a}\times \mathbf{b}=\mathbf{0},\] then [IIT Screening 1989; MNR 1988; UPSEAT 2000, 01] |
| A. | a is parallel to b |
| B. | a is perpendicular to b |
| C. | Either a or b is a null vector |
| D. | None of these |
| Answer» D. None of these | |
| 7372. |
The area of the triangle having vertices as \[\mathbf{i}-2\mathbf{j}+3\mathbf{k},\] \[\,-2\mathbf{i}+3\mathbf{j}+\mathbf{k}\] , \[4\mathbf{i}-7\mathbf{j}+7\mathbf{k}\] is [MP PET 2004] |
| A. | 26 |
| B. | 11 |
| C. | 36 |
| D. | 0 |
| Answer» E. | |
| 7373. |
The area of a parallelogram whose adjacent sides are \[\mathbf{i}-2\mathbf{j}+3\mathbf{k}\] and \[2\mathbf{i}+\mathbf{j}-4\mathbf{k},\] is [MP PET 1996, 2000] |
| A. | \[5\sqrt{3}\] |
| B. | \[10\sqrt{3}\] |
| C. | \[5\sqrt{6}\] |
| D. | \[10\sqrt{6}\] |
| Answer» D. \[10\sqrt{6}\] | |
| 7374. |
The position vectors of the points A, B and C are \[\mathbf{i}+\mathbf{j},\,\,\mathbf{j}+\mathbf{k}\] and \[\mathbf{k}+\mathbf{i}\] respectively. The vector area of the \[\Delta ABC=\pm \,\frac{1}{2}\overrightarrow{\alpha }\] where \[\overrightarrow{\alpha }=\] [MP PET 1989] |
| A. | \[-\mathbf{i}+\mathbf{j}+\mathbf{k}\] |
| B. | \[\mathbf{i}-\mathbf{j}+\mathbf{k}\] |
| C. | \[\mathbf{i}+\mathbf{j}-\mathbf{k}\] |
| D. | \[\mathbf{i}+\mathbf{j}+\mathbf{k}\] |
| Answer» E. | |
| 7375. |
The area of a triangle whose vertices are \[A\,(1,\,-1,\,2),\] \[B\,(2,\,1,\,-1)\] and \[C\,(3,\,-1,\,2)\] is [MNR 1983; IIT 1983] |
| A. | 13 |
| B. | \[\sqrt{13}\] |
| C. | 6 |
| D. | \[\sqrt{6}\] |
| Answer» C. 6 | |
| 7376. |
The unit vector perpendicular to both \[\mathbf{i}+\mathbf{j}\] and \[\mathbf{j}+\mathbf{k}\] is [Kerala (Engg.) 2002] |
| A. | i ? j + k |
| B. | i + j + k |
| C. | \[\frac{\mathbf{i}+\mathbf{j}-\mathbf{k}}{\sqrt{3}}\] |
| D. | \[\frac{\mathbf{i}-\mathbf{j}+\mathbf{k}}{\sqrt{3}}\] |
| Answer» E. | |
| 7377. |
The unit vector perpendicular to both the vectors i ? 2j + 3k and i + 2j ? k is [DCE 2001] |
| A. | \[\frac{1}{\sqrt{3}}(-i+j+k)\] |
| B. | \[(-\mathbf{i}+\mathbf{j}+\mathbf{k})\] |
| C. | \[\frac{(i+j-k)}{\sqrt{3}}\] |
| D. | None of these |
| Answer» B. \[(-\mathbf{i}+\mathbf{j}+\mathbf{k})\] | |
| 7378. |
If \[{{(\mathbf{a}\times \mathbf{b})}^{2}}+{{(\mathbf{a}\,\,.\,\,\mathbf{b})}^{2}}=144\] and \[|\mathbf{a}|\,=4,\] then \[|\mathbf{b}|\,=\] [EAMCET 1994] |
| A. | 16 |
| B. | 8 |
| C. | 3 |
| D. | 12 |
| Answer» D. 12 | |
| 7379. |
A unit vector perpendicular to vector c and coplanar with vectors a and b is [MP PET 1999] |
| A. | \[\frac{\mathbf{a}\times (\mathbf{b}\times \mathbf{c})}{|\mathbf{a}\times (\mathbf{b}\times \mathbf{c})|}\] |
| B. | \[\frac{\mathbf{b}\times (\mathbf{c}\times \mathbf{a})}{|\mathbf{b}\times (\mathbf{c}\times \mathbf{a})|}\] |
| C. | \[\frac{\mathbf{c}\times (\mathbf{a}\times \mathbf{b})}{|\mathbf{c}\times (\mathbf{a}\times \mathbf{b})|}\] |
| D. | None of these |
| Answer» D. None of these | |
| 7380. |
If \[\mathbf{a}=2\mathbf{i}+2\mathbf{j}-\mathbf{k}\] and \[\mathbf{b}=6\mathbf{i}-3\mathbf{j}+2\mathbf{k},\] then the value of \[\mathbf{a}\times \mathbf{b}\] is [MNR 1978; RPET 2001] |
| A. | \[2\mathbf{i}+2\mathbf{j}-\mathbf{k}\] |
| B. | \[6\mathbf{i}-3\mathbf{j}+2\mathbf{k}\] |
| C. | \[\mathbf{i}-10\mathbf{j}-18\mathbf{k}\] |
| D. | \[\mathbf{i}+\mathbf{j}+\mathbf{k}\] |
| Answer» D. \[\mathbf{i}+\mathbf{j}+\mathbf{k}\] | |
| 7381. |
Let a and b be two non-collinear unit vectors. If \[\mathbf{u}=\mathbf{a}-(\mathbf{a}\,.\,\mathbf{b})\,\mathbf{b}\] and \[\mathbf{v}=\mathbf{a}\times \mathbf{b},\] then | v | is [IIT 1999] |
| A. | | u | |
| B. | | u |+| u . a | |
| C. | | u |+| u . b | |
| D. | | u |+ u . (a+b) |
| Answer» D. | u |+ u . (a+b) | |
| 7382. |
The components of a vector \[\vec{a}\] along and perpendicular to a non-zero vector \[\vec{b}\] are |
| A. | \[\left( \frac{\vec{a}.\vec{b}}{{{\left| {\vec{b}} \right|}^{2}}} \right)\vec{b}\And \vec{a}-\left( \frac{\vec{a}.\vec{b}}{{{\left| {\vec{b}} \right|}^{2}}} \right)\vec{b}\] |
| B. | \[\left( \frac{\vec{a}.\vec{b}}{{{\left| {\vec{a}} \right|}^{2}}} \right)\vec{b}\And \vec{a}+\left( \frac{\vec{a}.\vec{b}}{{{\left| {\vec{a}} \right|}^{2}}} \right)\vec{b}\] |
| C. | \[\left( \frac{\vec{a}.\vec{b}}{{{\left| {\vec{a}} \right|}^{2}}} \right)\vec{a}-\left( \frac{\vec{a}.\vec{b}}{{{\left| {\vec{b}} \right|}^{2}}} \right)\vec{a}\] |
| D. | None of these |
| Answer» B. \[\left( \frac{\vec{a}.\vec{b}}{{{\left| {\vec{a}} \right|}^{2}}} \right)\vec{b}\And \vec{a}+\left( \frac{\vec{a}.\vec{b}}{{{\left| {\vec{a}} \right|}^{2}}} \right)\vec{b}\] | |
| 7383. |
Resolved part of vector \[\vec{a}\] along vector \[\vec{b}\] is \[{{\vec{a}}_{1}}\] and that perpendicular to \[\vec{b}\] is \[{{\vec{a}}_{2}}\] then \[{{\vec{a}}_{1}}\times {{\vec{a}}_{2}}\] is equal to |
| A. | \[\frac{(\vec{a}\times \vec{b})\cdot \vec{b}}{{{\left| {\vec{b}} \right|}^{2}}}\] |
| B. | \[\frac{(\vec{a}\cdot \vec{b})\vec{a}}{{{\left| {\vec{a}} \right|}^{2}}}\] |
| C. | \[\frac{(\vec{a}\cdot \vec{b})(\vec{b}\times \vec{a})}{{{\left| {\vec{b}} \right|}^{2}}}\] |
| D. | \[\frac{(\vec{a}\cdot \vec{b})(\vec{b}\times \vec{a})}{\left| \vec{b}\times \vec{a} \right|}\] |
| Answer» D. \[\frac{(\vec{a}\cdot \vec{b})(\vec{b}\times \vec{a})}{\left| \vec{b}\times \vec{a} \right|}\] | |
| 7384. |
If \[\overset{\to }{\mathop{p}}\,\] and \[\overset{\to }{\mathop{q}}\,\] are two unit vectors inclined at an angle \[\alpha \] to each other than \[|\overset{\to }{\mathop{p}}\,+\overset{\to }{\mathop{q}}\,| |
| A. | \[\frac{2\pi }{3}<\alpha <\frac{4\pi }{3}\] |
| B. | \[\frac{4\pi }{3}<\alpha <2\pi \] |
| C. | \[0<\alpha <\frac{\pi }{3}\] |
| D. | \[\alpha =\frac{\pi }{2}\] |
| Answer» B. \[\frac{4\pi }{3}<\alpha <2\pi \] | |
| 7385. |
What is the vector equally inclined to the vectors\[\hat{i}+3\hat{j}\] and\[3\hat{i}+\hat{j}\]? |
| A. | \[\hat{i}+\hat{j}\] |
| B. | \[2\hat{i}-\hat{j}\] |
| C. | \[2\hat{i}+\hat{j}\] |
| D. | None of theses |
| Answer» B. \[2\hat{i}-\hat{j}\] | |
| 7386. |
If \[\overset{\to }{\mathop{c}}\,\] is the unit vector perpendicular to both the vectors \[\overset{\to }{\mathop{a}}\,\] and \[\overset{\to }{\mathop{b}}\,\], then what is another unit vector perpendicular to both the vectors \[\overset{\to }{\mathop{a}}\,\] and \[\overset{\to }{\mathop{b}}\,?\] |
| A. | \[\overset{\to }{\mathop{c}}\,\times \overset{\to }{\mathop{a}}\,\] |
| B. | \[\overset{\to }{\mathop{c}}\,\times \overset{\to }{\mathop{b}}\,\] |
| C. | \[-\frac{\left( \overset{\to }{\mathop{a}}\,\times \overset{\to }{\mathop{b}}\, \right)}{\left| \overset{\to }{\mathop{a}}\,\times \overset{\to }{\mathop{b}}\, \right|}\] |
| D. | \[\frac{\left( \overset{\to }{\mathop{a}}\,\times \overset{\to }{\mathop{b}}\, \right)}{\left| \overset{\to }{\mathop{a}}\,\times \overset{\to }{\mathop{b}}\, \right|}\] |
| Answer» E. | |
| 7387. |
For any vector\[\vec{\alpha }\], what is\[\left( \overset{\to }{\mathop{\alpha }}\,.\widehat{i} \right)\widehat{i}+\left( \overset{\to }{\mathop{\alpha }}\,.\widehat{j} \right)\widehat{j}+\left( \overset{\to }{\mathop{a}}\,.\widehat{k} \right)\widehat{k}\] equal to? |
| A. | \[\overset{\to }{\mathop{\alpha }}\,\] |
| B. | \[3\overset{\to }{\mathop{\alpha }}\,\] |
| C. | \[-\overset{\to }{\mathop{\alpha }}\,\] |
| D. | \[\overset{\to }{\mathop{0}}\,\] |
| Answer» B. \[3\overset{\to }{\mathop{\alpha }}\,\] | |
| 7388. |
If \[\overset{\to }{\mathop{a}}\,,\text{ }\overset{\to }{\mathop{b}}\,,\text{ }\overset{\to }{\mathop{c}}\,\] are the position vectors of corners A, B, C of a parallelogram ABCD, then what is the position vector of the corner D? |
| A. | \[\overset{\to }{\mathop{a}}\,+\overset{\to }{\mathop{b}}\,+\overset{\to }{\mathop{c}}\,\] |
| B. | \[\overset{\to }{\mathop{a}}\,+\overset{\to }{\mathop{b}}\,-\overset{\to }{\mathop{c}}\,\] |
| C. | \[\overset{\to }{\mathop{a}}\,-\overset{\to }{\mathop{b}}\,+\overset{\to }{\mathop{c}}\,\] |
| D. | \[-\overset{\to }{\mathop{a}}\,+\overset{\to }{\mathop{b}}\,+\overset{\to }{\mathop{c}}\,\] |
| Answer» D. \[-\overset{\to }{\mathop{a}}\,+\overset{\to }{\mathop{b}}\,+\overset{\to }{\mathop{c}}\,\] | |
| 7389. |
If \[\vec{a}.\,\vec{b}=0\] and \[\vec{a}+\vec{b}\] makes an angle of \[60{}^\circ \] with \[\vec{a}\], then |
| A. | \[|\vec{a}|=2|\vec{b}|\] |
| B. | \[2|\vec{a}|=|\vec{b}|\] |
| C. | \[|\vec{a}|=\sqrt{3}|\vec{b}|\] |
| D. | \[|\vec{b}|=\sqrt{3}|\vec{a}|\] |
| Answer» E. | |
| 7390. |
Let \[\overrightarrow{A}={{a}_{1}}\hat{i}+{{a}_{2}}\hat{j}+{{a}_{3}}\hat{k},\text{ }\overrightarrow{B}={{b}_{1}}\hat{i}+{{b}_{2}}\hat{j}+{{b}_{3}}\hat{k}\] and \[\overrightarrow{C}={{c}_{1}}\hat{i}+{{c}_{2}}\hat{j}+{{c}_{3}}\hat{k}\] 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 \[\frac{\pi }{6}\], then. |
| A. | 0 |
| B. | 1 |
| C. | \[\frac{1}{4}(a_{1}^{2}+a_{2}^{2}+a_{3}^{2})(b_{1}^{2}+b_{3}^{2})\] |
| D. | \[\frac{3}{4}(a_{1}^{2}+a_{2}^{2}+a_{3}^{2})(b_{1}^{2}+b_{2}^{2}+b_{3}^{2})(c_{1}^{2}+c_{3}^{2})\] |
| Answer» D. \[\frac{3}{4}(a_{1}^{2}+a_{2}^{2}+a_{3}^{2})(b_{1}^{2}+b_{2}^{2}+b_{3}^{2})(c_{1}^{2}+c_{3}^{2})\] | |
| 7391. |
If \[\overrightarrow{OA}=\overrightarrow{a};\,\overrightarrow{OB}=\overrightarrow{b};\,\overrightarrow{OC}=2\overrightarrow{a}+3\overrightarrow{b};\] \[\overrightarrow{OD}=\overset{\to }{\mathop{a}}\,-2\text{ }\overset{\to }{\mathop{b}}\,,\]the length of \[\overrightarrow{OA}\] is three times the length of \[\overrightarrow{OB}\] and \[\overrightarrow{OA}\] is perpendicular to \[\overrightarrow{DB}\] then \[(\overrightarrow{BD}\times \overrightarrow{AC}).(\overrightarrow{OD}\times \overrightarrow{OC})\] is |
| A. | \[7|\overset{\to }{\mathop{a}}\,\times \overset{\to }{\mathop{b}}\,{{|}^{2}}\] |
| B. | \[42|\overset{\to }{\mathop{a}}\,\times \overset{\to }{\mathop{b}}\,{{|}^{2}}\] |
| C. | 0 |
| D. | None of these |
| Answer» C. 0 | |
| 7392. |
If the middle points of sides BC, CA & AB of triangle ABC are respectively D, E, F then position vector of centre of triangle DEF, when position vector of A, B, C are respectively \[\hat{i}+\hat{j},\hat{j}+\hat{k},\hat{k}+\hat{i}\]is |
| A. | \[\frac{1}{3}(\hat{i}+\hat{j}+\hat{k})\] |
| B. | \[(\hat{i}+\hat{j}+\hat{k})\] |
| C. | \[2(\hat{i}+\hat{j}+\hat{k})\] |
| D. | \[\frac{2}{3}(\hat{i}+\hat{j}+\hat{k})\] |
| Answer» E. | |
| 7393. |
The vector \[\overset{\to }{\mathop{c}}\,\] directed along the bisectors of the angle between the vectors \[\overset{\to }{\mathop{a}}\,=7\hat{i}-4\hat{j}-4\hat{k},\] \[\overset{\to }{\mathop{b}}\,=-2\hat{i}-\hat{j}+2\hat{k},\] and \[|\overset{\to }{\mathop{c}}\,|=3\sqrt{6}\] is given by |
| A. | \[\hat{i}-7\hat{j}+2\hat{k}\] |
| B. | \[\hat{i}+7\hat{j}-2\hat{k}\] |
| C. | \[\hat{i}+7\hat{j}+2\hat{k}\] |
| D. | \[\hat{i}+7\hat{j}+3\hat{k}\] |
| Answer» B. \[\hat{i}+7\hat{j}-2\hat{k}\] | |
| 7394. |
If \[\vec{u},\vec{v},\vec{w}\] are non-coplanar vectors and p, q are real numbers, then the equality \[[3\vec{u}\,\,p\vec{v}\,\,p\vec{w}]-[p\vec{v}\,\vec{\omega }\,q\vec{u}]-[2\vec{\omega }\,q\vec{v}\,q\vec{u}]=0\] holds for: |
| A. | Exactly two values of (p, q) |
| B. | More than two but not all values of (p, q) |
| C. | All values of (p, q) |
| D. | Exactly one value of (p, q) |
| Answer» E. | |
| 7395. |
Let \[\overset{\to }{\mathop{a}}\,,\text{ }\overset{\to }{\mathop{b}}\,\] and \[\overset{\to }{\mathop{c}}\,\] be three non-coplanar vectors, and let \[\overset{\to }{\mathop{p}}\,,\text{ }\overset{\to }{\mathop{q}}\,\] and \[\overset{\to }{\mathop{r}}\,\] be the vectors defined by the relations \[\overset{\to }{\mathop{p}}\,=\frac{\overset{\to }{\mathop{b}}\,\times \overset{\to }{\mathop{c}}\,}{[\overset{\to }{\mathop{a}}\,\,\overset{\to }{\mathop{b}}\,\,\overset{\to }{\mathop{c}}\,]},\overset{\to }{\mathop{q}}\,=\frac{\overset{\to }{\mathop{c}}\,\times \overset{\to }{\mathop{a}}\,}{[\overset{\to }{\mathop{a}}\,\,\overset{\to }{\mathop{b}}\,\,\overset{\to }{\mathop{c}}\,]}\] and \[\overset{\to }{\mathop{r}}\,=\frac{\overset{\to }{\mathop{a}}\,\times \overset{\to }{\mathop{b}}\,}{[\overset{\to }{\mathop{a}}\,\,\overset{\to }{\mathop{b}}\,\,\overset{\to }{\mathop{c}}\,]}.\] Then the value of the expression \[(\overset{\to }{\mathop{a}}\,+\overset{\to }{\mathop{b}}\,).\overset{\to }{\mathop{p}}\,+(\overset{\to }{\mathop{b}}\,+\overset{\to }{\mathop{c}}\,).\overset{\to }{\mathop{q}}\,+(\overset{\to }{\mathop{c}}\,+\overset{\to }{\mathop{a}}\,).\overset{\to }{\mathop{r}}\,\] is equal to |
| A. | 0 |
| B. | 1 |
| C. | 2 |
| D. | 3 |
| Answer» E. | |
| 7396. |
If the vectors \[\alpha \hat{i}+\alpha \hat{j}+\gamma \hat{k},\text{ }\hat{i}+\hat{k}\] and \[\gamma \hat{i}+\gamma \hat{j}+\beta \hat{k}\] lie on a plane, where \[\alpha ,\beta \] and \[\gamma \] are distinct non-negative numbers, then \[\gamma \] is |
| A. | Arithmetic mean of \[\alpha \] and \[\beta \] |
| B. | Geometric mean of \[\alpha \] and \[\beta \] |
| C. | Harmonic mean of \[\alpha \] and \[\beta \] |
| D. | None of the above |
| Answer» C. Harmonic mean of \[\alpha \] and \[\beta \] | |
| 7397. |
What is a vector of unit length orthogonal to both the vectors \[\hat{i}+\hat{j}+\hat{k}\] and\[2\hat{i}+3\hat{j}-\hat{k}\]? |
| A. | \[\frac{-4\hat{i}+3\hat{j}-\hat{k}}{\sqrt{26}}\] |
| B. | \[\frac{-4\hat{i}+3\hat{j}+\hat{k}}{\sqrt{26}}\] |
| C. | \[\frac{-3\hat{i}+2\hat{j}-\hat{k}}{\sqrt{14}}\] |
| D. | \[\frac{-3\hat{i}+2\hat{j}+\hat{k}}{\sqrt{14}}\] |
| Answer» C. \[\frac{-3\hat{i}+2\hat{j}-\hat{k}}{\sqrt{14}}\] | |
| 7398. |
A force \[\vec{F}=3\hat{i}+2\hat{j}-4\hat{k}\] is applied at the point (1, -1, 2). What is the moment of the force about the point (2, -1, 3)? |
| A. | \[\hat{i}+4\hat{j}+4\hat{k}\] |
| B. | \[2\hat{i}+\hat{j}+2\hat{k}\] |
| C. | \[2\hat{i}-7\hat{j}-2\hat{k}\] |
| D. | \[2\hat{i}+4\hat{j}-\hat{k}\] |
| Answer» D. \[2\hat{i}+4\hat{j}-\hat{k}\] | |
| 7399. |
If \[\vec{p}\] and \[\vec{q}\] are non-collinear unit vectors and \[\left| \vec{p}+\vec{q} \right|=\sqrt{3}\], then \[(2\vec{p}-3\vec{q})\cdot (3\vec{p}+\vec{q})\] is equal to |
| A. | 0 |
| B. | \[\frac{1}{3}\] |
| C. | \[-\frac{1}{3}\] |
| D. | \[-\frac{1}{2}\] |
| Answer» E. | |
| 7400. |
Let \[\overset{\to }{\mathop{p}}\,,\overset{\to }{\mathop{q}}\,,\overset{\to }{\mathop{r}}\,\] be three mutually perpendicular vectors of the same magnitude. If a vector \[\vec{x}\] satisfies the equation \[\overset{\to }{\mathop{p}}\,\times \{(\overset{\to }{\mathop{x}}\,-\overset{\to }{\mathop{q}}\,)\times \overset{\to }{\mathop{p}}\,\}+\overset{\to }{\mathop{q}}\,\times \{(\overset{\to }{\mathop{x}}\,-\overset{\to }{\mathop{r}}\,))\times \overset{\to }{\mathop{q}}\,\}\]\[+\overset{\to }{\mathop{r}}\,\times \{(\overset{\to }{\mathop{x}}\,-\overset{\to }{\mathop{p}}\,)\times \overset{\to }{\mathop{r}}\,\}=\overset{\to }{\mathop{0}}\,\] then \[\overset{\to }{\mathop{x}}\,\] is given by |
| A. | \[\frac{1}{2}(\overset{\to }{\mathop{p}}\,+\overset{\to }{\mathop{q}}\,-2\overset{\to }{\mathop{r}}\,)\] |
| B. | \[\frac{1}{2}(\overset{\to }{\mathop{p}}\,+\overset{\to }{\mathop{q}}\,+\overset{\to }{\mathop{r}}\,)\] |
| C. | \[\frac{1}{3}(\overset{\to }{\mathop{p}}\,+\overset{\to }{\mathop{q}}\,+\overset{\to }{\mathop{r}}\,)\] |
| D. | \[\frac{1}{3}(2\overset{\to }{\mathop{p}}\,+\overset{\to }{\mathop{q}}\,-\overset{\to }{\mathop{r}}\,)\] |
| Answer» C. \[\frac{1}{3}(\overset{\to }{\mathop{p}}\,+\overset{\to }{\mathop{q}}\,+\overset{\to }{\mathop{r}}\,)\] | |