Solved papers for JEE Main & Advanced AIEEE Solved Paper-2005

done AIEEE Solved Paper-2005 Total Questions - 75

• question_answer1) If C is the mid-point of AB and P is any point outside AB, then     AIEEE  Solved  Paper-2005

A)
$PA+PB+PC=0$

B)
$PA+PB+2PC=0$

C)
$PA+PB=PC$

D)
PA + PB = 2PC

• question_answer2) Let P be the point (1, 0) and Q be a point on the locus${{y}^{2}}=8x.$The locus of mid-point of PQ is     AIEEE  Solved  Paper-2005

A)
${{x}^{2}}-4y+2=0$

B)
${{x}^{2}}+4y+2=0$

C)
${{y}^{2}}+4x+2=0$

D)
${{y}^{2}}-4x+2=0$

• question_answer3) If in a frequency distribution, the mean and median are 21 and 22 respectively, then its mode is approximately     AIEEE  Solved  Paper-2005

A)
24.0

B)
25.5

C)
20.5

D)
22.0

• question_answer4) Let$R=\{(3,3),(6,6),(9,9),(12,12),$$(6,12),$$(3,9),(3,12),(3,6)\}$be a relation on the set A$=\{3,6,9,12\}$. The relation is     AIEEE  Solved  Paper-2005

A)
reflexive and symmetric only

B)
an equivalence relation

C)
reflexive only

D)
reflexive and transitive only

• question_answer5) If ${{A}^{2}}-A+I=O,$ then the inverse of A is       AIEEE  Solved  Paper-2005

A)
$l-A$

B)
$A-l$

C)
A

D)
$A+l$

• question_answer6) If the cube roots of unity are$1,\omega ,{{\omega }^{2}}$then the roots of the equation ${{(x-1)}^{3}}+8=0,$ are     AIEEE  Solved  Paper-2005

A)
$-1,1+2\omega ,1+2{{\omega }^{2}}$

B)
$-1,1-2\omega ,1-2{{\omega }^{2}}$

C)
$-1,-1,-1$

D)
$-1,-1+2\omega ,-1-2{{\omega }^{2}}$

• question_answer7) $\underset{n\to \infty }{\mathop{\lim }}\,\left[ \frac{1}{{{n}^{2}}}{{\sec }^{2}}\frac{1}{{{n}^{2}}}+\frac{2}{{{n}^{2}}}{{\sec }^{2}}\frac{4}{{{n}^{2}}} \right.$ $\left. +....+\frac{n}{{{n}^{2}}}{{\sec }^{2}}1 \right]$ equals     AIEEE  Solved  Paper-2005

A)
$\frac{1}{2}\text{ }tan\text{ }1$

B)
$tan\text{ }1$

C)
$\frac{1}{2}cosec\text{ }1$

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

• question_answer8) Area of the greatest rectangle that can be inscribed in the ellipse$\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1$is     AIEEE  Solved  Paper-2005

A)
$\frac{a}{b}$

B)
$\sqrt{ab}$

C)
$ab$

D)
$2ab$

• question_answer9) The differential equation representing the family of curves${{y}^{2}}=2c(x+\sqrt{c}),$where$c>0,$is a parameter, is of order and degree as follows       AIEEE  Solved  Paper-2005

A)
order 2, degree 2

B)
order 1, degree 3

C)
order 1, degree 1

D)
order 1, degree 2

• question_answer10) ABC is a triangle. Forces P,Q,R acting along $I,A,IB$and$IC$respectively are in equilibrium, where I is the incentre of$\Delta ABC.$Then, P : Q : R is     AIEEE  Solved  Paper-2005

A)
$\cos A:\cos B:\cos \,C$

B)
$\cos \frac{A}{2}:\cos \frac{B}{2}:\cos \,\frac{C}{2}$

C)
$\sin \frac{A}{2}:\sin \frac{B}{2}:\sin \,\frac{C}{2}$

D)
$\sin A:\sin B:\sin C$

• question_answer11) If the coefficients of rth,$(r+1)th$ and$(r+2)th$terms in the binomial expansion of${{(1+y)}^{m}}$are in AP, then m and r satisfy the equation     AIEEE  Solved  Paper-2005

A)
${{m}^{2}}-m(4r-1)+4{{r}^{2}}+2=0$

B)
${{m}^{2}}-m(4r+1)+4{{r}^{2}}-2=0$

C)
${{m}^{2}}-m(4r+1)+4{{r}^{2}}+2=0$

D)
${{m}^{2}}-m(4r-1)+4{{r}^{2}}-2=0$

• question_answer12) In a$\Delta PQR,\angle R=\frac{\pi }{2}$If$\tan \left( \frac{P}{2} \right)$and$\tan \left( \frac{Q}{2} \right)$are the roots of$a{{x}^{2}}+bx+c=0,a\ne 0,$then     AIEEE  Solved  Paper-2005

A)
$d=a+c$

B)
$b=c$

C)
$c=a+b$

D)
$a=o+c,$

• question_answer13) If the letters of the word SACHIN are arranged in all possible ways and these words are written out as in dictionary, then the word SACHIN appears at serial number     AIEEE  Solved  Paper-2005

A)
602

B)
603

C)
600

D)
601

• question_answer14) The value of $^{50}{{C}_{4}}+\sum\limits_{r=1}^{6}{^{56-r}{{C}_{3}}}$is     AIEEE  Solved  Paper-2005

A)
$^{56}{{C}_{4}}$

B)
$^{56}{{C}_{3}}$

C)
$^{55}{{C}_{3}}$

D)
$^{55}{{C}_{4}}$

• question_answer15) If $A=\left[ \begin{matrix} 1 & 0 \\ 1 & 1 \\ \end{matrix} \right]$and$I=\left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \right],$then which one of the following holds for all$n\ge 1,$the principle of mathematical induction?     AIEEE  Solved  Paper-2005

A)
${{A}^{n}}={{2}^{n-1}}A+(n-1)l$

B)
${{A}^{n}}=nA+(n-1)l$

C)
${{A}^{n}}={{2}^{n-1}}A+(n-1)l$

D)
${{A}^{n}}=nA+(n-1)l$

• question_answer16) If the coefficient of${{x}^{7}}$in${{\left[ a{{x}^{2}}+\left( \frac{1}{bx} \right) \right]}^{11}}$equals  the  coefficient  of${{x}^{-7}}$in${{\left[ ax-\left( \frac{1}{b{{x}^{2}}} \right) \right]}^{11}},$then a and b satisfy the relation     AIEEE  Solved  Paper-2005

A)
$ab=1$

B)
$\frac{a}{b}=1$

C)
$a+b=1$

D)
$a-d=1$

• question_answer17) Let$f:(-1,1)\to B$ be a function defined by$f(x)={{\tan }^{-1}}\frac{2x}{1-{{x}^{2}}},$ then f is both one-one and onto when B is the interval     AIEEE  Solved  Paper-2005

A)
$\left( -\frac{\pi }{2},\frac{\pi }{2} \right)$

B)
$\left[ -\frac{\pi }{2},\frac{\pi }{2} \right]$

C)
$\left[ 0,\frac{\pi }{2} \right)$

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

• question_answer18) If${{z}_{1}}$and${{z}_{2}}$are two non-zero complex numbers such that$|{{z}_{1}}+{{z}_{2}}|=|{{z}_{1}}|+|{{z}_{2}}|$then $\arg ({{z}_{1}})-\arg ({{z}_{2}})$is equal to

A)
$-\frac{\pi }{2}$

B)
0

C)
$-\pi$

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

• question_answer19) If$W=\frac{z}{z-\frac{1}{3}i}$and$|w|=1,$then z lies on     AIEEE  Solved  Paper-2005

A)
a parabola

B)
a straight line

C)
a circle

D)
an ellipse

• question_answer20) If${{a}^{2}}+{{b}^{2}}+{{c}^{2}}=-2$and $f(x)=\left| \begin{matrix} 1+{{a}^{2}}x & (1+{{b}^{2}})x & (1+{{c}^{2}})x \\ {{(1+a)}^{2}}x & 1+{{b}^{2}}x & (1+{{c}^{2}})x \\ (1+{{a}^{2}})x & (1+{{b}^{2}})x & 1+{{c}^{2}}x \\ \end{matrix} \right|,$ then$f(x)$is a polynomial of degree     AIEEE  Solved  Paper-2005

A)
2

B)
3

C)
0

D)
1

• question_answer21) The system of equations $\alpha x+y+z=\alpha -1,x+\alpha \,y+z=\alpha -l$ $x+y+\alpha z=\alpha -1$has no solution, if $\alpha$ is     AIEEE  Solved  Paper-2005

A)
1

B)
not-2

C)
either - 2 or 1

D)
- 2

• question_answer22) The value of a for which the sum of the squares of the roots of the equation${{x}^{2}}-(a-2)x-a-1=0$assume the least value is     AIEEE  Solved  Paper-2005

A)
2

B)
3

C)
0

D)
1

• question_answer23) If the roots of the equation${{x}^{2}}-bx+c=0$are two consecutive integers, then${{b}^{2}}-4c$equals     AIEEE  Solved  Paper-2005

A)
1

B)
2

C)
3

D)
-2

• question_answer24) Suppose$f(x)$is differentiable at$x=1$and$\underset{h\to 0}{\mathop{\lim }}\,\frac{1}{h}f(1+h)=5,$then$f'(1)$equals     AIEEE  Solved  Paper-2005

A)
6

B)
5

C)
4

D)
3

• question_answer25) Let f be differentiable for all$x$. If$f(1)=-2$and $f'(x)\ge 2$for$x\in [1,6],$then     AIEEE  Solved  Paper-2005

A)
$f(6)=5$

B)
$f(6)<5$

C)
$f(6)<8$

D)
$f(6)\ge 8$

• question_answer26) If$f$is a real-valued differentiable function satisfying$|f(x)-f(y)|\le {{(x-y)}^{2}},x,y\in R$and $f(0)=0,$then$f(1)$equals     AIEEE  Solved  Paper-2005

A)
1

B)
2

C)
0

D)
-1

• question_answer27) If$x$is so small that${{x}^{3}}$and higher powers of$x$   may  be   neglected, then $\frac{{{(1+x)}^{3/2}}-{{\left( 1+\frac{1}{2}x \right)}^{3}}}{{{(1-x)}^{1/2}}}$may be approximated as     AIEEE  Solved  Paper-2005

A)
$\frac{x}{2}-\frac{3}{8}{{x}^{2}}$

B)
$-\frac{3}{8}{{x}^{2}}$

C)
$3x+\frac{3}{8}{{x}^{2}}$

D)
$1-\frac{3}{8}{{x}^{2}}$

• question_answer28) If$x=\sum\limits_{n=0}^{\infty }{{{a}^{n}}},y=\sum\limits_{n=0}^{\infty }{{{b}^{n}}},z=\sum\limits_{n=0}^{\infty }{{{c}^{n}}},$where a, b, c are in AP and$|a|<1,|b|<1,|c|<1,$then$x,\text{ }y,\text{ }z$are in     AIEEE  Solved  Paper-2005

A)
HP

B)
AGP

C)
AP

D)
GP

• question_answer29) In a$\Delta ABC,$let $\angle C=\pi /2,$ if r is the inradius and R is the circumradius of the$\Delta ABC,$then $2(r+R)$equals     AIEEE  Solved  Paper-2005

A)
$c+a$

B)
$a+b+c$

C)
$a+b$

D)
$b+c$

• question_answer30) If${{\cos }^{-1}}x-{{\cos }^{-1}}\frac{y}{2}=\alpha ,$then$4{{x}^{2}}-4xy\,\cos \alpha +{{y}^{2}}$ is equal to     AIEEE  Solved  Paper-2005

A)
$-4\text{ }{{\sin }^{2}}\alpha$

B)
$4\text{ }{{\sin }^{2}}\alpha$

C)
4

D)
$2\text{ }{{\sin }^{2}}\alpha$

• question_answer31) If in a$\Delta ABC,$the altitudes from the vertices A,B,C on opposite sides are in HP, then sin A, sin B, sin C are in     AIEEE  Solved  Paper-2005

A)
HP

B)
Arithmetico-Geometric Progression

C)
AP

D)
GP

• question_answer32) The normal to the curve $x=a(\cos \theta +\theta \sin \theta ),y=a(\sin \theta -\theta \cos \theta )$at any point$'\theta '$is such that     AIEEE  Solved  Paper-2005

A)
it is at a constant distance from the origin

B)
it passes through$(a\text{ }\pi /2,-a)$

C)
it makes angle$\pi /2+\theta$with the X-axis

D)
it passes through the origin

• question_answer33) A function is matched below against an interval, where it is supposed to be increasing. Which of the following pairs is incorrectly matched? Interval                Function     AIEEE  Solved  Paper-2005

A)
$(-\infty ,\,\,-4]$           ${{x}^{3}}+6{{x}^{2}}+6$

B)
$\left( -\infty ,\frac{1}{3} \right]$           $3{{x}^{2}}-2x+1$

C)
$[2,\,\infty )$                  $2{{x}^{3}}-3{{x}^{2}}-12{{x}^{4}}-6$

D)
$(-\infty ,\infty )$                          ${{x}^{3}}-3{{x}^{2}}+3x+3$

• question_answer34) Let $\alpha$ and $\beta$ be the distinct roots of$a{{x}^{2}}+bx+c=0,$then $\underset{x\to \alpha }{\mathop{\lim }}\,\frac{1-\cos (a{{x}^{2}}+bx+c)}{{{(x-\alpha )}^{2}}}$is equal to     AIEEE  Solved  Paper-2005

A)
$\frac{1}{2}{{(\alpha -\beta )}^{2}}$

B)
$-\frac{{{a}^{2}}}{2}{{(\alpha -\beta )}^{2}}$

C)
0

D)
$\frac{{{a}^{2}}}{2}{{(\alpha -\beta )}^{2}}$

• question_answer35) If$x\frac{dy}{dx}=y(\log \text{ }y-\log \text{ }x+1),$then the solution of the equation is     AIEEE  Solved  Paper-2005

A)
$\log \left( \frac{x}{y} \right)=Cy$

B)
$\log \left( \frac{y}{x} \right)=Cx$

C)
$x\log \left( \frac{y}{x} \right)=Cy$

D)
$y\log \left( \frac{x}{y} \right)=Cx$

• question_answer36) The line parallel to the X-axis and passing through the intersection of the lines $ax+2\,by+3b=0$and$bx-2ay-3a=0,$where $(a,b)\ne (0,0)$is     AIEEE  Solved  Paper-2005

A)
above the X-axis at a distance of (2/3) from it

B)
above the X-axis at a distance of (3/2) from it

C)
below the X-axis at a distance of (2/3) from it

D)
below the X-axis at a distance of (3/2) from it

• question_answer37) A spherical iron ball 10 cm in radius is coated with a layer of ice of uniform thickness that melts at a rate of$50\text{ }c{{m}^{3}}/\min$. When the thickness of ice is 15 cm, then the rate at which the thickness of ice decreases, is     AIEEE  Solved  Paper-2005

A)
$\frac{5}{6\pi }cm/\min$

B)
$\frac{1}{54\pi }cm/\min$

C)
$\frac{1}{18\pi }cm/\min$

D)
$\frac{1}{36\pi }cm/\min$

• question_answer38) ${{\int{\left\{ \frac{(\log x-1)}{1+{{(\log \,x)}^{2}}} \right\}}}^{2}}dx$is equal to     AIEEE  Solved  Paper-2005

A)
$\frac{x}{{{(\log \,x)}^{2}}+1}+C$

B)
$\frac{x{{e}^{x}}}{1+\,{{x}^{2}}}+C$

C)
$\frac{x}{\,{{x}^{2}}+1}+C$

D)
$\frac{\log \,\,x}{\,{{(\log x)}^{2}}+1}+C$

• question_answer39) Let$f:\text{ }R\to R$be a differentiable function having$f(2)=6,f'(2)=\left( \frac{1}{48} \right)$ Then,$\underset{x\to 2}{\mathop{\lim }}\,\int_{6}^{f(x)}{\frac{4{{t}^{3}}}{x-2}}dt$ equals       AIEEE  Solved  Paper-2005

A)
18

B)
12

C)
36

D)
24

• question_answer40) Let$f(x)$be a non-negative continuous function such that the area bounded by the curve$y=f(x),$X-axis and the ordinates$x=\pi /4$ and $x=\beta >\pi /4$ is$\left( \beta \sin \beta +\frac{\pi }{4}\cos \beta +\sqrt{2}\beta \right)$.Then$f\left( \frac{\pi }{2} \right)$,is     AIEEE  Solved  Paper-2005

A)
$\left( 1-\frac{\pi }{4}+\sqrt{2} \right)$

B)
$\left( 1-\frac{\pi }{4}-\sqrt{2} \right)$

C)
$\left( \frac{\pi }{4}-\sqrt{2}+1 \right)$

D)
$\left( \frac{\pi }{4}+\sqrt{2}-1 \right)$

• question_answer41) If${{I}_{1}}=\int_{0}^{1}{{{2}^{{{x}^{2}}}}}dx$${{I}_{2}}=\int_{0}^{1}{{{2}^{{{x}^{3}}}}}dx,$ ${{I}_{3}}=\int_{1}^{2}{{{2}^{{{x}^{2}}}}}dx$and${{I}_{4}}=\int_{1}^{2}{{{2}^{{{x}^{3}}}}}dx,$then     AIEEE  Solved  Paper-2005

A)
${{l}_{3}}>{{l}_{4}}$

B)
${{l}_{3}}={{l}_{4}}$

C)
${{l}_{1}}>{{l}_{2}}$

D)
${{l}_{2}}>{{l}_{1}}$

• question_answer42) The area enclosed between the curve$y={{\log }_{e}}(x+e)$and the coordinate axes is     AIEEE  Solved  Paper-2005

A)
4

B)
3

C)
2

D)
1

• question_answer43) The parabolas${{y}^{2}}=4x$and${{x}^{2}}=4y$divide the square region bounded by the lines $x=4,\text{ }y=4$and the coordinate axes. If ${{S}_{1}},{{S}_{2}},{{S}_{3}}$are respectively the areas of these parts numbered from top to bottom, then${{S}_{1}}:{{S}_{2}}:{{S}_{3}}$     AIEEE  Solved  Paper-2005

A)
1 : 1 : 1

B)
2 : 1 : 2

C)
1 : 2 : 3

D)
1 : 2 : 1

• question_answer44) If the plane$2ax-3ay+4az+6=0$passes through the mid-point of the line joining the    centres    of    the    spheres${{x}^{2}}+{{y}^{2}}+{{z}^{2}}+6x-8y-2z=13$ and${{x}^{2}}+{{y}^{2}}+{{z}^{2}}-10x+4y-2z=8,$then a equals     AIEEE  Solved  Paper-2005

A)
2

B)
$-2$

C)
1

D)
$-1$

• question_answer45) The distance between the line$r=2\hat{i}-2\hat{j}+3\hat{k}+\lambda (\hat{i}-\hat{j}+4\hat{k})$and the plane$r.(\hat{i}+5\hat{j}+\hat{k})=5$is     AIEEE  Solved  Paper-2005

A)
$\frac{10}{3}$

B)
$\frac{3}{10}$

C)
$\frac{10}{3\sqrt{3}}$

D)
$\frac{10}{9}$

• question_answer46) For  any  vector a,   the  value of${{(a\times \hat{i})}^{2}}+{{(a\times \hat{j})}^{2}}+{{(a\times \hat{k})}^{2}}$is equal to     AIEEE  Solved  Paper-2005

A)
$4{{a}^{2}}$

B)
$2{{a}^{2}}$

C)
${{a}^{2}}$

D)
$3{{a}^{2}}$

• question_answer47) If non-zero numbers a, b, c are in HP, then the straight line $\frac{x}{a}+\frac{y}{b}+\frac{1}{c}=0$always passes through a fixed point. That point is     AIEEE  Solved  Paper-2005

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

B)
$(1,-2)$

C)
$(-1,-2)$

D)
$(-1,2)$

• question_answer48) It a vertex of a triangle is (1, 1) and the mid-points of two sides through this vertex are (-1, 2) and (3, 2), then the centroid of the triangle is     AIEEE  Solved  Paper-2005

A)
$\left( \frac{1}{3},\frac{7}{3} \right)$

B)
$\left( 1,\frac{7}{3} \right)$

C)
$\left( -\frac{1}{3},\frac{7}{3} \right)$

D)
$\left( -1,\frac{7}{3} \right)$

• question_answer49) If the circles${{x}^{2}}+{{y}^{2}}+2ax+cy+a=0$and ${{x}^{2}}+{{y}^{2}}-3\text{ }ax+dy-1=0$intersect in two distinct points P and 0, then the line $5x+by-a=0$passes through P and Q for     AIEEE  Solved  Paper-2005

A)
exactly two values of a

B)
infinitely many values of a

C)
no value of a

D)
exactly one value of a

• question_answer50) A circle touches the X-axis and also touches the circle with centre at (0, 3) and radius 2. The locus of the centre of the circle is       AIEEE  Solved  Paper-2005

A)
a parabola

B)
a hyperbola

C)
a circle

D)
an ellipse

• question_answer51) If a circle passes through the point (a, b) and  cuts  the  circle${{x}^{2}}+{{y}^{2}}={{p}^{2}}$orthogonally, then the equation of the locus of its centre is     AIEEE  Solved  Paper-2005

A)
$2ax+2by-({{a}^{2}}+{{b}^{2}}+{{p}^{2}})=0$

B)
${{x}^{2}}+{{y}^{2}}-2ax-3by+({{a}^{2}}-{{b}^{2}}-{{p}^{2}})=0$

C)
$2ax+2by-({{a}^{2}}-{{b}^{2}}+{{p}^{2}})=0$

D)
${{x}^{2}}+{{y}^{2}}-3ax-4by+({{a}^{2}}+{{b}^{2}}-{{p}^{2}})=0$

• question_answer52) An ellipse has OB as semi-minor axis, F and F' its foci and the angle FBF' is a right angle. Then, the eccentricity of the ellipse is     AIEEE  Solved  Paper-2005

A)
$1/\sqrt{3}$

B)
$1/4$

C)
$1/2$

D)
$1/\sqrt{2}$

• question_answer53) The locus of a point $P(\alpha ,\beta )$ moving under the condition that the line$y=\alpha x+\beta$is a tangent to the hyperbola$\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1$is     AIEEE  Solved  Paper-2005

A)
a hyperbola

B)
a parabola

C)
a circle

D)
an ellipse

• question_answer54) If the angle$\theta$between the line$\frac{x+1}{1}=\frac{y-1}{2}=\frac{z-2}{2}$and  the  plane $2x-y+\sqrt{\lambda }z+4=0$is such that $\sin \theta =\frac{1}{3}$The value of$\lambda$is     AIEEE  Solved  Paper-2005

A)
$-\frac{4}{3}$

B)
$\frac{3}{4}$

C)
$-\frac{3}{5}$

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

• question_answer55) The angle between the lines$2x=3y=-z$and $6x=-y=-4z$is     AIEEE  Solved  Paper-2005

A)
$30{}^\circ$

B)
$45{}^\circ$

C)
$90{}^\circ$

D)
$0{}^\circ$

• question_answer56) Let A and B be two events such that $P\overline{(A\cup B)}=\frac{1}{6},P(A\cap B)=\frac{1}{4}$and$P(\overline{A})=\frac{1}{4},$where $\overline{A}$ stands for complement of event A. Then, events A and B are     AIEEE  Solved  Paper-2005

A)
mutually exclusive and independent

B)
independent but not equally likely

C)
equally likely but not independent

D)
equally likely and mutuaiiy exclusive

• question_answer57) Three houses are available in a locality. Three persons apply for the houses. Each applies for one house without consulting others. The probability that all the three apply for the same house, is     AIEEE  Solved  Paper-2005

A)
7/9

B)
8/9

C)
1/9

D)
2/9

• question_answer58) A random variable X has Poisson distribution with mean 2. Then,$P(X>1.5)$equals     AIEEE  Solved  Paper-2005

A)
$\frac{3}{{{e}^{2}}}$

B)
$1-\frac{3}{{{e}^{2}}}$

C)
0

D)
$\frac{2}{{{e}^{2}}}$

• question_answer59) Two points A and B move from rest along a straight line with constant acceleration f and f?, respectively. If A takes m second more than B and describes n unit more than B in acquiring the same speed, then     AIEEE  Solved  Paper-2005

A)
$(f'-f)n=\frac{1}{2}ff'{{m}^{2}}$

B)
$\frac{1}{2}(f+f')m=ff'{{n}^{2}}$

C)
$(f+f'){{m}^{2}}=ff'n$

D)
$(f+f'){{m}^{2}}=ff'n$

• question_answer60) A lizard, at an initial distance of 21 cm behind an insect, moves from rest with an acceleration of$2\text{ }cm/{{s}^{2}}$and pursues the insect which is crawling uniformly along a straight line at a speed of 20 cm/s. Then, the lizard will catch the insect after     AIEEE  Solved  Paper-2005

A)
24 s

B)
21 s

C)
1 s

D)
20 s

• question_answer61) The resultant R of two forces acting on a particle is at right angles to one of them and its magnitude is one-third of the other force. The ratio of larger force to smaller one is     AIEEE  Solved  Paper-2005

A)
$3:2\sqrt{2}$

B)
$3:2$

C)
$3:\sqrt{2}$

D)
$2:1$

• question_answer62) Let$a=\hat{i}-\hat{k},b=x\hat{i}+\hat{j}+(1-x)\hat{k}$and $c=y\hat{i}+x\hat{j}+(1+x-y)\hat{k}$. Then,$[a\,\,b\,\,c]$depends on     AIEEE  Solved  Paper-2005

A)
neither$x$nor y

B)
both$x$and y

C)
only $x$

D)
only y

• question_answer63) Let a, b and c be distinct non-negative numbers. If the vectors$a\hat{i}+a\hat{j}+c\hat{k},\hat{i}+\hat{k}$and $c\hat{i}+c\hat{j}+b\hat{k}$ lie in a plane, then c is     AIEEE  Solved  Paper-2005

A)
the harmonic mean of a and b

B)
equal to zero

C)
the arithmetic mean of a and b

D)
the geometric mean of a and b

• question_answer64) If a, b, b are non-coplanar vectors and $\lambda$, is a real number, then$[\lambda (a+b){{\lambda }^{2}}\,b\,\,\lambda c]=[ab+cd]$for     AIEEE  Solved  Paper-2005

A)
exactly two values of $\lambda$

B)
exactly three values of $\lambda$

C)
no value of$\lambda$

D)
exactly one value of$\lambda$

• question_answer65) A and B are two like parallel forces. A couple of moment H lies in the plane of A and B and is contained with them. The resultant of A and B after combining is displaced through a distance     AIEEE  Solved  Paper-2005

A)
$\frac{H}{A-B}$

B)
$\frac{H}{2(A+B)}$

C)
$\frac{H}{A+B}$

D)
$\frac{2H}{A-B}$

• question_answer66) The sum of the series$1+\frac{1}{4.2!}+\frac{1}{16.4!}+\frac{1}{64.6!}+.....\infty$is     AIEEE  Solved  Paper-2005

A)
$\frac{e+1}{2\sqrt{e}}$

B)
$\frac{e-1}{2\sqrt{e}}$

C)
$\frac{e+1}{\sqrt{e}}$

D)
$\frac{e-1}{\sqrt{e}}$

• question_answer67) Let${{x}_{1}},{{x}_{2}},.....,{{x}_{n}}$be n observations such that $\Sigma x_{i}^{2}=400$and$\Sigma {{x}_{i}}=80$. Then, a possible value of n among the following is     AIEEE  Solved  Paper-2005

A)
12

B)
9

C)
18

D)
15

• question_answer68) A particle is projected from a point O with velocity u at an angle of$60{}^\circ$with the horizontal. When it is moving in a direction at right angle to its direction at O, then its velocity is given by     AIEEE  Solved  Paper-2005

A)
$\frac{u}{\sqrt{3}}$

B)
$\frac{2u}{3}$

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

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

• question_answer69) If both the roots of the quadratic equation${{x}^{2}}-2kx+{{k}^{2}}+k-5=0$are less than 5, then k lies in the interval     AIEEE  Solved  Paper-2005

A)
[4, 5]

B)
$(-\infty ,4)$

C)
$(6,\infty )$

D)
(5, 6]

• question_answer70) If${{a}_{1}},{{a}_{2}},{{a}_{3}},....,{{a}_{n}},....$are in GP, then the determinant $\Delta =\left| \begin{matrix} \log {{a}_{n}} & \log {{a}_{n+1}} & \log {{a}_{n+2}} \\ \log {{a}_{n+3}} & \log {{a}_{n+4}} & \log {{a}_{n+5}} \\ \log {{a}_{n+6}} & \log {{a}_{n+7}} & {{\log }_{n+8}} \\ \end{matrix} \right|$is equal to     AIEEE  Solved  Paper-2005

A)
2

B)
4

C)
0

D)
1

• question_answer71) A real valued function$f(x)$satisfies the functional equation $f(x-y)=f(x)\text{ }f(y)-f(a-x)\text{ }f(a+y)$ where, a is a given constant and$f(0)=1$$f(2\text{ }a-x)$is equal to     AIEEE  Solved  Paper-2005

A)
$f(-\text{ }x)$

B)
$f(a)+f(a-x)$

C)
$f(x)$

D)
$-f(x)$

• question_answer72) If the equation ${{a}_{n}}{{X}^{n}}+{{a}_{n-1}}{{X}^{n-1}}+....+{{a}_{1}}x=0,$ ${{a}_{1}}\ne 0,n\ge 2,$has a positive root $x=\alpha ,$ then the equation $n{{a}_{n}}{{x}^{n-1}}+(n-1){{a}_{n-1}}{{X}^{n-2}}+....+{{a}_{1}}=0$ has a positive root, which is     AIEEE  Solved  Paper-2005

A)
equal to$\alpha$

B)
greater than or equal to$\alpha$

C)
smaller than$\alpha$

D)
greater than $\alpha$

• question_answer73) The value of $\int_{-\pi }^{\pi }{\frac{{{\cos }^{2}}x}{1+{{a}^{x}}}}dx,a>0,$is     AIEEE  Solved  Paper-2005

A)
$2\pi$

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

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

D)
$a\pi$

• question_answer74) The plane$x+2y-z=4$cuts the sphere${{x}^{2}}+{{y}^{2}}+{{z}^{2}}-x+z-2=0$in a circle of radius     AIEEE  Solved  Paper-2005

A)
$\sqrt{2}$

B)
2

C)
1

D)
3

• question_answer75) If the pair of linesa${{x}^{2}}+2(a+b)xy+b{{y}^{2}}=0$ lie along diameters of a circle and divide the circle into four sectors such that the area of one of the sector is thrice the area of another sector, then     AIEEE  Solved  Paper-2005

A)
$3{{a}^{2}}+2ab+3{{b}^{2}}=0$

B)
$3{{a}^{2}}+10ab+3{{b}^{2}}=0$

C)
$3{{a}^{2}}-2ab+3{{b}^{2}}=0$

D)
$3{{a}^{2}}-10ab+3{{b}^{2}}=0$