Solved papers for JEE Main & Advanced AIEEE Solved Paper-2006

done AIEEE Solved Paper-2006 Total Questions - 40

• question_answer1) If the roots of the quadratic equation \[{{x}^{2}}+px+q=0\]are tan\[30{}^\circ \]and tan\[15{}^\circ \] respectively, then the value of\[2+q-p\]is     AIEEE  Solved  Paper-2006
  A)
  3                             
  done clear
  B)
         0                             
  done clear
  C)
         1                             
  done clear
  D)
         2
  done clear
  View Answer play_arrow

• question_answer2) The    value    of    the    integral\[\int_{3}^{6}{\frac{\sqrt{x}}{\sqrt{9-x}+\sqrt{x}}}dx\]is     AIEEE  Solved  Paper-2006
  A)
  \[\frac{3}{2}\]                   
  done clear
  B)
         2                             
  done clear
  C)
         1                             
  done clear
  D)
         \[\frac{1}{2}\]
  done clear
  View Answer play_arrow

• question_answer3) Let W denotes the words in the English dictionary. Define the relation R by \[R=\{(x,y)\in W\times W|\] the words\[x\]and y have atleast one letter in common}. Then, R is     AIEEE  Solved  Paper-2006
  A)
  reflexive, symmetric and not transitive
  done clear
  B)
  reflexive, symmetric and transitive
  done clear
  C)
  reflexive, not symmetric and transitive
  done clear
  D)
  not reflexive, symmetric and transitive
  done clear
  View Answer play_arrow

• question_answer4) The number of values of\[x\]in the interval \[[0,3\pi ]\]satisfying the equation \[2\text{ }si{{n}^{2}}x+5\text{ }sin\text{ }x-3=0\]is     AIEEE  Solved  Paper-2006
  A)
  6                             
  done clear
  B)
         1                             
  done clear
  C)
         2                             
  done clear
  D)
         4
  done clear
  View Answer play_arrow

• question_answer5) If A and B are square matrices of size\[n\times n\]such that\[{{A}^{2}}-{{B}^{2}}=(A-B)(A+B),\]then which of the following will be always true?     AIEEE  Solved  Paper-2006
  A)
  \[AB=BA\]
  done clear
  B)
  Either of A or B is a zero matrix
  done clear
  C)
  Either of A or B is an identity matrix
  done clear
  D)
  \[A=B\]
  done clear
  View Answer play_arrow

• question_answer6) The value of\[\sum\limits_{k=1}^{10}{\left( \sin \frac{2k\pi }{11}+i\cos \frac{2k\pi }{11} \right)}\]is     AIEEE  Solved  Paper-2006
  A)
  1             
  done clear
  B)
         -1           
  done clear
  C)
         \[-i\]                     
  done clear
  D)
         \[i\]
  done clear
  View Answer play_arrow

• question_answer7) If \[(\vec{a}\times \vec{b})\times \vec{c}=\vec{a}\times (\vec{b}\times \vec{c}),\] where \[\vec{a},\,\vec{b}\] and \[\vec{c}\] are any three vectors such that \[\vec{a}.\vec{b}\ne 0,\vec{b}.\vec{c}\ne 0,\] then \[\vec{a}\] and \[\vec{c}\] are     AIEEE  Solved  Paper-2006
  A)
  inclined at an angle of\[\frac{\pi }{6}\]between them b
  done clear
  B)
  perpendicular    
  done clear
  C)
        parallel
  done clear
  D)
  inclined at an angle of\[\frac{\pi }{3}\]between them
  done clear
  View Answer play_arrow

• question_answer8) All the values of m for which both roots of the equation \[{{x}^{2}}-2mx+{{m}^{2}}-1=0\]are greater than -2 but less than 4 lie in the interval     AIEEE  Solved  Paper-2006
  A)
  \[m>3\]                               
  done clear
  B)
  \[-1<m<3\]        
  done clear
  C)
         \[1<m<4\]          
  done clear
  D)
  \[-2<m<0\]
  done clear
  View Answer play_arrow

• question_answer9) ABC is a triangle, right singled at A. The resultant of the forces acting along \[\overrightarrow{AB},\,\overrightarrow{AC}\] with magnitudes\[\frac{1}{AB}\]and\[\frac{1}{AC}\]respectively is the force along \[\overrightarrow{AD},\] where D is the foot of the perpendicular from A to BC. The magnitude of the resultant is     AIEEE  Solved  Paper-2006
  A)
  \[\frac{(AB)(AC)}{AB+AC}\]       
  done clear
  B)
         \[\frac{1}{AB}+\frac{1}{AC}\]    
  done clear
  C)
         \[\frac{1}{AD}\]                               
  done clear
  D)
         \[\frac{A{{B}^{2}}+A{{C}^{2}}}{(A{{B}^{2}}){{(AC)}^{2}}}\]
  done clear
  View Answer play_arrow

• question_answer10) Suppose, a population A has 100 observations 101, 102,..., 200 and another population B has 100 observations 151, 152, .... 250. If\[{{V}_{A}}\]and\[{{V}_{B}}\]represent the variances of the two populations respectively, then\[\frac{{{V}_{A}}}{{{V}_{B}}}\]is     AIEEE  Solved  Paper-2006
  A)
  \[\frac{9}{4}\]                   
  done clear
  B)
         \[\frac{4}{9}\]                   
  done clear
  C)
         \[\frac{2}{3}\]                   
  done clear
  D)
         1
  done clear
  View Answer play_arrow

• question_answer11) \[\int_{-3\pi /2}^{-\pi /2}{[{{(x+\pi )}^{3}}+{{\cos }^{2}}(x+3\pi )]}dx\]is equal to     AIEEE  Solved  Paper-2006
  A)
  \[\left( \frac{{{\pi }^{4}}}{32} \right)+\left( \frac{\pi }{2} \right)\]                              
  done clear
  B)
  \[\frac{\pi }{2}\]                              
  done clear
  C)
         \[\left( \frac{\pi }{4} \right)-1\] 
  done clear
  D)
         \[\frac{{{\pi }^{4}}}{32}\]
  done clear
  View Answer play_arrow

• question_answer12) In an ellipse, the distances between its foci is 6 and minor axis is 8. Then. Its eccentricity is     AIEEE  Solved  Paper-2006
  A)
  \[\frac{1}{2}\]               
  done clear
  B)
                         \[\frac{4}{5}\]                                   
  done clear
  C)
  \[\frac{1}{\sqrt{5}}\]                     
  done clear
  D)
         \[\frac{3}{5}\]
  done clear
  View Answer play_arrow

• question_answer13) The locus of the vertices of the family of parabolas\[y=\frac{{{a}^{3}}{{x}^{2}}}{3}+\frac{{{a}^{2}}x}{2}-2a\]is     AIEEE  Solved  Paper-2006
  A)
  \[xy=\frac{3}{4}\]           
  done clear
  B)
         \[xy=\frac{35}{16}\]       
  done clear
  C)
         \[xy=\frac{64}{105}\]    
  done clear
  D)
         \[xy=\frac{105}{64}\]
  done clear
  View Answer play_arrow

• question_answer14) A straight line through the point A (3, 4) is such that its intercept between the axes is bisected at A. Its equation is     AIEEE  Solved  Paper-2006
  A)
  \[3x-4y+7=0\]                   
  done clear
  B)
  \[4x+3y=24\]    
  done clear
  C)
         \[3x+4y=25\]      
  done clear
  D)
         \[x+y=7\]
  done clear
  View Answer play_arrow

• question_answer15) The value of a, for which the points A, B, C with position vectors\[2\hat{i}-\hat{j}+\hat{k},\hat{i}-3\hat{j}-5\hat{k}\]and \[a\hat{i}-3\hat{j}+\hat{k}\] respectively are the vertices of a right angled triangle with \[C=\frac{\pi }{2}\]are     AIEEE  Solved  Paper-2006
  A)
  -2 and -1         
  done clear
  B)
         -2 and 1               
  done clear
  C)
                         2 and -1          
  done clear
  D)
         2 and 1
  done clear
  View Answer play_arrow

• question_answer16) \[\int_{0}^{\pi }{x\,f\,(\sin \,x)\,dx}\] is equal to     AIEEE  Solved  Paper-2006
  A)
  \[\pi \int_{0}^{\pi }{f(\sin x)}dx\]
  done clear
  B)
         \[\frac{\pi }{2}\int_{0}^{\pi /2}{f(\sin x)}dx\]     
  done clear
  C)
  \[\pi \int_{0}^{\pi /2}{f(\cos x)}dx\]
  done clear
  D)
         \[\pi \int_{0}^{\pi /2}{f(\cos x)}dx\]
  done clear
  View Answer play_arrow

• question_answer17) The two lines\[x=ay+b,z=cy+d\]and\[x=a'y+b',z=c'y+d'\]are perpendicular to each other, if     AIEEE  Solved  Paper-2006
  A)
  \[aa'+cc'=1\]     
  done clear
  B)
         \[\frac{a}{a'}\,+\frac{c}{c'}\,=-1\]            
  done clear
  C)
         \[\frac{a}{a'}\,+\frac{c}{c'}\,=1\]              
  done clear
  D)
         \[aa'+cc'=-1\]
  done clear
  View Answer play_arrow

• question_answer18) At an election, a voter may vote for any number of candidates not greater than the number to be elected. There are 10 candidates and 4 are to be elected. If a voter votes for atleast one candidate, then the number of ways in which he can vote, is     AIEEE  Solved  Paper-2006
  A)
  6210          
  done clear
  B)
                         385                        
  done clear
  C)
         1110                      
  done clear
  D)
         5040
  done clear
  View Answer play_arrow

• question_answer19) If the expansion in powers of\[x\]of the function \[\frac{1}{(1-ax)(1-bx)}\]is\[{{a}_{0}}+{{a}_{1}}x+{{a}_{2}}{{x}^{2}}+{{a}_{2}}{{x}^{3}}+....,\]then\[{{a}_{n}}\]is     AIEEE  Solved  Paper-2006
  A)
  \[\frac{{{a}^{n}}-{{b}^{n}}}{b-a}\]           
  done clear
  B)
         \[\frac{{{a}^{n+1}}-{{b}^{n+1}}}{b-a}\] 
  done clear
  C)
         \[\frac{{{b}^{n+1}}-{{a}^{n+1}}}{b-a}\] 
  done clear
  D)
         \[\frac{{{b}^{n}}-{{a}^{n}}}{b-a}\]
  done clear
  View Answer play_arrow

• question_answer20) For  natural   numbers   m,   n,  if \[{{(1-y)}^{m}}{{(1+y)}^{n}}=1+{{a}_{1}}y+{{a}_{2}}{{y}^{2}}+.....\]and\[{{a}_{1}}={{a}_{2}}=10,\]then (m, n) is     AIEEE  Solved  Paper-2006
  A)
  (35, 20)         
  done clear
  B)
         (45, 35)
  done clear
  C)
         (35, 45)            
  done clear
  D)
         (20, 45)
  done clear
  View Answer play_arrow

• question_answer21) A particle has two velocities of equal magnitude inclined to each other at an angle \[\theta \]. If one of them is halved, the angle between the- other and the original resultant velocity is bisected by the new resultant. Then, 0 is     AIEEE  Solved  Paper-2006
  A)
  \[120{}^\circ \]                                 
  done clear
  B)
  \[45{}^\circ \]                   
  done clear
  C)
         \[60{}^\circ \]                   
  done clear
  D)
         \[90{}^\circ \]
  done clear
  View Answer play_arrow

• question_answer22) At a telephone enquiry system, the number of phone calls regarding relevant enquiry follow Poisson distribution with an average of 5 phone calls during 10 min time intervals. The probability that there is atmost one phone call during a 10 min time period, is     AIEEE  Solved  Paper-2006
  A)
  \[\frac{5}{6}\]           
  done clear
  B)
                         \[\frac{6}{55}\]
  done clear
  C)
  \[\frac{6}{{{e}^{5}}}\]                   
  done clear
  D)
         \[\frac{6}{{{5}^{e}}}\]
  done clear
  View Answer play_arrow

• question_answer23) A body falling .from rest under gravity passes a certain point P. It was at a distance of 400 m from P, 4 s prior to passing through P. If \[g=10m/{{s}^{2}},\]then the height above the point P from where the body began to fall is                    AIEEE  Solved  Paper-2006
  A)
  900m
  done clear
  B)
                                         320 m   
  done clear
  C)
         680 m                   
  done clear
  D)
         720 m      
  done clear
  View Answer play_arrow

• question_answer24) The set of points, where \[f(x)=\frac{x}{1+|x|}\]is differentiable, is     AIEEE  Solved  Paper-2006
  A)
  \[(-\infty ,-1)\cup (-1,\infty )\]  
  done clear
  B)
  \[(-\infty ,\infty )\]         
  done clear
  C)
         \[(0,\infty )\]    
  done clear
  D)
         \[(-\infty ,0\cup (0,\infty )\]
  done clear
  View Answer play_arrow

• question_answer25) Let\[A=\left[ \begin{matrix}    1 & 2  \\    3 & 4  \\ \end{matrix} \right]\]and\[B=\left[ \begin{matrix}    a & 0  \\    0 & b  \\ \end{matrix} \right],a,b,\in N\]. Then,     AIEEE  Solved  Paper-2006
  A)
  there exist more than one but finite number of B's such that\[AB=BA\]
  done clear
  B)
  there exists exactly one B such that\[AB=BA\]
  done clear
  C)
  there exist infinitely many 8's such that\[AB=BA\]
  done clear
  D)
  there cannot exist any B such that\[AB=BA\]
  done clear
  View Answer play_arrow

• question_answer26) Let \[{{a}_{1}},{{a}_{2}},{{a}_{3}},....\] be terms of an AP. If \[\frac{{{a}_{1}}+{{a}_{2}}+.....+{{a}_{p}}}{{{a}_{1}}+{{a}_{2}}+....{{a}_{q}}}=\frac{{{p}^{2}}}{{{q}^{2}}},\] \[p\ne q,\]then\[\frac{{{a}_{6}}}{{{a}_{21}}}\]equals     AIEEE  Solved  Paper-2006
  A)
  \[\frac{7}{2}\]                   
  done clear
  B)
         \[\frac{2}{7}\]                   
  done clear
  C)
  \[\frac{11}{41}\]                              
  done clear
  D)
         \[\frac{41}{11}\]
  done clear
  View Answer play_arrow

• question_answer27) The function\[f(x)=\frac{x}{2}+\frac{2}{x}\]has a local minimum at     AIEEE  Solved  Paper-2006
  A)
  \[x=-2\]               
  done clear
  B)
  \[x=0\]
  done clear
  C)
         \[x=1\]                
  done clear
  D)
         \[x=2\]
  done clear
  View Answer play_arrow

• question_answer28) Angle between the tangents to the curve\[y={{x}^{2}}-5x+6\]at the points (2, 0) and (3, 0) is     AIEEE  Solved  Paper-2006
  A)
  \[\pi /2\]             
  done clear
  B)
         \[\pi /6\]             
  done clear
  C)
         \[\pi /4\]                 
  done clear
  D)
         \[\pi /3\]
  done clear
  View Answer play_arrow

• question_answer29) If \[x\]is real, the maximum value of\[\frac{3{{x}^{2}}+9x+17}{3{{x}^{2}}+9x+7}\]is     AIEEE  Solved  Paper-2006
  A)
  41   
  done clear
  B)
                                         1                             
  done clear
  C)
         17/7              
  done clear
  D)
                         1/4
  done clear
  View Answer play_arrow

• question_answer30) A triangular park is enclosed on two sides by a fence and on the third side by a straight river bank. The two sides having fence are of same length\[x\]. The maximum area enclosed by the park is     AIEEE  Solved  Paper-2006
  A)
  \[\sqrt{\frac{{{x}^{3}}}{8}}\]                      
  done clear
  B)
  \[\frac{1}{2}{{x}^{2}}\] 
  done clear
  C)
         \[\pi {{x}^{2}}\]                
  done clear
  D)
         \[\frac{3}{2}{{x}^{2}}\]
  done clear
  View Answer play_arrow

• question_answer31) If\[(a,\text{ }{{a}^{2}})\]falls inside the angle made by the lines \[y=\frac{x}{2},\text{ }x>0\]and\[y=3x,\text{ }x>0,\]then a belongs to       AIEEE  Solved  Paper-2006
  A)
  \[(3,\infty )\]    
  done clear
  B)
         \[\left( \frac{1}{2},3 \right)\]     
  done clear
  C)
         \[\left( -3,-\frac{1}{2} \right)\]  
  done clear
  D)
         \[\left( 0,\frac{1}{2} \right)\]
  done clear
  View Answer play_arrow

• question_answer32) If\[{{x}^{m}}{{y}^{n}}={{(x+y)}^{m+n}},\]then\[\frac{dy}{dx}\]is       AIEEE  Solved  Paper-2006
  A)
  \[\frac{x+y}{xy}\]
  done clear
  B)
                         \[xy\]                                   
  done clear
  C)
  \[\frac{x}{y}\]                   
  done clear
  D)
         \[\frac{y}{x}\]
  done clear
  View Answer play_arrow

• question_answer33) If the lines\[3x-4y-7=0\]and\[2x-3y-5=0\]are two diameters of a circle of area\[49\,\pi \,sq\]units, the equation of the circle is       AIEEE  Solved  Paper-2006
  A)
  \[{{x}^{2}}+{{y}^{2}}+2x-2y-62=0\]
  done clear
  B)
  \[{{x}^{2}}+{{y}^{2}}-2x+2y-62=0\]
  done clear
  C)
  \[{{x}^{2}}+{{y}^{2}}-2x+2y-47=0\]
  done clear
  D)
  \[{{x}^{2}}+{{y}^{2}}+2x-2y-47=0\]
  done clear
  View Answer play_arrow

• question_answer34) The image of the point (-1, 3, 4) in the plane \[x-2y=0\]is       AIEEE  Solved  Paper-2006
  A)
  (15, 11, 4)         
  done clear
  B)
  \[\left( -\frac{17}{3},-\frac{19}{3},1 \right)\]       
  done clear
  C)
   (8, 4, 4)          
  done clear
  D)
         \[\left( -\frac{17}{3},-\frac{19}{3},4 \right)\]
  done clear
  View Answer play_arrow

• question_answer35) The differential equation whose solution is \[A{{x}^{2}}+B{{y}^{2}}=1,\]where A and B are arbitrary constants, is of       AIEEE  Solved  Paper-2006
  A)
  first order and second degree
  done clear
  B)
  first order and first degree
  done clear
  C)
  second order and first degree
  done clear
  D)
  second order and second degree
  done clear
  View Answer play_arrow

• question_answer36) The value of \[\int_{I}^{a}{[x]\,f'\,(x)\,dx,\,\,a>1,}\] where\[[x]\] denotes  the  greatest  integer  not exceeding \[x,\]is       AIEEE  Solved  Paper-2006
  A)
  \[[a]f(a)-\{f(1)+f(2)+....+f([a])\}\]
  done clear
  B)
  \[[a]f([a])-\{f(1)+f(2)+....+f(a)\}\]
  done clear
  C)
  \[af([a])-\{f(1)+f(2)+....+f(a)\}\]
  done clear
  D)
  \[af(a)-\{f(1)+f(2)+....+f([a])\}\]
  done clear
  View Answer play_arrow

• question_answer37) Let C be the circle with centre (0, 0) and radius 3 units. The equation of the locus of the mid-points of the chords of the circle C 'that subtend an angle of\[\frac{2\pi }{3}\]at its centre, is       AIEEE  Solved  Paper-2006
  A)
  \[{{x}^{2}}+{{y}^{2}}=1\]             
  done clear
  B)
  \[{{x}^{2}}+{{y}^{2}}=\frac{27}{4}\]
  done clear
  C)
                         \[{{x}^{2}}+{{y}^{2}}=\frac{9}{4}\]           
  done clear
  D)
         \[{{x}^{2}}+{{y}^{2}}=\frac{3}{2}\]
  done clear
  View Answer play_arrow

• question_answer38) If\[{{a}_{1}},{{a}_{2}},....{{a}_{n}}\]are in HP, then the expression\[{{a}_{1}}{{a}_{2}}+{{a}_{2}}{{a}_{3}}+,....+{{a}_{n-1}}{{a}_{n}}\]is equal to       AIEEE  Solved  Paper-2006
  A)
  \[(n-1)({{a}_{1}}-{{a}_{n}})\]
  done clear
  B)
         \[n{{a}_{1}}{{a}_{n}}\]  
  done clear
  C)
         \[(n-1){{a}_{1}}{{a}_{n}}\]           
  done clear
  D)
         \[n({{a}_{1}}-{{a}_{n}})\]
  done clear
  View Answer play_arrow

• question_answer39) If\[{{z}^{2}}+z+1=0,\]where z is complex number, then the value of \[{{\left( z+\frac{1}{z} \right)}^{2}}+{{\left( {{z}^{2}}+\frac{1}{{{z}^{2}}} \right)}^{2}}+{{\left( {{z}^{3}}+\frac{1}{{{z}^{3}}} \right)}^{2}}\] \[+......+{{\left( {{z}^{6}}+\frac{1}{{{z}^{6}}} \right)}^{2}}\]is       AIEEE  Solved  Paper-2006
  A)
  54                          
  done clear
  B)
  6                             
  done clear
  C)
         12          
  done clear
  D)
         18
  done clear
  View Answer play_arrow

• question_answer40) If\[0<x<\pi \]and \[\cos x+\sin x=\frac{1}{2},\]then \[tan\text{ }x\]is       AIEEE  Solved  Paper-2006
  A)
  \[\frac{(4-\sqrt{7})}{3}\]
  done clear
  B)
                         \[-\frac{(4+\sqrt{7})}{3}\]           
  done clear
  C)
         \[\frac{(1+\sqrt{7})}{4}\]             
  done clear
  D)
         \[\frac{(1-\sqrt{7})}{4}\]
  done clear
  View Answer play_arrow