MCQOPTIONS
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MCQOPTIONS
This section includes 21 Mcqs, each offering curated multiple-choice questions to sharpen your Mathematics knowledge and support exam preparation. Choose a topic below to get started.
| 1. |
In a \[\Delta ABC,\]if \[3a=b+c,\]then the value of \[\cot \frac{B}{2}\cot \frac{C}{2}\] is [Pb. CET 2003; Roorkee 1986; MP PET 1990, 97, 98, 2003; EAMCET 2003; Orissa JEE 2005] |
| A. | 1 |
| B. | 2 |
| C. | \[\sqrt{3}\] |
| D. | \[\sqrt{2}\] |
| Answer» C. \[\sqrt{3}\] | |
| 2. |
If the two angle on the base of a triangle are \[{{\left( 22\frac{1}{2} \right)}^{o}}\] and \[{{\left( 112\frac{1}{2} \right)}^{o}}\], then the ratio of the height of the triangle to the length of the base is [MP PET 1993; Pb CET 2002] |
| A. | 1 : 2 |
| B. | 2 : 1 |
| C. | 2 : 3 |
| D. | 1: 1 |
| Answer» B. 2 : 1 | |
| 3. |
In triangle \[ABC\]if \[\frac{\cos A}{a}=\frac{\cos B}{b}=\frac{\cos C}{c}\], then the triangle is [Karnataka 1991; Pb. CET 1989] |
| A. | Right angled |
| B. | Obtuse angled |
| C. | Equilateral |
| D. | Isosceles |
| Answer» D. Isosceles | |
| 4. |
If a,b are different values of \[x\] satisfying \[a\cos x+b\sin x=c,\] then \[\tan \text{ }\left( \frac{\alpha +\beta }{2} \right)=\] [EAMCET 1986; Orissa JEE 2003] |
| A. | \[a+b\] |
| B. | \[a-b\] |
| C. | \[\frac{b}{a}\] |
| D. | \[\frac{a}{b}\] |
| Answer» D. \[\frac{a}{b}\] | |
| 5. |
In a triangle \[ABC\], \[\tan \frac{A}{2}=\frac{5}{6}\] and \[\tan \frac{C}{2}=\frac{2}{5},\] then[EAMCET 1994] |
| A. | \[a,\ b,\ c\]are in A.P. |
| B. | \[\cos A,\ \cos B,\ \cos C\]are in A.P. |
| C. | \[\sin A,\ \sin B,\ \sin C\]are in A.P. |
| D. | (a) and (c) both |
| Answer» E. | |
| 6. |
\[\Delta ABC,\] if \[\cos \frac{A}{2}=\sqrt{\frac{b+c}{2c}}\], then [MP PET 1990] |
| A. | \[{{a}^{2}}+{{b}^{2}}={{c}^{2}}\] |
| B. | \[{{b}^{2}}+{{c}^{2}}={{a}^{2}}\] |
| C. | \[{{c}^{2}}+{{a}^{2}}={{b}^{2}}\] |
| D. | \[b-c=c-a\] |
| Answer» B. \[{{b}^{2}}+{{c}^{2}}={{a}^{2}}\] | |
| 7. |
\[\frac{a\cos A+b\cos B+c\cos C}{a+b+c}=\] [Orissa JEE 2004] |
| A. | 1/r |
| B. | r/R |
| C. | R/r |
| D. | 1/R |
| Answer» C. R/r | |
| 8. |
If \[\sin 2x+\sin 4x=2\sin 3x,\]then \[x\]= [EAMCET 1989] |
| A. | \[\frac{n\pi }{3}\] |
| B. | \[n\pi +\frac{\pi }{3}\] |
| C. | \[2n\pi \pm \frac{\pi }{3}\] |
| D. | None of these |
| Answer» B. \[n\pi +\frac{\pi }{3}\] | |
| 9. |
If \[\sec 4\theta -\sec 2\theta =2\], then the general value of \[\theta \] is [IIT 1963] |
| A. | \[(2n+1)\frac{\pi }{4}\] |
| B. | \[(2n+1)\frac{\pi }{10}\] |
| C. | \[n\pi +\frac{\pi }{2}\]or \[\frac{n\pi }{5}+\frac{\pi }{10}\] |
| D. | None of these |
| Answer» D. None of these | |
| 10. |
The general solution of \[a\cos x+b\sin x=c,\] where\[a,\,\,b,\,\,c\] are constants |
| A. | \[x=n\pi +{{\cos }^{-1}}\left( \frac{c}{\sqrt{{{a}^{2}}+{{b}^{2}}}} \right)\] |
| B. | \[x=2n\pi -{{\tan }^{-1}}\left( \frac{b}{a} \right)\] |
| C. | \[x=2n\pi -{{\tan }^{-1}}\left( \frac{b}{a} \right)\pm {{\cos }^{-1}}\left( \frac{c}{\sqrt{{{a}^{2}}+{{b}^{2}}}} \right)\] |
| D. | \[x=2n\pi +{{\tan }^{-1}}\left( \frac{b}{a} \right)\pm {{\cos }^{-1}}\left( \frac{c}{\sqrt{{{a}^{2}}+{{b}^{2}}}} \right)\] |
| Answer» E. | |
| 11. |
A man from the top of a 100 meters high tower sees a car moving towards the tower at an angle of depression of 30 °. After some time, the angle of depression becomes\[{{60}^{o}}\]. The distance (in meters) travelled by the car during the time is [IIT Screening 2001] |
| A. | \[100\sqrt{3}\] |
| B. | \[\frac{200\sqrt{3}}{3}\] |
| C. | \[\frac{100\sqrt{3}}{3}\] |
| D. | \[200\sqrt{3}\] |
| Answer» C. \[\frac{100\sqrt{3}}{3}\] | |
| 12. |
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. \[50(\sqrt{3}+1)\] | |
| 13. |
If the sides of a \[\Delta \]be\[({{x}^{2}}+x+1),\,(2x+1)\] and \[({{x}^{2}}-1),\]then the greatest angle is [EAMCET 1987; Kerala (Engg.) 2001] |
| A. | \[{{105}^{o}}\] |
| B. | \[{{120}^{o}}\] |
| C. | \[{{135}^{o}}\] |
| D. | None |
| Answer» C. \[{{135}^{o}}\] | |
| 14. |
In a \[\Delta ABC,\]if \[a=2x,\]\[b=2y\]and \[\angle C=120{}^\circ \], then the area of the triangle is [MP PET 1986, 2002] |
| A. | \[xy\] |
| B. | \[xy\sqrt{3}\] |
| C. | \[3xy\] |
| D. | \[2xy\] |
| Answer» C. \[3xy\] | |
| 15. |
In a \[\Delta ABC,\] if \[(\sin A+\sin B+\sin C)\] \[(\sin A+\sin B-\sin C)\] = \[3\sin A\sin B,\]then the angle C is equal to [AMU 1999] |
| A. | \[\frac{\pi }{2}\] |
| B. | \[\frac{\pi }{3}\] |
| C. | \[\frac{\pi }{4}\] |
| D. | \[\frac{\pi }{6}\] |
| Answer» C. \[\frac{\pi }{4}\] | |
| 16. |
In \[\Delta ABC,\ a(b\cos C-c\cos B)=\] [EAMCET 1981] |
| A. | \[{{a}^{2}}-{{b}^{2}}\] |
| B. | \[{{b}^{2}}-{{c}^{2}}\] |
| C. | \[{{c}^{2}}-{{a}^{2}}\] |
| D. | \[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}\] |
| Answer» C. \[{{c}^{2}}-{{a}^{2}}\] | |
| 17. |
If \[a,\ b,\ c\]are the sides of a triangle ABC, then which of the following inequalities is not true [Kurukshetra CEE 1996] |
| A. | \[8abc\le (a+b)(b+c)(c+a)\] |
| B. | \[3abc\le {{a}^{3}}+{{b}^{3}}+{{c}^{3}}\] |
| C. | \[6abc\le bc(b+c)+ca(c+a)+ab(a+b)\] |
| D. | \[abc\le (a+b-c)(b+c-a)(c+a-b)\] |
| Answer» E. | |
| 18. |
The number of pairs (x, y) satisfying the equations \[\sin x+\sin y=\sin (x+y)\] and \[|x|+|y|=1\]is |
| A. | 2 |
| B. | 4 |
| C. | 6 |
| D. | \[\infty \] |
| Answer» D. \[\infty \] | |
| 19. |
In an equilateral triangle of side \[2\sqrt{3}\]cm, the circum-radius is [EAMCET 1978] |
| A. | 1 cm |
| B. | \[\sqrt{3}\]cm |
| C. | 2 cm |
| D. | \[2\sqrt{3}\]cm |
| Answer» D. \[2\sqrt{3}\]cm | |
| 20. |
Which is true in the following [UPSEAT 1999] |
| A. | \[a\cos A+b\cos B+c\cos C=R\sin A\sin B\sin C\] |
| B. | \[a\cos A+b\cos B+c\cos C=2R\sin A\sin B\sin C\] |
| C. | \[a\cos A+b\cos B+c\cos C=4R\sin A\sin B\sin C\] |
| D. | \[a\cos A+b\cos B+c\cos C=8R\sin A\sin B\sin C\] |
| Answer» D. \[a\cos A+b\cos B+c\cos C=8R\sin A\sin B\sin C\] | |
| 21. |
In \[\Delta ABC,\ \,2{{R}^{2}}\sin A\sin B\sin C=\] |
| A. | \[{{s}^{2}}\] |
| B. | \[ab+bc+ca\] |
| C. | \[\Delta \] |
| D. | None of these |
| Answer» D. None of these | |