MCQOPTIONS
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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.
| 5451. |
If the ratio of gradients of the lines represented by \[a{{x}^{2}}+2hxy+b{{y}^{2}}=0\]is 1 : 3, then the value of the ratio \[{{h}^{2}}:ab\]is [MP PET 1998] |
| A. | \[\frac{1}{3}\] |
| B. | \[\frac{3}{4}\] |
| C. | \[\frac{4}{3}\] |
| D. | 1 |
| Answer» D. 1 | |
| 5452. |
If \[L{{x}^{2}}-10xy+12{{y}^{2}}\]\[+5x-16y-3=0\] represents a pair of straight lines, then L is [MP PET 2001] |
| A. | 1 |
| B. | 2 |
| C. | 3 |
| D. | -1 |
| Answer» C. 3 | |
| 5453. |
If the point (2,-3) lies on \[k{{x}^{2}}-3{{y}^{2}}+2x+y-2=0\], then k is equal to |
| A. | \[\frac{1}{7}\] |
| B. | 16 |
| C. | 7 |
| D. | 12 |
| Answer» D. 12 | |
| 5454. |
The gradient of one of the lines \[{{x}^{2}}+hxy+2{{y}^{2}}=0\] is twice that of the other, then h = [MP PET 1996] |
| A. | \[\pm \,3\] |
| B. | \[\pm \,\frac{3}{2}\] |
| C. | \[\pm \,2\] |
| D. | \[\pm \,1\] |
| Answer» B. \[\pm \,\frac{3}{2}\] | |
| 5455. |
\[2{{x}^{2}}+7xy+3{{y}^{2}}+8x+14y+\lambda =0\] will represent a pair of straight lines, when \[\lambda \]= [MP PET 1996] |
| A. | 2 |
| B. | 4 |
| C. | 6 |
| D. | 8 |
| Answer» E. | |
| 5456. |
The value of \[\lambda \] for which the equation \[{{x}^{2}}-\lambda xy+2{{y}^{2}}+3x-5y+2=0\] may represent a pair of straight lines is [Kurukshetra CEE 1996] |
| A. | 2 |
| B. | 3 |
| C. | 4 |
| D. | 1 |
| Answer» C. 4 | |
| 5457. |
The joint equation of the straight lines \[x+y=1\]and \[x-y=4\]is [Karnataka CET 1993] |
| A. | \[{{x}^{2}}-{{y}^{2}}=-4\] |
| B. | \[{{x}^{2}}-{{y}^{2}}=4\] |
| C. | \[(x+y-1)\,(x-y-4)=0\] |
| D. | \[(x+y+1)(x-y+4)=0\] |
| Answer» D. \[(x+y+1)(x-y+4)=0\] | |
| 5458. |
The equation \[{{x}^{2}}+k{{y}^{2}}+4xy=0\]represents two coincident lines, if k = [MP PET 1995] |
| A. | 0 |
| B. | 1 |
| C. | 4 |
| D. | 16 |
| Answer» D. 16 | |
| 5459. |
The condition of representing the coincident lines by the general quadratic equation \[f(x,\,y)=0\], is |
| A. | \[\Delta =0\] and \[{{h}^{2}}=ab\] |
| B. | \[\Delta =0\] and \[a+b=0\] |
| C. | \[\Delta =0\] and \[{{h}^{2}}=ab\], \[{{g}^{2}}=ac\], \[{{f}^{2}}=bc\] |
| D. | \[{{h}^{2}}=ab\], \[{{g}^{2}}=ac\] and \[{{f}^{2}}=bc\] |
| Answer» 2 , 3. \[\Delta =0\] and \[{{h}^{2}}=ab\], \[{{g}^{2}}=ac\], \[{{f}^{2}}=bc\] | |
| 5460. |
If in general quadratic equation \[f(x,\,y)=0\], \[\Delta =0\] and \[{{h}^{2}}=ab\], then the equation represents |
| A. | Two parallel straight lines |
| B. | Two perpendicular straight lines |
| C. | Two coincident lines |
| D. | None of these |
| Answer» B. Two perpendicular straight lines | |
| 5461. |
If the equation \[hxy+gx+fy+c=0\] represents a pair of straight lines, then |
| A. | \[fh=cg\] |
| B. | \[fg=ch\] |
| C. | \[{{h}^{2}}=gf\] |
| D. | \[fgh=c\] |
| Answer» C. \[{{h}^{2}}=gf\] | |
| 5462. |
The pair of straight lines passes through the point (1, 2) and perpendicular to the pair of straight lines \[3{{x}^{2}}-8xy+5{{y}^{2}}=0\], is |
| A. | \[(5x+3y+11)(x+y+3)=0\] |
| B. | \[(5x+3y-11)(x+y-3)=0\] |
| C. | \[(3x+5y-11)(x+y+3)=0\] |
| D. | \[(3x-5y+11)(x+y-3)=0\] |
| Answer» C. \[(3x+5y-11)(x+y+3)=0\] | |
| 5463. |
The equation of one of the line represented by the equation \[pq({{x}^{2}}-{{y}^{2}})+({{p}^{2}}-{{q}^{2}})xy=0\], is |
| A. | \[px+qy=0\] |
| B. | \[px-qy=0\] |
| C. | \[{{p}^{2}}x+{{q}^{2}}y=0\] |
| D. | \[{{q}^{2}}x-{{p}^{2}}y=0\] |
| Answer» B. \[px-qy=0\] | |
| 5464. |
The equation of one of the line represented by the equation \[{{x}^{2}}+2xy\cot \theta -{{y}^{2}}=0\], is |
| A. | \[x-y\cot \theta =0\] |
| B. | \[x+y\tan \theta =0\] |
| C. | \[x\sin \theta +y(\cos \theta +1)=0\] |
| D. | \[x\cos \theta +y(\sin \theta +1)=0\] |
| Answer» D. \[x\cos \theta +y(\sin \theta +1)=0\] | |
| 5465. |
If the equation \[\lambda {{x}^{2}}+2{{y}^{2}}-5xy+5x-7y+3=0\] represents two straight lines, then the value of l will be [RPET 1989] |
| A. | 3 |
| B. | 2 |
| C. | 8 |
| D. | - 8 |
| Answer» C. 8 | |
| 5466. |
The equation \[{{x}^{2}}-7xy+12{{y}^{2}}=0\] represents [BIT Ranchi 1991] |
| A. | Circle |
| B. | Pair of parallel straight lines |
| C. | Pair of perpendicular straight lines |
| D. | Pair of non-perpendicular intersecting straight lines |
| Answer» E. | |
| 5467. |
The equations of the lines represented by the equation \[a{{x}^{2}}+(a+b)xy+b{{y}^{2}}+x+y=0\] are |
| A. | \[ax+by+1=0\], \[x+y=0\] |
| B. | \[ax+by-1=0\], \[x+y=0\] |
| C. | \[ax+by+1=0\], \[x-y=0\] |
| D. | None of these |
| Answer» B. \[ax+by-1=0\], \[x+y=0\] | |
| 5468. |
If the equation \[a{{x}^{2}}+b{{y}^{2}}+cx+cy=0\] represents a pair of straight lines, then |
| A. | \[a(b+c)=0\] |
| B. | \[b(c+a)=0\] |
| C. | \[c(a+b)=0\] |
| D. | \[a+b+c=0\] |
| Answer» D. \[a+b+c=0\] | |
| 5469. |
The equation of the lines represented by the equation \[{{x}^{2}}-5xy+6{{y}^{2}}=0\] are |
| A. | \[y+2x=0\], \[y-3x=0\] |
| B. | \[y-2x=0\], \[y-3x=0\] |
| C. | \[y+2x=0\], \[y+3x=0\] |
| D. | None of these |
| Answer» E. | |
| 5470. |
Two lines represented by equation \[{{x}^{2}}+xy+{{y}^{2}}=0\] are |
| A. | Coincident |
| B. | Parallel |
| C. | Mutually perpendicular |
| D. | Imaginary |
| Answer» E. | |
| 5471. |
The equation \[2{{y}^{2}}-xy-{{x}^{2}}+6x-8=0\] represents [MP PET 1992] |
| A. | A pair of straight lines |
| B. | A circle |
| C. | An ellipse |
| D. | A parabola |
| Answer» B. A circle | |
| 5472. |
The equation \[4{{x}^{2}}+12xy+9{{y}^{2}}+2gx+2fy+c=0\] will represents two real parallel straight lines, if |
| A. | g = 4, f = 9, c = 0 |
| B. | g = 2, f = 3, c = 1 |
| C. | g = 2, f = 3, c is any number |
| D. | g = 4, f = 9, c > 1 |
| Answer» D. g = 4, f = 9, c > 1 | |
| 5473. |
If the slope of one of the line represented by the equation \[a{{x}^{2}}+2hxy+b{{y}^{2}}=0\] be l times that of the other, then |
| A. | \[4\lambda h=ab(1+\lambda )\] |
| B. | \[\lambda h=ab{{(1+\lambda )}^{2}}\] |
| C. | \[4\lambda {{h}^{2}}=ab{{(1+\lambda )}^{2}}\] |
| D. | None of these |
| Answer» D. None of these | |
| 5474. |
The equation \[xy+{{a}^{2}}=a(x+y)\] represents [MP PET 1991] |
| A. | A parabola |
| B. | A pair of straight lines |
| C. | An ellipse |
| D. | Two parallel straight lines |
| Answer» C. An ellipse | |
| 5475. |
The equation of the lines passing through the origin and having slopes 3 and \[-\frac{1}{3}\] is |
| A. | \[3{{y}^{2}}+8xy-3{{x}^{2}}=0\] |
| B. | \[3{{x}^{2}}+8xy-3{{y}^{2}}=0\] |
| C. | \[3{{y}^{2}}-8xy+3{{x}^{2}}=0\] |
| D. | \[3{{x}^{2}}+8xy+3{{y}^{2}}=0\] |
| Answer» C. \[3{{y}^{2}}-8xy+3{{x}^{2}}=0\] | |
| 5476. |
If the equation \[A{{x}^{2}}+2Bxy+C{{y}^{2}}+Dx+Ey+F=0\] represents a pair of straight lines, then \[{{B}^{2}}-AC\] [MP PET 1992] |
| A. | < 0 |
| B. | = 0 |
| C. | > 0 |
| D. | None of these |
| Answer» E. | |
| 5477. |
The equation of the lines passing through the origin and parallel to the lines represented by the equation \[2{{x}^{2}}-xy-6{{y}^{2}}+7x+21y-15=0\], is |
| A. | \[2{{x}^{2}}-xy-6{{y}^{2}}=0\] |
| B. | \[6{{x}^{2}}-xy+2{{y}^{2}}=0\] |
| C. | \[6{{x}^{2}}-xy-2{{y}^{2}}=0\] |
| D. | \[2{{x}^{2}}+xy-6{{y}^{2}}=0\] |
| Answer» B. \[6{{x}^{2}}-xy+2{{y}^{2}}=0\] | |
| 5478. |
If one of the lines represented by the equation \[a{{x}^{2}}+2hxy+b{{y}^{2}}=0\] be \[y=mx\], then [UPSEAT 1999] |
| A. | \[b{{m}^{2}}+2hm+a=0\] |
| B. | \[b{{m}^{2}}+2hm-a=0\] |
| C. | \[a{{m}^{2}}+2hm+b=0\] |
| D. | \[b{{m}^{2}}-2hm+a=0\] |
| Answer» B. \[b{{m}^{2}}+2hm-a=0\] | |
| 5479. |
If the slope of one of the lines represented by \[a{{x}^{2}}+2hxy+b{{y}^{2}}=0\] be the square of the other, then |
| A. | \[{{a}^{2}}b+a{{b}^{2}}-6abh+8{{h}^{3}}=0\] |
| B. | \[{{a}^{2}}b+a{{b}^{2}}+6abh+8{{h}^{3}}=0\] |
| C. | \[{{a}^{2}}b+a{{b}^{2}}-3abh+8{{h}^{3}}=0\] |
| D. | \[{{a}^{2}}b+a{{b}^{2}}-6abh-8{{h}^{3}}=0\] |
| Answer» B. \[{{a}^{2}}b+a{{b}^{2}}+6abh+8{{h}^{3}}=0\] | |
| 5480. |
. The lines \[{{a}^{2}}{{x}^{2}}+bc{{y}^{2}}=a(b+c)xy\] will be coincident, if |
| A. | \[a=0\] or \[b=c\] |
| B. | \[a=b\] or \[a=c\] |
| C. | \[c=0\] or \[a=b\] |
| D. | \[a=b+c\] |
| Answer» B. \[a=b\] or \[a=c\] | |
| 5481. |
Which of the following second degree equation represented a pair of straight lines [MP PET 1990] |
| A. | \[{{x}^{2}}-xy-{{y}^{2}}=1\] |
| B. | \[-{{x}^{2}}+xy-{{y}^{2}}=1\] |
| C. | \[4{{x}^{2}}-4xy+{{y}^{2}}=4\] |
| D. | \[{{x}^{2}}+{{y}^{2}}=4\] |
| Answer» D. \[{{x}^{2}}+{{y}^{2}}=4\] | |
| 5482. |
The nature of straight lines represented by the equation \[4{{x}^{2}}+12xy+9{{y}^{2}}=0\] is [MP PET 1988] |
| A. | Real and coincident |
| B. | Real and different |
| C. | Imaginary and different |
| D. | None of the above |
| Answer» B. Real and different | |
| 5483. |
If the equation \[a{{x}^{2}}+2hxy+b{{y}^{2}}=0\] represents two lines \[y={{m}_{1}}x\] and \[y={{m}_{2}}x\], then [CEE 1993; MP PET 1988] |
| A. | \[{{m}_{1}}+{{m}_{2}}=\frac{-2h}{b}\] and \[{{m}_{1}}{{m}_{2}}=\frac{a}{b}\] |
| B. | \[{{m}_{1}}+{{m}_{2}}=\frac{2h}{b}\] and \[{{m}_{1}}{{m}_{2}}=\frac{-a}{b}\] |
| C. | \[{{m}_{1}}+{{m}_{2}}=\frac{2h}{b}\] and \[{{m}_{1}}{{m}_{2}}=\frac{a}{b}\] |
| D. | \[{{m}_{1}}+{{m}_{2}}=\frac{2h}{b}\] and \[{{m}_{1}}{{m}_{2}}=-ab\] |
| Answer» B. \[{{m}_{1}}+{{m}_{2}}=\frac{2h}{b}\] and \[{{m}_{1}}{{m}_{2}}=\frac{-a}{b}\] | |
| 5484. |
The lines represented by the equation \[a{{x}^{2}}(b-c)-xy(ab-bc)+c{{y}^{2}}(a-b)=0\] are |
| A. | \[a(b-c)x-c(a-b)y=0\], \[x+y=0\] |
| B. | \[x+y=0\], \[x-y=0\] |
| C. | \[a(b-c)x-c(a-b)y=0\], \[x-y=0\] |
| D. | None of these |
| Answer» D. None of these | |
| 5485. |
A second degree homogenous equation in x and y always represents |
| A. | A pair of straight lines |
| B. | A circle |
| C. | A conic section |
| D. | None of these |
| Answer» E. | |
| 5486. |
If \[6{{x}^{2}}+11xy-10{{y}^{2}}+x+31y+k=0\] represents a pair of straight lines, then \[k=\] [MP PET 1991] |
| A. | - 15 |
| B. | 6 |
| C. | - 10 |
| D. | - 4 |
| Answer» B. 6 | |
| 5487. |
If \[4ab=3{{h}^{2}}\], then the ratio of slopes of the lines represented by the equation \[a{{x}^{2}}+2hxy+b{{y}^{2}}=0\] will be |
| A. | \[\sqrt{2}:1\] |
| B. | \[\sqrt{3}:1\] |
| C. | \[2:1\] |
| D. | \[1:3\] |
| Answer» E. | |
| 5488. |
The lines joining the points of intersection of the curve \[{{(x-h)}^{2}}+{{(y-k)}^{2}}-{{c}^{2}}=0\] and the line \[kx+hy=2hk\] to the origin are perpendicular, then |
| A. | \[c=h\pm k\] |
| B. | \[{{c}^{2}}={{h}^{2}}+{{k}^{2}}\] |
| C. | \[{{c}^{2}}={{(h+k)}^{2}}\] |
| D. | \[4{{c}^{2}}={{h}^{2}}+{{k}^{2}}\] |
| Answer» C. \[{{c}^{2}}={{(h+k)}^{2}}\] | |
| 5489. |
The equation of pair of straight lines joining the point of intersection of the curve \[{{x}^{2}}+{{y}^{2}}=4\] and \[y-x=2\] to the origin, is |
| A. | \[{{x}^{2}}+{{y}^{2}}={{(y-x)}^{2}}\] |
| B. | \[{{x}^{2}}+{{y}^{2}}+{{(y-x)}^{2}}=0\] |
| C. | \[{{x}^{2}}+{{y}^{2}}=4{{(y-x)}^{2}}\] |
| D. | \[{{x}^{2}}+{{y}^{2}}+4{{(y-x)}^{2}}=0\] |
| Answer» B. \[{{x}^{2}}+{{y}^{2}}+{{(y-x)}^{2}}=0\] | |
| 5490. |
The lines joining the points of intersection of line \[x+y=1\] and curve \[{{x}^{2}}+{{y}^{2}}-2y+\lambda =0\] to the origin are perpendicular, then the value of \[1/\sqrt{10}\] will be |
| A. | 1/2 |
| B. | -1/2 |
| C. | \[1/\sqrt{2}\] |
| D. | 0 |
| Answer» E. | |
| 5491. |
The equation of second degree \[{{x}^{2}}+2\sqrt{2}xy+2{{y}^{2}}+4x+4\sqrt{2}y+1=0\] represents a pair of straight lines. The distance between them is [MNR 1984; UPSEAT 2000] |
| A. | 4 |
| B. | \[4/\sqrt{3}\] |
| C. | 2 |
| D. | \[2\sqrt{3}\] |
| Answer» D. \[2\sqrt{3}\] | |
| 5492. |
The equation of the line joining origin to the points of intersection of the curve \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\] and \[{{x}^{2}}+{{y}^{2}}-ax-ay=0\] is |
| A. | \[{{x}^{2}}-{{y}^{2}}=0\] |
| B. | \[xy=0\] |
| C. | \[xy-{{x}^{2}}=0\] |
| D. | \[{{y}^{2}}+xy=0\] |
| Answer» C. \[xy-{{x}^{2}}=0\] | |
| 5493. |
The equation of pair of lines joining origin to the points of intersection of \[{{x}^{2}}+{{y}^{2}}=9\]and \[x+y=3\] is [MP PET 2004] |
| A. | \[{{(x+y)}^{2}}=9\] |
| B. | \[{{x}^{2}}+{{(3-x)}^{2}}=9\] |
| C. | \[xy=0\] |
| D. | \[{{(3-x)}^{2}}+{{y}^{2}}=9\] |
| Answer» D. \[{{(3-x)}^{2}}+{{y}^{2}}=9\] | |
| 5494. |
Distance between the lines represented by the equation \[{{x}^{2}}+2\sqrt{3}xy+3{{y}^{2}}-3x-3\sqrt{3}y-4=0\]is [Roorkee 1989] |
| A. | 5/2 |
| B. | 5/4 |
| C. | 5 |
| D. | 0 |
| Answer» B. 5/4 | |
| 5495. |
Distance between the pair of lines represented by the equation \[{{x}^{2}}-6xy+9{{y}^{2}}+3x-9y-4=0\]is [Kerala (Engg,) 2002] |
| A. | \[\frac{15}{\sqrt{10}}\] |
| B. | \[\frac{1}{2}\] |
| C. | \[\sqrt{\frac{5}{2}}\] |
| D. | \[\frac{1}{\sqrt{10}}\] |
| Answer» D. \[\frac{1}{\sqrt{10}}\] | |
| 5496. |
The equation \[8{{x}^{2}}+8xy+2{{y}^{2}}+26x+13y+15=0\] represents a pair of straight lines. The distance between them is [UPSEAT 2001] |
| A. | \[7/\sqrt{5}\] |
| B. | \[7/2\sqrt{5}\] |
| C. | \[\sqrt{7}/5\] |
| D. | None of these |
| Answer» C. \[\sqrt{7}/5\] | |
| 5497. |
The pair of straight lines joining the origin to the points of intersection of the line \[y=2\sqrt{2}x+c\]and the circle \[{{x}^{2}}+{{y}^{2}}=2\]are at right angles, if [MP PET 1996] |
| A. | \[{{c}^{2}}-4=0\] |
| B. | \[{{c}^{2}}-8=0\] |
| C. | \[{{c}^{2}}-9=0\] |
| D. | \[{{c}^{2}}-10=0\] |
| Answer» D. \[{{c}^{2}}-10=0\] | |
| 5498. |
Two lines are given by\[{{(x-2y)}^{2}}+k(x-2y)=0\]. The value of k so that the distance between them is 3, is |
| A. | \[\frac{1}{\sqrt{5}}\] |
| B. | \[\pm \frac{2}{\sqrt{5}}\] |
| C. | \[\pm 3\sqrt{5}\] |
| D. | None of these |
| Answer» D. None of these | |
| 5499. |
The distance between the parallel lines \[9{{x}^{2}}-6xy+{{y}^{2}}+18x-6y+8=0\] is [EAMCET 1994] |
| A. | \[1/\sqrt{10}\] |
| B. | \[2/\sqrt{10}\] |
| C. | \[4/\sqrt{10}\] |
| D. | \[\sqrt{10}\] |
| Answer» C. \[4/\sqrt{10}\] | |
| 5500. |
The lines joining the origin to the points of intersection of the line \[3x-2y=1\] and the curve \[3{{x}^{2}}+5xy-3{{y}^{2}}+2x+3y=0\], are |
| A. | Parallel to each other |
| B. | Perpendicular to each other |
| C. | Inclined at \[{{45}^{o}}\]to each other |
| D. | None of these |
| Answer» C. Inclined at \[{{45}^{o}}\]to each other | |