# 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

B)
0

C)
1

D)
2

• 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)
$\frac{3}{2}$

B)
2

C)
1

D)
$\frac{1}{2}$

• 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

B)
reflexive, symmetric and transitive

C)
reflexive, not symmetric and transitive

D)
not reflexive, symmetric and transitive

• 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

B)
1

C)
2

D)
4

• 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)
$AB=BA$

B)
Either of A or B is a zero matrix

C)
Either of A or B is an identity matrix

D)
$A=B$

• 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

B)
-1

C)
$-i$

D)
$i$

• 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

B)
perpendicular

C)
parallel

D)
inclined at an angle of$\frac{\pi }{3}$between them

• 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)
$m>3$

B)
$-1<m<3$

C)
$1<m<4$

D)
$-2<m<0$

• 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)
$\frac{(AB)(AC)}{AB+AC}$

B)
$\frac{1}{AB}+\frac{1}{AC}$

C)
$\frac{1}{AD}$

D)
$\frac{A{{B}^{2}}+A{{C}^{2}}}{(A{{B}^{2}}){{(AC)}^{2}}}$

• 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)
$\frac{9}{4}$

B)
$\frac{4}{9}$

C)
$\frac{2}{3}$

D)
1

• 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)
$\left( \frac{{{\pi }^{4}}}{32} \right)+\left( \frac{\pi }{2} \right)$

B)
$\frac{\pi }{2}$

C)
$\left( \frac{\pi }{4} \right)-1$

D)
$\frac{{{\pi }^{4}}}{32}$

• 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)
$\frac{1}{2}$

B)
$\frac{4}{5}$

C)
$\frac{1}{\sqrt{5}}$

D)
$\frac{3}{5}$

• 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)
$xy=\frac{3}{4}$

B)
$xy=\frac{35}{16}$

C)
$xy=\frac{64}{105}$

D)
$xy=\frac{105}{64}$

• 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)
$3x-4y+7=0$

B)
$4x+3y=24$

C)
$3x+4y=25$

D)
$x+y=7$

• 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

B)
-2 and 1

C)
2 and -1

D)
2 and 1

• question_answer16) $\int_{0}^{\pi }{x\,f\,(\sin \,x)\,dx}$ is equal to     AIEEE  Solved  Paper-2006

A)
$\pi \int_{0}^{\pi }{f(\sin x)}dx$

B)
$\frac{\pi }{2}\int_{0}^{\pi /2}{f(\sin x)}dx$

C)
$\pi \int_{0}^{\pi /2}{f(\cos x)}dx$

D)
$\pi \int_{0}^{\pi /2}{f(\cos x)}dx$

• 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)
$aa'+cc'=1$

B)
$\frac{a}{a'}\,+\frac{c}{c'}\,=-1$

C)
$\frac{a}{a'}\,+\frac{c}{c'}\,=1$

D)
$aa'+cc'=-1$

• 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

B)
385

C)
1110

D)
5040

• 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)
$\frac{{{a}^{n}}-{{b}^{n}}}{b-a}$

B)
$\frac{{{a}^{n+1}}-{{b}^{n+1}}}{b-a}$

C)
$\frac{{{b}^{n+1}}-{{a}^{n+1}}}{b-a}$

D)
$\frac{{{b}^{n}}-{{a}^{n}}}{b-a}$

• 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)

B)
(45, 35)

C)
(35, 45)

D)
(20, 45)

• 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)
$120{}^\circ$

B)
$45{}^\circ$

C)
$60{}^\circ$

D)
$90{}^\circ$

• 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)
$\frac{5}{6}$

B)
$\frac{6}{55}$

C)
$\frac{6}{{{e}^{5}}}$

D)
$\frac{6}{{{5}^{e}}}$

• 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

B)
320 m

C)
680 m

D)
720 m

• question_answer24) The set of points, where $f(x)=\frac{x}{1+|x|}$is differentiable, is     AIEEE  Solved  Paper-2006

A)
$(-\infty ,-1)\cup (-1,\infty )$

B)
$(-\infty ,\infty )$

C)
$(0,\infty )$

D)
$(-\infty ,0\cup (0,\infty )$

• 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$AB=BA$

B)
there exists exactly one B such that$AB=BA$

C)
there exist infinitely many 8's such that$AB=BA$

D)
there cannot exist any B such that$AB=BA$

• 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)
$\frac{7}{2}$

B)
$\frac{2}{7}$

C)
$\frac{11}{41}$

D)
$\frac{41}{11}$

• question_answer27) The function$f(x)=\frac{x}{2}+\frac{2}{x}$has a local minimum at     AIEEE  Solved  Paper-2006

A)
$x=-2$

B)
$x=0$

C)
$x=1$

D)
$x=2$

• 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)
$\pi /2$

B)
$\pi /6$

C)
$\pi /4$

D)
$\pi /3$

• 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

B)
1

C)
17/7

D)
1/4

• 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)
$\sqrt{\frac{{{x}^{3}}}{8}}$

B)
$\frac{1}{2}{{x}^{2}}$

C)
$\pi {{x}^{2}}$

D)
$\frac{3}{2}{{x}^{2}}$

• 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)
$(3,\infty )$

B)
$\left( \frac{1}{2},3 \right)$

C)
$\left( -3,-\frac{1}{2} \right)$

D)
$\left( 0,\frac{1}{2} \right)$

• question_answer32) If${{x}^{m}}{{y}^{n}}={{(x+y)}^{m+n}},$then$\frac{dy}{dx}$is       AIEEE  Solved  Paper-2006

A)
$\frac{x+y}{xy}$

B)
$xy$

C)
$\frac{x}{y}$

D)
$\frac{y}{x}$

• 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)
${{x}^{2}}+{{y}^{2}}+2x-2y-62=0$

B)
${{x}^{2}}+{{y}^{2}}-2x+2y-62=0$

C)
${{x}^{2}}+{{y}^{2}}-2x+2y-47=0$

D)
${{x}^{2}}+{{y}^{2}}+2x-2y-47=0$

• 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)

B)
$\left( -\frac{17}{3},-\frac{19}{3},1 \right)$

C)
(8, 4, 4)

D)
$\left( -\frac{17}{3},-\frac{19}{3},4 \right)$

• 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

B)
first order and first degree

C)
second order and first degree

D)
second order and second degree

• 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)
$[a]f(a)-\{f(1)+f(2)+....+f([a])\}$

B)
$[a]f([a])-\{f(1)+f(2)+....+f(a)\}$

C)
$af([a])-\{f(1)+f(2)+....+f(a)\}$

D)
$af(a)-\{f(1)+f(2)+....+f([a])\}$

• 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)
${{x}^{2}}+{{y}^{2}}=1$

B)
${{x}^{2}}+{{y}^{2}}=\frac{27}{4}$

C)
${{x}^{2}}+{{y}^{2}}=\frac{9}{4}$

D)
${{x}^{2}}+{{y}^{2}}=\frac{3}{2}$

• 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)
$(n-1)({{a}_{1}}-{{a}_{n}})$

B)
$n{{a}_{1}}{{a}_{n}}$

C)
$(n-1){{a}_{1}}{{a}_{n}}$

D)
$n({{a}_{1}}-{{a}_{n}})$

• 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

B)
6

C)
12

D)
18

• question_answer40) If$0<x<\pi$and $\cos x+\sin x=\frac{1}{2},$then $tan\text{ }x$is       AIEEE  Solved  Paper-2006

A)
$\frac{(4-\sqrt{7})}{3}$

B)
$-\frac{(4+\sqrt{7})}{3}$

C)
$\frac{(1+\sqrt{7})}{4}$

D)
$\frac{(1-\sqrt{7})}{4}$