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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.
| 4401. |
The solution of the equation \[\frac{dy}{dx}=\frac{y}{x}\left( \log \frac{y}{x}+1 \right)\] is |
| A. | \[\log \left( \frac{y}{x} \right)=cx\] |
| B. | \[\frac{y}{x}=\log y+c\] |
| C. | \[y=\log y+1\] |
| D. | \[y=xy+c\] |
| Answer» B. \[\frac{y}{x}=\log y+c\] | |
| 4402. |
Solution of differential equation \[\frac{dy}{dx}=\frac{y-x}{y+x}\]is [MP PET 1997] |
| A. | \[{{\log }_{e}}({{x}^{2}}+{{y}^{2}})+2{{\tan }^{-1}}\frac{y}{x}+c=0\] |
| B. | \[\frac{{{y}^{2}}}{2}+xy=xy-\frac{{{x}^{2}}}{2}+c\] |
| C. | \[\left( 1+\frac{x}{y} \right)\text{ }y=\left( 1-\frac{x}{y} \right)\text{ }x+c\] |
| D. | \[y=x-2{{\log }_{e}}y+c\] |
| Answer» B. \[\frac{{{y}^{2}}}{2}+xy=xy-\frac{{{x}^{2}}}{2}+c\] | |
| 4403. |
The solution of the differential equation \[{{x}^{2}}\frac{dy}{dx}={{x}^{2}}+xy+{{y}^{2}}\] is |
| A. | \[{{\tan }^{-1}}\left( \frac{y}{x} \right)=\log x+c\] |
| B. | \[{{\tan }^{-1}}\left( \frac{y}{x} \right)=-\log x+c\] |
| C. | \[{{\sin }^{-1}}\left( \frac{y}{x} \right)=\log x+c\] |
| D. | \[{{\tan }^{-1}}\left( \frac{x}{y} \right)=\log x+c\] |
| Answer» B. \[{{\tan }^{-1}}\left( \frac{y}{x} \right)=-\log x+c\] | |
| 4404. |
The solution of the equation \[\frac{dy}{dx}=\frac{x}{2y-x}\]is |
| A. | \[(x-y){{(x+2y)}^{2}}=c\] |
| B. | \[y=x+c\] |
| C. | \[y=(2y-x)+c\] |
| D. | \[y=\frac{x}{2y-x}+c\] |
| Answer» B. \[y=x+c\] | |
| 4405. |
A house of height 100 metres subtends a right angle at the window of an opposite house. If the height of the window be 64 metres, then the distance between the two houses is |
| A. | 48 m |
| B. | 36 m |
| C. | 54 m |
| D. | 72 m |
| Answer» B. 36 m | |
| 4406. |
From a 60 meter high tower angles of depression of the top and bottom of a house are a and b respectively. If the height of the house is \[\frac{60\,\sin \,(\beta -\alpha )}{x},\] then x = |
| A. | \[\sin \,\,\alpha \,\,\sin \,\,\beta \] |
| B. | \[\cos \,\,\alpha \,\,\cos \,\,\beta \] |
| C. | \[\sin \,\,\alpha \,\,\cos \,\,\beta \] |
| D. | \[\cos \,\,\alpha \,\,\sin \,\,\beta \] |
| Answer» E. | |
| 4407. |
\[ABCD\] is a rectangular field. A vertical lamp post of height 12m stands at the corner A. If the angle of elevation of its top from B is \[{{60}^{o}}\] and from C is\[{{45}^{o}}\], then the area of the field is [Kerala (Engg.) 2005] |
| A. | \[48\sqrt{2}sq.m\] |
| B. | \[48\sqrt{3}sq.m\] |
| C. | \[48sq.m\] |
| D. | \[12\sqrt{2}sq.m\] |
| E. | \[12\sqrt{3}sq.m\] |
| Answer» B. \[48\sqrt{3}sq.m\] | |
| 4408. |
A ladder rests against a wall making an angle \[\alpha \]with the horizontal. The foot of the ladder is pulled away from the wall through a distance x, so that it slides a distance y down the wall making an angle\[\beta \]with the horizontal. The correct relation is [IIT 1985] |
| A. | \[x=y\tan \frac{\alpha +\beta }{2}\] |
| B. | \[y=x\tan \frac{\alpha +\beta }{2}\] |
| C. | \[x=y\tan (\alpha +\beta )\] |
| D. | \[y=x\tan (\alpha +\beta )\] |
| Answer» B. \[y=x\tan \frac{\alpha +\beta }{2}\] | |
| 4409. |
Two pillars of equal height stand on either side of a roadway which is 60 metres wide. At a point in the roadway between the pillars, the elevation of the top of pillars are 60° and 30°. The height of the pillars is [UPSEAT 2004] |
| A. | \[15\sqrt{3}m\] |
| B. | \[\frac{15}{\sqrt{3}}m\] |
| C. | \[15m\] |
| D. | \[20m\] |
| Answer» B. \[\frac{15}{\sqrt{3}}m\] | |
| 4410. |
A tower subtends angles \[\alpha ,\,2\alpha ,\,3\alpha \]respectively at points A, B and \[C\], all lying on a horizontal line through the foot of the tower. Then \[AB/BC=\] [EAMCET 2003] |
| A. | \[\frac{\sin 3\alpha }{\sin 2\alpha }\] |
| B. | \[1+2\cos 2\alpha \] |
| C. | \[2+\cos 3\alpha \] |
| D. | \[\frac{\sin 2\alpha }{\sin \alpha }\] |
| Answer» C. \[2+\cos 3\alpha \] | |
| 4411. |
20 metre high flag pole is fixed on a 80 metre high pillar, 50 metre away from it, on a point on the base of pillar the flag pole makes and angle \[\alpha \], then the value of \[\tan \alpha \], is [MP PET 2003] |
| A. | \[\frac{2}{11}\] |
| B. | \[\frac{2}{21}\] |
| C. | \[\frac{21}{2}\] |
| D. | \[\frac{21}{4}\] |
| Answer» C. \[\frac{21}{2}\] | |
| 4412. |
A vertical pole consists of two parts, the lower part being one third of the whole. At a point in the horizontal plane through the base of the pole and distance 20 meters from it, the upper part of the pole subtends an angle whose tangent is \[\frac{1}{2}\]. The possible heights of the pole are [IIT 1964] |
| A. | 20 m and \[20\sqrt{3}\,m\] |
| B. | 20 m and 60 m |
| C. | 16 m and 48 m |
| D. | None of these |
| Answer» C. 16 m and 48 m | |
| 4413. |
For a man, the angle of elevation of the highest point of the temple situated east of him is\[{{60}^{o}}\]. On walking 240 metres to north, the angle of elevation is reduced to\[{{30}^{o}}\], then the height of the temple is [MP PET 2003] |
| A. | \[60\sqrt{6}m\] |
| B. | \[60m\] |
| C. | \[50\sqrt{3}m\] |
| D. | \[30\sqrt{6}m\] |
| Answer» B. \[60m\] | |
| 4414. |
If the angle of elevation of the top of tower at a distance 500 m from its foot is\[{{30}^{o}}\], then height of the tower is [Kerala (Engg.) 2002] |
| A. | \[\frac{1}{\sqrt{3}}\] |
| B. | \[\frac{500}{\sqrt{3}}\] |
| C. | \[\sqrt{3}\] |
| D. | \[\frac{1}{500}\] |
| Answer» C. \[\sqrt{3}\] | |
| 4415. |
The shadow of a tower standing on a level ground is found to be 60 m longer when the sun's altitude is\[{{30}^{o}}\]than when it is\[{{45}^{o}}\]. The height of the tower is [EAMCET 2001] |
| A. | 60 m |
| B. | 30 m |
| C. | \[60\sqrt{3}m\] |
| D. | \[30(\sqrt{3}+1)m\] |
| Answer» E. | |
| 4416. |
The angle of elevation of the top of a pillar at any point A on the ground is\[{{15}^{o}}\]. On walking 40 metres towards the pillar, the angle become\[{{30}^{o}}\]. The height of the pillar is [MP PET 2001] |
| A. | 40 metres |
| B. | 20 metres |
| C. | \[20\sqrt{3}metres\] |
| D. | \[\frac{40}{3}\sqrt{3}metres\] |
| Answer» C. \[20\sqrt{3}metres\] | |
| 4417. |
The top of a hill observed from the top and bottom of a building of height h is at the angle of elevation p and q respectively. The height of the hills is [UPSEAT 2001; EAMCET 1989] |
| A. | \[\frac{h\cot q}{\cot q-\cot p}\] |
| B. | \[\frac{h\cot p}{\cot p-\cot q}\] |
| C. | \[\frac{h\tan p}{\tan p-\tan q}\] |
| D. | None of these |
| Answer» C. \[\frac{h\tan p}{\tan p-\tan q}\] | |
| 4418. |
A ladder 5 metre long leans against a vertical wall. The bottom of the ladder is 3 metre from the wall. If the bottom of the ladder is pulled 1 metre farther from the wall, how much does the top of the ladder slide down the wall [AMU 2000] |
| A. | 1 m |
| B. | 7 m |
| C. | 2 m |
| D. | None of these |
| Answer» B. 7 m | |
| 4419. |
Two men are on the opposite side of a tower. They measure the angles of elevation of the top of the tower\[{{45}^{o}}\]and\[{{30}^{o}}\]respectively. If the height of the tower is 40 m, find the distance between the men [Karnataka CET 1998] |
| A. | 40 m |
| B. | \[40\sqrt{3}\,m\] |
| C. | 68.280 m |
| D. | 109.28 m |
| Answer» E. | |
| 4420. |
The angles of elevation of the top of a tower from the top and bottom at a building of height a are\[{{30}^{o}}\]and\[{{45}^{o}}\]respectively. If the tower and the building stand at the same level, then the height of the tower is [Karnataka CET 2000] |
| A. | \[a\sqrt{3}\] |
| B. | \[\frac{a\sqrt{3}}{\sqrt{3}-1}\] |
| C. | \[\frac{a\,(3+\sqrt{3})}{2}\] |
| D. | \[a\,(\sqrt{3}-1)\] |
| Answer» D. \[a\,(\sqrt{3}-1)\] | |
| 4421. |
A person standing on the bank of a river observes that the angle subtended by a tree on the opposite bank is 60°. When he retires 40 meters from the bank, he finds the angle to be\[{{30}^{o}}\]. The breadth of the river is [IIT 1975; AIEEE 2004] |
| A. | 20 m |
| B. | 40 m |
| C. | 30 m |
| D. | 60 m |
| Answer» B. 40 m | |
| 4422. |
A person is standing on a tower of height \[15(\sqrt{3}+1)\,m\] and observing a car coming towards the tower. He observed that angle of depression changes from\[{{30}^{o}}\]to\[{{45}^{o}}\]in 3 sec. What is the speed of the car [Karnataka CET 1998] |
| A. | 36 km/hr |
| B. | 72 km/hr |
| C. | 18 km/hr |
| D. | 30 km/hr |
| Answer» B. 72 km/hr | |
| 4423. |
The angular elevation of a tower CD at a point A due south of it is\[{{60}^{o}}\]and at a point B due west of A, the elevation is\[{{30}^{o}}\]. If AB = 3 km, the height of the tower is [MP PET 1998] |
| A. | \[2\sqrt{3}\,km\] |
| B. | \[2\sqrt{6}\,km\] |
| C. | \[\frac{3\sqrt{3}}{2}km\] |
| D. | \[\frac{3\sqrt{6}}{4}km\] |
| Answer» E. | |
| 4424. |
The angle of depression of a point situated at a distance of 70 metres from the base of a tower is\[{{45}^{o}}\]. The height of the tower is [MP PET 1997] |
| A. | 70 m |
| B. | \[70\sqrt{2}\] m |
| C. | \[\frac{70}{\sqrt{2}}m\] |
| D. | 35 m |
| Answer» B. \[70\sqrt{2}\] m | |
| 4425. |
A flag-staff of 5 m high stands on a building of 25 m high. At an observer at a height of 30 m. The flag-staff and the building subtend equal angles. The distance of the observer from the top of the flag-staff is [EAMCET 1993] |
| A. | \[\frac{5\sqrt{3}}{2}\] |
| B. | \[5\sqrt{\frac{3}{2}}\] |
| C. | \[5\sqrt{\frac{2}{3}}\] |
| D. | None of these |
| Answer» C. \[5\sqrt{\frac{2}{3}}\] | |
| 4426. |
If a flagstaff of 6 metres high placed on the top of a tower throws a shadow of \[2\sqrt{3}\,metres\] along the ground, then the angle (in degrees) that the sun makes with the ground is [EAMCET 1990] |
| A. | \[{{60}^{o}}\] |
| B. | \[{{80}^{o}}\] |
| C. | \[{{75}^{o}}\] |
| D. | None of these |
| Answer» B. \[{{80}^{o}}\] | |
| 4427. |
The angle of elevation of a cliff at a point A on the ground and a point B, 100 m vertically at A are a and b respectively. The height of the cliff is [EAMCET 1986] |
| A. | \[\frac{100\,\,\cot \,\alpha }{\cot \alpha -\cot \beta }\] |
| B. | \[\frac{100\,\,\cot \beta }{\cot \,\alpha -\cot \,\beta }\] |
| C. | \[\frac{100\,\,\cot \beta }{\cot \beta -\cot \alpha }\] |
| D. | \[\frac{100\,\,\cot \beta }{\cot \beta +\cot \alpha }\] |
| Answer» D. \[\frac{100\,\,\cot \beta }{\cot \beta +\cot \alpha }\] | |
| 4428. |
Two straight roads intersect at an angle of\[{{60}^{o}}\]. A bus on one road is 2 km away from the intersection and a car on the other road is 3 km away from the intersection. Then the direct distance between the two vehicles is [BIT Ranchi 1993] |
| A. | 1 km |
| B. | \[\sqrt{2}\,\,km\] |
| C. | 4 km |
| D. | \[\sqrt{7}\,\,km\] |
| Answer» E. | |
| 4429. |
AB is a vertical pole resting at the end A on the level ground. P is a point on the level ground such that AP = 3 AB. If C is the mid-point of AB and CB subtends an angle b at P, the value of is [Bihar CEE 1994] |
| A. | \[\frac{18}{19}\] |
| B. | \[\frac{3}{19}\] |
| C. | \[\frac{1}{6}\] |
| D. | None of these |
| Answer» C. \[\frac{1}{6}\] | |
| 4430. |
A person standing on the bank of a river finds that the angle of elevation of the top of a tower on the opposite bank is\[{{45}^{o}}\]. Then which of the following statements is correct [MP PET 1994] |
| A. | Breadth of the river is twice the height of the tower |
| B. | Breadth of the river and the height of the tower are the same |
| C. | Breadth of the river is half of the height of the tower |
| D. | None of the above |
| Answer» C. Breadth of the river is half of the height of the tower | |
| 4431. |
The angle of elevation of the top of a tower from a point A due south of the tower is \[\alpha \]and from a point B due east of the tower is \[\beta \]. If AB =d, then the height of the tower is [Roorkee 1979; Kurukshetra CEE 1998] |
| A. | \[\frac{d}{\sqrt{{{\tan }^{2}}\alpha -{{\tan }^{2}}\beta }}\] |
| B. | \[\frac{d}{\sqrt{{{\tan }^{2}}\alpha +{{\tan }^{2}}\beta }}\] |
| C. | \[\frac{d}{\sqrt{{{\cot }^{2}}\alpha +{{\cot }^{2}}\beta }}\] |
| D. | \[\frac{d}{\sqrt{{{\cot }^{2}}\alpha -{{\cot }^{2}}\beta }}\] |
| Answer» D. \[\frac{d}{\sqrt{{{\cot }^{2}}\alpha -{{\cot }^{2}}\beta }}\] | |
| 4432. |
A balloon is coming down at the rate of 4 m/min. and its angle of elevation is 45o from a point on the ground which has been reduced to 30o after 10 minutes. Balloon will be on the ground at a distance of how many meters from the observer |
| A. | \[20\,\sqrt{3}\,m\] |
| B. | \[20\,(3+\sqrt{3})\,m\] |
| C. | \[10\,(3+\sqrt{3})\,m\] |
| D. | None of these |
| Answer» C. \[10\,(3+\sqrt{3})\,m\] | |
| 4433. |
A flag-post 20m high standing on the top of a house subtends an angle whose tangent is \[\frac{1}{6}\] at a distance 70 m from the foot of the house. The height of the house is |
| A. | 30 m |
| B. | 60 m |
| C. | 50 m |
| D. | None of these |
| Answer» D. None of these | |
| 4434. |
A vertical pole (more than 100 m high) consists of two portions, the lower being one-third of the whole. If the upper portion subtends an angle \[{{\tan }^{-1}}\frac{1}{2}\] at a point in a horizontal plane through the foot of the pole and distance 40 ft from it, then the height of the pole is [AMU 1981] |
| A. | 100 ft |
| B. | 120 ft |
| C. | 150 ft |
| D. | None of these |
| Answer» C. 150 ft | |
| 4435. |
A balloon is observed simultaneously from three points A, B and C on a straight road directly under it. The angular elevation at B is twice and at C is thrice that of A. If the distance between A and B is 200 metres and the distance between B and C is 100 metres, then the height of balloon is given by [Roorkee 1989] |
| A. | 50 metres |
| B. | \[50\,\sqrt{3}\] metres |
| C. | \[50\,\sqrt{2}\] metres |
| D. | None of these |
| Answer» E. | |
| 4436. |
The angle of elevation of a stationary cloud from a point 2500 m above a lake is\[{{15}^{o}}\]and the angle of depression of its reflection in the lake is\[{{45}^{o}}\]. The height of cloud above the lake level is [IIT 1976] |
| A. | \[2500\,\sqrt{3}\,metres\] |
| B. | 2500 metres |
| C. | \[500\,\sqrt{3}\,metres\] |
| D. | None of these |
| Answer» B. 2500 metres | |
| 4437. |
The angle of depression of a ship from the top of a tower 30 metre high is\[{{60}^{o}}\], then the distance of ship from the base of tower is [MP PET 1988; Pb. CET 2003] |
| A. | 30 m |
| B. | \[30\,\sqrt{3}\,\,m\] |
| C. | \[10\sqrt{3}\,m\] |
| D. | 10 m |
| Answer» D. 10 m | |
| 4438. |
A tree is broken by wind, its upper part touches the ground at a point 10 metres from the foot of the tree and makes an angle of\[{{45}^{o}}\]with the ground. The total length of tree is [BIT Ranchi 1992] |
| A. | 15 metres |
| B. | 20 metres |
| C. | \[10\,(1+\sqrt{2})\]metres |
| D. | \[10\,\left( 1+\frac{\sqrt{3}}{2} \right)\]metres |
| Answer» D. \[10\,\left( 1+\frac{\sqrt{3}}{2} \right)\]metres | |
| 4439. |
A tower of height b subtends an angle at a point O on the level of the foot of the tower and at a distance a from the foot of the tower. If a pole mounted on the tower also subtends an equal angle at O, the height of the pole is [MP PET 1993, 2004] |
| A. | \[b\,\left( \frac{{{a}^{2}}-{{b}^{2}}}{{{a}^{2}}+{{b}^{2}}} \right)\] |
| B. | \[b\,\left( \frac{{{a}^{2}}+{{b}^{2}}}{{{a}^{2}}-{{b}^{2}}} \right)\] |
| C. | \[a\,\left( \frac{{{a}^{2}}-{{b}^{2}}}{{{a}^{2}}+{{b}^{2}}} \right)\] |
| D. | \[a\,\left( \frac{{{a}^{2}}+{{b}^{2}}}{{{a}^{2}}-{{b}^{2}}} \right)\] |
| Answer» C. \[a\,\left( \frac{{{a}^{2}}-{{b}^{2}}}{{{a}^{2}}+{{b}^{2}}} \right)\] | |
| 4440. |
A tower subtends an angle of\[{{30}^{o}}\]at a point distant d from the foot of the tower and on the same level as the foot of the tower. At a second point h vertically above the first, the depression of the foot of the tower is\[{{60}^{o}}\]. The height of the tower is [MP PET 1993] |
| A. | h/3 |
| B. | h/3d |
| C. | 3h |
| D. | \[\frac{3h}{d}\] |
| Answer» B. h/3d | |
| 4441. |
The angle of elevation of the top of the tower observed from each of the three points \[A,B,C\]on the ground, forming a triangle is the same angle \[\alpha \]. If R is the circum-radius of the triangle ABC, then the height of the tower is [EAMCET 1994] |
| A. | \[R\sin \alpha \] |
| B. | \[R\cos \alpha \] |
| C. | \[R\cot \alpha \] |
| D. | \[R\tan \alpha \] |
| Answer» E. | |
| 4442. |
A man whose eye level is 1.5 metres above the ground observes the angle of elevation of a tower to be\[{{60}^{o}}\]. If the distance of the man from the tower be 10 meters, the height of the tower is |
| A. | \[(1.5+10\sqrt{3})\,m\] |
| B. | \[10\,\sqrt{3}\,m\] |
| C. | \[\left( 1.5+\frac{10}{\sqrt{3}} \right)\,m\] |
| D. | None of these |
| Answer» B. \[10\,\sqrt{3}\,m\] | |
| 4443. |
The angle of elevation of the top of a tower from the top of a house is\[{{60}^{o}}\]and the angle of depression of its base is\[{{30}^{o}}\]. If the horizontal distance between the house and the tower be 12 m, then the height of the tower is |
| A. | \[48\,\sqrt{3}\,\,m\] |
| B. | \[16\,\sqrt{3}\,\,m\] |
| C. | \[24\,\sqrt{3}\,\,m\] |
| D. | \[16/\,\sqrt{3}\,\,m\] |
| Answer» C. \[24\,\sqrt{3}\,\,m\] | |
| 4444. |
Some portion of a 20 meters long tree is broken by the wind and the top struck the ground at an angle of\[{{30}^{o}}\]. The height of the point where the tree is broken is [MNR 1974] |
| A. | 10 m |
| B. | \[(2\sqrt{3}-3)\,20\,\,m\] |
| C. | \[\frac{20}{3}m\] |
| D. | None of these |
| Answer» D. None of these | |
| 4445. |
At a point on the ground the angle of elevation of a tower is such that its cotangent is 3/5. On walking 32 metres towards the tower the cotangent of the angle of elevation is 2/5. The height of the tower is |
| A. | 160 m |
| B. | 120 m |
| C. | 64 m |
| D. | None of these |
| Answer» B. 120 m | |
| 4446. |
If the angles of elevation of two towers from the middle point of the line joining their feet be\[{{60}^{o}}\]and\[{{30}^{o}}\]respectively, then the ratio of their heights is [EAMCET 1987] |
| A. | 0.0840277777777778 |
| B. | \[1\,\,:\,\,\sqrt{2}\] |
| C. | 0.125694444444444 |
| D. | \[1\,\,:\,\,\sqrt{3}\] |
| Answer» D. \[1\,\,:\,\,\sqrt{3}\] | |
| 4447. |
A ladder rests against a wall so that its top touches the roof of the house. If the ladder makes an angle of\[{{60}^{o}}\]with the horizontal and height of the house be \[6\sqrt{3}\] meters, then the length of the ladder is |
| A. | \[12\sqrt{3}\] |
| B. | 12 m |
| C. | \[12/\sqrt{3}\,\,m\] |
| D. | None of these |
| Answer» C. \[12/\sqrt{3}\,\,m\] | |
| 4448. |
The angle of elevation of the sun, when the shadow of the pole is\[\sqrt{3}\]times the height of the pole, is [MP PET 1991, 96] |
| A. | \[{{60}^{o}}\] |
| B. | \[{{30}^{o}}\] |
| C. | \[{{45}^{o}}\] |
| D. | \[{{15}^{o}}\] |
| Answer» C. \[{{45}^{o}}\] | |
| 4449. |
A house subtends a right angle at the window of an opposite house and the angle of elevation of the window from the bottom of the first house is\[{{60}^{o}}\]. If the distance between the two houses be 6 metres, then the height of the first house is [MNR 1978] |
| A. | \[6\sqrt{3}\,\,m\] |
| B. | \[8\sqrt{3}\,\,m\] |
| C. | \[4\sqrt{3}\,\,m\] |
| D. | None of these |
| Answer» C. \[4\sqrt{3}\,\,m\] | |
| 4450. |
At a distance 2h from the foot of a tower of height h, the tower and a pole at the top of the tower subtend equal angles. Height of the pole should be |
| A. | \[\frac{5h}{3}\] |
| B. | \[\frac{4h}{3}\] |
| C. | \[\frac{7h}{5}\] |
| D. | \[\frac{3h}{2}\] |
| Answer» B. \[\frac{4h}{3}\] | |