MCQOPTIONS
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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.
| 6501. |
\[\underset{x\to 1}{\mathop{\lim }}\,(1-x)\tan \left( \frac{\pi x}{2} \right)=\] [IIT 1978, 84; RPET 1997, 2001; UPSEAT 2003; Pb. CET 2003] |
| A. | \[\frac{\pi }{2}\] |
| B. | \[\pi \] |
| C. | \[\frac{2}{\pi }\] |
| D. | 0 |
| Answer» D. 0 | |
| 6502. |
P is a fixed point \[(a,\,a,\,a)\] on a line through the origin equally inclined to the axes, then any plane through P perpendicular to OP, makes intercepts on the axes, the sum of whose reciprocals is equal to |
| A. | a |
| B. | \[\frac{3}{2a}\] |
| C. | \[\frac{3a}{2}\] |
| D. | None of these |
| Answer» E. | |
| 6503. |
If \[\mathbf{a}\times \mathbf{r}=\mathbf{b}+\lambda \mathbf{a}\] and \[\mathbf{a}\,\,.\,\,\mathbf{r}=3,\] where \[\mathbf{a}=2\mathbf{i}+\mathbf{j}-\mathbf{k}\] and \[\mathbf{b}=-\mathbf{i}-2\mathbf{j}+\mathbf{k},\] then r and l are equal to |
| A. | \[\mathbf{r}=\frac{7}{6}\mathbf{i}+\frac{2}{3}\mathbf{j},\,\,\lambda =\frac{6}{5}\] |
| B. | \[\mathbf{r}=\frac{7}{6}\mathbf{i}+\frac{2}{3}\mathbf{j},\,\,\lambda =\frac{5}{6}\] |
| C. | \[\mathbf{r}=\frac{6}{7}\mathbf{i}+\frac{2}{3}\mathbf{j},\,\,\lambda =\frac{6}{5}\] |
| D. | None of these |
| Answer» C. \[\mathbf{r}=\frac{6}{7}\mathbf{i}+\frac{2}{3}\mathbf{j},\,\,\lambda =\frac{6}{5}\] | |
| 6504. |
If the straight line through the point \[P(3,\,4)\]makes an angle \[\frac{\pi }{6}\]with the x-axis and meets the line \[12x+5y+10=0\] at Q, then the length \[PQ\]is |
| A. | \[\frac{132}{12\sqrt{3}+5}\] |
| B. | \[\frac{132}{12\sqrt{3}-5}\] |
| C. | \[\frac{132}{5\sqrt{3}+12}\] |
| D. | \[\frac{132}{5\sqrt{3}-12}\] |
| Answer» B. \[\frac{132}{12\sqrt{3}-5}\] | |
| 6505. |
\[\int_{{}}^{{}}{{{(\log x)}^{2}}\ dx=}\] [IIT 1971, 77] |
| A. | \[x{{(\log x)}^{2}}-2x\log x-2x+c\] |
| B. | \[x{{(\log x)}^{2}}-2x\log x-x+c\] |
| C. | \[x{{(\log x)}^{2}}-2x\log x+2x+c\] |
| D. | \[x{{(\log x)}^{2}}-2x\log x+x+c\] |
| Answer» D. \[x{{(\log x)}^{2}}-2x\log x+x+c\] | |
| 6506. |
If \[x=\sin t\] and \[y=\sin pt\], then the value of \[\left( 1-{{x}^{2}} \right)\frac{{{d}^{2}}y}{d{{x}^{2}}}-x\frac{dy}{dx}+{{p}^{2}}y\]is equal to [Pb. CET 2002] |
| A. | 0 |
| B. | 1 |
| C. | -1 |
| D. | \[\sqrt{2}\] |
| Answer» B. 1 | |
| 6507. |
If \[f:R\to R\]and \[g:R\to R\] are defined by \[f(x)=2x+3\]and \[g(x)={{x}^{2}}+7\], then the values of x such that \[g(f(x))=8\] are [EAMCET 2000, 03] |
| A. | 1, 2 |
| B. | -1, 2 |
| C. | -1, -2 |
| D. | 1, -2 |
| Answer» D. 1, -2 | |
| 6508. |
A variable plane at a constant distance p from origin meets the co-ordinates axes in \[A,B,C\]. Through these points planes are drawn parallel to co-ordinate planes. Then locus of the point of intersection is |
| A. | \[\frac{1}{{{x}^{2}}}+\frac{1}{{{y}^{2}}}+\frac{1}{{{z}^{2}}}=\frac{1}{{{p}^{2}}}\] |
| B. | \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}={{p}^{2}}\] |
| C. | \[x+y+z=p\] |
| D. | \[\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=p\] |
| Answer» B. \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}={{p}^{2}}\] | |
| 6509. |
If \[\mathbf{a}=2\mathbf{i}+\mathbf{k},\,\,\mathbf{b}=\mathbf{i}+\mathbf{j}+\mathbf{k}\] and \[\mathbf{c}=4\mathbf{i}-3\mathbf{j}+7\mathbf{k}.\] If \[\mathbf{d}\times \mathbf{b}=\mathbf{c}\times \mathbf{b}\] and \[\mathbf{d}\,.\,\mathbf{a}=0,\] then d will be [IIT 1990] |
| A. | \[\mathbf{i}+8\mathbf{j}+2\mathbf{k}\] |
| B. | \[\mathbf{i}-8\mathbf{j}+2\mathbf{k}\] |
| C. | \[-\mathbf{i}+8\mathbf{j}-\mathbf{k}\] |
| D. | \[-\mathbf{i}-8\mathbf{j}+2\mathbf{k}\] |
| Answer» E. | |
| 6510. |
In \[\Delta ABC,\ \frac{1}{a}{{\cos }^{2}}\frac{A}{2}+\frac{1}{b}{{\cos }^{2}}\frac{B}{2}+\frac{1}{c}{{\cos }^{2}}\frac{C}{2}=\] |
| A. | \[s\] |
| B. | \[\frac{s}{abc}\] |
| C. | \[\frac{{{s}^{2}}}{abc}\] |
| D. | \[\frac{{{s}^{3}}}{abc}\] |
| Answer» D. \[\frac{{{s}^{3}}}{abc}\] | |
| 6511. |
Given vertices \[A(1,\,1),B(4,\,-2)\]and \[C(5,\,5)\]of a triangle, then the equation of the perpendicular dropped from C to the interior bisector of the angle A is [Roorkee 1994] |
| A. | \[y-5=0\] |
| B. | \[x-5=0\] |
| C. | \[y+5=0\] |
| D. | \[x+5=0\] |
| Answer» C. \[y+5=0\] | |
| 6512. |
The lines joining the origin to the points of intersection of the line \[y=mx+c\]and the circle \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\]will be mutually perpendicular, if [Roorkee 1977] |
| A. | \[{{a}^{2}}({{m}^{2}}+1)={{c}^{2}}\] |
| B. | \[{{a}^{2}}({{m}^{2}}-1)={{c}^{2}}\] |
| C. | \[{{a}^{2}}({{m}^{2}}+1)={{c}^{2}}\] |
| D. | \[{{a}^{2}}({{m}^{2}}-1)=2{{c}^{2}}\] |
| Answer» D. \[{{a}^{2}}({{m}^{2}}-1)=2{{c}^{2}}\] | |
| 6513. |
If one of the lines of the pair \[a{{x}^{2}}+2hxy+b{{y}^{2}}=0\] bisects the angle between positive directions of the axes, then a, b, h satisfy the relation [Roorkee 1992] |
| A. | \[a+b=2|h|\] |
| B. | \[a+b=-2h\] |
| C. | \[a-b=2|h|\] |
| D. | \[{{(a-b)}^{2}}=4{{h}^{2}}\] |
| Answer» C. \[a-b=2|h|\] | |
| 6514. |
If integrating factor of \[x(1-{{x}^{2}})dy+(2{{x}^{2}}y-y-a{{x}^{3}})dx=0\] is \[{{e}^{\int_{{}}^{{}}{Pdx}}},\] then P is equal to [MP PET 1999] |
| A. | \[\frac{2{{x}^{2}}-a{{x}^{3}}}{x(1-{{x}^{2}})}\] |
| B. | \[(2{{x}^{2}}-1)\] |
| C. | \[\frac{2{{x}^{2}}-1}{a{{x}^{3}}}\] |
| D. | \[\frac{(2{{x}^{2}}-1)}{x(1-{{x}^{2}})}\] |
| Answer» E. | |
| 6515. |
\[\int_{{}}^{{}}{\frac{{{x}^{2}}+1}{{{x}^{4}}-{{x}^{2}}+1}\ dx=}\] [MP PET 1991] |
| A. | \[{{\tan }^{-1}}\left( \frac{1+{{x}^{2}}}{x} \right)+c\] |
| B. | \[{{\cot }^{-1}}\left( \frac{1+{{x}^{2}}}{x} \right)+c\] |
| C. | \[{{\tan }^{-1}}\left( \frac{{{x}^{2}}-1}{x} \right)+c\] |
| D. | \[{{\cot }^{-1}}\left( \frac{{{x}^{2}}-1}{x} \right)+c\] |
| Answer» D. \[{{\cot }^{-1}}\left( \frac{{{x}^{2}}-1}{x} \right)+c\] | |
| 6516. |
If \[f(x)=3x+10\], \[g(x)={{x}^{2}}-1\], then \[{{(fog)}^{-1}}\] is equal to [UPSEAT 2001] |
| A. | \[{{\left( \frac{x-7}{3} \right)}^{1/2}}\] |
| B. | \[{{\left( \frac{x+7}{3} \right)}^{1/2}}\] |
| C. | \[{{\left( \frac{x-3}{7} \right)}^{1/2}}\] |
| D. | \[{{\left( \frac{x+3}{7} \right)}^{1/2}}\] |
| Answer» B. \[{{\left( \frac{x+7}{3} \right)}^{1/2}}\] | |
| 6517. |
Two middle terms in the expansion of \[{{\left( x-\frac{1}{x} \right)}^{11}}\] are |
| A. | 231x and \[\frac{231}{x}\] |
| B. | \[462x\] and \[\frac{462}{x}\] |
| C. | \[-462x\] and \[\frac{462}{x}\] |
| D. | None of these |
| Answer» D. None of these | |
| 6518. |
AB is a vertical tower. The point A is on the ground and C is the middle point of AB. The part CB subtend an angle \[\alpha \]at a point P on the ground. If \[AP=n\,AB,\]then the correct relation is [MNR 1989; IIT 1980] |
| A. | \[n=({{n}^{2}}+1)\tan \alpha \] |
| B. | \[n=(2{{n}^{2}}-1)\tan \alpha \] |
| C. | \[{{n}^{2}}=(2{{n}^{2}}+1)\tan \alpha \] |
| D. | \[n=(2{{n}^{2}}+1)\tan \alpha \] |
| Answer» E. | |
| 6519. |
\[2{{\tan }^{-1}}(\cos x)={{\tan }^{-1}}(\text{cose}{{\text{c}}^{2}}x),\] then x = [UPSEAT 2002] |
| A. | \[\frac{\pi }{2}\] |
| B. | \[\pi \] |
| C. | \[\frac{\pi }{6}\] |
| D. | \[\frac{\pi }{3}\] |
| Answer» E. | |
| 6520. |
The sides \[AB,BC,CD\] and \[DA\]of a quadrilateral are \[x+2y=3,\,x=1,\] \[x-3y=4,\,\] \[\,5x+y+12=0\] respectively. The angle between diagonals \[AC\]and \[BD\]is [Roorkee 1993] |
| A. | \[{{45}^{o}}\] |
| B. | \[{{60}^{o}}\] |
| C. | \[{{90}^{o}}\] |
| D. | \[{{30}^{o}}\] |
| Answer» D. \[{{30}^{o}}\] | |
| 6521. |
The solution of the given differential equation \[\frac{dy}{dx}+2xy=y\] is [Roorkee 1995] |
| A. | \[y=c{{e}^{x-{{x}^{2}}}}\] |
| B. | \[y=c{{e}^{{{x}^{2}}-x}}\] |
| C. | \[y=c{{e}^{x}}\] |
| D. | \[y=c{{e}^{-{{x}^{2}}}}\] |
| Answer» B. \[y=c{{e}^{{{x}^{2}}-x}}\] | |
| 6522. |
\[\int_{{}}^{{}}{\frac{dx}{4{{\sin }^{2}}x+5{{\cos }^{2}}x}=}\] [AISSE 1986] |
| A. | \[\frac{1}{\sqrt{5}}{{\tan }^{-1}}\left( \frac{2\tan x}{\sqrt{5}} \right)+c\] |
| B. | \[\frac{1}{\sqrt{5}}{{\tan }^{-1}}\left( \frac{\tan x}{\sqrt{5}} \right)+c\] |
| C. | \[\frac{1}{2\sqrt{5}}{{\tan }^{-1}}\left( \frac{2\tan x}{\sqrt{5}} \right)+c\] |
| D. | None of these |
| Answer» D. None of these | |
| 6523. |
Let \[f(x)\]and \[g(x)\]be two functions having finite non-zero 3rd order derivatives \[{f}'''(x)\]and \[{g}'''(x)\] for all, \[x\in R\]. If \[f(x)g(x)=1\]for all \[x\in R\], then \[\frac{{{f}'''}}{{{f}'}}-\frac{{{g}'''}}{{{g}'}}\]is equal to |
| A. | \[3\text{ }\left( \frac{{{f}''}}{g}-\frac{{{g}''}}{f} \right)\] |
| B. | \[3\text{ }\left( \frac{{{f}''}}{f}-\frac{{{g}''}}{g} \right)\] |
| C. | \[3\text{ }\left( \frac{g''}{g}-\frac{f''}{g} \right)\] |
| D. | \[3\text{ }\left( \frac{{{f}''}}{f}-\frac{{{g}''}}{f} \right)\] |
| Answer» C. \[3\text{ }\left( \frac{g''}{g}-\frac{f''}{g} \right)\] | |
| 6524. |
If \[g(f(x))=|\sin x|\] and \[f(g(x))={{(\sin \sqrt{x})}^{2}}\], then [IIT 1998] |
| A. | \[f(x)={{\sin }^{2}}x,\ g(x)=\sqrt{x}\] |
| B. | \[f(x)=\sin x,\ g(x)=|x|\] |
| C. | \[f(x)={{x}^{2}},\ g(x)=\sin \sqrt{x}\] |
| D. | f and g cannot be determined |
| Answer» B. \[f(x)=\sin x,\ g(x)=|x|\] | |
| 6525. |
Let R be a relation on the set N be defined by {(x, y)| x, y \[\overset{\hat{\ }}{\mathop{i}}\,\]N, 2x + y = 41}. Then R is |
| A. | Reflexive |
| B. | Symmetric |
| C. | Transitive |
| D. | None of these |
| Answer» E. | |
| 6526. |
Middle term in the expansion of \[{{(1+3x+3{{x}^{2}}+{{x}^{3}})}^{6}}\]is [MP PET 1997] |
| A. | \[{{4}^{th}}\] |
| B. | \[{{3}^{rd}}\] |
| C. | \[{{10}^{th}}\] |
| D. | None of these |
| Answer» D. None of these | |
| 6527. |
The equation \[{{x}^{2}}-3xy+\lambda {{y}^{2}}+3x-5y+2=0\] when \[\lambda \]is a real number, represents a pair of straight lines. If \[\theta \] is the angle between the lines, then \[\text{cose}{{\text{c}}^{2}}\theta \]= [EAMCET 1992] |
| A. | 3 |
| B. | 9 |
| C. | 10 |
| D. | 100 |
| Answer» D. 100 | |
| 6528. |
The solution of the equation \[\frac{dy}{dx}=\frac{1}{x+y+1}\] is |
| A. | \[x=c{{e}^{y}}-y-2\] |
| B. | \[y=x+c{{e}^{y}}-2\] |
| C. | \[x+c{{e}^{y}}-y-2=0\] |
| D. | None of these |
| Answer» B. \[y=x+c{{e}^{y}}-2\] | |
| 6529. |
If \[\int_{{}}^{{}}{f(x)\sin x\cos x\ dx=\frac{1}{2({{b}^{2}}-{{a}^{2}})}\log (f(x))}+c\], then \[f(x)=\] |
| A. | \[\frac{1}{{{a}^{2}}{{\sin }^{2}}x+{{b}^{2}}{{\cos }^{2}}x}\] |
| B. | \[\frac{1}{{{a}^{2}}{{\sin }^{2}}x-{{b}^{2}}{{\cos }^{2}}x}\] |
| C. | \[\frac{1}{{{a}^{2}}{{\cos }^{2}}x+{{b}^{2}}{{\sin }^{2}}x}\] |
| D. | \[\frac{1}{{{a}^{2}}{{\cos }^{2}}x-{{b}^{2}}{{\sin }^{2}}x}\] |
| Answer» B. \[\frac{1}{{{a}^{2}}{{\sin }^{2}}x-{{b}^{2}}{{\cos }^{2}}x}\] | |
| 6530. |
If \[{{y}^{2}}=p(x)\]is a polynomial of degree three, then \[2\frac{d}{dx}\left\{ {{y}^{3}}.\frac{{{d}^{2}}y}{d{{x}^{2}}} \right\}\]= [IIT 1988; RPET 2000] |
| A. | \[{p}'''(x)+p'(x)\] |
| B. | \[{p}''(x).{p}'''(x)\] |
| C. | \[p(x).{p}'''(x)\] |
| D. | Constant |
| Answer» D. Constant | |
| 6531. |
If \[f(x)={{\sin }^{2}}x+{{\sin }^{2}}\left( x+\frac{\pi }{3} \right)+\cos x\cos \left( x+\frac{\pi }{3} \right)\] and \[g\left( \frac{5}{4} \right)=1\], then \[(gof)(x)=\] [IIT 1996] |
| A. | -2 |
| B. | -1 |
| C. | 2 |
| D. | 1 |
| Answer» E. | |
| 6532. |
Let a relation R be defined by R = {(4, 5); (1, 4); (4, 6); (7, 6); (3, 7)} then \[{{R}^{-1}}oR\] is |
| A. | {(1, 1), (4, 4), (4, 7), (7, 4), (7, 7), (3, 3)} |
| B. | {(1, 1), (4, 4), (7, 7), (3, 3)} |
| C. | {(1, 5), (1, 6), (3, 6)} |
| D. | None of these |
| Answer» B. {(1, 1), (4, 4), (7, 7), (3, 3)} | |
| 6533. |
The value of \[\left( \begin{matrix} 30 \\ 0 \\\end{matrix} \right)\left( \begin{matrix} 30 \\ 10 \\\end{matrix} \right)-\left( \begin{matrix} 30 \\ 1 \\\end{matrix} \right)\left( \begin{matrix} 30 \\ 11 \\\end{matrix} \right)+\left( \begin{matrix} 30 \\ 2 \\\end{matrix} \right)\left( \begin{matrix} 30 \\ 12 \\\end{matrix} \right)+......+\left( \begin{matrix} 30 \\ 20 \\\end{matrix} \right)\left( \begin{matrix} 30 \\ 30 \\\end{matrix} \right)\][IIT Screening 2005] |
| A. | \[^{60}{{C}_{20}}\] |
| B. | \[^{30}{{C}_{10}}\] |
| C. | \[^{60}{{C}_{30}}\] |
| D. | \[^{40}{{C}_{30}}\] |
| Answer» C. \[^{60}{{C}_{30}}\] | |
| 6534. |
Two fixed points are \[A(a,0)\]and\[B(-a,0)\]. If\[\angle A-\angle B=\theta \], then the locus of point C of triangle ABC will be [Roorkee 1982] |
| A. | \[{{x}^{2}}+{{y}^{2}}+2xy\tan \theta ={{a}^{2}}\] |
| B. | \[{{x}^{2}}-{{y}^{2}}+2xy\tan \theta ={{a}^{2}}\] |
| C. | \[{{x}^{2}}+{{y}^{2}}+2xy\cot \theta ={{a}^{2}}\] |
| D. | \[{{x}^{2}}-{{y}^{2}}+2xy\cot \theta ={{a}^{2}}\] |
| Answer» E. | |
| 6535. |
\[2{{\tan }^{-1}}\left[ \sqrt{\frac{a-b}{a+b}}\tan \frac{\theta }{2} \right]=\] [Dhanbad Engg. 1976] |
| A. | \[{{\cos }^{-1}}\left( \frac{a\cos \theta +b}{a+b\cos \theta } \right)\] |
| B. | \[{{\cos }^{-1}}\left( \frac{a+b\cos \theta }{a\cos \theta +b} \right)\] |
| C. | \[{{\cos }^{-1}}\left( \frac{a\cos \theta }{a+b\cos \theta } \right)\] |
| D. | \[{{\cos }^{-1}}\left( \frac{a\cos +b\theta }{a+b\cos \theta } \right)\] |
| Answer» B. \[{{\cos }^{-1}}\left( \frac{a+b\cos \theta }{a\cos \theta +b} \right)\] | |
| 6536. |
In what direction a line be drawn through the point (1, 2) so that its points of intersection with the line \[x+y=4\] is at a distance \[\frac{\sqrt{6}}{3}\] from the given point [IIT 1966; MNR 1987] |
| A. | \[{{30}^{o}}\] |
| B. | \[{{45}^{o}}\] |
| C. | \[{{60}^{o}}\] |
| D. | \[{{75}^{o}}\] |
| Answer» E. | |
| 6537. |
The figure formed by the lines \[{{x}^{2}}+4xy+{{y}^{2}}=0\] and \[x-y=4,\] is [Roorkee 1980] |
| A. | A right angled triangle |
| B. | An isosceles triangle |
| C. | An equilateral triangle |
| D. | None of these |
| Answer» D. None of these | |
| 6538. |
\[\int_{{}}^{{}}{x\sqrt{\frac{1-{{x}^{2}}}{1+{{x}^{2}}}}}\ dx=\] |
| A. | \[\frac{1}{2}[{{\sin }^{-1}}{{x}^{2}}+\sqrt{1-{{x}^{4}}}]+c\] |
| B. | \[\frac{1}{2}[{{\sin }^{-1}}{{x}^{2}}+\sqrt{1-{{x}^{2}}}]+c\] |
| C. | \[{{\sin }^{-1}}{{x}^{2}}+\sqrt{1-{{x}^{4}}}+c\] |
| D. | \[{{\sin }^{-1}}{{x}^{2}}+\sqrt{1-{{x}^{2}}}+c\] |
| Answer» B. \[\frac{1}{2}[{{\sin }^{-1}}{{x}^{2}}+\sqrt{1-{{x}^{2}}}]+c\] | |
| 6539. |
If f is an even function defined on the interval (-5, 5), then four real values of x satisfying the equation \[f(x)=f\left( \frac{x+1}{x+2} \right)\] are [IIT 1996] |
| A. | \[\frac{-3-\sqrt{5}}{2},\ \frac{-3+\sqrt{5}}{2},\ \frac{3-\sqrt{5}}{2},\ \frac{3+\sqrt{5}}{2}\] |
| B. | \[\frac{-5+\sqrt{3}}{2},\ \frac{-3+\sqrt{5}}{2},\ \frac{3+\sqrt{5}}{2},\ \frac{3-\sqrt{5}}{2}\] |
| C. | \[\frac{3-\sqrt{5}}{2},\ \frac{3+\sqrt{5}}{2},\ \frac{-3-\sqrt{5}}{2},\ \frac{5+\sqrt{3}}{2}\] |
| D. | \[-3-\sqrt{5},\ -3+\sqrt{5},\ 3-\sqrt{5},\ 3+\sqrt{5}\] |
| Answer» B. \[\frac{-5+\sqrt{3}}{2},\ \frac{-3+\sqrt{5}}{2},\ \frac{3+\sqrt{5}}{2},\ \frac{3-\sqrt{5}}{2}\] | |
| 6540. |
A line with direction cosines proportional to 2,1, 2 meets each of the lines \[x=y+a=z\]and \[x+a=2y=2z\]. The co-ordinates of each of the points of intersection are given by [AIEEE 2004] |
| A. | \[(2a,\,\,a,\,3a),(2a,\,a,\,a)\] |
| B. | \[(3a,\,2a,\,3a),\ (a,\,a,\,a)\] |
| C. | \[(3a,\,2a,\,3a),(a,\,a,\,2a)\] |
| D. | \[(3a,\,3a,\,3a),(a,\,a,\,a)\] |
| Answer» C. \[(3a,\,2a,\,3a),(a,\,a,\,2a)\] | |
| 6541. |
Let \[\mathbf{b}=4\mathbf{i}+3\mathbf{j}\] and c be two vectors perpendicular to each other in the xy-plane. All vectors in the same plane having projections 1 and 2 along b and c respectively, are given by [IIT 1987] |
| A. | \[2\mathbf{i}-\mathbf{j},\,\,\frac{2}{5}\mathbf{i}+\frac{11}{5}\mathbf{j}\] |
| B. | \[2\mathbf{i}+\mathbf{j},\,\,-\frac{2}{5}\mathbf{i}+\frac{11}{5}\mathbf{j}\] |
| C. | \[2\mathbf{i}+\mathbf{j},\,-\frac{2}{5}\mathbf{i}-\frac{11}{5}\mathbf{j}\] |
| D. | \[2\mathbf{i}-\mathbf{j},\,\,-\frac{2}{5}\mathbf{i}+\frac{11}{5}\mathbf{j}\] |
| Answer» E. | |
| 6542. |
Let R and S be two non-void relations on a set A. Which of the following statements is false |
| A. | R and S are transitive \[\Rightarrow \text{ }R\text{ }\cup \text{ }S\] is transitive |
| B. | R and S are transitive \[\Rightarrow \text{ }R\text{ }\cap \text{ }S\] is transitive |
| C. | R and S are symmetric \[\Rightarrow \text{ }R\text{ }\cup \text{ }S\] is symmetric |
| D. | R and S are reflexive \[\Rightarrow \text{ }R\text{ }\cap \text{ }S\] is reflexive |
| Answer» B. R and S are transitive \[\Rightarrow \text{ }R\text{ }\cap \text{ }S\] is transitive | |
| 6543. |
The number of solution of the given equation \[a\sin x+b\cos x=c\] , where \[|c|\,>\,\sqrt{{{a}^{2}}+{{b}^{2}}},\]is [DCE 1998] |
| A. | 1 |
| B. | 2 |
| C. | Infinite |
| D. | None of these |
| Answer» E. | |
| 6544. |
If \[{{\tan }^{-1}}x+{{\tan }^{-1}}y+{{\tan }^{-1}}z=\pi \], then \[x+y+z\] is equal to [Kerala (Engg.) 2002] |
| A. | xyz |
| B. | 0 |
| C. | 1 |
| D. | 2xyz |
| Answer» B. 0 | |
| 6545. |
A line \[4x+y=1\]passes through the point \[A(2,\ -\ 7)\] meets the line BC whose equation is \[3x-4y+1=0\] at the point B. The equation to the line AC so that AB = AC, is [IIT 1971] |
| A. | \[52x+89y+519=0\] |
| B. | \[\beta \] |
| C. | \[89x+52y+519=0\] |
| D. | \[89x+52y-519=0\] |
| Answer» B. \[\beta \] | |
| 6546. |
The pair of lines represented by \[3a{{x}^{2}}+5xy+({{a}^{2}}-2){{y}^{2}}=0\] are perpendicular to each other for [AIEEE 2002] |
| A. | Two values of \[a\] |
| B. | \[\forall a\] |
| C. | For one value of \[a\] |
| D. | For no value of \[a\] |
| Answer» B. \[\forall a\] | |
| 6547. |
Let \[f(x)={{(x+1)}^{2}}-1,\ \ (x\ge -1)\]. Then the set \[S=\{x:f(x)={{f}^{-1}}(x)\}\] is [IIT 1995] |
| A. | Empty |
| B. | {0, -1} |
| C. | {0, 1, -1} |
| D. | \[\left\{ 0,\ -1,\ \frac{-3+i\sqrt{3}}{2},\ \frac{-3-i\sqrt{3}}{2} \right\}\] |
| Answer» E. | |
| 6548. |
A square \[ABCD\] of diagonal 2a is folded along the diagonal \[AC\] so that the planes \[DAC\] and \[BAC\] are at right angle. The shortest distance between \[DC\] and \[AB\] is [Kurukshetra CEE 1998] |
| A. | \[\sqrt{2}a\] |
| B. | \[2a/\sqrt{3}\] |
| C. | \[2a/\sqrt{5}\] |
| D. | [(\sqrt{3}/2)a\] |
| Answer» C. \[2a/\sqrt{5}\] | |
| 6549. |
If \[\overrightarrow{A}=\mathbf{i}+2\mathbf{j}+3\mathbf{k},\,\,\,\overrightarrow{B}=-\mathbf{i}+2\mathbf{j}+\mathbf{k}\] and \[\overrightarrow{C}=3\mathbf{i}+\mathbf{j},\] then the value of t such that \[\overrightarrow{A}+t\overrightarrow{B}\] is at right angle to vector \[3\mathbf{i}+4\mathbf{j}\] is [RPET 2002] |
| A. | 2 |
| B. | 4 |
| C. | 5 |
| D. | 6 |
| Answer» D. 6 | |
| 6550. |
Let R be a relation on the set N of natural numbers defined by nRm \[\Leftrightarrow \] n is a factor of m (i.e., n|m). Then R is |
| A. | Reflexive and symmetric |
| B. | Transitive and symmetric |
| C. | Equivalence |
| D. | Reflexive, transitive but not symmetric |
| Answer» E. | |