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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.
| 6351. |
If \[\int{\frac{(2{{x}^{2}}+1)\,\,dx}{({{x}^{2}}-4)\,\,({{x}^{2}}-1)}=\log \left[ {{\left( \frac{x+1}{x-1} \right)}^{a}}\,\,{{\left( \frac{x-2}{x+2} \right)}^{b}} \right]}+C,\] then the values of a and b are respectively [Roorkee 2000] |
| A. | 1/2, ¾ |
| B. | -1, 3/2 |
| C. | 1, 3/2 |
| D. | -1/2, ¾ |
| Answer» B. -1, 3/2 | |
| 6352. |
The function \[f(x)\,=\,|x|+|x-1|\] is [RPET 1996; Kurukshetra CEE 2002] |
| A. | Continuous at \[x=1,\] but not differentiable at \[x=1\] |
| B. | Both continuous and differentiable at \[x=1\] |
| C. | Not continuous at \[x=1\] |
| D. | Not differentiable at \[x=1\] |
| Answer» B. Both continuous and differentiable at \[x=1\] | |
| 6353. |
The point of intersection of \[\mathbf{r}\times \mathbf{a}=\mathbf{b}\times \mathbf{a}\] and \[\mathbf{r}\times \mathbf{b}=\mathbf{a}\times \mathbf{b}\], where \[\mathbf{a}=\mathbf{i}+\mathbf{j}\] and \[\mathbf{b}=2\mathbf{i}-\mathbf{k}\] is [Orissa JEE 2004] |
| A. | \[3\mathbf{i}+\mathbf{j}-\mathbf{k}\] |
| B. | \[3\mathbf{i}-\mathbf{k}\] |
| C. | \[3\mathbf{i}+2\mathbf{j}+\mathbf{k}\] |
| D. | None of these |
| Answer» B. \[3\mathbf{i}-\mathbf{k}\] | |
| 6354. |
If \[\int_{{}}^{{}}{\frac{2x+3}{(x-1)({{x}^{2}}+1)}\ dx={{\log }_{e}}\left\{ {{(x-1)}^{\frac{5}{2}}}{{({{x}^{2}}+1)}^{a}} \right\}}-\frac{1}{2}{{\tan }^{-1}}x+A\] , where A is any arbitrary constant, then the value of ?a? is [MP PET 1998] |
| A. | 44291 |
| B. | -1.66666666666667 |
| C. | -0.833333333333333 |
| D. | -1.25 |
| Answer» E. | |
| 6355. |
Let p, q, r be three mutually perpendicular vectors of the same magnitude. If a vector x satisfies equation \[\mathbf{p}\times \{(\mathbf{x}-\mathbf{q})\times \mathbf{p}\}+\mathbf{q}\times \{(\mathbf{x}-\mathbf{r})\times \mathbf{q}\}+\mathbf{r}\times \{(\mathbf{x}-\mathbf{p})\times \mathbf{r}\}=0,\] then x is given by [IIT 1997 Cancelled] |
| A. | \[\frac{1}{2}\,(\mathbf{p}+\mathbf{q}-2\mathbf{r})\] |
| B. | \[\frac{1}{2}(\mathbf{p}+\mathbf{q}+\mathbf{r})\] |
| C. | \[\frac{1}{3}(\mathbf{p}+\mathbf{q}+\mathbf{r})\] |
| D. | \[\frac{1}{3}(2\mathbf{p}+\mathbf{q}-\mathbf{r})\] |
| Answer» C. \[\frac{1}{3}(\mathbf{p}+\mathbf{q}+\mathbf{r})\] | |
| 6356. |
\[\int_{{}}^{{}}{\frac{dx}{(\sin x+\sin 2x)}=}\] [IIT 1984] |
| A. | \[\frac{1}{6}\log (1-\cos x)+\frac{1}{2}\log (1+\cos x)-\frac{2}{3}\log (1+2\cos x)\] |
| B. | \[6\log (1-\cos x)+2\log (1+\cos x)-\frac{2}{3}\log (1+2\cos x)\] |
| C. | \[6\log (1-\cos x)+\frac{1}{2}\log (1+\cos x)+\frac{2}{3}\log (1+2\cos x)\] |
| D. | None of these |
| Answer» B. \[6\log (1-\cos x)+2\log (1+\cos x)-\frac{2}{3}\log (1+2\cos x)\] | |
| 6357. |
Let \[g(x)=x.\,f(x),\]where \[f(x)=\left\{ \begin{align} & x\sin \frac{1}{x},\,x\ne 0 \\ & \,\,\,\,\,\,\,\,\,\,\,\,0,\,x=0 \\ \end{align} \right.\] at \[x=0\] [IIT Screening 1994; UPSEAT 2004] |
| A. | g is differentiable but g' is not continuous |
| B. | g is differentiable while f is not |
| C. | Both f and g are differentiable |
| D. | g is differentiable and g' is continuous |
| Answer» B. g is differentiable while f is not | |
| 6358. |
Let \[\mathbf{a}=2\mathbf{i}+\mathbf{j}+\mathbf{k},\,\,\mathbf{b}=\mathbf{i}+2\mathbf{j}-\mathbf{k}\]and a unit vector c be coplanar. If c is perpendicular to a, then c = [IIT 1999; Pb. CET 2003; DCE 2005] |
| A. | \[\frac{1}{\sqrt{2}}(-\mathbf{j}+\mathbf{k})\] |
| B. | \[\frac{1}{\sqrt{3}}(-\mathbf{i}-\mathbf{j}-\mathbf{k})\] |
| C. | \[\frac{1}{\sqrt{5}}\,(\mathbf{i}-2\mathbf{j})\] |
| D. | \[\frac{1}{\sqrt{3}}(\mathbf{i}-\mathbf{j}-\mathbf{k})\] |
| Answer» B. \[\frac{1}{\sqrt{3}}(-\mathbf{i}-\mathbf{j}-\mathbf{k})\] | |
| 6359. |
\[\int_{{}}^{{}}{\frac{x}{{{x}^{4}}+{{x}^{2}}+1}dx}\] equal to [MP PET 2004] |
| A. | \[\frac{1}{3}{{\tan }^{-1}}\left( \frac{2{{x}^{2}}+1}{3} \right)\] |
| B. | \[\frac{1}{\sqrt{3}}{{\tan }^{-1}}\left( \frac{2{{x}^{2}}+1}{\sqrt{3}} \right)\] |
| C. | \[\frac{1}{\sqrt{3}}{{\tan }^{-1}}(2{{x}^{2}}+1)\] |
| D. | None of these |
| Answer» C. \[\frac{1}{\sqrt{3}}{{\tan }^{-1}}(2{{x}^{2}}+1)\] | |
| 6360. |
Let \[h(x)=f(x)-{{(f(x))}^{2}}+{{(f(x))}^{3}}\] for every real number x. Then [IIT 1998] |
| A. | h is increasing whenever f is increasing |
| B. | h is increasing whenever f is decreasing |
| C. | h is decreasing whenever f is decreasing |
| D. | Nothing can be said in general |
| Answer» B. h is increasing whenever f is decreasing | |
| 6361. |
Let a, b, c are three non-coplanar vectors such that \[{{\mathbf{r}}_{1}}=\mathbf{a}-\mathbf{b}+\mathbf{c},\,\,{{\mathbf{r}}_{2}}=\mathbf{b}+\mathbf{c}-\mathbf{a},\,\,{{\mathbf{r}}_{3}}=\mathbf{c}+\mathbf{a}+\mathbf{b},\] \[\mathbf{r}=2\mathbf{a}-3\mathbf{b}+4\mathbf{c}.\] If \[\mathbf{r}={{\lambda }_{1}}{{\mathbf{r}}_{1}}+{{\lambda }_{2}}{{\mathbf{r}}_{2}}+{{\lambda }_{3}}{{\mathbf{r}}_{3}},\] then |
| A. | \[{{\lambda }_{1}}=7\] |
| B. | \[{{\lambda }_{1}}+{{\lambda }_{3}}=3\] |
| C. | \[{{\lambda }_{1}}+{{\lambda }_{2}}+{{\lambda }_{3}}=4\] |
| D. | \[{{\lambda }_{3}}+{{\lambda }_{2}}=2\] |
| Answer» C. \[{{\lambda }_{1}}+{{\lambda }_{2}}+{{\lambda }_{3}}=4\] | |
| 6362. |
From the bottom of a pole of height h, the angle of elevation of the top of a tower is \[\alpha \]and the pole subtends angle \[\beta \]at the top of the tower. The height of the tower is [Roorkee 1988] |
| A. | \[\frac{h\tan (\alpha -\beta )}{\tan (\alpha -\beta )-\tan \alpha }\] |
| B. | \[\frac{h\cot (\alpha -\beta )}{\cot (\alpha -\beta )-\cot \alpha }\] |
| C. | \[\frac{\cot (\alpha -\beta )}{\cot (\alpha -\beta )-\cot \alpha }\] |
| D. | None of these |
| Answer» C. \[\frac{\cot (\alpha -\beta )}{\cot (\alpha -\beta )-\cot \alpha }\] | |
| 6363. |
\[\int_{{}}^{{}}{\frac{dx}{2+\cos x}=}\] |
| A. | \[2{{\tan }^{-1}}\left( \frac{1}{\sqrt{3}}\tan \frac{x}{2} \right)+c\] |
| B. | \[\frac{2}{\sqrt{3}}{{\tan }^{-1}}\left( \frac{1}{\sqrt{3}}\tan \frac{x}{2} \right)+c\] |
| C. | \[\frac{1}{\sqrt{3}}{{\tan }^{-1}}\left( \frac{1}{\sqrt{3}}\tan \frac{x}{2} \right)+c\] |
| D. | None of these |
| Answer» C. \[\frac{1}{\sqrt{3}}{{\tan }^{-1}}\left( \frac{1}{\sqrt{3}}\tan \frac{x}{2} \right)+c\] | |
| 6364. |
The function \[f(x)={{\sin }^{4}}x+{{\cos }^{4}}x\] increases, if [IIT 1999; Pb. CET 2001] |
| A. | \[0<x<\frac{\pi }{8}\] |
| B. | \[\frac{\pi }{4}<x<\frac{3\pi }{8}\] |
| C. | \[\frac{3\pi }{8}<x<\frac{5\pi }{8}\] |
| D. | \[\frac{5\pi }{8}<x<\frac{3\pi }{4}\] |
| Answer» C. \[\frac{3\pi }{8}<x<\frac{5\pi }{8}\] | |
| 6365. |
If \[f(x)\,=\,\left\{ \begin{matrix} x{{e}^{-\,\left( \frac{1}{|\,x\,|}\,+\,\frac{1}{x} \right)}}, & x\ne 0 \\ 0\,\,\,\,\,\,\,\,\,\,\,\,\,, & x=0 \\ \end{matrix} \right.\] , then \[f(x)\,\] is [AIEEE 2003] |
| A. | Continuous as well as differentiable for all x |
| B. | Continuous for all x but not differentiable at \[x=0\] |
| C. | Neither differentiable nor continuous at \[x=0\] |
| D. | Discontinuous every where |
| Answer» C. Neither differentiable nor continuous at \[x=0\] | |
| 6366. |
The distance of the point \[B\,(\mathbf{i}+2\mathbf{j}+3\mathbf{k})\] from the line which is passing through \[A\,(4\mathbf{i}+2\mathbf{j}+2\mathbf{k})\] and which is parallel to the vector \[\overrightarrow{C}=2\mathbf{i}+3\mathbf{j}+6\mathbf{k}\] is [Roorkee 1993] |
| A. | 10 |
| B. | \[\sqrt{10}\] |
| C. | 100 |
| D. | None of these |
| Answer» C. 100 | |
| 6367. |
\[{{(1+x)}^{n}}-nx-1\] is divisible by (where \[n\in N\]) |
| A. | \[2x\] |
| B. | \[{{x}^{2}}\] |
| C. | \[2{{x}^{3}}\] |
| D. | All of these |
| Answer» C. \[2{{x}^{3}}\] | |
| 6368. |
ABC is triangular park with AB = AC = 100 m. A clock tower is situated at the mid-point of BC. The angles of elevation of the top of the tower at\[A\]and\[B\]are \[{{\cot }^{-1}}3.2\] and \[\text{cose}{{\text{c}}^{-1}}2.6\]respectively. The height of the tower is[EAMCET 1992] |
| A. | 50 m |
| B. | 25 m |
| C. | 40 m |
| D. | None of these |
| Answer» C. 40 m | |
| 6369. |
If \[\int_{{}}^{{}}{\frac{2x+3}{{{x}^{2}}-5x+6}}\ dx=9\ \ln (x-3)-7\ln (x-2)+A\], then \[A=\] [MP PET 1992] |
| A. | \[5\ln (x-2)+\]Constant |
| B. | \[-4\ln (x-3)+\]constant |
| C. | Constant |
| D. | None of these |
| Answer» D. None of these | |
| 6370. |
\[\int_{{}}^{{}}{{{e}^{x/2}}\sin \left( \frac{x}{2}+\frac{\pi }{4} \right)\ dx=}\] [Roorkee 1982] |
| A. | \[{{e}^{x/2}}\cos \frac{x}{2}+c\] |
| B. | \[\sqrt{2}{{e}^{x/2}}\cos \frac{x}{2}+c\] |
| C. | \[{{e}^{x/2}}\sin \frac{x}{2}+c\] |
| D. | \[\sqrt{2}{{e}^{x/2}}\sin \frac{x}{2}+c\] |
| Answer» E. | |
| 6371. |
The function \[f(x)=\frac{\text{ln}(\pi +x)}{\text{ln}(e+x)}\] is [IIT 1995] |
| A. | Increasing on \[\left[ 0,\,\infty \right)\] |
| B. | Decreasing on \[\left[ 0,\,\infty \right)\] |
| C. | Decreasing on \[\left[ 0,\frac{\pi }{e} \right)\]and increasing on \[\left[ \frac{\pi }{e},\infty \right)\] |
| D. | Increasing on \[\left[ 0,\frac{\pi }{e} \right)\] and decreasing on \[\left[ \frac{\pi }{e},\infty \right)\] |
| Answer» C. Decreasing on \[\left[ 0,\frac{\pi }{e} \right)\]and increasing on \[\left[ \frac{\pi }{e},\infty \right)\] | |
| 6372. |
The function \[f(x)={{[x]}^{2}}-[{{x}^{2}}]\], (where [y] is the greatest integer less than or equal to y),is discontinuous at [IIT 1999] |
| A. | All integers |
| B. | All integers except 0 and 1 |
| C. | All integers except 0 |
| D. | All integers except 1 |
| Answer» E. | |
| 6373. |
The edge of a cube is of length -a- then the shortest distance between the diagonal of a cube and an edge skew to it is |
| A. | \[a\sqrt{2}\] |
| B. | a |
| C. | \[\sqrt{2}/a\] |
| D. | \[a/\sqrt{2}\] |
| Answer» E. | |
| 6374. |
The vector c directed along the internal bisector of the angle between the vectors \[\mathbf{a}=7\mathbf{i}-4\mathbf{j}-4\mathbf{k}\] and \[\mathbf{b}=-2\mathbf{i}-\mathbf{j}+2\mathbf{k}\] with \[|\mathbf{c}|\,=5\sqrt{6},\] is |
| A. | \[\frac{5}{3}\,(\mathbf{i}-7\mathbf{j}+2\mathbf{k})\] |
| B. | \[\frac{5}{3}\,(5\mathbf{i}+5\mathbf{j}+2\mathbf{k})\] |
| C. | \[\frac{5}{3}\,(\mathbf{i}+7\mathbf{j}+2\mathbf{k})\] |
| D. | \[\frac{5}{3}\,(-5\mathbf{i}+5\mathbf{j}+2\mathbf{k})\] |
| Answer» B. \[\frac{5}{3}\,(5\mathbf{i}+5\mathbf{j}+2\mathbf{k})\] | |
| 6375. |
Let P(n) be a statement and let P(n) Þ p(n + 1) for all natural numbers n, then P(n) is true |
| A. | For all n |
| B. | For all n > 1 |
| C. | For all n > m, m being a fixed positive integer |
| D. | Nothing can be said |
| Answer» E. | |
| 6376. |
A tower is situated on horizontal plane. From two points, the line joining three points passes through the base and which are \[a\]and \[b\]distance from the base. The angle of elevation of the top are \[\alpha \]and \[90{}^\circ -\alpha \]and \[\theta \]is that angle which two points joining the line makes at the top, the height of tower will be [UPSEAT 1999] |
| A. | \[\frac{a+b}{a-b}\] |
| B. | \[\frac{a-b}{a+b}\] |
| C. | \[\sqrt{ab}\] |
| D. | \[{{(ab)}^{1/3}}\] |
| Answer» D. \[{{(ab)}^{1/3}}\] | |
| 6377. |
If the function \[f(x)=2{{x}^{3}}-9a{{x}^{2}}\] \[+12{{a}^{2}}x+1,\]where \[a>0\] attains its maximum and minimum at p and q respectively such that \[{{p}^{2}}=q\], then a equals [AIEEE 2003] |
| A. | 3 |
| B. | 1 |
| C. | 2 |
| D. | \[\frac{1}{2}\] |
| Answer» D. \[\frac{1}{2}\] | |
| 6378. |
The value of \[p\] for which the function \[f(x)=\left\{ \begin{align} & \frac{{{({{4}^{x}}-1)}^{3}}}{\sin \frac{x}{p}\log \left[ 1+\frac{{{x}^{2}}}{3} \right]},\,x\ne 0 \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,12{{(\log 4)}^{3}},\,\,x=0 \\ \end{align} \right.\]may be continuous at \[x=0\], is [Orissa JEE 2004] |
| A. | 1 |
| B. | 2 |
| C. | 3 |
| D. | None of these |
| Answer» E. | |
| 6379. |
If \[{{P}_{1}}\] and \[{{P}_{2}}\] are the lengths of the perpendiculars from the points (2,3,4) and (1,1,4) respectively from the plane \[3x-6y+2z+11=0\], then \[{{P}_{1}}\] and \[{{P}_{2}}\] are the roots of the equation |
| A. | \[{{P}^{2}}-23P+7=0\] |
| B. | \[7{{P}^{2}}-23P+16=0\] |
| C. | \[{{P}^{2}}-17P+16=0\] |
| D. | \[{{P}^{2}}-16P+7=0\] |
| Answer» C. \[{{P}^{2}}-17P+16=0\] | |
| 6380. |
The distance between two points P and Q is d and the length of their projections of PQ on the co-ordinate planes are \[{{d}_{1}},{{d}_{2}},{{d}_{3}}\]. Then \[d_{1}^{2}+d_{2}^{2}+d_{3}^{2}=k{{d}^{2}}\] where . k- is |
| A. | 1 |
| B. | 5 |
| C. | 3 |
| D. | 2 |
| Answer» E. | |
| 6381. |
If \[\mathbf{x}\] is parallel to \[\mathbf{y}\] and \[\mathbf{z}\] where \[\mathbf{x}=2\mathbf{i}+\mathbf{j}+\alpha \mathbf{k}\], \[\mathbf{y}=\alpha \mathbf{i}+\mathbf{k}\] and \[\mathbf{z}=5\mathbf{i}-\mathbf{j}\], then \[\alpha \] is equal to [J & K 2005] |
| A. | \[\pm \sqrt{5}\] |
| B. | \[\pm \sqrt{6}\] |
| C. | \[\pm \sqrt{7}\] |
| D. | None of these |
| Answer» D. None of these | |
| 6382. |
The value of the natural numbers n such that the inequality \[{{2}^{n}}>2n+1\] is valid is [MNR 1994] |
| A. | For \[n\ge 3\] |
| B. | For n < 3 |
| C. | For mn |
| D. | For any n |
| Answer» B. For n < 3 | |
| 6383. |
In a \[\Delta ABC,\]let \[\angle C=\frac{\pi }{2}.\]If \[r\]and \[R\]are in radius and the circum-radius respectively of the triangle, then \[2(r+R)\] is equal to [IIT Screening 2000; AIEEE 2005] |
| A. | \[a+b\] |
| B. | \[b+c\] |
| C. | \[c+a\] |
| D. | \[a+b+c\] |
| Answer» B. \[b+c\] | |
| 6384. |
If \[{{I}_{n}}=\int{{{(\log x)}^{n}}\,\,dx},\] then \[{{I}_{n}}+n{{I}_{n-1}}=\] [Karnataka CET 2003] |
| A. | \[x{{(\log x)}^{n}}\] |
| B. | \[{{(x\log x)}^{n}}\] |
| C. | \[{{(\log x)}^{n-1}}\] |
| D. | \[n{{(\log x)}^{n}}\] |
| Answer» B. \[{{(x\log x)}^{n}}\] | |
| 6385. |
The radius of the circular section of the sphere \[|\mathbf{r}|\,=5\]by the plane \[\mathbf{r}\,.\,(\mathbf{i}+\mathbf{j}+\mathbf{k})=3\sqrt{3}\] is [DCE 1999] |
| A. | 1 |
| B. | 2 |
| C. | 3 |
| D. | 4 |
| Answer» E. | |
| 6386. |
The least remainder when \[{{17}^{30}}\] is divided by 5 is [Karnataka CET 2003] |
| A. | 1 |
| B. | 2 |
| C. | 3 |
| D. | 4 |
| Answer» E. | |
| 6387. |
In \[\Delta ABC,\]if \[8{{R}^{2}}={{a}^{2}}+{{b}^{2}}+{{c}^{2}},\]then the triangle is |
| A. | Right angled |
| B. | Equilateral |
| C. | Acute angled |
| D. | Obtuse angled |
| Answer» B. Equilateral | |
| 6388. |
If \[u=\int_{{}}^{{}}{{{e}^{ax}}\cos bx\ dx}\] and \[v=\int_{{}}^{{}}{{{e}^{ax}}\sin bx\ dx}\], then \[({{a}^{2}}+{{b}^{2}})({{u}^{2}}+{{v}^{2}})=\] |
| A. | \[2{{e}^{ax}}\] |
| B. | \[({{a}^{2}}+{{b}^{2}}){{e}^{2ax}}\] |
| C. | \[{{e}^{2ax}}\] |
| D. | \[({{a}^{2}}-{{b}^{2}}){{e}^{2ax}}\] |
| Answer» D. \[({{a}^{2}}-{{b}^{2}}){{e}^{2ax}}\] | |
| 6389. |
The function \[f(x)=\int\limits_{-1}^{x}{t({{e}^{t}}-1)(t-1){{(t-2)}^{3}}{{(t-3)}^{5}}}dt\] has a local minimum at x = [IIT 1999] |
| A. | 0 |
| B. | 1 |
| C. | 2 |
| D. | 3 |
| Answer» C. 2 | |
| 6390. |
The function\[f(x)=[x]\cos \left[ \frac{2x-1}{2} \right]\pi ,\,\]where\[[.]\] denotes the greatest integer function, is discontinuous at [IIT 1995] |
| A. | All x |
| B. | No x |
| C. | All integer points |
| D. | x which is not an integer |
| Answer» D. x which is not an integer | |
| 6391. |
The plane \[lx+my=0\] is rotated an angle \[\alpha \] about its line of intersection with the plane \[z=0\], then the equation to the plane in its new position is |
| A. | \[lx+my\pm z\sqrt{({{l}^{2}}+{{m}^{2}})}\tan \alpha =0\] |
| B. | \[lx-my\pm z\sqrt{({{l}^{2}}+{{m}^{2}})}\tan \alpha =0\] |
| C. | \[lx+my\pm z\sqrt{({{l}^{2}}+{{m}^{2}})}\cos \alpha =0\] |
| D. | \[lx-my\pm z\sqrt{({{l}^{2}}+{{m}^{2}})}\cos \alpha =0\] |
| Answer» B. \[lx-my\pm z\sqrt{({{l}^{2}}+{{m}^{2}})}\tan \alpha =0\] | |
| 6392. |
Unit vectors a, b and c are coplanar. A unit vector d is perpendicular to them. If \[(\mathbf{a}\times \mathbf{b})\times (\mathbf{c}\times \mathbf{d})=\frac{1}{6}\mathbf{i}-\frac{1}{3}\mathbf{j}+\frac{1}{3}\mathbf{k}\] and the angle between a and b is \[{{30}^{o}}\], then c is [Roorkee Qualifying 1998] |
| A. | \[\frac{(\mathbf{i}-2\mathbf{j}+2\mathbf{k})}{3}\] |
| B. | \[\frac{(2\mathbf{i}+\mathbf{j}-\mathbf{k})}{3}\] |
| C. | \[\frac{(-\mathbf{i}+2\mathbf{j}-2\mathbf{k})}{3}\] |
| D. | \[\frac{(-\mathbf{i}+2\mathbf{j}+\mathbf{k})}{3}\] |
| Answer» B. \[\frac{(2\mathbf{i}+\mathbf{j}-\mathbf{k})}{3}\] | |
| 6393. |
For every natural number n, \[{{3}^{2n+2}}-8n-9\] is divisible by [IIT 1977] |
| A. | 16 |
| B. | 128 |
| C. | 256 |
| D. | None of these |
| Answer» B. 128 | |
| 6394. |
\[\int_{{}}^{{}}{\frac{{{x}^{2}}}{{{(x\sin x+\cos x)}^{2}}}\ dx=}\] [MNR 1989; RPET 2000] |
| A. | \[\frac{\sin x+\cos x}{x\sin x+\cos x}\] |
| B. | \[\frac{x\sin x-\cos x}{x\sin x+\cos x}\] |
| C. | \[\frac{\sin x-x\cos x}{x\sin x+\cos x}\] |
| D. | None of these |
| Answer» D. None of these | |
| 6395. |
On the interval [0, 1], the function \[{{x}^{25}}{{(1-x)}^{75}}\] takes its maximum value at the point [IIT 1995] |
| A. | 0 |
| B. | 44228 |
| C. | 44256 |
| D. | ¼ |
| Answer» E. | |
| 6396. |
If \[f(x)=\left\{ \begin{align} & {{x}^{2}}-3,\ 2 |
| A. | \[{{x}^{2}}-7x+3=0\] |
| B. | \[{{x}^{2}}-20x+66=0\] |
| C. | \[{{x}^{2}}-17x+66=0\] |
| D. | \[{{x}^{2}}-18x+60=0\] |
| Answer» D. \[{{x}^{2}}-18x+60=0\] | |
| 6397. |
The equation of motion of a rocket are: \[x=2t,\,y=-4t,\] \[\,z=4t\] where the time 't' is given in seconds, and the co-ordinates of a moving point in kilometers. What is the path of the rocket. At what distance will be the rocket be from the starting point 0(0, 0, 0) in 10 seconds |
| A. | Straight line, 60 km |
| B. | Straight line, 30 km |
| C. | Parabola, 60 km |
| D. | Ellipse, 60 km |
| Answer» B. Straight line, 30 km | |
| 6398. |
\[[(\mathbf{a}\times \mathbf{b})\times (\mathbf{b}\times \mathbf{c})\,(\mathbf{b}\times \mathbf{c})\times (\mathbf{c}\times \mathbf{a})\,(\mathbf{c}\times \mathbf{a})\times (\mathbf{a}\times \mathbf{b})]=\,\] |
| A. | \[{{[\mathbf{a}\,\,\mathbf{b}\,\,\mathbf{c}]}^{2}}\] |
| B. | \[{{[\mathbf{a}\,\,\mathbf{b}\,\,\mathbf{c}]}^{3}}\] |
| C. | \[{{[\mathbf{a}\,\,\mathbf{b}\,\,\mathbf{c}]}^{4}}\] |
| D. | None of these |
| Answer» D. None of these | |
| 6399. |
The coefficient of \[{{x}^{5}}\] in the expansion of \[{{(1+{{x}^{2}})}^{5}}{{(1+x)}^{4}}\] is [EAMCET 1996; UPSEAT 2001; Pb. CET 2002] |
| A. | 30 |
| B. | 60 |
| C. | 40 |
| D. | None of these |
| Answer» C. 40 | |
| 6400. |
If \[\tan \theta =-\frac{1}{\sqrt{3}}\]and \[\sin \theta =\frac{1}{2}\], \[\cos \theta =-\frac{\sqrt{3}}{2}\], then the principal value of \[\theta \] will be [MP PET 1983, 84] |
| A. | \[\frac{\pi }{6}\] |
| B. | \[\frac{5\pi }{6}\] |
| C. | \[\frac{7\pi }{6}\] |
| D. | \[-\frac{\pi }{6}\] |
| Answer» C. \[\frac{7\pi }{6}\] | |