MCQOPTIONS
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MCQOPTIONS
This section includes 29 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 the eccentricity of the two ellipse \[\frac{{{x}^{2}}}{169}+\frac{{{y}^{2}}}{25}=1\]and \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] are equal, then the value of \[a/b\] is [UPSEAT 2001] |
| A. | 5/13 |
| B. | 6/13 |
| C. | 13/5 |
| D. | 13/6 |
| Answer» D. 13/6 | |
| 2. |
C the centre of the hyperbola\[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\]. The tangents at any point P on this hyperbola meets the straight lines \[bx-ay=0\]and \[bx+ay=0\] in the points Q and R respectively. Then \[CQ\ .\ CR=\] |
| A. | \[{{a}^{2}}+{{b}^{2}}\] |
| B. | \[{{a}^{2}}-{{b}^{2}}\] |
| C. | \[\frac{1}{{{a}^{2}}}+\frac{1}{{{b}^{2}}}\] |
| D. | \[\frac{1}{{{a}^{2}}}-\frac{1}{{{b}^{2}}}\] |
| Answer» B. \[{{a}^{2}}-{{b}^{2}}\] | |
| 3. |
If the two tangents drawn on hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] in such a way that the product of their gradients is \[{{c}^{2}}\], then they intersects on the curve |
| A. | \[{{y}^{2}}+{{b}^{2}}={{c}^{2}}({{x}^{2}}-{{a}^{2}})\] |
| B. | \[{{y}^{2}}+{{b}^{2}}={{c}^{2}}({{x}^{2}}+{{a}^{2}})\] |
| C. | \[a{{x}^{2}}+b{{y}^{2}}={{c}^{2}}\] |
| D. | None of these |
| Answer» B. \[{{y}^{2}}+{{b}^{2}}={{c}^{2}}({{x}^{2}}+{{a}^{2}})\] | |
| 4. |
Equation \[\frac{1}{r}=\frac{1}{8}+\frac{3}{8}\cos \theta \] represents [EAMCET 2002] |
| A. | A rectangular hyperbola |
| B. | A hyperbola |
| C. | An ellipse |
| D. | A parabola |
| Answer» C. An ellipse | |
| 5. |
Tangent is drawn to ellipse \[\frac{{{x}^{2}}}{27}+{{y}^{2}}=1\] at \[(3\sqrt{3}\cos \theta ,\ \sin \theta )\] where \[\theta \in (0,\ \pi /2)\]. Then the value of \[\theta \] such that sum of intercepts on axes made by this tangent is minimum, is [IIT Screening 2003] |
| A. | \[\pi /3\] |
| B. | \[\pi /6\] |
| C. | \[\pi /8\] |
| D. | \[\pi /4\] |
| Answer» C. \[\pi /8\] | |
| 6. |
The area of the quadrilateral formed by the tangents at the end points of latus rectum to the ellipse \[\frac{{{x}^{2}}}{9}+\frac{{{y}^{2}}}{5}=1\], is [IIT Screening 2003] |
| A. | 27/4 sq. unit |
| B. | 9 sq. unit |
| C. | 27/2 sq. unit |
| D. | 27 sq. unit |
| Answer» E. | |
| 7. |
The eccentricity of an ellipse, with its centre at the origin, is \[\frac{1}{2}\]. If one of the directrices is \[x=4\], then the equation of the ellipse is [AIEEE 2004] |
| A. | \[4{{x}^{2}}+3{{y}^{2}}=1\] |
| B. | \[3{{x}^{2}}+4{{y}^{2}}=12\] |
| C. | \[4{{x}^{2}}+3{{y}^{2}}=12\] |
| D. | \[3{{x}^{2}}+4{{y}^{2}}=1\] |
| Answer» C. \[4{{x}^{2}}+3{{y}^{2}}=12\] | |
| 8. |
The co-ordinates of the foci of the ellipse \[3{{x}^{2}}+4{{y}^{2}}-12x-8y+4=0\] are |
| A. | (1, 2), (3, 4) |
| B. | (1, 4), (3, 1) |
| C. | (1, 1), (3, 1) |
| D. | (2, 3), (5, 4) |
| Answer» D. (2, 3), (5, 4) | |
| 9. |
The equation \[14{{x}^{2}}-4xy+11{{y}^{2}}-44x-58y+71=0\] represents [BIT Ranchi 1986] |
| A. | A circle |
| B. | An ellipse |
| C. | A hyperbola |
| D. | A rectangular hyperbola |
| Answer» C. A hyperbola | |
| 10. |
The centre of the ellipse \[4{{x}^{2}}+9{{y}^{2}}-16x-54y+61=0\] is [MP PET 1992] |
| A. | (1, 3) |
| B. | (2, 3) |
| C. | (3, 2) |
| D. | (3, 1) |
| Answer» C. (3, 2) | |
| 11. |
P is any point on the ellipse\[9{{x}^{2}}+36{{y}^{2}}=324\]., whose foci are S and S?. Then \[SP+S'P\] equals [DCE 1999] |
| A. | 3 |
| B. | 12 |
| C. | 36 |
| D. | 324 |
| Answer» C. 36 | |
| 12. |
The length of the latus rectum of the ellipse \[9{{x}^{2}}+4{{y}^{2}}=1\], is [MP PET 1999] |
| A. | \[\frac{3}{2}\] |
| B. | \[\frac{8}{3}\] |
| C. | \[\frac{4}{9}\] |
| D. | \[\frac{8}{9}\] |
| Answer» D. \[\frac{8}{9}\] | |
| 13. |
For the ellipse \[3{{x}^{2}}+4{{y}^{2}}=12\], the length of latus rectum is [MNR 1973] |
| A. | \[\frac{3}{2}\] |
| B. | 3 |
| C. | \[\frac{8}{3}\] |
| D. | \[\sqrt{\frac{3}{2}}\] |
| Answer» C. \[\frac{8}{3}\] | |
| 14. |
If the centre, one of the foci and semi-major axis of an ellipse be (0, 0), (0, 3) and 5 then its equation is [AMU 1981] |
| A. | \[\frac{{{x}^{2}}}{16}+\frac{{{y}^{2}}}{25}=1\] |
| B. | \[\frac{{{x}^{2}}}{25}+\frac{{{y}^{2}}}{16}=1\] |
| C. | \[\frac{{{x}^{2}}}{9}+\frac{{{y}^{2}}}{25}=1\] |
| D. | None of these |
| Answer» B. \[\frac{{{x}^{2}}}{25}+\frac{{{y}^{2}}}{16}=1\] | |
| 15. |
The equation of the ellipse whose foci are \[(\pm 5,\ 0)\] and one of its directrix is \[5x=36\], is |
| A. | \[\frac{{{x}^{2}}}{36}+\frac{{{y}^{2}}}{11}=1\] |
| B. | \[\frac{{{x}^{2}}}{6}+\frac{{{y}^{2}}}{\sqrt{11}}=1\] |
| C. | \[\frac{{{x}^{2}}}{6}+\frac{{{y}^{2}}}{11}=1\] |
| D. | None of these |
| Answer» B. \[\frac{{{x}^{2}}}{6}+\frac{{{y}^{2}}}{\sqrt{11}}=1\] | |
| 16. |
The eccentricity of the ellipse \[25{{x}^{2}}+16{{y}^{2}}-150x-175=0\] is[Kerala (Engg.) 2005] |
| A. | 2/5 |
| B. | 2/3 |
| C. | 4/5 |
| D. | 3/4 |
| E. | 3/5 |
| Answer» F. | |
| 17. |
Consider a circle with its centre lying on the focus of the parabola \[{{y}^{2}}=2px\] such that it touches the directrix of the parabola. Then, a point of intersection of the circle and the parabola is [IIT 1995] |
| A. | \[\left( \frac{p}{2},\ p \right)\] |
| B. | \[\left( \frac{p}{2},\ -p \right)\] |
| C. | \[\left( \frac{-p}{2},\ p \right)\] |
| D. | \[\left( \frac{-p}{2},\ -p \right)\] |
| Answer» C. \[\left( \frac{-p}{2},\ p \right)\] | |
| 18. |
The centre of the circle passing through the point (0, 1) and touching the curve \[y={{x}^{2}}\]at (2, 4) is [IIT 1983] |
| A. | \[\left( \frac{-16}{5},\ \frac{27}{10} \right)\] |
| B. | \[\left( \frac{-16}{7},\ \frac{5}{10} \right)\] |
| C. | \[\left( \frac{-16}{5},\ \frac{53}{10} \right)\] |
| D. | None of these |
| Answer» D. None of these | |
| 19. |
The line \[x-1=0\] is the directrix of the parabola \[{{y}^{2}}-kx+8=0\]. Then one of the values of k is [IIT Screening 2000] |
| A. | \[\frac{1}{8}\] |
| B. | 8 |
| C. | 4 |
| D. | \[\frac{1}{4}\] |
| Answer» D. \[\frac{1}{4}\] | |
| 20. |
The locus of the midpoint of the line segment joining the focus to a moving point on the parabola \[{{y}^{2}}=4ax\] is another parabola with the directrix [IIT Screening 2002] |
| A. | \[x=-a\] |
| B. | \[x=-\frac{a}{2}\] |
| C. | \[x=0\] |
| D. | \[x=\frac{a}{2}\] |
| Answer» D. \[x=\frac{a}{2}\] | |
| 21. |
If the chord joining the points \[(at_{1}^{2},\ 2a{{t}_{1}})\] and \[(at_{2}^{2},\ 2a{{t}_{2}})\] of the parabola \[{{y}^{2}}=4ax\] passes through the focus of the parabola, then [MP PET 1993] |
| A. | \[{{t}_{1}}{{t}_{2}}=-1\] |
| B. | \[{{t}_{1}}{{t}_{2}}=1\] |
| C. | \[{{t}_{1}}+{{t}_{2}}=-1\] |
| D. | \[{{t}_{1}}-{{t}_{2}}=1\] |
| Answer» B. \[{{t}_{1}}{{t}_{2}}=1\] | |
| 22. |
The number of points of intersection of the two curves\[y=2\sin x\] and \[y=5{{x}^{2}}+2x+3\] is [IIT 1994] |
| A. | 0 |
| B. | 1 |
| C. | 2 |
| D. | \[\infty \] |
| Answer» B. 1 | |
| 23. |
On the ellipse \[4{{x}^{2}}+9{{y}^{2}}=1\], the points at which the tangents are parallel to the line \[8x=9y\] are [IIT 1999] |
| A. | \[\left( \frac{2}{5},\ \frac{1}{5} \right)\] |
| B. | \[\left( -\frac{2}{5},\ \frac{1}{5} \right)\] |
| C. | \[\left( -\frac{2}{5},\ -\frac{1}{5} \right)\] |
| D. | \[\left( \frac{2}{5},\ -\frac{1}{5} \right)\] |
| Answer» C. \[\left( -\frac{2}{5},\ -\frac{1}{5} \right)\] | |
| 24. |
If the angle between the lines joining the end points of minor axis of an ellipse with its foci is \[{{x}^{2}}-{{y}^{2}}=25\], then the eccentricity of the ellipse is [IIT Screening 1997; Pb. CET 2001; DCE 2002] |
| A. | 1/2 |
| B. | \[1/\sqrt{2}\] |
| C. | \[\sqrt{3}/2\] |
| D. | \[1/2\sqrt{2}\] |
| Answer» C. \[\sqrt{3}/2\] | |
| 25. |
A man running round a race-course notes that the sum of the distance of two flag-posts from him is always 10 metres and the distance between the flag-posts is 8 metres. The area of the path he encloses in square metres is [MNR 1991; UPSEAT 2000] |
| A. | \[15\pi \] |
| B. | \[12\pi \] |
| C. | \[18\pi \] |
| D. | \[8\pi \] |
| Answer» B. \[12\pi \] | |
| 26. |
The equation of the parabola whose focus is the point (0, 0) and the tangent at the vertex is \[x-y+1=0\]is [Orissa JEE 2002] |
| A. | \[{{x}^{2}}+{{y}^{2}}-2xy-4x+4y-4=0\] |
| B. | \[{{x}^{2}}+{{y}^{2}}-2xy+4x-4y-4=0\] |
| C. | \[{{x}^{2}}+{{y}^{2}}+2xy-4x+4y-4=0\] |
| D. | \[{{x}^{2}}+{{y}^{2}}+2xy-4x-4y+4=0\] |
| Answer» D. \[{{x}^{2}}+{{y}^{2}}+2xy-4x-4y+4=0\] | |
| 27. |
The equation of the common tangent to the curves \[{{y}^{2}}=8x\] and \[xy=-1\] is [IIT Screening 2002] |
| A. | \[3y=9x+2\] |
| B. | \[y=2x+1\] |
| C. | \[2y=x+8\] |
| D. | \[y=x+2\] |
| Answer» E. | |
| 28. |
The angle of intersection of the curves \[{{y}^{2}}=2x/\pi \] and \[y=\sin x\], is [Roorkee 1998] |
| A. | \[{{\cot }^{-1}}(-1/\pi )\] |
| B. | \[{{\cot }^{-1}}\pi \] |
| C. | \[{{\cot }^{-1}}(-\pi )\] |
| D. | \[{{\cot }^{-1}}(1/\pi )\] |
| Answer» C. \[{{\cot }^{-1}}(-\pi )\] | |
| 29. |
Equation\[\sqrt{{{(x-2)}^{2}}+{{y}^{2}}}+\sqrt{{{(x+2)}^{2}}+{{y}^{2}}}=4\]represents [Orissa JEE 2004] |
| A. | Parabola |
| B. | Ellipse |
| C. | Circle |
| D. | Pair of straight lines |
| Answer» E. | |