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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.
| 4451. |
The angle of elevation of a tower at a point distant d meters from its base is 30°. If the tower is 20 meters high, then the value of d is [MP PET 1982, 88] |
| A. | \[10\sqrt{3}m\] |
| B. | \[\frac{20}{\sqrt{3}}m\] |
| C. | \[20\sqrt{3}m\] |
| D. | \[10\]m |
| Answer» D. \[10\]m | |
| 4452. |
Two vertical poles of equal heights are 120 m apart. On the line joining their bottoms, A and B are two points. Angle of elevation of the top of one pole from A is 45o and that of the other pole from B is also 45o. If AB = 30 m, then the height of each pole is |
| A. | 40 m |
| B. | 45 m |
| C. | 50 m |
| D. | 42 m |
| Answer» C. 50 m | |
| 4453. |
If the angle of depression of a point A on the ground from the top of a tower be\[{{30}^{o}}\], then the angle of elevation of the top of the tower from the point A will be |
| A. | \[{{60}^{o}}\] |
| B. | \[{{45}^{o}}\] |
| C. | \[{{30}^{o}}\] |
| D. | None of these |
| Answer» D. None of these | |
| 4454. |
From a point a metre above a lake the angle of elevation of a cloud is a and the angle of depression of its reflection is b. The height of the cloud is [Roorkee 1983; EAMCET 1983, 85] |
| A. | \[\frac{a\sin \,(\alpha +\beta )}{\sin \,(\alpha -\beta )}\] metre |
| B. | \[\frac{a\sin \,(\alpha +\beta )}{\sin \,(\beta -\alpha )}\] metre |
| C. | \[\frac{a\sin \,(\beta -\alpha )}{\sin \,(\alpha +\beta )}\] metre |
| D. | None of these |
| Answer» C. \[\frac{a\sin \,(\beta -\alpha )}{\sin \,(\alpha +\beta )}\] metre | |
| 4455. |
An aeroplane flying horizontally 1 km above the ground is observed at an elevation of 60o and after 10 seconds the elevation is observed to be\[{{30}^{o}}\]. The uniform speed of the aeroplane in \[km/h\] is [IIT 1965] |
| A. | 240 |
| B. | \[240\sqrt{3}\] |
| C. | \[60\sqrt{3}\] |
| D. | None of these |
| Answer» C. \[60\sqrt{3}\] | |
| 4456. |
The angle of elevation of a tower from a point A due south of it is 30o and from a point B due west of it is 45o. If the height of the tower be 100 metres, then AB = |
| A. | 150 m |
| B. | 200 m |
| C. | 173.2 m |
| D. | 141.4 m |
| Answer» C. 173.2 m | |
| 4457. |
A tower subtends an angle a at a point A in the plane of its base and the angle of depression of the foot of the tower at a point l meters just above A is b. The height of the tower is [MP PET 1990; RPET 1990] |
| A. | \[l\,\,\tan \,\beta \,\cot \,\alpha \] |
| B. | \[l\,\,\tan \,\alpha \,\cot \,\beta \] |
| C. | \[l\,\,\tan \,\,\alpha \,\,\tan \,\beta \] |
| D. | \[l\,\,\cot \,\,\alpha \,\,\cot \,\beta \] |
| Answer» C. \[l\,\,\tan \,\,\alpha \,\,\tan \,\beta \] | |
| 4458. |
An observer in a boat finds that the angle of elevation of a tower standing on the top of a cliff is 60o and that of the top of cliff is 30o. If the height of the tower be 60 meters, then the height of the cliff is [Roorkee 1982] |
| A. | 30 m |
| B. | \[60\sqrt{3}\,\,m\] |
| C. | \[20\sqrt{3}\,\,m\] |
| D. | None of these |
| Answer» B. \[60\sqrt{3}\,\,m\] | |
| 4459. |
From the top of a light house 60 meters high with its base at the sea level, the angle of depression of a boat is 15o. The distance of the boat from the foot of light house is [MNR 1988; IIT 1983; MP PET 1994, 2001; UPSEAT 2000] |
| A. | \[\left( \frac{\sqrt{3}-1}{\sqrt{3}+1} \right)\,60\,\,m\] |
| B. | \[\left( \frac{\sqrt{3}+1}{\sqrt{3}-1} \right)\,60\,\,m\] |
| C. | \[\left( \frac{\sqrt{3}+1}{\sqrt{3}-1} \right)\,\,\,m\] |
| D. | None of these |
| Answer» C. \[\left( \frac{\sqrt{3}+1}{\sqrt{3}-1} \right)\,\,\,m\] | |
| 4460. |
The angle of elevation of the top of a tower from a point 20 metre away from its base is 45o. The height of the tower is [MP PET 1984, 89] |
| A. | 10 m |
| B. | 20 m |
| C. | 40 m |
| D. | \[20\sqrt{3}\] m |
| Answer» C. 40 m | |
| 4461. |
The length of the shadow of a pole inclined at 10o to the vertical towards the sun is 2.05 metres, when the elevation of the sun is 38o. The length of the pole is [Roorkee 1976] |
| A. | \[\frac{2.05\,\,\sin \,\,{{38}^{o}}}{\sin \,{{42}^{o}}}\] |
| B. | \[\frac{2.05\,\,\sin \,\,{{42}^{o}}}{\sin \,{{38}^{o}}}\] |
| C. | \[\frac{2.05\,\,\cos \,\,{{38}^{o}}}{\cos \,\,{{42}^{o}}}\] |
| D. | None of these |
| Answer» B. \[\frac{2.05\,\,\sin \,\,{{42}^{o}}}{\sin \,{{38}^{o}}}\] | |
| 4462. |
The angle of elevation of the top of a tower at point on the ground is\[{{30}^{o}}\]. If on walking 20 metres toward the tower, the angle of elevation become\[{{60}^{o}}\], then the height of the tower is [MNR 1975; IIT 1967] |
| A. | 10 metre |
| B. | \[\frac{10}{\sqrt{3}}metre\] |
| C. | \[10\sqrt{3}metre\] |
| D. | None of these |
| Answer» D. None of these | |
| 4463. |
If \[a,\ b,\ c\] are three distinct positive real numbers which are in H.P., then \[\frac{3a+2b}{2a-b}+\frac{3c+2b}{2c-b}\] is |
| A. | Greater than or equal to 10 |
| B. | Less than or equal to 10 |
| C. | Only equal to 10 |
| D. | None of these |
| Answer» E. | |
| 4464. |
The first term of a harmonic progression is 1/7 and the second term is 1/9. The \[{{12}^{th}}\] term is [MP PET 1994] |
| A. | 43466 |
| B. | 47119 |
| C. | 42736 |
| D. | 46388 |
| Answer» C. 42736 | |
| 4465. |
If \[{{a}_{1}},\ {{a}_{2}},\ {{a}_{3}},...............,\ {{a}_{n}}\] are in H.P., then \[{{a}_{1}}{{a}_{2}}+{{a}_{2}}{{a}_{3}}+\] \[..........+{{a}_{n-1}}{{a}_{n}}\] will be equal to [IIT 1975] |
| A. | \[{{a}_{1}}{{a}_{n}}\] |
| B. | \[n{{a}_{1}}{{a}_{n}}\] |
| C. | \[(n-1){{a}_{1}}{{a}_{n}}\] |
| D. | None of these |
| Answer» D. None of these | |
| 4466. |
If \[{{5}^{th}}\] term of a H.P. is \[\frac{1}{45}\]and \[{{11}^{th}}\] term is \[\frac{1}{69}\], then its \[{{16}^{th}}\] term will be [RPET 1987, 97] |
| A. | 32509 |
| B. | 31048 |
| C. | 29221 |
| D. | 28856 |
| Answer» B. 31048 | |
| 4467. |
If \[x,\ y,\ z\] are in H.P., then the value of expression \[\log (x+z)+\log (x-2y+z)\] will be [RPET 1985, 2000] |
| A. | \[\log (x-z)\] |
| B. | \[2\log (x-z)\] |
| C. | \[3\log (x-z)\] |
| D. | \[4\log (x-z)\] |
| Answer» C. \[3\log (x-z)\] | |
| 4468. |
If \[a,\ b,\ c,\ d\] are in H.P., then [RPET 1991] |
| A. | \[a+d>b+c\] |
| B. | \[ad>bc\] |
| C. | Both (a) and (b) |
| D. | None of these |
| Answer» D. None of these | |
| 4469. |
The fifth term of the H.P., \[2,\ 2\frac{1}{2},\ 3\frac{1}{3},.............\] will be [MP PET 1984] |
| A. | \[5\frac{1}{5}\] |
| B. | \[3\frac{1}{5}\] |
| C. | 44470 |
| D. | 10 |
| Answer» E. | |
| 4470. |
If the harmonic mean between \[a\] and \[b\] be \[H\], then \[\frac{H+a}{H-a}+\frac{H+b}{H-b}=\] [AMU 1998] |
| A. | 4 |
| B. | 2 |
| C. | 1 |
| D. | \[a+b\] |
| Answer» C. 1 | |
| 4471. |
If \[a,\ b,\ c\] be in H.P., then |
| A. | \[{{a}^{2}}+{{c}^{2}}>{{b}^{2}}\] |
| B. | \[{{a}^{2}}+{{b}^{2}}>2{{c}^{2}}\] |
| C. | \[{{a}^{2}}+{{c}^{2}}>2{{b}^{2}}\] |
| D. | \[{{a}^{2}}+{{b}^{2}}>{{c}^{2}}\] |
| Answer» D. \[{{a}^{2}}+{{b}^{2}}>{{c}^{2}}\] | |
| 4472. |
If \[\frac{{{a}^{n+1}}+{{b}^{n+1}}}{{{a}^{n}}+{{b}^{n}}}\] be the harmonic mean between \[a\] and \[b\], then the value of \[n\] is [Assam PET 1986] |
| A. | 1 |
| B. | \[-1\] |
| C. | 0 |
| D. | 2 |
| Answer» C. 0 | |
| 4473. |
The sixth H.M. between 3 and \[\frac{6}{13}\] is [RPET 1996] |
| A. | \[\frac{63}{120}\] |
| B. | \[\frac{63}{12}\] |
| C. | \[\frac{126}{105}\] |
| D. | \[\frac{120}{63}\] |
| Answer» B. \[\frac{63}{12}\] | |
| 4474. |
Which number should be added to the numbers 13, 15, 19 so that the resulting numbers be the consecutive terms of a H.P. |
| A. | 7 |
| B. | 6 |
| C. | \[-6\] |
| D. | \[-7\] |
| Answer» E. | |
| 4475. |
The harmonic mean of \[\frac{a}{1-ab}\] and \[\frac{a}{1+ab}\] is [MP PET 1996; Pb. CET 2001] |
| A. | \[\frac{a}{\sqrt{1-{{a}^{2}}{{b}^{2}}}}\] |
| B. | \[\frac{a}{1-{{a}^{2}}{{b}^{2}}}\] |
| C. | \[a\] |
| D. | \[\frac{1}{1-{{a}^{2}}{{b}^{2}}}\] |
| Answer» D. \[\frac{1}{1-{{a}^{2}}{{b}^{2}}}\] | |
| 4476. |
H.M. between the roots of the equation \[{{x}^{2}}-10x+11=0\] is [MP PET 1995] |
| A. | \[\frac{1}{5}\] |
| B. | \[\frac{5}{21}\] |
| C. | \[\frac{21}{20}\] |
| D. | \[\frac{11}{5}\] |
| Answer» E. | |
| 4477. |
If the harmonic mean between \[a\] and \[b\] be \[H\], then the value of \[\frac{1}{H-a}+\frac{1}{H-b}\] is |
| A. | \[a+b\] |
| B. | \[ab\] |
| C. | \[\frac{1}{a}+\frac{1}{b}\] |
| D. | \[\frac{1}{a}-\frac{1}{b}\] |
| Answer» D. \[\frac{1}{a}-\frac{1}{b}\] | |
| 4478. |
If \[H\] is the harmonic mean between \[p\] and \[q\], then the value of \[\frac{H}{p}+\frac{H}{q}\] is [MNR 1990; UPSEAT 2000, 01] |
| A. | 2 |
| B. | \[\frac{pq}{p+q}\] |
| C. | \[\frac{p+q}{pq}\] |
| D. | None of these |
| Answer» B. \[\frac{pq}{p+q}\] | |
| 4479. |
The 4th term of a H.P. is \[\frac{3}{5}\] and 8th term is \[\frac{1}{3},\] then its 6th term is [MP PET 2003] |
| A. | \[\frac{1}{6}\] |
| B. | \[\frac{3}{7}\] |
| C. | \[\frac{1}{7}\] |
| D. | \[\frac{3}{5}\] |
| Answer» C. \[\frac{1}{7}\] | |
| 4480. |
In a H.P., pth term is q and the qth term is p. Then pqth term is [Karnataka CET 2002] |
| A. | 0 |
| B. | 1 |
| C. | pq |
| D. | \[pq(p+q)\] |
| Answer» C. pq | |
| 4481. |
If sixth term of a H.P. is \[\frac{1}{61}\] and its tenth term is \[\frac{1}{105},\] then first term of that H.P. is [Karnataka CET 2001] |
| A. | \[\frac{1}{28}\] |
| B. | \[\frac{1}{39}\] |
| C. | \[\frac{1}{6}\] |
| D. | \[\frac{1}{17}\] |
| Answer» D. \[\frac{1}{17}\] | |
| 4482. |
If the \[{{7}^{th}}\] term of a H.P. is \[\frac{1}{10}\] and the \[{{12}^{th}}\] term is \[\frac{1}{25}\], then the \[{{20}^{th}}\] term is [MP PET 1997] |
| A. | \[\frac{1}{37}\] |
| B. | \[\frac{1}{41}\] |
| C. | \[\frac{1}{45}\] |
| D. | \[\frac{1}{49}\] |
| Answer» E. | |
| 4483. |
If the \[{{7}^{th}}\] term of a harmonic progression is 8 and the \[{{8}^{th}}\]term is 7, then its \[{{15}^{th}}\] term is [MP PET 1996] |
| A. | 16 |
| B. | 14 |
| C. | \[\frac{27}{14}\] |
| D. | \[\frac{56}{15}\] |
| Answer» E. | |
| 4484. |
If the vertices of a quadrilateral be \[A=1+2i,\] \[B=-3+i,\] \[C=-2-3i\] and \[D=2-2i\], then the quadrilateral is |
| A. | Parallelogram |
| B. | Rectangle |
| C. | Square |
| D. | Rhombus |
| Answer» D. Rhombus | |
| 4485. |
If \[|z|=2\], then the points representing the complex numbers \[-1+5z\] will lie on a |
| A. | Circle |
| B. | Straight line |
| C. | Parabola |
| D. | None of these |
| Answer» B. Straight line | |
| 4486. |
If \[z=\sqrt{2}-i\sqrt{2}\] is rotated through an angle \[45{}^\circ \] in the anti-clockwise direction about the origin, then the coordinates of its new position are [Kerala (Engg.) 2005] |
| A. | (2, 0) |
| B. | (\[\sqrt{2},\,\sqrt{2}\]) |
| C. | \[(\sqrt{2},\,-\sqrt{2}\]) |
| D. | \[(\sqrt{2},0)\] |
| E. | (4, 0) |
| Answer» E. (4, 0) | |
| 4487. |
If \[|z-2-3i|+|z+2-6i|=4\], where \[i=\sqrt{-1}\], then locus of \[P(z)\] is [DCE 2005] |
| A. | An ellipse |
| B. | \[\varphi \] |
| C. | Line segment joining of point \[2+3i\] and \[-2+6i\] |
| D. | None of these |
| Answer» C. Line segment joining of point \[2+3i\] and \[-2+6i\] | |
| 4488. |
The number of solutions for the equations \[|z-1|=|z-2|=\] \[|z-i|\] is [Orissa JEE 2005] |
| A. | One solution |
| B. | 3 solutions |
| C. | 2 solutions |
| D. | No solution |
| Answer» B. 3 solutions | |
| 4489. |
PQ and PR are two infinite rays. QAR is an arc. Point lying in the shaded region excluding the boundary satisfies [IIT Screening 2005] |
| A. | \[|z-1|>2;|\arg (z-1)|\,<\frac{\pi }{4}\] |
| B. | \[|z-1|>2;|\arg (z-1)|\,<\frac{\pi }{2}\] |
| C. | \[|z+1|>2;|\arg (z+1)|\,<\frac{\pi }{4}\] |
| D. | \[|z+1|>2;|\arg (z+1)|\,<\frac{\pi }{2}\] |
| Answer» D. \[|z+1|>2;|\arg (z+1)|\,<\frac{\pi }{2}\] | |
| 4490. |
If \[|8+z|+|z-8|=16\] where z is a complex number, then the point z will lie on [J & K 2005] |
| A. | A circle |
| B. | An ellipse |
| C. | A straight line |
| D. | None of these |
| Answer» C. A straight line | |
| 4491. |
If \[a\] and \[b\] are real numbers between 0 and 1 such that the points \[{{z}_{1}}=a+i,{{z}_{2}}=1+bi\] and \[{{z}_{3}}=0\] form an equilateral triangle, then [IIT 1989] |
| A. | \[a=b=2+\sqrt{3}\] |
| B. | \[a=b=2-\sqrt{3}\] |
| C. | \[a=2-\sqrt{3},b=2+\sqrt{3}\] |
| D. | None of these |
| Answer» C. \[a=2-\sqrt{3},b=2+\sqrt{3}\] | |
| 4492. |
If \[w=\frac{z}{z-\frac{1}{3}i}\] and \[|w|=1\], then z lies on [AIEEE 2005] |
| A. | A straight line |
| B. | A parabola |
| C. | An ellipse |
| D. | A circle |
| Answer» B. A parabola | |
| 4493. |
The equation \[|z-5i|\div |z+5i|\,=12,\] where \[z=x+iy,\] represents a/an [AMU 1999] |
| A. | Circle |
| B. | Ellipse |
| C. | Parabola |
| D. | No real curve |
| Answer» B. Ellipse | |
| 4494. |
If \[z=x+iy\] and \[arg\,\left( \frac{z-2}{z+2} \right)=\frac{\pi }{6}\], then locus of z is [RPET 2002] |
| A. | A straight line |
| B. | A circle |
| C. | A parabola |
| D. | An ellipse |
| Answer» C. A parabola | |
| 4495. |
If \[|{{z}^{2}}-1|\,=\,|z{{|}^{2}}+1\], then \[z\]lies on [AIEEE 2004] |
| A. | An ellipse |
| B. | The imaginary axis |
| C. | A circle |
| D. | The real axis |
| Answer» C. A circle | |
| 4496. |
The locus of the point z satisfying \[arg\left( \frac{z-1}{z+1} \right)=k,\] (where k is non zero) is [Orissa JEE 2002] |
| A. | Circle with centre on y-axis |
| B. | Circle with centre on x-axis |
| C. | A straight line parallel to x-axis |
| D. | A straight line making an angle \[{{60}^{o}}\] with the x-axis |
| Answer» B. Circle with centre on x-axis | |
| 4497. |
Locus of the point z satisfying the equation \[|iz-1|\]+ \[|z-i|=2\] is [Roorkee 1999] |
| A. | A straight line |
| B. | A circle |
| C. | An ellipse |
| D. | A pair of straight lines |
| Answer» B. A circle | |
| 4498. |
If \[z=x+iy\] is a complex number satisfying \[{{\left| z+\frac{i}{2} \right|}^{2}}=\] \[\,\,{{\left| z-\frac{i}{2} \right|}^{2}},\] then the locus of z is [EAMCET 2002] |
| A. | \[2y=x\] |
| B. | \[y=x\] |
| C. | y-axis |
| D. | x-axis |
| Answer» E. | |
| 4499. |
If \[z=x+iy\] and \[|z-2+i|\,=\,|z-3-i|,\] then locus of z is [RPET 1999] |
| A. | \[2x+4y-5=0\] |
| B. | \[2x-4y-5=0\] |
| C. | \[x+2y=0\] |
| D. | \[x-2y+5=0\] |
| Answer» B. \[2x-4y-5=0\] | |
| 4500. |
If three complex numbers are in A.P., then they lie on [IIT 1985; DCE 2001; Pb. CET 2003] |
| A. | A circle in the complex plane |
| B. | A straight line in the complex plane |
| C. | A parabola in the complex plane |
| D. | None of these |
| Answer» C. A parabola in the complex plane | |