MCQOPTIONS
Saved Bookmarks
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.
| 7501. |
One root of the equation \[\cos x-x+\frac{1}{2}=0\]lies in the interval [Kurukshetra CEE 1996] |
| A. | \[\left[ 0,\,\frac{\pi }{2} \right]\] |
| B. | \[\left[ -\frac{\pi }{2},\,0 \right]\] |
| C. | \[\left[ \frac{\pi }{2},\,\pi \right]\] |
| D. | \[\left[ \pi ,\frac{3\pi }{2} \right]\] |
| Answer» B. \[\left[ -\frac{\pi }{2},\,0 \right]\] | |
| 7502. |
The equation \[\sin x\cos x=2\]has |
| A. | One solution |
| B. | Two solutions |
| C. | Infinite solutions |
| D. | No solutions |
| Answer» E. | |
| 7503. |
If \[{{\sin }^{2}}\theta -2\cos \theta +\frac{1}{4}=0,\]then the general value of \[\theta \]is [MP PET 1984] |
| A. | \[n\pi \pm \frac{\pi }{3}\] |
| B. | \[2n\pi \pm \frac{\pi }{3}\] |
| C. | \[2n\pi \pm \frac{\pi }{6}\] |
| D. | \[n\pi \pm \frac{\pi }{6}\] |
| Answer» C. \[2n\pi \pm \frac{\pi }{6}\] | |
| 7504. |
If \[2{{\cos }^{2}}x+3\sin x-3=0,\,\,0\le x\le {{180}^{o}}\], then x = [MP PET 1986] |
| A. | \[{{30}^{o}},{{90}^{o}},{{150}^{o}}\] |
| B. | \[{{60}^{o}},{{120}^{o}},{{180}^{o}}\] |
| C. | \[{{0}^{o}},{{30}^{o}},{{150}^{o}}\] |
| D. | \[{{45}^{o}},{{90}^{o}},{{135}^{o}}\] |
| Answer» B. \[{{60}^{o}},{{120}^{o}},{{180}^{o}}\] | |
| 7505. |
The solution of the equation \[\left| \,\begin{matrix} \cos \theta & \sin \theta & \cos \theta \\ -\sin \theta & \cos \theta & \sin \theta \\ -\cos \theta & -\sin \theta & \cos \theta \\ \end{matrix}\, \right|=0\], is [AMU 2002] |
| A. | \[\theta =n\pi \] |
| B. | \[\theta =2n\pi \pm \frac{\pi }{2}\] |
| C. | \[\theta =n\pi \pm {{(-1)}^{n}}\frac{\pi }{4}\] |
| D. | \[\theta =2n\pi \pm \frac{\pi }{4}\] |
| Answer» C. \[\theta =n\pi \pm {{(-1)}^{n}}\frac{\pi }{4}\] | |
| 7506. |
If \[\sin 2\theta =\cos 3\theta \]and \[\theta \]is an acute angle, then \[\sin \theta \]is equal to [EAMCET 1980] |
| A. | \[\frac{\sqrt{5}-1}{4}\] |
| B. | \[\frac{-\sqrt{5}-1}{4}\] |
| C. | 0 |
| D. | None of these |
| Answer» B. \[\frac{-\sqrt{5}-1}{4}\] | |
| 7507. |
The general value \[\theta \] is obtained from the equation \[\cos 2\theta =\sin \alpha ,\] is [MP PET 1996] |
| A. | \[2\theta =\frac{\pi }{2}-\alpha \] |
| B. | \[\theta =2n\pi \pm \left( \frac{\pi }{2}-\alpha \right)\] |
| C. | \[\theta =\frac{n\pi +{{(-1)}^{n}}\alpha }{2}\] |
| D. | \[\theta =n\pi \pm \left( \frac{\pi }{4}-\frac{\alpha }{2} \right)\] |
| Answer» E. | |
| 7508. |
The solution of the equation \[4{{\cos }^{2}}x+6\]\[{{\sin }^{2}}x=5\] [AI CBSE 1983] |
| A. | \[x=n\pi \pm \frac{\pi }{2}\] |
| B. | \[x=n\pi \pm \frac{\pi }{4}\] |
| C. | \[x=n\pi \pm \frac{3\pi }{2}\] |
| D. | None of these |
| Answer» C. \[x=n\pi \pm \frac{3\pi }{2}\] | |
| 7509. |
If cot \[(\alpha +\beta )=0,\]then \[\sin (\alpha +2\beta )=\] [Kerala (Engg.) 2001] |
| A. | \[\sin \alpha \] |
| B. | \[\cos \alpha \] |
| C. | \[\sin \beta \] |
| D. | \[\cos 2\beta \] |
| Answer» B. \[\cos \alpha \] | |
| 7510. |
If \[\sin \theta +\cos \theta =\sqrt{2}\cos \alpha \], then the general value of \[\theta \] is |
| A. | \[2n\pi -\frac{\pi }{4}\pm \,\,\alpha \] |
| B. | \[2n\pi +\frac{\pi }{4}\pm \alpha \] |
| C. | \[n\pi -\frac{\pi }{4}\pm \alpha \] |
| D. | \[n\pi +\frac{\pi }{4}\pm \alpha \] |
| Answer» C. \[n\pi -\frac{\pi }{4}\pm \alpha \] | |
| 7511. |
If \[12{{\cot }^{2}}\theta -31\,\text{cosec }\theta +\text{32}=\text{0}\], then the value of \[\sin \theta \] is [Karnataka CET 2005] |
| A. | \[\frac{3}{5}\] or 1 |
| B. | \[-\sin (B+2C)=\frac{1}{2}\] or \[\frac{-2}{3}\] |
| C. | \[\frac{4}{5}\] or \[\frac{3}{4}\] |
| D. | \[\pm \frac{1}{2}\] |
| Answer» D. \[\pm \frac{1}{2}\] | |
| 7512. |
If \[\sqrt{3}\tan 2\theta +\sqrt{3}\tan 3\theta +\tan 2\theta \tan 3\theta =1\], then the general value of\[\theta \]is |
| A. | \[n\pi +\frac{\pi }{5}\] |
| B. | \[\left( n+\frac{1}{6} \right)\frac{\pi }{5}\] |
| C. | \[\left( 2n\pm \frac{1}{6} \right)\frac{\pi }{5}\] |
| D. | \[\left( n+\frac{1}{3} \right)\frac{\pi }{5}\] |
| Answer» C. \[\left( 2n\pm \frac{1}{6} \right)\frac{\pi }{5}\] | |
| 7513. |
If in any \[\Delta ABC\], \[\cot \frac{A}{2},\cot \frac{B}{2},\cos \frac{C}{2}\]are in A. P. then [MP PET 2003] |
| A. | \[\cot \frac{A}{2}\cot \frac{B}{2}=4\] |
| B. | \[\cot \frac{A}{2}\cot \frac{C}{2}=3\] |
| C. | \[\cot \frac{B}{2}\cot \frac{C}{2}=1\] |
| D. | \[\cot \frac{B}{2}\tan \frac{C}{2}=0\] |
| Answer» B. \[\cot \frac{A}{2}\cot \frac{C}{2}=3\] | |
| 7514. |
The number of triangles ABC that can be formed with \[a=3,b=8\] and \[\sin A=\frac{5}{13}\]is [Roorkee Qualifying 1998] |
| A. | 0 |
| B. | 1 |
| C. | 2 |
| D. | 3 |
| Answer» B. 1 | |
| 7515. |
If angles of a triangle are in the ratio of 2 : 3: 7, then the sides are in the ratio of [MP PET 1996] |
| A. | \[\sqrt{2}:2:(\sqrt{3}+1)\] |
| B. | \[2:\sqrt{2}:(\sqrt{3}+1)\] |
| C. | \[\sqrt{2}:(\sqrt{3}+1):2\] |
| D. | \[2:(\sqrt{3}+1):\sqrt{2}\] |
| Answer» B. \[2:\sqrt{2}:(\sqrt{3}+1)\] | |
| 7516. |
In a \[\Delta ABC\], if \[A={{30}^{o}}\]\[b=2,c=\sqrt{3}+1\], then \[\frac{C-B}{2}=\] |
| A. | \[{{15}^{o}}\] |
| B. | \[{{30}^{o}}\] |
| C. | \[{{45}^{o}}\] |
| D. | None of these |
| Answer» C. \[{{45}^{o}}\] | |
| 7517. |
The sides of a triangle are three consecutive natural numbers and its largest angle is twice the smallest one. Then the sides of the triangle are |
| A. | 1, 2, 3 |
| B. | 2, 3, 4 |
| C. | 3, 4, 5 |
| D. | 4, 5, 6 |
| Answer» E. | |
| 7518. |
If \[A={{30}^{o}},a=7,b=8\]in \[\Delta ABC\], then B has |
| A. | One solution |
| B. | Two solutions |
| C. | No solution |
| D. | None of these |
| Answer» C. No solution | |
| 7519. |
In \[\Delta ABC,\frac{\sin B}{\sin (A+B)}=\] [MP PET 1989] |
| A. | \[\frac{b}{a+b}\] |
| B. | \[\frac{b}{c}\] |
| C. | \[\frac{c}{b}\] |
| D. | None of these |
| Answer» C. \[\frac{c}{b}\] | |
| 7520. |
In a triangle \[ABC\], if \[B=3C\], then the values of \[\sqrt{\left( \frac{b+c}{4c} \right)}\] and \[\left( \frac{b-c}{2c} \right)\] are |
| A. | \[\sin C,\sin \frac{A}{2}\] |
| B. | \[\cos C,\sin \frac{A}{2}\] |
| C. | \[\sin C,\cos \frac{A}{2}\] |
| D. | None of these |
| Answer» C. \[\sin C,\cos \frac{A}{2}\] | |
| 7521. |
If \[a=9,b=8\]and \[c=x\]satisfies \[3\cos C=2,\]then [MP PET 1984] |
| A. | \[x=5\] |
| B. | \[x=6\] |
| C. | \[x=4\] |
| D. | \[x=7\] |
| Answer» E. | |
| 7522. |
In a \[\Delta ABC\], if \[\angle C={{30}^{o}}\], \[a=47cm\]and \[b=94\]cm, then the triangle is [MP PET 1986] |
| A. | Right angled |
| B. | Right angled isosceles |
| C. | Isosceles |
| D. | Obtuse angled |
| Answer» E. | |
| 7523. |
In \[\Delta ABC\], \[{{a}^{2}}({{\cos }^{2}}B-{{\cos }^{2}}C)+\] \[{{b}^{2}}({{\cos }^{2}}C-{{\cos }^{2}}A)+\] \[{{c}^{2}}({{\cos }^{2}}A-{{\cos }^{2}}B)=\] |
| A. | 0 |
| B. | 1 |
| C. | \[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}\] |
| D. | \[2({{a}^{2}}+{{b}^{2}}+{{c}^{2}})\] |
| Answer» B. 1 | |
| 7524. |
In a triangle ABC, \[{{a}^{3}}\cos (B-C)+{{b}^{3}}\cos (C-A)+{{c}^{3}}\cos (A-B)=\] [Kerala (Engg.) 2002] |
| A. | \[abc\] |
| B. | \[3abc\] |
| C. | \[a+b+c\] |
| D. | None of these |
| Answer» C. \[a+b+c\] | |
| 7525. |
If \[\tan \frac{B-C}{2}=x\cot \frac{A}{2},\]then \[x=\] [MP PET 1992, 2002] |
| A. | \[\frac{c-a}{c+a}\] |
| B. | \[\frac{a-b}{a+b}\] |
| C. | \[\frac{b-c}{b+c}\] |
| D. | None of these |
| Answer» D. None of these | |
| 7526. |
In \[\Delta ABC,\] if \[(a+b+c)(a-b+c)\]=3ac, then [AMU 1996] |
| A. | \[\angle B={{60}^{o}}\] |
| B. | \[\angle B={{30}^{o}}\] |
| C. | \[\angle C={{60}^{o}}\] |
| D. | \[\angle A+\angle C={{90}^{o}}\] |
| Answer» B. \[\angle B={{30}^{o}}\] | |
| 7527. |
In a \[\Delta ABC,\]if \[\frac{\sin A}{\sin C}=\frac{\sin (A-B)}{\sin (B-C)},\]then \[{{a}^{2}},\ {{b}^{2}},\ {{c}^{2}}\] are in [Pb. CET 2001; Karnataka CET 1999] |
| A. | A.P. |
| B. | G.P. |
| C. | H.P. |
| D. | None of these |
| Answer» B. G.P. | |
| 7528. |
In a \[\Delta ABC\] if the sides are \[a=3,\,b=5\] and \[c=4\], then \[\sin \frac{B}{2}+\cos \frac{B}{2}\] is equal to [Karnataka CET 2005] |
| A. | \[\sqrt{2}\] |
| B. | \[\frac{\sqrt{3}+1}{2}\] |
| C. | \[\frac{\sqrt{3}-1}{2}\] |
| D. | 1 |
| Answer» B. \[\frac{\sqrt{3}+1}{2}\] | |
| 7529. |
The sides of triangle are \[3x+4y,\ 4x+3y\]and \[5x+5y\]units, where \[x,\ y>0.\]The triangle is [AIEEE 2002] |
| A. | Right angled |
| B. | Equilateral |
| C. | Obtuse angled |
| D. | None of these |
| Answer» D. None of these | |
| 7530. |
In any triangle ABC, the value of \[a({{b}^{2}}+{{c}^{2}})\cos A+b({{c}^{2}}+{{a}^{2}})\cos B+c({{a}^{2}}+{{b}^{2}})\cos C\]is [MP PET 1994] |
| A. | \[3ab{{c}^{2}}\] |
| B. | \[3{{a}^{2}}bc\] |
| C. | \[3abc\] |
| D. | \[3a{{b}^{2}}c\] |
| Answer» D. \[3a{{b}^{2}}c\] | |
| 7531. |
If in a triangle ABC the sides \[AB\]and AC are perpendicular, then the true equation is |
| A. | \[\tan A+\tan B=0\] |
| B. | \[\tan B+\tan C=0\] |
| C. | \[\tan A+2\tan C=0\] |
| D. | \[\tan B.\tan C=1\] |
| Answer» E. | |
| 7532. |
In \[\Delta ABC,\]if \[{{\sin }^{2}}\frac{A}{2},{{\sin }^{2}}\frac{B}{2},{{\sin }^{2}}\frac{C}{2}\] be in H. P. then a, b, c will be in |
| A. | A. P. |
| B. | G. P. |
| C. | H. P. |
| D. | None of these |
| Answer» D. None of these | |
| 7533. |
In any \[\Delta ABC\]if \[a\cos B=b\cos A\], then the triangle is [MP PET 1984] |
| A. | Equilateral triangle |
| B. | Isosceles triangle |
| C. | Scalene |
| D. | Right angled |
| Answer» C. Scalene | |
| 7534. |
In \[\Delta ABC\] if \[a=2,b=4\]and \[\angle C={{60}^{o}}\], then \[\angle A\]and \[\angle B\] are equal to |
| A. | \[{{90}^{o}},{{30}^{o}}\] |
| B. | \[{{60}^{o}},{{60}^{o}}\] |
| C. | \[{{30}^{o}},{{90}^{o}}\] |
| D. | \[{{60}^{o}},{{45}^{o}}\] |
| Answer» D. \[{{60}^{o}},{{45}^{o}}\] | |
| 7535. |
Let D be the middle point of the side BC of a triangle ABC. If the triangle ADC is equilateral, then \[{{a}^{2}}:{{b}^{2}}:{{c}^{2}}\]is equal to [Pb. CET 2004] |
| A. | \[1:4:3\] |
| B. | \[4:1:3\] |
| C. | \[4:3:1\] |
| D. | \[3:4:1\] |
| Answer» C. \[4:3:1\] | |
| 7536. |
The period of \[{{\sin }^{4}}x+{{\cos }^{4}}x\]is [RPET 1997] |
| A. | \[\pi /2\] |
| B. | \[\pi \] |
| C. | \[2\pi \] |
| D. | \[3\pi /2\] |
| Answer» B. \[\pi \] | |
| 7537. |
Period of \[\cos (7x-5)\]is |
| A. | \[\frac{2\pi -5}{7}\] |
| B. | \[2\pi -5\] |
| C. | \[\frac{2\pi }{7}\] |
| D. | \[\frac{\pi }{7}\] |
| Answer» D. \[\frac{\pi }{7}\] | |
| 7538. |
Which of the following functions has period \[2\pi \] [Pb. CET 2004] |
| A. | \[y=\sin \left( 2\pi t+\frac{\pi }{3} \right)+\]\[2\sin \left( 3\pi t+\frac{\pi }{4} \right)+3\sin 5\pi t\] |
| B. | \[y=\sin \frac{\pi }{3}t+\sin \frac{\pi }{4}t\] |
| C. | \[y=\sin t+\cos 2t\] |
| D. | None of these |
| Answer» D. None of these | |
| 7539. |
The angular depressions of the top and the foot of a chimney as seen from the top of a second chimney, which is 150 m high and standing on the same level as the first are \[\theta \] and \[\varphi \] respectively, then the distance between their tops when \[\tan \theta =\frac{4}{3}\]and \[\tan \varphi =\frac{5}{2}\], is [Pb. CET 2004; IIT 1965] |
| A. | \[\frac{150}{\sqrt{3}}metres\] |
| B. | \[100\sqrt{3}metres\] |
| C. | \[150metres\] |
| D. | \[100metres\] |
| Answer» E. | |
| 7540. |
A person observes the angle of elevation of a building as 30°. The person proceeds towards the building with a speed of \[25(\sqrt{3}-1)m/hour.\]After \[2\,hours\], he observes the angle of elevation as 45°. The height of the building (in meter) is [UPSEAT 2003] |
| A. | 100 |
| B. | 50 |
| C. | \[50(\sqrt{3}+1)\] |
| D. | \[50(\sqrt{3}-1)\] |
| Answer» C. \[50(\sqrt{3}+1)\] | |
| 7541. |
A tower subtends an angle \[\alpha \]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\[\beta \]. The height of the tower is |
| 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. \[l\tan \alpha \tan \beta \] | |
| 7542. |
The top of a hill when observed form the top and bottom of a building of height h is at angles of elevation p and q respectively. What is the height of the hill? |
| A. | \[\frac{h\cot q}{\cot q-\cot p}\] |
| B. | \[\frac{h\cot p}{\cot p-\cot q}\] |
| C. | \[\frac{2h\tan p}{\tan p-\tan q}\] |
| D. | \[\frac{2h\tan q}{\tan q-\tan p}\] |
| Answer» C. \[\frac{2h\tan p}{\tan p-\tan q}\] | |
| 7543. |
If \[A+B+C=\pi ,\] then\[\cos 2A+\cos 2B+\cos 2C+4\sin A\sin B\sin C\] is equal to: |
| A. | 0 |
| B. | 1 |
| C. | 2 |
| D. | 3 |
| Answer» C. 2 | |
| 7544. |
If the angles of a triangle are \[30{}^\circ \] and \[45{}^\circ \] and the included side is \[(\sqrt{3}+1),\] then what is the area of the triangle? |
| A. | \[\frac{\sqrt{3}+1}{2}\] |
| B. | \[2(\sqrt{3}+1)\] |
| C. | \[\frac{\sqrt{3}+1}{3}\] |
| D. | \[\frac{\sqrt{3}-1}{2}\] |
| Answer» B. \[2(\sqrt{3}+1)\] | |
| 7545. |
Angles of a triangle are in the ratio \[4:1:1.\] The ratio between its greatest side and perimeter is |
| A. | \[\frac{3}{2+\sqrt{3}}\] |
| B. | \[\frac{1}{2+\sqrt{3}}\] |
| C. | \[\frac{\sqrt{3}}{\sqrt{3}+2}\] |
| D. | \[\frac{2}{2+\sqrt{3}}\] |
| Answer» D. \[\frac{2}{2+\sqrt{3}}\] | |
| 7546. |
In a \[\Delta ABC,\]\[\frac{\sin A}{\sin C}=\frac{\sin (A-B)}{\sin (B-C)},\] then \[{{a}^{2}},{{b}^{2}},{{c}^{2}}\] are such that |
| A. | \[{{b}^{2}}=ac\] |
| B. | \[{{b}^{2}}=\frac{{{a}^{2}}{{c}^{2}}}{{{a}^{2}}+{{c}^{2}}}\] |
| C. | They are in A.P. |
| D. | \[{{b}^{2}}={{a}^{2}}+{{c}^{2}}\] |
| Answer» D. \[{{b}^{2}}={{a}^{2}}+{{c}^{2}}\] | |
| 7547. |
If the co-ordinates of the points \[A,B,C\]be \[(-1,\,3,\,2),\,\,(2,\,3,\,5)\] and (3, 5,?2) respectively, then \[\angle A=\] |
| A. | \[0{}^\circ \] |
| B. | \[45{}^\circ \] |
| C. | \[60{}^\circ \] |
| D. | \[90{}^\circ \] |
| Answer» E. | |
| 7548. |
The co-ordinates of points \[A,B,C,D\]are (a, 2, 1), (1, ?1, 1), (2, ? 3, 4) and (a+1, a+2, a+3) respectively. If \[AB=5\]and \[CD=6\], then \[a=\] |
| A. | 2 |
| B. | 3 |
| C. | ? 2 |
| D. | ? 3 |
| Answer» E. | |
| 7549. |
The projection of a line on a co-ordinate axes are 2, 3, 6. Then the length of the line is [Orissa JEE 2002] |
| A. | 7 |
| B. | 5 |
| C. | 1 |
| D. | 11 |
| Answer» B. 5 | |
| 7550. |
If \[A(1,\,2,\,3),\,B(-1,-1,-1)\] be the points, then the distance AB is [MP PET 2001; Pb. CET 2001] |
| A. | \[\sqrt{5}\] |
| B. | \[\sqrt{21}\] |
| C. | \[\sqrt{29}\] |
| D. | None of these |
| Answer» D. None of these | |