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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.
| 7451. |
Locus of the point P, for which \[\overrightarrow{OP}\] represents a vector with direction cosine \[\cos \,\,\alpha =\frac{1}{2}\] (where O is the origin) is |
| A. | a circle parallel to the y-z plane with centre on the x-axis |
| B. | a cone concentric with the positive x-axis having vertex at the origin and the slant height equal to the magnitude of the vector |
| C. | a ray emanating from the origin and making an angle of \[60{}^\circ \]with the x-axis |
| D. | a disc parallel to the y-z plane with centre on the x-axis and radius equal to \[\left| \overrightarrow{OP} \right|\] sin 60° |
| Answer» C. a ray emanating from the origin and making an angle of \[60{}^\circ \]with the x-axis | |
| 7452. |
A point O is the centre of a circle circumscribed about a triangle ABC. Then \[\overrightarrow{OA}\] sin 2A+\[\overrightarrow{OB}\] sin 2B + \[\overrightarrow{OC}\] sin 2C is equal to |
| A. | \[(\overrightarrow{OA}+\overrightarrow{OB}+\overrightarrow{OC})sin2A\] |
| B. | \[3\,\overrightarrow{OG}\], where G is the centroid of triangle ABC |
| C. | \[\overrightarrow{0}\] |
| D. | none of these |
| Answer» D. none of these | |
| 7453. |
The position vector of a point at a distance of \[3\sqrt{11}\] units from \[\mathbf{i}-\mathbf{j}+2\mathbf{k}\] on a line passing through the points \[\mathbf{i}-\mathbf{j}+2\mathbf{k}\] and \[3\mathbf{i}+\mathbf{j}+\mathbf{k}\] is |
| A. | \[10\mathbf{i}+2\mathbf{j}-5\mathbf{k}\] |
| B. | \[-8\mathbf{i}-4\mathbf{j}-\mathbf{k}\] |
| C. | \[8\mathbf{i}+4\mathbf{j}+\mathbf{k}\] |
| D. | \[-10\mathbf{i}-2\mathbf{j}-5\mathbf{k}\] |
| Answer» C. \[8\mathbf{i}+4\mathbf{j}+\mathbf{k}\] | |
| 7454. |
The centre of the circle given by \[\mathbf{r}.(\mathbf{i}+2\mathbf{j}+2\mathbf{k})=15\] and \[|\mathbf{r}-(\mathbf{j}+2\mathbf{k})|=4\]is |
| A. | (0, 1, 2) |
| B. | (1, 3, 4) |
| C. | (?1, 3, 4) |
| D. | None of these |
| Answer» C. (?1, 3, 4) | |
| 7455. |
If b and c are any two non-collinear unit vectors and a is any vector, then \[(\mathbf{a}\,.\,\mathbf{b})\,\mathbf{b}+(\mathbf{a}\,.\,\mathbf{c})\,\mathbf{c}+\frac{\mathbf{a}\,.\,(\mathbf{b}\times \mathbf{c})}{|\mathbf{b}\times \mathbf{c}|}\,(\mathbf{b}\times \mathbf{c})=\] [IIT 1996] |
| A. | a |
| B. | b |
| C. | c |
| D. | 0 |
| Answer» B. b | |
| 7456. |
If a, b, c are non-coplanar unit vectors such that \[\mathbf{a}\times (\mathbf{b}\times \mathbf{c})=\frac{\mathbf{b}+\mathbf{c}}{\sqrt{2}}\], then the angle between a and b is [IIT 1995] |
| A. | \[\frac{\pi }{4}\] |
| B. | \[\frac{\pi }{2}\] |
| C. | \[\frac{3\pi }{4}\] |
| D. | \[\pi \] |
| Answer» D. \[\pi \] | |
| 7457. |
\[{{\cos }^{2}}\alpha +{{\cos }^{2}}(\alpha +120{}^\circ )+{{\cos }^{2}}(\alpha -120{}^\circ )\] is equal to [MP PET 1993] |
| A. | 3/2 |
| B. | 1 |
| C. | 1/2 |
| D. | 0 |
| Answer» B. 1 | |
| 7458. |
\[1+\cos 2x+\cos 4x+\cos 6x=\] [Roorkee 1974] |
| A. | \[2\cos x\cos 2x\cos 3x\] |
| B. | \[4\sin x\,\cos 2x\cos 3x\] |
| C. | \[4\cos x\cos 2x\cos 3x\] |
| D. | None of these |
| Answer» D. None of these | |
| 7459. |
The expression \[\frac{\cos 6x+6\cos 4x+15\cos 2x+10}{\cos 5x+5\cos 3x+10\cos x}\] is equal to |
| A. | \[\cos 2x\] |
| B. | \[2\cos x\] |
| C. | \[{{\cos }^{2}}x\] |
| D. | \[1+\cos x\] |
| Answer» C. \[{{\cos }^{2}}x\] | |
| 7460. |
\[\frac{\sin (B+A)+\cos (B-A)}{\sin (B-A)+\cos (B+A)}=\] [Roorkee 1970; IIT 1966] |
| A. | \[\frac{\cos B+\sin B}{\cos B-\sin B}\] |
| B. | \[\frac{\cos A+\sin A}{\cos A-\sin A}\] |
| C. | \[\frac{\cos A-\sin A}{\cos A+\sin A}\] |
| D. | None of these |
| Answer» C. \[\frac{\cos A-\sin A}{\cos A+\sin A}\] | |
| 7461. |
\[{{\cos }^{2}}\left( \frac{\pi }{4}-\beta \right)-{{\sin }^{2}}\left( \alpha -\frac{\pi }{4} \right)=\] |
| A. | \[\sin (\alpha +\beta )\sin (\alpha -\beta )\] |
| B. | \[\cos (\alpha +\beta )\cos (\alpha -\beta )\] |
| C. | \[\sin (\alpha -\beta )\cos (\alpha +\beta )\] |
| D. | \[\sin (\alpha +\beta )\cos (\alpha -\beta )\] |
| Answer» E. | |
| 7462. |
If \[x=\cos 10{}^\circ \cos 20{}^\circ \cos 40{}^\circ ,\]then the value of \[x\] is [Roorkee 1995] |
| A. | \[\frac{1}{4}\tan 10{}^\circ \] |
| B. | \[\frac{1}{8}\cot 10{}^\circ \] |
| C. | \[\frac{1}{8}\text{cosec}10{}^\circ \] |
| D. | \[\frac{1}{8}\sec 10{}^\circ \] |
| Answer» C. \[\frac{1}{8}\text{cosec}10{}^\circ \] | |
| 7463. |
The equations \[(b-c)x+(c-a)y+(a-b)=0\] and \[({{b}^{3}}-{{c}^{3}})x+({{c}^{3}}-{{a}^{3}})y+{{a}^{3}}-{{b}^{3}}=0\] will represent the same line, if |
| A. | b = c |
| B. | c = a |
| C. | a = b |
| D. | a + b + c = 0 |
| Answer» E. | |
| 7464. |
\[\frac{\cos 12{}^\circ -\sin 12{}^\circ }{\cos 12{}^\circ +\sin 12{}^\circ }+\frac{\sin 147{}^\circ }{\cos 147{}^\circ }=\] [MP PET 1991] |
| A. | 1 |
| B. | -1 |
| C. | 0 |
| D. | None of these |
| Answer» D. None of these | |
| 7465. |
If \[\tan A-\tan B=x\] and \[\cot B-\cot A=y,\]then \[\cot (A-B)=\] |
| A. | \[\frac{1}{x}+y\] |
| B. | \[\frac{1}{xy}\] |
| C. | \[\frac{1}{x}-\frac{1}{y}\] |
| D. | \[\frac{1}{x}+\frac{1}{y}\] |
| Answer» E. | |
| 7466. |
\[\frac{\cos {{10}^{o}}+\sin {{10}^{o}}}{\cos {{10}^{o}}-\sin {{10}^{o}}}=\] [MP PET 2002] |
| A. | \[\tan \,{{55}^{o}}\] |
| B. | \[\cot {{55}^{o}}\] |
| C. | \[-\tan {{35}^{o}}\] |
| D. | \[-\cot {{35}^{o}}\] |
| Answer» B. \[\cot {{55}^{o}}\] | |
| 7467. |
\[\frac{\cos 17{}^\circ +\sin 17{}^\circ }{\cos 17{}^\circ -\sin 17{}^\circ }=\] [MP PET 1998] |
| A. | \[\tan 62{}^\circ \] |
| B. | \[\tan 56{}^\circ \] |
| C. | \[\tan 54{}^\circ \] |
| D. | \[\tan 73{}^\circ \] |
| Answer» B. \[\tan 56{}^\circ \] | |
| 7468. |
The expression\[{{\cos }^{2}}(A-B)+{{\cos }^{2}}B-2\cos (A-B)\cos A\cos B\] is |
| A. | Dependent on B |
| B. | Dependent on A and B |
| C. | Dependent on A |
| D. | Independent of A and B |
| Answer» D. Independent of A and B | |
| 7469. |
If \[\cos (\alpha +\beta )=\frac{4}{5},\sin (\alpha -\beta )=\frac{5}{13}\] and \[\alpha ,\beta \] lie between 0 and \[\frac{\pi }{4},\]then \[\tan 2\alpha =\] [IIT 1979; EAMCET 2002] |
| A. | \[\frac{16}{63}\] |
| B. | \[\frac{56}{33}\] |
| C. | \[\frac{28}{33}\] |
| D. | None of these |
| Answer» C. \[\frac{28}{33}\] | |
| 7470. |
If \[\tan A=2\tan B+\cot B,\]then \[2\tan (A-B)=\] |
| A. | \[\tan B\] |
| B. | \[2\tan B\] |
| C. | \[\cot B\] |
| D. | \[2\cot B\] |
| Answer» D. \[2\cot B\] | |
| 7471. |
\[\frac{1}{\sin 10{}^\circ }-\frac{\sqrt{3}}{\cos 10{}^\circ }\]= [IIT 1974] |
| A. | 0 |
| B. | 1 |
| C. | 2 |
| D. | 4 |
| Answer» E. | |
| 7472. |
\[\frac{\sin 2A}{1+\cos 2A}.\frac{\cos A}{1+\cos A}=\] |
| A. | \[\tan \frac{A}{2}\] |
| B. | \[\cot \frac{A}{2}\] |
| C. | \[\sec \frac{A}{2}\] |
| D. | \[\text{cosec}\frac{A}{2}\] |
| Answer» B. \[\cot \frac{A}{2}\] | |
| 7473. |
\[\frac{\sin 3\theta -\cos 3\theta }{\sin \theta +\cos \theta }+1=\] |
| A. | \[2\sin 2\theta \] |
| B. | \[2\cos 2\theta \] |
| C. | \[\tan 2\theta \] |
| D. | \[\cot 2\theta \] |
| Answer» B. \[2\cos 2\theta \] | |
| 7474. |
\[2{{\cos }^{2}}\theta -2{{\sin }^{2}}\theta =1\],then \[\theta \]= [Karnataka CET 1998] |
| A. | \[15{}^\circ \] |
| B. | \[30{}^\circ \] |
| C. | \[45{}^\circ \] |
| D. | \[60{}^\circ \] |
| Answer» C. \[45{}^\circ \] | |
| 7475. |
. Given that \[\cos \left( \frac{\alpha -\beta }{2} \right)=2\cos \left( \frac{\alpha +B}{2} \right)\], then \[\tan \frac{\alpha }{2}\tan \frac{\beta }{2}\]is equal to [AMU 2001] |
| A. | \[\frac{1}{2}\] |
| B. | \[\frac{1}{3}\] |
| C. | \[\frac{1}{4}\] |
| D. | \[\frac{1}{8}\] |
| Answer» C. \[\frac{1}{4}\] | |
| 7476. |
If \[\sin \theta +\cos \theta =x,\] then \[{{\sin }^{6}}\theta +{{\cos }^{6}}\theta =\frac{1}{4}[4-3{{({{x}^{2}}-1)}^{2}}]\] for |
| A. | All real x |
| B. | \[{{x}^{2}}\le 2\] |
| C. | \[{{x}^{2}}\ge 2\] |
| D. | None of these |
| Answer» C. \[{{x}^{2}}\ge 2\] | |
| 7477. |
If \[\tan \frac{A}{2}=\frac{3}{2},\]then \[\frac{1+\cos A}{1-\cos A}=\] |
| A. | \[-5\] |
| B. | \[5\] |
| C. | \[9/4\] |
| D. | \[4/9\] |
| Answer» E. | |
| 7478. |
If \[\theta \]and \[\varphi \]are angles in the 1st quadrant such that \[\tan \theta =1/7\]and \[\sin \varphi =1/\sqrt{10}\].Then [Kurukshetra CEE 1998; AMU 2001] |
| A. | \[\theta +2\varphi =90{}^\circ \] |
| B. | \[\theta +2\varphi =60{}^\circ \] |
| C. | \[\theta +2\varphi =30{}^\circ \] |
| D. | \[\theta +2\varphi =45{}^\circ \] |
| Answer» E. | |
| 7479. |
If \[\tan \alpha =\frac{1}{7},\ \tan \beta =\frac{1}{3},\]then \[\cos 2\alpha =\] [CET 1986] |
| A. | \[\sin 2\beta \] |
| B. | \[\sin 4\beta \] |
| C. | \[\sin 3\beta \] |
| D. | None of these |
| Answer» C. \[\sin 3\beta \] | |
| 7480. |
\[\frac{\sec 8A-1}{\sec 4A-1}=\] [MP PET 1995] |
| A. | \[\frac{\tan 2A}{\tan 8A}\] |
| B. | \[\frac{\tan 8A}{\tan 2A}\] |
| C. | \[\frac{\cot 8A}{\cot 2A}\] |
| D. | None of these |
| Answer» C. \[\frac{\cot 8A}{\cot 2A}\] | |
| 7481. |
If \[\tan \alpha =\frac{1}{7}\]and \[\sin \beta =\frac{1}{\sqrt{10}}\left( 0 |
| A. | \[\frac{\pi }{4}-\alpha \] |
| B. | \[\frac{3\pi }{4}-\alpha \] |
| C. | \[\frac{\pi }{8}-\frac{\alpha }{2}\] |
| D. | \[\frac{3\pi }{8}-\frac{\alpha }{2}\] |
| Answer» B. \[\frac{3\pi }{4}-\alpha \] | |
| 7482. |
If \[\cos 3\theta =\alpha \cos \theta +\beta {{\cos }^{3}}\theta ,\]then \[(\alpha ,\beta )=\] |
| A. | \[(3,\,4)\] |
| B. | \[(4,\,3)\] |
| C. | \[(-3,\,4)\] |
| D. | \[(3,\,-4)\] |
| Answer» D. \[(3,\,-4)\] | |
| 7483. |
If \[(\sec \alpha +\tan \alpha )(\sec \beta +\tan \beta )(\sec \gamma +\tan \gamma )\] \[=\tan \,\alpha \tan \beta \tan \gamma ,\] then expression \[(\sec \alpha -\tan \alpha )\,(sec\beta -tan\beta )(sec\gamma -tan\gamma )\]is equal to |
| A. | \[\cot \alpha \,\,\cot \beta \,\,\cot \gamma \] |
| B. | \[\tan \alpha \,\,tan\beta \,\,tan\gamma \] |
| C. | \[\cot \alpha +\cot \beta +\cot \gamma \] |
| D. | \[tan\alpha +tan\beta +tan\gamma \] |
| Answer» B. \[\tan \alpha \,\,tan\beta \,\,tan\gamma \] | |
| 7484. |
General solution of the equation \[2{{\cot }^{2}}\theta +2\sqrt{3}\cot \theta +4\operatorname{cosec}+8=0\] is |
| A. | \[\theta =n\pi \pm \frac{\pi }{6},n\in I\] |
| B. | \[n\pi +\frac{\pi }{6},n\in I\] |
| C. | \[2n\pi +\frac{\pi }{6},n\in I\] |
| D. | \[2n\pi +\frac{11\pi }{6},n\in I\] |
| Answer» E. | |
| 7485. |
The solution set of the system of equation \[x+y=2\pi /3,\] \[\cos x+\cos y=3/2,\] where x and y are real, is |
| A. | \[x=\frac{\pi }{3}-n\pi ,y=n\pi \] |
| B. | \[\phi \] |
| C. | \[x=n\pi ,y=\frac{\pi }{3}-n\pi \] |
| D. | None of these |
| Answer» C. \[x=n\pi ,y=\frac{\pi }{3}-n\pi \] | |
| 7486. |
If \[\sin A\,(60{}^\circ -A)\,\sin (60{}^\circ +A)=k\sin 3A,\] then what is k equal to? |
| A. | \[1/4\] |
| B. | \[1/2\] |
| C. | \[1\] |
| D. | \[4\] |
| Answer» B. \[1/2\] | |
| 7487. |
On simplifying \[\frac{{{\sin }^{3}}A+\sin 3A}{\sin A}+\frac{{{\cos }^{3}}A-\cos 3A}{\cos A},\] we get |
| A. | \[\sin 3A\] |
| B. | \[\cos 3A\] |
| C. | \[\sin A+\cos A\] |
| D. | 3 |
| Answer» E. | |
| 7488. |
The value of \[\frac{\sin 8x+7\sin 6x+18\sin 4x+12\sin 2x}{\sin 7x+6\sin 5x+12\sin 3x}\] equal to? |
| A. | \[2\cos x\] |
| B. | \[\cos x\] |
| C. | \[2sinx\] |
| D. | \[sinx\] |
| Answer» B. \[\cos x\] | |
| 7489. |
If \[\alpha +\beta +\gamma =\pi \]then the minimum value of \[cos\text{ }A+cos\text{ }B+cos\text{ }C\] |
| A. | is zero |
| B. | is positive |
| C. | lies between \[-2\] and \[-3\] |
| D. | is \[-3\] |
| Answer» E. | |
| 7490. |
What is \[\frac{\cot 224{}^\circ -\cot 134{}^\circ }{\cot 226{}^\circ +\cot 316{}^\circ }\] equal to? |
| A. | \[-\text{cosec }88{}^\circ \] |
| B. | \[-\text{cosec 2}{}^\circ \] |
| C. | \[-\text{cosec 44}{}^\circ \] |
| D. | \[-\text{cosec 46}{}^\circ \] |
| Answer» C. \[-\text{cosec 44}{}^\circ \] | |
| 7491. |
What is \[\cos 20{}^\circ +\cos 100{}^\circ +\cos 140{}^\circ \] equal to? |
| A. | 2 |
| B. | 1 |
| C. | \[1/2\] |
| D. | 0 |
| Answer» E. | |
| 7492. |
If \[\sin (y+z-x)\],\[\sin (z+x-y)\], \[\sin (x+y-z)\]are in A.P., then \[\tan x,\tan y,\tan z\]are in |
| A. | A.P. |
| B. | G.P. |
| C. | H.P. |
| D. | None of these |
| Answer» B. G.P. | |
| 7493. |
The sum of all the solutions of \[\cot \theta =\sin 2\theta (\theta \ne n\pi ,n\,integer)\], \[0\le \theta \le \pi \]is |
| A. | \[3\pi /2\] |
| B. | \[\pi \] |
| C. | \[3\pi /4\] |
| D. | \[2\pi \] |
| Answer» B. \[\pi \] | |
| 7494. |
If \[\cot \,\theta +\tan \theta =m\]and \[\sec \theta -\cos \theta =n,\]then which of the following is correct |
| A. | \[m{{(m{{n}^{2}})}^{1/3}}-n{{(n{{m}^{2}})}^{1/3}}=1\] |
| B. | \[m{{({{m}^{2}}n)}^{1/3}}-n{{(m{{n}^{2}})}^{1/3}}=1\] |
| C. | \[n{{(m{{n}^{2}})}^{1/3}}-m{{(n{{m}^{2}})}^{1/3}}=1\] |
| D. | \[n{{({{m}^{2}}n)}^{1/3}}-m{{(m{{n}^{2}})}^{1/3}}=1\] |
| Answer» B. \[m{{({{m}^{2}}n)}^{1/3}}-n{{(m{{n}^{2}})}^{1/3}}=1\] | |
| 7495. |
\[\tan \left( \frac{\pi }{4}+\theta \right)-\tan \left( \frac{\pi }{4}-\theta \right)=\] |
| A. | \[2\tan 2\theta \] |
| B. | \[2\cot 2\theta \] |
| C. | \[\tan 2\theta \] |
| D. | \[\cot 2\theta \] |
| Answer» B. \[2\cot 2\theta \] | |
| 7496. |
If \[\sin x=\frac{-24}{25},\] then the value of \[\tan x\] is [UPSEAT 2003] |
| A. | \[\frac{24}{25}\] |
| B. | \[\frac{-24}{7}\] |
| C. | \[\frac{25}{24}\] |
| D. | None of these |
| Answer» C. \[\frac{25}{24}\] | |
| 7497. |
\[{{\sin }^{2}}\frac{\pi }{8}+{{\sin }^{2}}\frac{3\pi }{8}+{{\sin }^{2}}\frac{5\pi }{8}+{{\sin }^{2}}\frac{7\pi }{8}=\] |
| A. | 1 |
| B. | -1 |
| C. | 0 |
| D. | 2 |
| Answer» E. | |
| 7498. |
The smallest positive angle which satisfies the equation \[2{{\sin }^{2}}\theta +\sqrt{3}\cos \theta +1=0\], is [ISM Dhanbad 1972; MP PET 1993] |
| A. | \[\frac{5\pi }{6}\] |
| B. | \[\frac{2\pi }{3}\] |
| C. | \[\frac{\pi }{3}\] |
| D. | \[\frac{\pi }{6}\] |
| Answer» B. \[\frac{2\pi }{3}\] | |
| 7499. |
The solution of equation \[{{\cos }^{2}}\theta +\sin \theta +1=0\] lies in the interval [UPSEAT 2004; IIT 1992] |
| A. | \[\left( -\frac{\pi }{4},\frac{\pi }{4} \right)\] |
| B. | \[\left( \frac{\pi }{4},\frac{3\pi }{4} \right)\] |
| C. | \[\left( \frac{3\pi }{4},\frac{5\pi }{4} \right)\] |
| D. | \[\left( \frac{5\pi }{4},\frac{7\pi }{4} \right)\] |
| Answer» E. | |
| 7500. |
The values of \[\theta \] satisfying \[\sin 7\theta =\sin 4\theta -\sin \theta \] and \[0 |
| A. | \[\frac{\pi }{9},\frac{\pi }{4}\] |
| B. | \[\frac{\pi }{3},\frac{\pi }{9}\] |
| C. | \[\frac{\pi }{6},\frac{\pi }{9}\] |
| D. | \[\frac{\pi }{3},\frac{\pi }{4}\] |
| Answer» B. \[\frac{\pi }{3},\frac{\pi }{9}\] | |