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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.
| 1301. |
Let \[f(x)=\left\{ \begin{align} & \frac{1-{{\sin }^{3}}x}{3{{\cos }^{2}}x},x\frac{\pi }{2} \\ \end{align} \right.\] If f(x) is continuous at \[x=\frac{\pi }{2},(p,q)=\] |
| A. | \[(1,4)\] |
| B. | \[\left( \frac{1}{2},2 \right)\] |
| C. | \[\left( \frac{1}{2},4 \right)\] |
| D. | None of these |
| Answer» D. None of these | |
| 1302. |
If \[f(x)=\left\{ \begin{align} & \frac{x\log \cos x}{\log (1+{{x}^{2}})},x\ne 0 \\ & \,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,,x=0 \\ \end{align} \right.\text{then}\,\,\text{f(x)is}\] is |
| A. | Continuous as well as differentiable at x = 0 |
| B. | Continuous but not differentiable at x = 0 |
| C. | Differentiable but not continuous at x = 0 |
| D. | Neither continuous nor differentiable at x = 0 |
| Answer» B. Continuous but not differentiable at x = 0 | |
| 1303. |
Let \[f(x)=g(x).\frac{{{e}^{1/x}}-{{e}^{-1/x}}}{{{e}^{1/x}}+{{e}^{-1/x}}}\], where g is a continuous function then \[\underset{x\to 0}{\mathop{\lim }}\,f(x)\] does not exist if |
| A. | g(x) is any constant function |
| B. | g(x)=x |
| C. | \[g(x)={{x}^{2}}\] |
| D. | g(x) = x h (x), where h(x) is a polynomial. |
| Answer» B. g(x)=x | |
| 1304. |
If\[y={{\log }^{n}}x\], where \[{{\log }^{n}}\] means log log log... (repeated n time), then \[x\,log\text{ }x\text{ }log\text{ }x\text{ }lo{{g}^{2}}x\text{ }lo{{g}^{3}}x\]\[....{{\log }^{n-1}}x{{\log }^{n}}x\frac{dy}{dx}\] is equal to |
| A. | \[\log x\] |
| B. | \[{{\log }^{n}}x\] |
| C. | \[\frac{1}{\log \,x}\] |
| D. | 1 |
| Answer» C. \[\frac{1}{\log \,x}\] | |
| 1305. |
Which one of the following is correct? The eccentricity of the conic \[\frac{{{x}^{2}}}{{{a}^{2}}+\lambda }+\frac{{{y}^{2}}}{{{b}^{2}}+\lambda }=1,(\lambda \ge 0)\] |
| A. | Increases with increase in \[\lambda \] |
| B. | Decreases with increase in \[\lambda \] |
| C. | Does not change with \[\lambda \] |
| D. | None of these |
| Answer» C. Does not change with \[\lambda \] | |
| 1306. |
Let E be the ellipse \[\frac{{{x}^{2}}}{9}+\frac{{{y}^{2}}}{4}=1\] and C be the circle \[{{x}^{2}}+{{y}^{2}}=9.\] Let \[P=(1,2)\] and \[Q=(2,1)\] Which one of the following is correct? |
| A. | Q lies inside C but outside E |
| B. | Q lies outside both C and E |
| C. | P lies inside both C and E |
| D. | P lies inside both C but outside E. |
| Answer» E. | |
| 1307. |
The locus of the point of intersection of two tangents of the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] which are inclined at angles \[{{\theta }_{1}}\], and \[{{\theta }_{2}}\] with the major axis such that \[{{\tan }^{2}}{{\theta }_{1}}+{{\tan }^{2}}{{\theta }_{2}}\] is constant, is |
| A. | \[4{{x}^{2}}{{y}^{2}}+2({{x}^{2}}-{{a}^{2}})({{y}^{2}}-{{b}^{2}})=k{{({{x}^{2}}-{{a}^{2}})}^{2}}\] |
| B. | \[4{{x}^{2}}{{y}^{2}}-2({{x}^{2}}-{{a}^{2}})({{y}^{2}}-{{b}^{2}})=k{{({{x}^{2}}-{{a}^{2}})}^{2}}\] |
| C. | \[4{{x}^{2}}{{y}^{2}}-2({{x}^{2}}-{{a}^{2}})({{y}^{2}}-{{b}^{2}})=k{{({{x}^{2}}+{{a}^{2}})}^{2}}\] |
| D. | None of these |
| Answer» C. \[4{{x}^{2}}{{y}^{2}}-2({{x}^{2}}-{{a}^{2}})({{y}^{2}}-{{b}^{2}})=k{{({{x}^{2}}+{{a}^{2}})}^{2}}\] | |
| 1308. |
The curve described parametrically by \[x=2-3\sec t,y=1+4\tan t\] represents: |
| A. | An ellipse centred at (2, 1) and of eccentricity \[\frac{3}{5}\] |
| B. | A circle centred at (2, 1) and of radius 5 units |
| C. | A hyperbola centred at (2, 1) & of eccentricity \[\frac{8}{5}\] |
| D. | A hyperbola centred at \[(2,1)\]& of eccentricity \[\frac{5}{3}\] |
| Answer» E. | |
| 1309. |
The sum of the squares of the perpendiculars on any tangent to the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] from two points on the minor axis each at a distance \[\sqrt{{{a}^{2}}-{{b}^{2}}}\] from the centre is |
| A. | \[2{{a}^{2}}\] |
| B. | \[2{{b}^{2}}\] |
| C. | \[{{a}^{2}}+{{b}^{2}}\] |
| D. | \[{{a}^{2}}-{{b}^{2}}\] |
| Answer» B. \[2{{b}^{2}}\] | |
| 1310. |
If the focal distance of an end of the minor axis of any ellipse (referred to its axis as the axes of x and y respectively) is k and the distance between the foci is 2h, then its equation is |
| A. | \[\frac{{{x}^{2}}}{{{k}^{2}}}+\frac{{{y}^{2}}}{{{k}^{2}}+{{h}^{2}}}=1\] |
| B. | \[\frac{{{x}^{2}}}{{{k}^{2}}}+\frac{{{y}^{2}}}{{{h}^{2}}-{{k}^{2}}}=1\] |
| C. | \[\frac{{{x}^{2}}}{{{k}^{2}}}+\frac{{{y}^{2}}}{{{k}^{2}}-{{h}^{2}}}=1\] |
| D. | \[\frac{{{x}^{2}}}{{{k}^{2}}}+\frac{{{y}^{2}}}{{{h}^{2}}}=1\] |
| Answer» D. \[\frac{{{x}^{2}}}{{{k}^{2}}}+\frac{{{y}^{2}}}{{{h}^{2}}}=1\] | |
| 1311. |
Under which one of the following conditions does the circle \[{{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0\]meet the x-axis in two points on opposite sides of the origin? |
| A. | \[c>0\] |
| B. | \[c<0\] |
| C. | \[c=0\] |
| D. | \[c\le 0\] |
| Answer» C. \[c=0\] | |
| 1312. |
If the angle between the straight lines joining the foci to an extremity of minor axis in an ellipse be \[90{}^\circ \]; then the eccentricity of the ellipse is |
| A. | \[\frac{1}{2}\] |
| B. | \[\frac{1}{\sqrt{3}}\] |
| C. | \[\frac{1}{\sqrt{2}}\] |
| D. | \[\frac{1}{3}\] |
| Answer» D. \[\frac{1}{3}\] | |
| 1313. |
If the line \[y=mx+\sqrt{{{a}^{2}}{{m}^{2}}-{{b}^{2}}}\] touches the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] at the point\[\varphi \]. Then \[\varphi \]= |
| A. | \[{{\sin }^{-1}}(m)\] |
| B. | \[{{\sin }^{-1}}\left( \frac{a}{bm} \right)\] |
| C. | \[{{\sin }^{-1}}\left( \frac{b}{am} \right)\] |
| D. | \[{{\sin }^{-1}}\left( \frac{bm}{a} \right)\] |
| Answer» D. \[{{\sin }^{-1}}\left( \frac{bm}{a} \right)\] | |
| 1314. |
The equation of one of the common tangents to the parabola \[{{y}^{2}}=8x\] and \[{{x}^{2}}+{{y}^{2}}-12x+4=0\]is |
| A. | \[y=-x+2\] |
| B. | \[y=x-2\] |
| C. | \[y=x+2\] |
| D. | None of these |
| Answer» D. None of these | |
| 1315. |
If a Point \[P(x,y)\] moves along the ellipse \[\frac{{{x}^{2}}}{25}+\frac{{{y}^{2}}}{16}=1\] and if C is the centre of the ellipse, then, 4 max \[\{CP\}+5min\{CP\}=\] |
| A. | 25 |
| B. | 40 |
| C. | 45 |
| D. | 54 |
| Answer» C. 45 | |
| 1316. |
The equation of the parabola whose focus is (0, 0) and the tangent at the vertex is \[8x-y+1=0\] is |
| A. | \[{{x}^{2}}+{{y}^{2}}+2xy-4x+4y-4=0\] |
| B. | \[{{x}^{2}}-4x+4y-4=0\] |
| C. | \[{{y}^{2}}-4x+4y-4=0\] |
| D. | \[2{{x}^{2}}+2{{y}^{2}}-4xy-x+y-4=0\] |
| Answer» B. \[{{x}^{2}}-4x+4y-4=0\] | |
| 1317. |
If \[x=9\] is the chord of contact of the hyperbola \[{{x}^{2}}-{{y}^{2}}=9\], then the equation of the corresponding pair of tangents is |
| A. | \[9{{x}^{2}}-8{{y}^{2}}+18x-9=0\] |
| B. | \[9{{x}^{2}}-8{{y}^{2}}-18x+9=0\] |
| C. | \[9{{x}^{2}}-8{{y}^{2}}-18x-9=0\] |
| D. | \[9{{x}^{2}}-8{{y}^{2}}+18x+9=0\] |
| Answer» C. \[9{{x}^{2}}-8{{y}^{2}}-18x-9=0\] | |
| 1318. |
If the tangents at P and Q on a parabola meet in T, Then SP, ST and SQ are in |
| A. | A.P |
| B. | GP. |
| C. | H.P. |
| D. | None of these |
| Answer» C. H.P. | |
| 1319. |
The point \[(a,2a)\] is an interior point of region bounded by the parabola \[{{x}^{2}}=16y\] and the double ordinate through focus then |
| A. | a<4 |
| B. | 0<a<4 |
| C. | 0<a<2 |
| D. | a>4 |
| Answer» D. a>4 | |
| 1320. |
If the chords of contact of tangents from two points \[(\alpha ,\beta )\] and \[(\gamma ,\delta )\] to the ellipse \[\frac{{{x}^{2}}}{5}+\frac{{{y}^{2}}}{2}=1\] are perpendicular, then \[\frac{\alpha \gamma }{\beta \delta }=\] |
| A. | \[\frac{4}{25}\] |
| B. | \[\frac{-4}{25}\] |
| C. | \[\frac{25}{4}\] |
| D. | \[\frac{-25}{4}\] |
| Answer» E. | |
| 1321. |
If a variable point P on an ellipse of eccentricity e lines joining the foci \[{{S}_{1}}\] and \[{{S}_{2}}\] then the in centre of the triangle \[P{{S}_{1}}{{S}_{2}}\] lies on |
| A. | The major axis of the ellipse |
| B. | The circle with radius e |
| C. | Another ellipse of eccentricity \[\sqrt{\frac{3+{{e}^{2}}}{4}}\] |
| D. | None of these |
| Answer» D. None of these | |
| 1322. |
The line joining (5, 0) to is divided internally in the ratio 2 : 3 at P. If \[\theta \] varies, then the locus of P is |
| A. | A pair of straight lines |
| B. | A circle |
| C. | A straight line |
| D. | None of these |
| Answer» C. A straight line | |
| 1323. |
What is the area of the triangle formed by the lines joining the vertex of the parabola \[{{x}^{2}}=12y\] to the ends of the latus rectum? |
| A. | 9 square units |
| B. | 12 square units |
| C. | 14 square units |
| D. | 18 square units |
| Answer» E. | |
| 1324. |
Through the vertex O at a parabola \[{{y}^{2}}=4x,\] chords OP and OQ are drawn at right angles to one another. The locus of the middle point of PQ is |
| A. | \[{{y}^{2}}=2x+8\] |
| B. | \[{{y}^{2}}=x+8\] |
| C. | \[{{y}^{2}}=2x-8\] |
| D. | \[{{y}^{2}}=x-8\] |
| Answer» D. \[{{y}^{2}}=x-8\] | |
| 1325. |
The locus of the point of intersection of two tangents to the parabola \[{{y}^{2}}=4ax,\] which are at right angle to one another is |
| A. | \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\] |
| B. | \[a{{y}^{2}}=x\] |
| C. | \[x+a=0\] |
| D. | \[x+y\pm a=0\] |
| Answer» D. \[x+y\pm a=0\] | |
| 1326. |
If tangents are drawn from any point on the line \[x+4a=0\] to the parabola \[{{y}^{2}}=4ax,\] then their chord of contact subtends angle at the vertex equal to |
| A. | \[\frac{\pi }{4}\] |
| B. | \[\frac{\pi }{3}\] |
| C. | \[\frac{\pi }{2}\] |
| D. | \[\frac{\pi }{6}\] |
| Answer» D. \[\frac{\pi }{6}\] | |
| 1327. |
If the pair of lines \[a{{x}^{2}}+2(a+b)xy+b{{y}^{2}}=0\] lie along diameters of a circle and divide the circle into four sectors such that the area of one of the sectors is thrice the area of another sector then |
| A. | \[3{{a}^{2}}-10ab+3{{b}^{2}}=0\] |
| B. | \[3{{a}^{2}}-2ab+3{{b}^{2}}=0\] |
| C. | \[3{{a}^{2}}+10ab+3{{b}^{2}}=0\] |
| D. | \[3{{a}^{2}}+2ab+3{{b}^{2}}=0\] |
| Answer» E. | |
| 1328. |
A tangent to the parabola \[{{y}^{2}}=8x,\] which makes an angle of \[45{}^\circ \] with the straight line \[y=3x+5\]is |
| A. | \[2x-y+1=0\] |
| B. | \[2x+y+1=0\] |
| C. | \[x-2y+8=0\] |
| D. | Both & |
| Answer» E. | |
| 1329. |
If p is the length of the perpendicular form the focus S of the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] to a tangent at a point P on the ellipse, then \[\frac{2a}{SP}-1=\] |
| A. | \[\frac{{{a}^{2}}}{{{p}^{2}}}\] |
| B. | \[\frac{{{b}^{2}}}{{{p}^{2}}}\] |
| C. | \[{{p}^{2}}\] |
| D. | \[\frac{{{a}^{2}}+{{b}^{2}}}{{{p}^{2}}}\] |
| Answer» C. \[{{p}^{2}}\] | |
| 1330. |
The common chord of \[{{x}^{2}}+{{y}^{2}}-4x-4y=0\] and \[{{x}^{2}}+{{y}^{2}}=16\] subtends at the origin an angle equal to |
| A. | \[\frac{\pi }{6}\] |
| B. | \[\frac{\pi }{4}\] |
| C. | \[\frac{\pi }{3}\] |
| D. | \[\frac{\pi }{2}\] |
| Answer» E. | |
| 1331. |
If \[{{\left( \frac{x}{a} \right)}^{2}}+\left( {{\frac{y}{b}}^{2}} \right)=1(a>b)\] and \[{{x}^{2}}-{{y}^{2}}={{c}^{2}}\] cut at right angles, then |
| A. | \[{{a}^{2}}+{{b}^{2}}=2{{c}^{2}}\] |
| B. | \[{{b}^{2}}-{{a}^{2}}=2{{c}^{2}}\] |
| C. | \[{{a}^{2}}-{{b}^{2}}=2{{c}^{2}}\] |
| D. | \[{{a}^{2}}-{{b}^{2}}={{c}^{2}}\] |
| Answer» D. \[{{a}^{2}}-{{b}^{2}}={{c}^{2}}\] | |
| 1332. |
If the ellipse \[9{{x}^{2}}+16{{y}^{2}}=144\] intercepts the line \[3x+4y=12,\] then what is the length of the chord so formed? |
| A. | 5 units |
| B. | 6 units |
| C. | 8 units |
| D. | 10 units |
| Answer» B. 6 units | |
| 1333. |
Area of the equilateral triangle inscribed in the circle \[{{x}^{2}}+{{y}^{2}}-7x+9y+5=0\] is |
| A. | \[\frac{155}{8}\sqrt{3}\] square units |
| B. | \[\frac{165}{8}\sqrt{3}\] square units |
| C. | \[\frac{175}{8}\sqrt{3}\] square units |
| D. | \[\frac{165}{8}\sqrt{3}\] square units |
| Answer» E. | |
| 1334. |
If AB is a double ordinate of the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] such that \[\Delta OAB\] is an equilateral triangle O being the origin, then the eccentricity of the hyperbola satisfies. |
| A. | \[e>\sqrt{3}\] |
| B. | \[1<e<\frac{2}{\sqrt{3}}\] |
| C. | \[e=\frac{2}{\sqrt{3}}\] |
| D. | \[e>\frac{2}{\sqrt{3}}\] |
| Answer» E. | |
| 1335. |
The hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] passes through the point \[(3\sqrt{5},1)\] and the length of its laths rectum is \[\frac{4}{3}\]units. The length of the conjugate axis is |
| A. | 2 units |
| B. | 3 units |
| C. | 4 units |
| D. | 5 units |
| Answer» D. 5 units | |
| 1336. |
The angle of intersection of the circles \[{{x}^{2}}+{{y}^{2}}=4\] and \[{{x}^{2}}+{{y}^{2}}=2x+2y\] is |
| A. | \[\frac{\pi }{2}\] |
| B. | \[\frac{\pi }{3}\] |
| C. | \[\frac{\pi }{6}\] |
| D. | \[\frac{\pi }{4}\] |
| Answer» E. | |
| 1337. |
In the given figure, the equation of the larger circle is \[{{x}^{2}}+{{y}^{2}}+4y-5=0\] and the distance between centres is 4. Then the equation of smaller circle is |
| A. | \[{{(x-\sqrt{7})}^{2}}+{{(y-1)}^{2}}=1\] |
| B. | \[{{(x+\sqrt{7})}^{2}}+{{(y-1)}^{2}}=1\] |
| C. | \[{{x}^{2}}+{{y}^{2}}=2\sqrt{7}x+2y\] |
| D. | None of these |
| Answer» B. \[{{(x+\sqrt{7})}^{2}}+{{(y-1)}^{2}}=1\] | |
| 1338. |
The value of m, for which the line \[y=mx+\frac{25\sqrt{3}}{3}\] is a normal to the conic \[\frac{{{x}^{2}}}{16}-\frac{{{y}^{2}}}{9}=1,\] is |
| A. | \[-\frac{2}{\sqrt{3}}\] |
| B. | \[\sqrt{3}\] |
| C. | \[-\frac{\sqrt{3}}{2}\] |
| D. | None of these |
| Answer» B. \[\sqrt{3}\] | |
| 1339. |
A hyperbola having the transverse axis of length \[2\sin \theta ,\] is confocal with the ellipse \[3{{x}^{2}}+4{{y}^{2}}=12.\] Then its equation is |
| A. | \[{{x}^{2}}\cos e{{c}^{2}}\theta -{{y}^{2}}{{\sec }^{2}}\theta =1\] |
| B. | \[{{x}^{2}}{{\sec }^{2}}\theta -{{y}^{2}}\cos e{{c}^{2}}\theta =1\] |
| C. | \[{{x}^{2}}{{\sin }^{2}}\theta -{{y}^{2}}{{\cos }^{2}}\theta =1\] |
| D. | \[{{x}^{2}}{{\cos }^{2}}\theta -{{y}^{2}}{{\sin }^{2}}\theta =1\] |
| Answer» B. \[{{x}^{2}}{{\sec }^{2}}\theta -{{y}^{2}}\cos e{{c}^{2}}\theta =1\] | |
| 1340. |
The equation of a circle which passes through the point (2, 0) and whose centre is the limit of the point of intersection of the lines \[3x+5y=1\] and \[(2+c)x+5{{c}^{2}}y=1\] as c tends to 1, is |
| A. | \[25({{x}^{2}}+{{y}^{2}})+20x+2y-60=0\] |
| B. | \[25({{x}^{2}}+{{y}^{2}})-20x+2y+60=0\] |
| C. | \[25({{x}^{2}}+{{y}^{2}})-20x+2y-60=0\] |
| D. | None of these |
| Answer» D. None of these | |
| 1341. |
Four distinct points \[(2k,3k),\,\,(1,0),\,\,(0,1)\] and \[(0,0)\] lie on a circle for |
| A. | Only one value of k |
| B. | \[0<k<1\] |
| C. | \[k<0\] |
| D. | All integral values of k |
| Answer» B. \[0<k<1\] | |
| 1342. |
Let \[P(a\,\,sec\theta ,b\,\,tan\,\theta )\] and Q \[Q(a\,\,sec\,\phi ,\,\,b\,tan\,\,\phi ),\]where \[\theta +\phi =\pi /2,\] be two points on the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1.\] If \[(h,k)\] is the point of intersection of the normal at P and Q, then kz is equal to |
| A. | \[\frac{{{a}^{2}}+{{b}^{2}}}{a}\] |
| B. | \[-\left( \frac{{{a}^{2}}+{{b}^{2}}}{a} \right)\] |
| C. | \[\frac{{{a}^{2}}+{{b}^{2}}}{b}\] |
| D. | \[-\left( \frac{{{a}^{2}}+{{b}^{2}}}{b} \right)\] |
| Answer» E. | |
| 1343. |
Tangents at any point on the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] cut the axes at A and B respectively. If the rectangle OAPB (where O is the origin)is completed, then locus of point P is given by : |
| A. | \[\frac{{{a}^{2}}}{{{x}^{2}}}-\frac{{{b}^{2}}}{{{y}^{2}}}=1\] |
| B. | \[\frac{{{a}^{2}}}{{{x}^{2}}}+\frac{{{b}^{2}}}{{{y}^{2}}}=1\] |
| C. | \[\frac{{{a}^{2}}}{{{y}^{2}}}-\frac{{{b}^{2}}}{{{x}^{2}}}=1\] |
| D. | None of these |
| Answer» B. \[\frac{{{a}^{2}}}{{{x}^{2}}}+\frac{{{b}^{2}}}{{{y}^{2}}}=1\] | |
| 1344. |
The equation of the image of circle \[{{x}^{2}}+{{y}^{2}}+16x-24y+183=0\] by the line mirror \[4x+7y+13=0\] is |
| A. | \[{{x}^{2}}+{{y}^{2}}+32x-4y+235=0\] |
| B. | \[{{x}^{2}}+{{y}^{2}}+32x+4y-235=0\] |
| C. | \[{{x}^{2}}+{{y}^{2}}+32x-4y-235=0\] |
| D. | \[{{x}^{2}}+{{y}^{2}}+32x+4y+235=0\] |
| Answer» E. | |
| 1345. |
The line \[y=mx+c\] intersects the circle \[{{x}^{2}}+{{y}^{2}}={{r}^{2}}\] at the two real distinct points if |
| A. | \[-r\sqrt{1+{{m}^{2}}}<c<r\sqrt{1+{{m}^{2}}}\] |
| B. | \[-r<c<r\] |
| C. | \[-r\sqrt{1-{{m}^{2}}}<c<r\sqrt{1+{{m}^{2}}}\] |
| D. | None of these |
| Answer» B. \[-r<c<r\] | |
| 1346. |
The point \[([P+1],[P])\](where [x] is the greatest integer less than or equal to x), lying inside the region bounded by the circle \[{{x}^{2}}+{{y}^{2}}-2x-15=0\] and \[{{x}^{2}}+{{y}^{2}}-2x-7=0,\]then |
| A. | \[P\in [-1,0)\cup [0,1)\cup [1,2)\] |
| B. | \[P\in [-1,\,\,2)-\{0,\,\,1\}\] |
| C. | \[P\in (-1,\,\,2)\] |
| D. | None of these |
| Answer» E. | |
| 1347. |
Consider any point P on the ellipse \[\frac{{{x}^{2}}}{25}+\frac{{{y}^{2}}}{9}=1\] in the first quadrant. Let r and s represent its distances from (4, 0) and (-4, 0) respectively, then (r + s) is equal to |
| A. | 10 unit |
| B. | 9 unit |
| C. | 8 unit |
| D. | 6 unit |
| Answer» B. 9 unit | |
| 1348. |
If OA and OB are the tangents form the origin to the circle \[{{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0\] and C is the centre of the circle, the area of the quadrilateral OACD is |
| A. | \[\frac{1}{2}\sqrt{c({{g}^{2}}+{{f}^{2}}-c)}\] |
| B. | \[\sqrt{c({{g}^{2}}+{{f}^{2}}-c)}\] |
| C. | \[c\sqrt{{{g}^{2}}+{{f}^{2}}-c}\] |
| D. | \[\frac{\sqrt{{{g}^{2}}+{{f}^{2}}-c}}{c}\] |
| Answer» C. \[c\sqrt{{{g}^{2}}+{{f}^{2}}-c}\] | |
| 1349. |
Let A be the centre of the circle \[{{x}^{2}}+{{y}^{2}}-2x-4y-20=0,\] and \[B(1,7)\] and \[D(4,-2)\] are points on the circle then, if tangents be drawn at B and D, which meet at C, then area of quadrilateral ABCD is- |
| A. | 150 |
| B. | 75 |
| C. | 75/2 |
| D. | None of these |
| Answer» C. 75/2 | |
| 1350. |
If the circle \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\] intersects the hyperbola \[xy={{c}^{2}}\] in four points \[P({{x}_{1}},{{y}_{1}}),Q({{x}_{2}},{{y}_{2}}),R({{x}_{3}},{{y}_{3}}),S({{x}_{4}},{{y}_{4}})\] Then |
| A. | \[{{x}_{1}}+{{x}_{2}}+{{x}_{3}}+{{x}_{4}}=0\] |
| B. | \[{{y}_{1}}+{{y}_{2}}+{{y}_{3}}+{{y}_{4}}=2\] |
| C. | \[{{x}_{1}}{{x}_{2}}{{x}_{3}}{{x}_{4}}=2{{c}^{4}}\] |
| D. | \[{{y}_{1}}{{y}_{2}}{{y}_{3}}{{y}_{4}}=2{{c}^{4}}\] |
| Answer» B. \[{{y}_{1}}+{{y}_{2}}+{{y}_{3}}+{{y}_{4}}=2\] | |