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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.
| 7551. |
The triangle formed by the points (0, 7, 10), (?1, 6, 6), (? 4, 9, 6) is [RPET 2001] |
| A. | Equilateral |
| B. | Isosceles |
| C. | Right angled |
| D. | Right angled Isosceles |
| Answer» E. | |
| 7552. |
The points \[A(5,\,-1,\,\,1)\]; \[B\,(7,-4,\,7);\] \[C(1,\,-6,\,10)\] and \[D(-1,-3,\,4)\] are vertices of a [RPET 2000] |
| A. | Square |
| B. | Rhombus |
| C. | Rectangle |
| D. | None of these |
| Answer» C. Rectangle | |
| 7553. |
The direction cosines of the normal to the plane \[3x+4y+12z=52\] will be [MP PET 1997] |
| A. | 3, 4, 12 |
| B. | ? 3, ? 4, ? 12 |
| C. | \[\frac{3}{13},\frac{4}{13},\frac{12}{13}\] |
| D. | \[\frac{3}{\sqrt{13}},\frac{4}{\sqrt{13}},\frac{12}{\sqrt{13}}\] |
| Answer» D. \[\frac{3}{\sqrt{13}},\frac{4}{\sqrt{13}},\frac{12}{\sqrt{13}}\] | |
| 7554. |
The co-ordinates of the point which divides the join of the points (2, ?1, 3) and (4, 3, 1) in the ratio 3 : 4 internally are given by [MP PET 1997] |
| A. | \[\frac{2}{7},\frac{20}{7},\frac{10}{7}\] |
| B. | \[\frac{15}{7},\frac{20}{7},\frac{3}{7}\] |
| C. | \[\frac{10}{7},\frac{15}{7},\frac{2}{7}\] |
| D. | \[\frac{20}{7},\frac{5}{7},\frac{15}{7}\] |
| Answer» E. | |
| 7555. |
The plane \[XOZ\] divides the join of \[(1,\,-1,\,\,5)\] and (2, 3, 4) in the ratio \[\lambda :1\], then \[\lambda \] is [JET 1988] |
| A. | ? 3 |
| B. | 3 |
| C. | \[-\frac{1}{3}\] |
| D. | \[\frac{1}{3}\] |
| Answer» E. | |
| 7556. |
The projection of the line segment joining the points (?1, 0, 3) and (2, 5, 1) on the line whose direction ratios are 6, 2, 3 is [AI CBSE 1985] |
| A. | \[\frac{10}{7}\] |
| B. | \[\frac{22}{7}\] |
| C. | \[\frac{18}{7}\] |
| D. | None of these |
| Answer» C. \[\frac{18}{7}\] | |
| 7557. |
If the sum of the squares of the distance of a point from the three co-ordinate axes be 36,then its distance from the origin is |
| A. | 6 |
| B. | \[3\sqrt{2}\] |
| C. | \[2\sqrt{3}\] |
| D. | None of these |
| Answer» C. \[2\sqrt{3}\] | |
| 7558. |
If \[A\,(1\,,\,\,2,\,\,-1)\] and \[B(-1,\,\,0,\,\,1)\] are given, then the co-ordinates of P which divides \[AB\] externally in the ratio\[1:2\], are [MP PET 1989] |
| A. | \[\frac{1}{3}(1,\,4,-1)\] |
| B. | (3, 4, ?3) |
| C. | \[\frac{1}{3}(3,\,4,-3)\] |
| D. | None of these |
| Answer» C. \[\frac{1}{3}(3,\,4,-3)\] | |
| 7559. |
The direction cosines of the line \[x=y=z\] are [MP PET 1989] |
| A. | \[\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}\] |
| B. | \[\frac{1}{3},\frac{1}{3},\frac{1}{3}\] |
| C. | 1, 1, 1 |
| D. | None of these |
| Answer» B. \[\frac{1}{3},\frac{1}{3},\frac{1}{3}\] | |
| 7560. |
The centre of sphere passes through four points (0, 0, 0), (0, 2, 0), (1, 0, 0) and (0, 0, 4) is [MP PET 2002] |
| A. | \[\left( \frac{1}{2},\,1,\,2 \right)\] |
| B. | \[\left( -\frac{1}{2},\,1,\,2 \right)\] |
| C. | \[\left( \frac{1}{2},\,1,-\,2 \right)\] |
| D. | \[\left( 1,\frac{1}{2},\,2 \right)\] |
| Answer» B. \[\left( -\frac{1}{2},\,1,\,2 \right)\] | |
| 7561. |
Equation \[a{{x}^{2}}+b{{y}^{2}}+c{{z}^{2}}+2fyz+2gxz+2hxy\] \[+2ux+2vy+2wz+d=0\]represents a sphere, if [MP PET 1990] |
| A. | \[a=b=c\] |
| B. | \[f=g=h=0\] |
| C. | \[v=u=w\] |
| D. | \[a=b=c\] and \[f=g=h=0\] |
| Answer» E. | |
| 7562. |
How many different sphere of radius ?r? can be drawn which touches all the three co-ordinate axes |
| A. | 4 |
| B. | 2 |
| C. | 6 |
| D. | 8 |
| Answer» E. | |
| 7563. |
Equation of the plane through the mid-point of the line segment joining the points P(4, 5, -10) and Q(-1, 2, 1) and perpendicular to PQ is |
| A. | \[\vec{r}.\left( \frac{3}{2}\hat{i}+\frac{7}{2}\hat{j}-\frac{9}{2}\hat{k} \right)=45\] |
| B. | \[\vec{r}.\left( -\hat{i}+2\hat{j}-\hat{k} \right)=\frac{135}{2}\] |
| C. | \[\vec{r}.(5\hat{i}+3\hat{j}-11\hat{k})+\frac{135}{2}=0\] |
| D. | \[\vec{r}.(5\hat{i}+3\hat{j}-11\hat{k})=\frac{135}{2}\] |
| Answer» E. | |
| 7564. |
The perpendicular distance of P (1, 2, 3) form the lie \[\frac{x-6}{3}=\frac{y-7}{2}=\frac{z-7}{-2}\] is |
| A. | 7 |
| B. | 5 |
| C. | 0 |
| D. | 6 |
| Answer» B. 5 | |
| 7565. |
What is the angle between the lines\[\frac{x-2}{1}=\frac{y+1}{-2}\] and \[\frac{x-1}{1}=\frac{2y+3}{3}=\frac{z+5}{2}?\] |
| A. | \[\frac{\pi }{2}\] |
| B. | \[\frac{\pi }{3}\] |
| C. | \[\frac{\pi }{6}\] |
| D. | None of the above |
| Answer» B. \[\frac{\pi }{3}\] | |
| 7566. |
If O, P are the points (0, 0, 0), (2, 3, -1) respectively, then what is the equation to the plane through P at right angles to OP? |
| A. | \[2x+3y+z=16\] |
| B. | \[2x+3y-z=14\] |
| C. | \[2x+3y+z=14\] |
| D. | \[2x+3y-z=0\] |
| Answer» C. \[2x+3y+z=14\] | |
| 7567. |
The plane \[2x-3y+6z-11=0\] makes an angle \[{{\sin }^{-1}}(a)\] with the x-axis. Then the value of a is- |
| A. | \[\frac{\sqrt{3}}{2}\] |
| B. | \[\frac{\sqrt{2}}{3}\] |
| C. | \[\frac{3}{7}\] |
| D. | \[\frac{2}{7}\] |
| Answer» E. | |
| 7568. |
A variable plane passes through a fixed point (1, 2, 3). The locus of the foot of the perpendicular from the origin to this plane is given by |
| A. | \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-14=0\] |
| B. | \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}+x+2y+3z=0\] |
| C. | \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-x-2y-3z=0\] |
| D. | None of these |
| Answer» D. None of these | |
| 7569. |
If \[{{l}_{1}},{{m}_{1}},{{n}_{1}}\] and \[{{l}_{2}},{{m}_{2}},{{n}_{2}}\] are direction consines of the two lines inclined to each other at an angle \[\theta \], the direction cosines of the bisector of the angle between these lines are |
| A. | \[\frac{{{l}_{1}}-{{l}_{2}}}{2\sin \frac{\theta }{2}},\frac{{{m}_{1}}-{{m}_{2}}}{2\sin \frac{\theta }{2}},\frac{{{n}_{1}}-{{n}_{2}}}{2\sin \frac{\theta }{2}}\] |
| B. | \[\frac{{{l}_{1}}-{{l}_{2}}}{2\cos \frac{\theta }{2}},\frac{{{m}_{1}}-{{m}_{2}}}{2\cos \frac{\theta }{2}},\frac{{{n}_{1}}-{{n}_{2}}}{2\cos \frac{\theta }{2}}\] |
| C. | \[\frac{{{l}_{1}}-{{l}_{2}}}{2\sin \frac{\theta }{2}},\frac{{{m}_{1}}-{{m}_{2}}}{2\sin \frac{\theta }{2}},\frac{{{n}_{1}}-{{n}_{2}}}{2\sin \frac{\theta }{2}}\] |
| D. | \[\frac{{{l}_{1}}-{{l}_{2}}}{2\cos \frac{\theta }{2}},\frac{{{m}_{1}}-{{m}_{2}}}{2\cos \frac{\theta }{2}},\frac{{{n}_{1}}-{{n}_{2}}}{2\cos \frac{\theta }{2}}\] |
| Answer» D. \[\frac{{{l}_{1}}-{{l}_{2}}}{2\cos \frac{\theta }{2}},\frac{{{m}_{1}}-{{m}_{2}}}{2\cos \frac{\theta }{2}},\frac{{{n}_{1}}-{{n}_{2}}}{2\cos \frac{\theta }{2}}\] | |
| 7570. |
The vector equation of the line of intersection of the planes \[\vec{r}=\vec{b}+{{\lambda }_{1}}(\vec{b}-\vec{a})+{{\mu }_{1}}(\vec{a}-\vec{c})\] and \[\vec{r}=\vec{b}+{{\lambda }_{2}}(\vec{b}-\vec{c})+{{\mu }_{2}}(\vec{a}+\vec{c})\vec{a},\vec{b},\vec{c}\] being non-coplanar vectors, is |
| A. | \[\vec{r}=\vec{b}+{{\mu }_{1}}(\vec{a}+\vec{c})\] |
| B. | \[\vec{r}=\vec{b}+{{\lambda }_{1}}(\vec{a}-\vec{c})\] |
| C. | \[\vec{r}=2\vec{b}+{{\lambda }_{2}}(\vec{a}-\vec{c})\] |
| D. | None of these |
| Answer» B. \[\vec{r}=\vec{b}+{{\lambda }_{1}}(\vec{a}-\vec{c})\] | |
| 7571. |
If lines \[x=y=z\] and \[x=\frac{y}{2}=\frac{z}{3}\] and third line passing through (1, 1, 1) form a triangle of area \[\sqrt{6}\] units, then the point of intersection of third line with the second line will be |
| A. | \[(1,2,3)\] |
| B. | \[(2,4,6)\] |
| C. | \[\left( \frac{4}{3},\frac{8}{3},\frac{12}{3} \right)\] |
| D. | None of these |
| Answer» C. \[\left( \frac{4}{3},\frac{8}{3},\frac{12}{3} \right)\] | |
| 7572. |
A mirror and a source of light are situated at the origin 0 and at a point on OX respectively. A ray of light from the source strikes the mirror and is reflected. If the direction ratios of the normal to the plane are 1, -1, 1, then direction consines of the reflected rays are |
| A. | \[\frac{1}{3},\frac{2}{3},\frac{2}{3}\] |
| B. | \[-\frac{1}{3},\frac{2}{3},\frac{2}{3}\] |
| C. | \[-\frac{1}{3},\frac{2}{3},-\frac{2}{3}\] |
| D. | \[-\frac{1}{3},-\frac{2}{3},\frac{2}{3}\] |
| Answer» E. | |
| 7573. |
What is the value of n so that the angle between the lines having direction ratios (1, 1, 1) and (1, -1, n) is \[60{}^\circ \]? |
| A. | \[\sqrt{3}\] |
| B. | \[\sqrt{6}\] |
| C. | 3 |
| D. | None of these |
| Answer» C. 3 | |
| 7574. |
A plane passing through (1, 1, 1) cuts positive direction of coordinate axes at A, B and C, then the volume of tetrahedron OABC satisfies |
| A. | \[V\le \frac{9}{2}\] |
| B. | \[V\ge \frac{9}{2}\] |
| C. | \[V=\frac{9}{2}\] |
| D. | None of these |
| Answer» C. \[V=\frac{9}{2}\] | |
| 7575. |
Under what condition does the equation \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}+2uc+2uy+2wz+d=0\] represent a real sphere? |
| A. | \[{{u}^{2}}+{{v}^{2}}+{{w}^{2}}={{d}^{2}}\] |
| B. | \[{{u}^{2}}+{{v}^{2}}+{{w}^{2}}>d\] |
| C. | \[{{u}^{2}}+{{v}^{2}}+{{w}^{2}}<d\] |
| D. | \[{{u}^{2}}+{{v}^{2}}+{{w}^{2}}<{{d}^{2}}\] |
| Answer» C. \[{{u}^{2}}+{{v}^{2}}+{{w}^{2}}<d\] | |
| 7576. |
A line makes \[45{}^\circ \] with positive x-axis and makes equal angles with positive y, z axes, respectively. What is the sum of the three angles which the line makes with positive x, y and z axes? |
| A. | \[180{}^\circ \] |
| B. | \[165{}^\circ \] |
| C. | \[150{}^\circ \] |
| D. | \[135{}^\circ \] |
| Answer» C. \[150{}^\circ \] | |
| 7577. |
What is the distance between the planes\[x-2y+z-1=0\] and\[-3x+6y-3z+2=0\]? |
| A. | 3 unit |
| B. | 1 unit |
| C. | 0 |
| D. | None of the above |
| Answer» E. | |
| 7578. |
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 a and b? |
| A. | \[\alpha =2,\beta =2\] |
| B. | \[\alpha =1,\beta =2\] |
| C. | \[\alpha =2,\beta =1\] |
| D. | \[\alpha =2,\beta =1\] |
| Answer» E. | |
| 7579. |
Consider the following relations among the angles\[\alpha \], \[\beta \] and \[\gamma \] made by a vector with the coordinate axes I. \[\cos 2\alpha +\cos 2\beta +\cos 2\gamma =-1\] II. \[{{\sin }^{2}}\alpha +{{\sin }^{2}}\beta +{{\sin }^{2}}\gamma =1\] Which of the above is/are correct? |
| A. | Only I |
| B. | Only II |
| C. | Both I and II |
| D. | Neither I nor II |
| Answer» B. Only II | |
| 7580. |
Under which one of the following condition will the two planes \[x+y+z=7\] and\[\alpha x+\beta y+\gamma z=3\], be parallel (but not coincident)? |
| A. | \[\alpha =\beta =\gamma =1only\] |
| B. | \[\alpha =\beta =\gamma =\frac{3}{7}only\] |
| C. | \[\alpha =\beta =\gamma \] |
| D. | None of the above |
| Answer» D. None of the above | |
| 7581. |
The foot of the perpendicular from the point (1, 6, 3) to the line \[\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}\] is |
| A. | (1, 2, 5) |
| B. | (-1, -1, -1) |
| C. | (2, 5, 8) |
| D. | (-2, -3, -4) |
| Answer» B. (-1, -1, -1) | |
| 7582. |
Under what condition do \[\left\langle \frac{1}{\sqrt{2}},\frac{1}{2},k \right\rangle \] represent direction cosines of a line? |
| A. | \[k=\frac{1}{2}\] |
| B. | \[k=-\frac{1}{2}\] |
| C. | \[k=\pm \frac{1}{2}\] |
| D. | K can take any value |
| Answer» D. K can take any value | |
| 7583. |
A line makes angles \[\theta ,\phi \] and \[\psi \] with x, y, z axes respectively. Consider the following 1. \[{{\sin }^{2}}\theta +{{\sin }^{2}}\phi ={{\cos }^{2}}\psi \] 2. \[{{\cos }^{2}}\theta +{{\cos }^{2}}\phi ={{\sin }^{2}}\psi \] 3. \[{{\sin }^{2}}\theta +{{\cos }^{2}}\phi ={{\cos }^{2}}\psi \] Which of the above is/are correct? |
| A. | 1 only |
| B. | 2 only |
| C. | 3 only |
| D. | 2 and 3 |
| Answer» C. 3 only | |
| 7584. |
Which one of the following is the plane containing the lien \[\frac{x-2}{2}=\frac{y-3}{3}=\frac{z-4}{5}\] and parallel to z axis? |
| A. | \[2x-3y=0\] |
| B. | \[5x-2z=0\] |
| C. | \[5y-3z=0\] |
| D. | \[3x-2y=0\] |
| Answer» E. | |
| 7585. |
The angle between the line \[\frac{x-2}{a}=\frac{y-2}{b}=\frac{z-2}{c}\] and the plane \[ax+by+cz+6=0\] is |
| A. | \[{{\sin }^{-1}}\left( \frac{1}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}} \right)\] |
| B. | \[45{}^\circ \] |
| C. | \[60{}^\circ \] |
| D. | \[90{}^\circ \] |
| Answer» E. | |
| 7586. |
The angle between the straight lines \[\vec{r}=(2-3t)\vec{i}+(1+2t)\vec{j}+(2+6t)\vec{k}\] and \[\vec{r}=(1+4s)\vec{i}+(2-s)\vec{j}+(8s-1)\vec{k}\]is |
| A. | \[{{\cos }^{-1}}\left( \frac{\sqrt{41}}{34} \right)\] |
| B. | \[{{\cos }^{-1}}\left( \frac{21}{34} \right)\] |
| C. | \[{{\cos }^{-1}}\left( \frac{43}{63} \right)\] |
| D. | \[{{\cos }^{-1}}\left( \frac{34}{63} \right)\] |
| Answer» E. | |
| 7587. |
The direction consines of two lines are related by\[l+m+n=0\]\[a{{l}^{2}}+b{{m}^{2}}+c{{n}^{2}}=0\]. The lines are parallel if |
| A. | \[a+b+c=0\] |
| B. | \[{{a}^{-1}}+{{b}^{-1}}+{{c}^{-1}}=0\] |
| C. | \[a=b=c\] |
| D. | None of these |
| Answer» C. \[a=b=c\] | |
| 7588. |
Value of\[\lambda \] such that the line\[\frac{x-1}{2}=\frac{y-1}{3}=\frac{z-1}{\lambda }\]Is perpendicular to normal to the plane\[\vec{r}.(2\vec{i}+3\vec{j}+4\vec{k})=0\] is |
| A. | \[-\frac{13}{4}\] |
| B. | \[-\frac{17}{4}\] |
| C. | \[4\] |
| D. | None of these |
| Answer» B. \[-\frac{17}{4}\] | |
| 7589. |
The equation of the plane which passes through the line of intersection of planes \[\vec{r}.{{\vec{n}}_{1}}={{q}_{1}},\vec{r}.{{\vec{n}}_{2}}=q\] And is parallel to the line of intersection of planes \[\vec{r}.{{\vec{n}}_{3}}={{q}_{3}}\] and \[\vec{r}.{{\vec{n}}_{4}}={{q}_{4}}\]is |
| A. | \[[{{\vec{n}}_{2}}{{\vec{n}}_{3}}{{\vec{n}}_{4}}](\vec{r}.{{\vec{n}}_{1}}-{{\vec{q}}_{1}})=[{{\vec{n}}_{1}}{{\vec{n}}_{3}}{{\vec{n}}_{4}}](\vec{r}.{{\vec{n}}_{2}}-{{\vec{q}}_{2}})\] |
| B. | \[[{{\vec{n}}_{1}}{{\vec{n}}_{2}}{{\vec{n}}_{4}}](\vec{r}.{{\vec{n}}_{4}}{{q}_{4}})=[{{\vec{n}}_{4}}{{\vec{n}}_{3}}{{\vec{n}}_{1}}](\vec{r}.{{\vec{n}}_{2}}-{{q}_{2}})\] |
| C. | \[[{{\vec{n}}_{4}}{{\vec{n}}_{3}}{{\vec{n}}_{1}}](\vec{r}.{{\vec{n}}_{4}}-{{q}_{4}})=[{{\vec{n}}_{1}}{{\vec{n}}_{2}}\vec{n} 3](\vec{r}.{{\vec{n}}_{2}}={{q}_{2}})\] |
| D. | None of these |
| Answer» B. \[[{{\vec{n}}_{1}}{{\vec{n}}_{2}}{{\vec{n}}_{4}}](\vec{r}.{{\vec{n}}_{4}}{{q}_{4}})=[{{\vec{n}}_{4}}{{\vec{n}}_{3}}{{\vec{n}}_{1}}](\vec{r}.{{\vec{n}}_{2}}-{{q}_{2}})\] | |
| 7590. |
If the straight line \[\frac{x-{{x}_{0}}}{\ell }=\frac{y-{{y}_{0}}}{m}=\frac{z-{{z}_{0}}}{n}\] is parallel to the plane \[ax+by+cz+d=0\]then which one of the following is correct? |
| A. | \[\ell +m+n=0\] |
| B. | \[a+b+c=0\] |
| C. | \[\frac{a}{\ell }+\frac{b}{m}+\frac{c}{n}=0\] |
| D. | \[a\ell +bm+cn=0\] |
| Answer» E. | |
| 7591. |
The foot of the perpendicular drawn from the origin to a plane is the point (1,-3, 1). What is the intercept cut on the x-axis by the plane? |
| A. | 1 |
| B. | 3 |
| C. | \[\sqrt{11}\] |
| D. | 11 |
| Answer» E. | |
| 7592. |
If the center of the sphere \[a{{x}^{2}}+b{{y}^{2}}+c{{z}^{2}}-2x+4y+2z-3=0\]is \[(1/2,-1,-1/2)\], what is the value of b ? |
| A. | 1 |
| B. | -1 |
| C. | 2 |
| D. | -2 |
| Answer» D. -2 | |
| 7593. |
The equation of the line which passes through the point (1, 1, 1) and intersect the lines \[\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}\] and \[\frac{x+2}{1}=\frac{y-3}{2}=\frac{z+1}{4}\] is |
| A. | \[\frac{x-1}{3}=\frac{y-1}{10}=\frac{z-1}{17}\] |
| B. | \[\frac{x-1}{3}=\frac{y-1}{3}=\frac{z-1}{-5}\] |
| C. | \[\frac{x-1}{-2}=\frac{y-1}{1}=\frac{z-1}{-4}\] |
| D. | \[\frac{x-1}{8}=\frac{y-1}{-2}=\frac{z-1}{3}\] |
| Answer» B. \[\frac{x-1}{3}=\frac{y-1}{3}=\frac{z-1}{-5}\] | |
| 7594. |
If the plane \[2ax-3ay+4az+6=0\] passes through the midpoint of the line joining the centres of the spheres \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}+6x-8y-2z=13\] and \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-10x+4y-2z=8\] then a equals |
| A. | -1 |
| B. | 1 |
| C. | -2 |
| D. | 2 |
| Answer» D. 2 | |
| 7595. |
The direction cosines l, m, n of two lines are connected by the relations l + m + n = 0, lm = 0, then the angle between them is: |
| A. | \[\pi /3\] |
| B. | \[\pi /4\] |
| C. | \[\pi /2\] |
| D. | 0 |
| Answer» B. \[\pi /4\] | |
| 7596. |
If \[\overset{\to }{\mathop{r}}\,=(\hat{i}+2\hat{j}+3\hat{k})+\lambda (\hat{i}+\hat{j}+\hat{k})\] and \[\overset{\to }{\mathop{r}}\,=(\hat{i}+2\hat{j}+3\hat{k})+\mu (\hat{i}+\hat{j}-\hat{k})\] are two lines, then the equation of acute angle bisector of two lines is |
| A. | \[\overset{\to }{\mathop{r}}\,=(\hat{i}+2\hat{j}+3\hat{k})+t(\hat{j}-\hat{k})\] |
| B. | \[\overset{\to }{\mathop{r}}\,=(\hat{i}+2\hat{j}+3\hat{k})+t(2\hat{i})\] |
| C. | \[\overset{\to }{\mathop{r}}\,=(\hat{i}+2\hat{j}+3\hat{k})+t(\hat{j}+\hat{k})\] |
| D. | None of these |
| Answer» B. \[\overset{\to }{\mathop{r}}\,=(\hat{i}+2\hat{j}+3\hat{k})+t(2\hat{i})\] | |
| 7597. |
What are the direction ratios of the line determined by the planes \[x-y+2z=1\] and\[x+y-z=3\]? |
| A. | (-1, 3, 2) |
| B. | (-1, -3, 2) |
| C. | (2, 1, 3) |
| D. | (2, 3, 2) |
| Answer» B. (-1, -3, 2) | |
| 7598. |
What are the direction cosines of a line which is equally inclined to the positive directions of the axes? |
| A. | \[\left\langle \frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}} \right\rangle \] |
| B. | \[\left\langle -\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}} \right\rangle \] |
| C. | \[\left\langle -\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}} \right\rangle \] |
| D. | \[\left\langle \frac{1}{3},\frac{1}{3},\frac{1}{3} \right\rangle \] |
| Answer» B. \[\left\langle -\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}} \right\rangle \] | |
| 7599. |
From a point \[P(\lambda ,\lambda ,\lambda ),\] perpendiculars PQ and PR are drawn, respectively, on the lines \[y=x,\text{ }z=1\] and \[y=-x,\text{ }z=-1\]. If \[\angle QPR\] is a right angle, then the possible value(s) of \[\lambda \] is/are |
| A. | 2 |
| B. | 1 |
| C. | -1 |
| D. | \[-\,\sqrt{2}\] |
| Answer» D. \[-\,\sqrt{2}\] | |
| 7600. |
A plane passes through a fixed point (a, b, c). The locus of the foot of the perpendicular to it from the origin is the sphere |
| A. | \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-ax-by-cz=0\] |
| B. | \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-2ax-2by-2cz=0\] |
| C. | \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-4ax-4by-4cz=0\] |
| D. | None of these |
| Answer» B. \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-2ax-2by-2cz=0\] | |