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

AIEEE Solved Paper-2005 Total Questions - 75

1) 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

View Answer
2) 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)

       \[{{y}^{2}}-4x+2=0\]

View Answer
3) 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

View Answer
4) 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

View Answer
5) 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)

       \[A+l\]

View Answer
6) 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)

\[-1,-1+2\omega ,-1-2{{\omega }^{2}}\]

View Answer
7) \[\underset{n\to \infty }{\mathop{\lim }}\,\left[ \frac{1}{{{n}^{2}}}{{\sec }^{2}}\frac{1}{{{n}^{2}}}+\frac{2}{{{n}^{2}}}{{\sec }^{2}}\frac{4}{{{n}^{2}}} \right.\] \[\left. +....+\frac{n}{{{n}^{2}}}{{\sec }^{2}}1 \right]\] 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)

       \[\frac{1}{2}sec\text{ }1\]

View Answer
8) 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)

       \[2ab\]

View Answer
9) 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

View Answer
10) 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)

\[\sin A:\sin B:\sin C\]

View Answer
11) 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)

\[{{m}^{2}}-m(4r-1)+4{{r}^{2}}-2=0\]

View Answer
12) 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)

\[a=o+c,\]

View Answer
13) 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

View Answer
14) 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)

       \[^{55}{{C}_{4}}\]

View Answer
15) 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)

\[{{A}^{n}}=nA+(n-1)l\]

View Answer
16) 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)

       \[a-d=1\]

View Answer
17) 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)

       \[\left( 0,\frac{\pi }{2} \right)\]

View Answer
18) 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)

       \[\frac{\pi }{2}\]

View Answer
19) 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

View Answer
20) 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

View Answer
21) 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

View Answer
22) 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

View Answer
23) 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

View Answer
24) 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

View Answer
25) 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)

       \[f(6)\ge 8\]

View Answer
26) 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

View Answer
27) 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)

       \[1-\frac{3}{8}{{x}^{2}}\]

View Answer
28) 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

View Answer
29) 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)

       \[b+c\]

View Answer
30) 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)

       \[2\text{ }{{\sin }^{2}}\alpha \]

View Answer
31) 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

View Answer
32) 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

View Answer
33) 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)

\[(-\infty ,\infty )\]                          \[{{x}^{3}}-3{{x}^{2}}+3x+3\]

View Answer
34) 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)

       \[\frac{{{a}^{2}}}{2}{{(\alpha -\beta )}^{2}}\]

View Answer
35) 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)

\[y\log \left( \frac{x}{y} \right)=Cx\]

View Answer
36) 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

View Answer
37) 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)

\[\frac{1}{36\pi }cm/\min \]

View Answer
38) \[{{\int{\left\{ \frac{(\log x-1)}{1+{{(\log \,x)}^{2}}} \right\}}}^{2}}dx\]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)

\[\frac{\log \,\,x}{\,{{(\log x)}^{2}}+1}+C\]

View Answer
39) 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

View Answer
40) 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)

       \[\left( \frac{\pi }{4}+\sqrt{2}-1 \right)\]

View Answer
41) 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)

       \[{{l}_{2}}>{{l}_{1}}\]

View Answer
42) 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

View Answer
43) 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

View Answer
44) 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)

       \[-1\]

View Answer
45) 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)

\[\frac{10}{9}\]

View Answer
46) 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)

       \[3{{a}^{2}}\]

View Answer
47) 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)

       \[(-1,2)\]

View Answer
48) 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)

       \[\left( -1,\frac{7}{3} \right)\]

View Answer
49) 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

View Answer
50) 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

View Answer
51) 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)

\[{{x}^{2}}+{{y}^{2}}-3ax-4by+({{a}^{2}}+{{b}^{2}}-{{p}^{2}})=0\]

View Answer
52) 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)

       \[1/\sqrt{2}\]

View Answer
53) 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

View Answer
54) 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)

       \[\frac{5}{3}\]

View Answer
55) 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)

       \[0{}^\circ \]

View Answer
56) 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

View Answer
57) 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

View Answer
58) 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)

       \[\frac{2}{{{e}^{2}}}\]

View Answer
59) 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)

\[(f+f'){{m}^{2}}=ff'n\]

View Answer
60) 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

View Answer
61) 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)

       \[2:1\]

View Answer
62) 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

View Answer
63) 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

View Answer
64) 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\[\lambda \]

View Answer
65) 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)

       \[\frac{2H}{A-B}\]

View Answer
66) 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)

       \[\frac{e-1}{\sqrt{e}}\]

View Answer
67) 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

View Answer
68) 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)

       \[\frac{u}{3}\]

View Answer
69) 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]

View Answer
70) 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

View Answer
71) 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)

\[-f(x)\]

View Answer
72) 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 \[\alpha \]

View Answer
73) 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)

       \[a\pi \]

View Answer
74) 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

View Answer
75) 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)

\[3{{a}^{2}}-10ab+3{{b}^{2}}=0\]

View Answer

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AIEEE Solved Paper-2005

AIEEE Solved Paper-2005 Total Questions - 75