MCQOPTIONS
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MCQOPTIONS
This section includes 18 Mcqs, each offering curated multiple-choice questions to sharpen your Mathematics knowledge and support exam preparation. Choose a topic below to get started.
| 1. |
If three non-zero vectors are \[\mathbf{a}={{a}_{1}}\mathbf{i}+{{a}_{2}}\mathbf{j}+{{a}_{3}}\mathbf{k},\] \[\mathbf{b}={{b}_{1}}\mathbf{i}+{{b}_{2}}\mathbf{j}+{{b}_{3}}\mathbf{k}\] and \[\mathbf{c}={{c}_{1}}\mathbf{i}+{{c}_{2}}\mathbf{j}+{{c}_{3}}\mathbf{k}.\] If c is the unit vector perpendicular to the vectors a and b and the angle between a and b is \[\frac{\pi }{6},\] then \[{{\left| \,\begin{matrix}{{a}_{1}} & {{a}_{2}} & {{a}_{3}}\\{{b}_{1}} & {{b}_{2}} & {{b}_{3}}\\ {{c}_{1}} & {{c}_{2}} & {{c}_{3}}\\ \end{matrix}\, \right|}^{2}}\] is equal to [IIT 1986] |
| A. | 0 |
| B. | \[\frac{3\,(\Sigma a_{1}^{2})\,(\Sigma b_{1}^{2})\,(\Sigma c_{1}^{2})}{4}\] |
| C. | 1 |
| D. | \[\frac{(\Sigma a_{1}^{2})\,(\Sigma b_{1}^{2})}{4}\] |
| Answer» E. | |
| 2. |
If a of magnitude 50 is collinear with the vector \[\mathbf{b}=6\,\mathbf{i}-8\,\mathbf{j}-\frac{15\,\mathbf{k}}{2},\] and makes an acute angle with the positive direction of z-axis, then the vector a is equal to [Pb. CET 2004] |
| A. | \[24\,\mathbf{i}-32\,\mathbf{j}+30\,\mathbf{k}\] |
| B. | \[-24\,\mathbf{i}+32\,\mathbf{j}+30\,\mathbf{k}\] |
| C. | \[16\,\mathbf{i}-16\,\mathbf{j}-15\,\mathbf{k}\] |
| D. | \[-12\,\mathbf{i}+16\,\mathbf{j}-30\,\mathbf{k}\] |
| Answer» C. \[16\,\mathbf{i}-16\,\mathbf{j}-15\,\mathbf{k}\] | |
| 3. |
If two vertices of a triangle are \[\mathbf{i}-\mathbf{j}\] and \[\mathbf{j}+\mathbf{k}\], then the third vertex can be [Roorkee 1995] |
| A. | \[\mathbf{i}+\mathbf{k}\] |
| B. | \[\mathbf{i}-2\mathbf{j}-\mathbf{k}\] |
| C. | \[\mathbf{i}-\mathbf{k}\] |
| D. | \[2\mathbf{i}-\mathbf{j}\] |
| E. | All the above |
| Answer» F. | |
| 4. |
The vectors \[\overrightarrow{AB}=3\mathbf{i}+5\mathbf{j}+4\mathbf{k}\] and \[\overrightarrow{AC}=5\mathbf{i}-5\mathbf{j}+2\mathbf{k}\] are the sides of a triangle ABC. The length of the median through A is [UPSEAT 2004] |
| A. | \[\sqrt{13}\] unit |
| B. | \[\theta \] unit |
| C. | 5 unit |
| D. | 10 unit |
| Answer» D. 10 unit | |
| 5. |
If a, b and c are unit vectors, then \[|\mathbf{a}-\mathbf{b}{{|}^{2}}+|\mathbf{b}-\mathbf{c}{{|}^{2}}+|\mathbf{c}-\mathbf{a}{{|}^{2}}\]does not exceed [IIT Screening 2001] |
| A. | 4 |
| B. | 9 |
| C. | 8 |
| D. | 6 |
| Answer» C. 8 | |
| 6. |
The vectors b and c are in the direction of north-east and north-west respectively and |b|=|c|= 4. The magnitude and direction of the vector d = c - b, are [Roorkee 2000] |
| A. | \[4\sqrt{2},\] towards north |
| B. | \[4\sqrt{2}\], towards west |
| C. | 4, towards east |
| D. | 4, towards south |
| Answer» C. 4, towards east | |
| 7. |
The value of 'a' so that the volume of parallelopiped formed by \[\mathbf{i}+a\mathbf{j}+\mathbf{k},\mathbf{j}+a\,\mathbf{k}\] and \[a\,\mathbf{i}+\mathbf{k}\] becomes minimum is [IIT Screening 2003] |
| A. | -3 |
| B. | 3 |
| C. | \[\frac{1}{\sqrt{3}}\] |
| D. | \[\sqrt{3}\] |
| Answer» D. \[\sqrt{3}\] | |
| 8. |
If \[\mathbf{u}=2\,\mathbf{i}+2\mathbf{j}-\mathbf{k}\]and \[\mathbf{v}=6\,\mathbf{i}-3\,\mathbf{j}+2\,\mathbf{k},\] then a unit vector perpendicular to both u and v is [MP PET 1987] |
| A. | \[\mathbf{i}-10\mathbf{j}-18\mathbf{k}\] |
| B. | \[\frac{1}{\sqrt{17}}\,\left( \frac{1}{5}\mathbf{i}-2\mathbf{j}-\frac{18}{5}\mathbf{k} \right)\] |
| C. | \[\frac{1}{\sqrt{473}}\,(7\mathbf{i}-10\mathbf{j}-18\mathbf{k})\] |
| D. | None of these |
| Answer» C. \[\frac{1}{\sqrt{473}}\,(7\mathbf{i}-10\mathbf{j}-18\mathbf{k})\] | |
| 9. |
A vector a has components 2p and 1 with respect to a rectangular cartesian system. The system is rotated through a certain angle about the origin in the anti-clockwise sense. If a has components p+1 and 1 with respect to the new system, then [IIT 1984] |
| A. | \[p=0\] |
| B. | \[p=1\] or \[-\frac{1}{3}\] |
| C. | \[p=-1\] or \[\frac{1}{3}\] |
| D. | \[p=1\] or \[-1\] |
| Answer» C. \[p=-1\] or \[\frac{1}{3}\] | |
| 10. |
Let \[\mathbf{a}=2\mathbf{i}-\mathbf{j}+\mathbf{k},\,\,\mathbf{b}=\mathbf{i}+2\mathbf{j}-\mathbf{k}\] and \[\mathbf{c}=\mathbf{i}+\mathbf{j}-2\mathbf{k}\] be three vectors. A vector in the plane of b and c whose projection on a is of magnitude \[\sqrt{2/3}\] is [IIT 1993; Pb. CET 2004] |
| A. | \[2\mathbf{i}+3\mathbf{j}-3\mathbf{k}\] |
| B. | \[2\mathbf{i}+3\mathbf{j}+3\mathbf{k}\] |
| C. | \[-\,2\mathbf{i}-\mathbf{j}+5\mathbf{k}\] |
| D. | \[2\mathbf{i}+\mathbf{j}+5\mathbf{k}\] |
| Answer» B. \[2\mathbf{i}+3\mathbf{j}+3\mathbf{k}\] | |
| 11. |
If the equation of a line through a point a and parallel to vector b is \[\mathbf{r}=\mathbf{a}+t\,\mathbf{b},\] where t is a parameter, then its perpendicular distance from the point c is [MP PET 1998] |
| A. | \[|(\mathbf{c}-\mathbf{b})\times \mathbf{a}|\div |\mathbf{a}|\] |
| B. | \[|(\mathbf{c}-\mathbf{a})\times \mathbf{b}|\div |\mathbf{b}|\] |
| C. | \[|(\mathbf{a}-\mathbf{b})\times \mathbf{c}|\div |\mathbf{c}|\] |
| D. | \[|(\mathbf{a}-\mathbf{b})\times \mathbf{c}|\div |\mathbf{a}+\mathbf{c}|\] |
| Answer» C. \[|(\mathbf{a}-\mathbf{b})\times \mathbf{c}|\div |\mathbf{c}|\] | |
| 12. |
A Plane meets the co-ordinate axes at P, Q and R such that the position vector of the centroid of \[\Delta PQR\] is \[2\mathbf{i}-5\mathbf{j}+8\mathbf{k}\]. Then the equation of the plane is [J & K 2005] |
| A. | \[\mathbf{r}.(20\mathbf{i}-8\mathbf{j}+5\mathbf{k})=120\] |
| B. | \[\mathbf{r}.(20\mathbf{i}-8\mathbf{j}+5\mathbf{k})=1\] |
| C. | \[\mathbf{r}.(20\mathbf{i}-8\mathbf{j}+5\mathbf{k})=2\] |
| D. | \[\mathbf{r}.(20\mathbf{i}-8\mathbf{j}+5\mathbf{k})=20\] |
| Answer» B. \[\mathbf{r}.(20\mathbf{i}-8\mathbf{j}+5\mathbf{k})=1\] | |
| 13. |
The position vector of the point where the line \[\mathbf{r}=\mathbf{i}-\mathbf{j}+\mathbf{k}+t(\mathbf{i}+\mathbf{j}+\mathbf{k})\]meets the plane \[\mathbf{r}.(\mathbf{i}+\mathbf{j}+\mathbf{k})=5\]is [Kerala (Engg.) 2005] |
| A. | \[5\mathbf{i}+\mathbf{j}-\mathbf{k}\] |
| B. | \[5\mathbf{i}+3\mathbf{j}-3\mathbf{k}\] |
| C. | \[2\mathbf{i}+\mathbf{j}+2\mathbf{k}\] |
| D. | \[5\mathbf{i}+\mathbf{j}+\mathbf{k}\] |
| E. | \[4\mathbf{i}+2\mathbf{j}-2\mathbf{k}\] |
| Answer» C. \[2\mathbf{i}+\mathbf{j}+2\mathbf{k}\] | |
| 14. |
The equation \[|\mathbf{r}{{|}^{2}}-\mathbf{r}.(2\mathbf{i}+4\mathbf{j}-2\mathbf{k})-10=0\] represents a |
| A. | Circle |
| B. | Plane |
| C. | Sphere of radius 4 |
| D. | Sphere of radius 3 |
| E. | None of these |
| Answer» D. Sphere of radius 3 | |
| 15. |
The line through \[\mathbf{i}+3\mathbf{j}+2\mathbf{k}\] and perpendicular to the lines \[\mathbf{r}=(\mathbf{i}+2\mathbf{j}-\mathbf{k})+\lambda (2\mathbf{i}+\mathbf{j}+\mathbf{k})\] and \[\mathbf{r}=(2\mathbf{i}+6\mathbf{j}+\mathbf{k})+\mu (\mathbf{i}+2\mathbf{j}+3\mathbf{k})\] is |
| A. | \[\mathbf{r}=(\mathbf{i}+2\mathbf{j}-\mathbf{k})+\lambda (-\mathbf{i}+5\mathbf{j}-3\mathbf{k})\] |
| B. | \[\mathbf{r}=\mathbf{i}+3\mathbf{j}+2\mathbf{k}+\lambda (\mathbf{i}-5\mathbf{j}+3\mathbf{k})\] |
| C. | \[\mathbf{r}=\mathbf{i}+3\mathbf{j}+2\mathbf{k}+\lambda (\mathbf{i}+5\mathbf{j}+3\mathbf{k})\] |
| D. | \[\mathbf{r}=\mathbf{i}+3\mathbf{j}+2\mathbf{k}+\lambda (-\mathbf{i}+5\mathbf{j}-3\mathbf{k})\] |
| Answer» E. | |
| 16. |
The position vector of a point at a distance of \[3\sqrt{11}\] units from \[\mathbf{i}-\mathbf{j}+2\mathbf{k}\] on a line passing through the points \[\mathbf{i}-\mathbf{j}+2\mathbf{k}\] and \[3\mathbf{i}+\mathbf{j}+\mathbf{k}\] is |
| A. | \[10\mathbf{i}+2\mathbf{j}-5\mathbf{k}\] |
| B. | \[-8\mathbf{i}-4\mathbf{j}-\mathbf{k}\] |
| C. | \[8\mathbf{i}+4\mathbf{j}+\mathbf{k}\] |
| D. | \[-10\mathbf{i}-2\mathbf{j}-5\mathbf{k}\] |
| Answer» C. \[8\mathbf{i}+4\mathbf{j}+\mathbf{k}\] | |
| 17. |
The equation of the plane containing the lines \[\mathbf{r}={{\mathbf{a}}_{1}}+\lambda {{\mathbf{a}}_{2}}\] and \[\mathbf{r}={{\mathbf{a}}_{2}}+\lambda {{\mathbf{a}}_{1}}\] is |
| A. | \[[\mathbf{r}\,\ {{\mathbf{a}}_{1}}\ \,{{\mathbf{a}}_{2}}]=0\] |
| B. | \[[\mathbf{r}\ \,{{\mathbf{a}}_{1}}\ \,{{\mathbf{a}}_{2}}]={{\mathbf{a}}_{1}}.\ {{\mathbf{a}}_{2}}\] |
| C. | \[[\mathbf{r}\ \,{{\mathbf{a}}_{2}}\ \,{{\mathbf{a}}_{1}}]={{\mathbf{a}}_{1}}.\ {{\mathbf{a}}_{2}}\] |
| D. | None of these |
| Answer» B. \[[\mathbf{r}\ \,{{\mathbf{a}}_{1}}\ \,{{\mathbf{a}}_{2}}]={{\mathbf{a}}_{1}}.\ {{\mathbf{a}}_{2}}\] | |
| 18. |
The centre of the circle given by \[\mathbf{r}.(\mathbf{i}+2\mathbf{j}+2\mathbf{k})=15\] and \[|\mathbf{r}-(\mathbf{j}+2\mathbf{k})|=4\]is |
| A. | (0, 1, 2) |
| B. | (1, 3, 4) |
| C. | (?1, 3, 4) |
| D. | None of these |
| Answer» C. (?1, 3, 4) | |