# Solved papers for JEE Main & Advanced AIEEE Solved Paper-2003

### done AIEEE Solved Paper-2003 Total Questions - 75

• question_answer1)              A function $f$ from the set of natural numbers to integers defined by             f(n)=\left\{ \begin{align} & \frac{n-1}{n},\text{ }when\text{ }n\text{ }is\text{ }odd \\ & \frac{-n}{2},\,\,when\text{ }n\text{ }is\text{ }even \\ \end{align} \right.  is     AIEEE  Solved  Paper-2003

A)
one-one but not onto

B)
onto but not one-one

C)
one-one and onto both

D)
neither one-one nor onto

• question_answer2) Let ${{z}_{1}}$ and ${{z}_{2}}$ be two roots of the equation ${{z}_{2}}+az+b=0,\,\,z$ being complex. Further, assume that the origin, ${{z}_{1}}$ and ${{z}_{2}}$ form an equilateral triangle. Then,     AIEEE  Solved  Paper-2003

A)
${{a}^{2}}=b$

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

C)
${{a}^{2}}=3b$

D)
${{a}^{2}}=4b$

• question_answer3) If z and $\omega$ are two non-zero complex numbers  such  that  $\left| z\,\omega \right|=1$ and arg (z) - arg $(\omega )=\frac{\pi }{2}$, then $\overline{z}\omega$ is equal to     AIEEE  Solved  Paper-2003

A)
1

B)
-1

C)
$i$

D)
$-i$

• question_answer4) If ${{\left( \frac{1+i}{1-i} \right)}^{x}}=1$ , then             AIEEE  Solved  Paper-2003

A)
$x=4\,n$, where n is any positive integer

B)
$x=2\,n$, where n is any positive integer

C)
$x=4\,n+1$, where n is any positive integer

D)
$x=2\,n+1$, where n is any positive integer

• question_answer5) If $\left| \begin{matrix} a & {{a}^{2}} & 1+{{a}^{3}} \\ b & {{b}^{2}} & 1+{{b}^{3}} \\ c & {{c}^{2}} & 1+{{c}^{3}} \\ \end{matrix} \right|=0$ and vectors $(1,\,\,a,\,\,{{a}^{2}})$, $(1,\,\,a,\,\,{{a}^{2}})$ and $(1,\,\,c,\,\,{{c}^{2}})$ are non-coplanar, then the product abc equals     AIEEE  Solved  Paper-2003

A)
2

B)
-1

C)
1

D)
0

• question_answer6) If the system of linear equations                 $x+2$ ay $+\,az=0$                 $x+3$ by $+\,bz=0$ and        $x+4$ cy $+cz=0$ has a non-zero solution, then a, b, c     AIEEE  Solved  Paper-2003

A)
are in AP

B)
are in GP

C)
are in HP

D)
satisfy a+2b+3c=0

• question_answer7) If the sum of the roots of the quadratic equation $a{{x}^{2}}+bx+c=0$ is equal to the sum of the squares f their reciprocals, then $\frac{a}{c},\frac{b}{a}$, and $\frac{c}{b}$ are in     AIEEE  Solved  Paper-2003

A)
arithmetic progression

B)
geometric progression

C)
harmonic progression

D)
arithmetico-geometric progression

• question_answer8) The number of the real solutions of the equation ${{x}^{2}}-3\left| x \right|+2=0$ is

A)
2

B)
4

C)
1

D)
3

• question_answer9) The value of a for which one root of the quadratic equation             $({{a}^{2}}-5a+3)\,{{x}^{2}}+(3a-1)\,x+2=0$ is twice as large as the other, is     AIEEE  Solved  Paper-2003

A)
2/3

B)
-2/3

C)
1/3

D)
-1/3

• question_answer10) If $A=\left| \begin{matrix} a & b \\ b & a \\ \end{matrix} \right|$ and ${{A}^{2}}=\left| \begin{matrix} \alpha & \beta \\ \beta & \alpha \\ \end{matrix} \right|$,  then     AIEEE  Solved  Paper-2003

A)
$\alpha ={{a}^{2}}+{{b}^{2}},\,\beta =ab$

B)
$\alpha ={{a}^{2}}+{{b}^{2}},\,\beta =2ab$

C)
$\alpha ={{a}^{2}}+{{b}^{2}},\,\beta ={{a}^{2}}-{{b}^{2}}$

D)
$\alpha =2ab,\,\,\beta ={{a}^{2}}+{{b}^{2}}$

• question_answer11) A student is to answer 10 out of 13 questions in an examination such that he must choose atleast 4 from the first five questions. The number of choices available to him is     AIEEE  Solved  Paper-2003

A)
140

B)
196

C)
280

D)
346

• question_answer12) The number of ways in which 6 men and 5 women can dine at a round table, if no two women are to sit together, is given by     AIEEE  Solved  Paper-2003

A)
$6!\,\,\times 5!$

B)
30

C)
$5!\,\,\times 4!$

D)
$7!\,\,\times 5!$

• question_answer13) If $1,\,\omega ,\,{{\omega }^{2}}$are the cube roots of unity, then                 $\Delta =\left| \begin{matrix} 1 & {{\omega }^{n}} & {{\omega }^{2n}} \\ {{\omega }^{n}} & {{\omega }^{2n}} & 1 \\ {{\omega }^{2n}} & 1 & {{\omega }^{n}} \\ \end{matrix} \right|$ is equal to     AIEEE  Solved  Paper-2003

A)
0

B)
1

C)
$\omega$

D)
${{\omega }^{2}}$

• question_answer14) If $^{n}{{C}_{r}}$ denotes the number of combinations of n things taken r at a time, then the expression $^{n}{{C}_{r+1}}{{+}^{n}}{{C}_{r-1}}+2{{\times }^{n}}{{C}_{r}}$ equals     AIEEE  Solved  Paper-2003

A)
$^{n+2}{{C}_{r}}$

B)
$^{n+2}{{C}_{r+1}}$

C)
$^{n+1}{{C}_{r}}$

D)
$^{n+1}{{C}_{r+1}}$

• question_answer15) The number of integral terms in the expansion of ${{(\sqrt{3}+\sqrt[8]{5})}^{256}}$ is     AIEEE  Solved  Paper-2003

A)
32

B)
33

C)
34

D)
35

• question_answer16) If $x$ is positive, the first negative term in the expansion of ${{(1+x)}^{27/5}}$ is     AIEEE  Solved  Paper-2003

A)
7th term

B)
5th term

C)
8th term

D)
6th term

• question_answer17) The     sum     of     the     series $\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{3.4}-$... upto $\infty$ is equal to     AIEEE  Solved  Paper-2003

A)
$2\,\,{{\log }_{e}}2$

B)
${{\log }_{e}}\,\,2-1$

C)
${{\log }_{e}}\,\,2$

D)
${{\log }_{e}}\,\,\left( \frac{4}{e} \right)$

• question_answer18) Let $f(x)$ be a polynomial function of second degree. If $f(1)=f(-1)$ and a, b, c are in AP, then $f'\,(a),\,\,f'(b)$ and $f'(c)$ are in     AIEEE  Solved  Paper-2003

A)
AP

B)
GP

C)
HP

D)
arithmetico-geometric progression

• question_answer19) If ${{x}_{1}},{{x}_{2}},{{x}_{3}}$ and ${{y}_{1}},{{y}_{2}},{{y}_{3}}$ are both in GP with the same common ratio, then the points $({{x}_{1}},{{y}_{1}}),\,({{x}_{2}},{{y}_{2}})$ and $({{x}_{3}},{{y}_{3}})$      AIEEE  Solved  Paper-2003

A)
lie on a straight

B)
line lie on an ellipse

C)
lie on a circle

D)
are vertices of a triangle

• question_answer20) The sum of the radii of inscribed and circumscribed circles for an n sided regular polygon of side a, is     AIEEE  Solved  Paper-2003

A)
$a\cot \,\left( \frac{\pi }{v} \right)$

B)
$\frac{a}{2}\cot \left( \frac{\pi }{2n} \right)$

C)
$a\cot \,\left( \frac{\pi }{2n} \right)$

D)
$\frac{a}{4}\cot \left( \frac{\pi }{2n} \right)$

• question_answer21) If a $\Delta ABC$,                 $a{{\cos }^{2}}\left( \frac{C}{2} \right)+c{{\cos }^{2}}\left( \frac{A}{2} \right)=\frac{3b}{2}$ then the sides a, b and c     AIEEE  Solved  Paper-2003

A)
are in AP

B)
are in GP

C)
are in HP

D)
satisfy a + b = c

• question_answer22) In $\Delta ABC$, medians AD and BE are drawn. If AD = 4, $\angle DAB=\frac{\pi }{6}$ and $\angle ABE=\frac{\pi }{3}$, then the area of the $\Delta ABC$ is     AIEEE  Solved  Paper-2003

A)
8/3

B)
16/3

C)
32/3

D)
64/3

• question_answer23) The trigonometric equation ${{\sin }^{-1}}x=2{{\sin }^{-1}}a$, has a solution for     AIEEE  Solved  Paper-2003

A)
$-\frac{1}{\sqrt{2}}<a<\frac{1}{\sqrt{2}}$

B)
all real values of a

C)
$\left| a \right|<\frac{1}{2}$

D)
$\left| a \right|\ge \frac{1}{\sqrt{2}}$

• question_answer24) The upper 3/4th portion of a vertical pole subtends an angle ${{\tan }^{-1}}3/5$ at a point in the horizontal plane through its foot and at a distance 40 m from the foot. A possible height of the vertical pole is     AIEEE  Solved  Paper-2003

A)
20m

B)
40m

C)
60m

D)
80 m

• question_answer25) The real number $x$ when added to its inverse gives the minimum sum at $x$ equals to     AIEEE  Solved  Paper-2003

A)
2

B)
1

C)
-1

D)
-2

• question_answer26) If $f:R\to R$ satisfies$f(x+y)=f(x)+f(y)$, for all $x,y\in R$ and $f(1)=7$, then $\sum\limits_{r=1}^{n}{f(r)}$ is     AIEEE  Solved  Paper-2003

A)
$\frac{7n}{2}$

B)
$\frac{7(n+1)}{2}$

C)
$7n\,(n+1)$

D)
$\frac{7n\,(n+1)}{2}$

• question_answer27) If $f(x)={{x}^{n}}$, then the value of $f(1)-\frac{f'(1)}{1!}+\frac{f''(1)}{2!}-\frac{f'''(1)}{3!}+...+\frac{{{(-1)}^{n}}{{f}^{n}}(1)}{n!}$     AIEEE  Solved  Paper-2003

A)
${{2}^{n}}$

B)
${{2}^{n-1}}$

C)
0

D)
1

• question_answer28) Domain of definition of the function $f(x)=\frac{3}{4-{{x}^{2}}}+{{\log }_{10}}({{x}^{3}}-x)$, is     AIEEE  Solved  Paper-2003

A)
(1, 2)

B)
$(-1,0)\cup (1,2)$

C)
$(1,2)\cup (2,\infty )$

D)
$(-1,0)\cup (1,2,)\cup (2,\infty )$

• question_answer29) $\underset{x\to \frac{\pi }{2}}{\mathop{\lim }}\,\frac{\left[ 1-\tan \left( \frac{x}{2} \right) \right][1-\sin x]}{\left[ 1+\tan \left( \frac{x}{2} \right) \right]{{[x-2x]}^{3}}}$  is     AIEEE  Solved  Paper-2003

A)
$\frac{1}{8}$

B)
0

C)
$\frac{1}{32}$

D)
$\infty$

• question_answer30) If $\underset{x\to 0}{\mathop{\lim }}\,\frac{\log \,(3+x)-\log \,(3-x)}{x}=k$ then value of k is     AIEEE  Solved  Paper-2003

A)
0

B)
-1/3

C)
2/3

D)
-2/3

• question_answer31) Let f(a) = g(a) = k and their nth derivatives ${{f}^{n}}(a),\,{{g}^{n}}(a)$ exist and are not equal for some n. Further, if $\underset{x\to a}{\mathop{\lim }}\,\frac{f(a)\,g(x)-i(a)-g(a)f(x)+g(a)}{g(x)-f(x)}=4$, then the value of k is equal to     AIEEE  Solved  Paper-2003

A)
4

B)
2

C)
1

D)
0

• question_answer32) The function $f(x)=\log \,(x+\sqrt{\,{{x}^{2}}+1})$, is     AIEEE  Solved  Paper-2003

A)
an even function

B)
an odd function

C)
a periodic function

D)
neither an even nor an odd function

• question_answer33) If $f(x)=\left\{ \begin{matrix} x{{e}^{-\left[ \frac{1}{\left| x \right|}+\frac{1}{x} \right]}} & ,x\ne 0 \\ 0 & ,x=0 \\ \end{matrix} \right.$ , then f(x) is     AIEEE  Solved  Paper-2003

A)
continuous as well as differentiable for all x

B)
continuous for all x but not differentiable at $x=0$

C)
neither differentiable nor continuous at $x=0$

D)
discontinuous everywhere

• question_answer34) If the function $f(x)=2{{x}^{3}}-9a{{x}^{2}}+12{{a}^{2}}x+1$, where a > 0, attains its maximum and minimum at p and q respectively such that ${{p}^{2}}=q$, then a equals     AIEEE  Solved  Paper-2003

A)
3

B)
1

C)
2

D)
1/2

• question_answer35) If $f(y)={{e}^{y}},g(y)=y;y>0$ and$F(t)=\int_{0}^{t}{f(t-y)\,g(y)\,dy}$, then      AIEEE  Solved  Paper-2003

A)
$F(t)=1-{{e}^{-t}}(1+t)$

B)
$F(t)\,={{e}^{t}}\,-(1+t)$

C)
$F(t)=t{{e}^{-t}}$

D)
$F(t)=t{{e}^{-t}}$

• question_answer36) If $f(a+b-x)=f(x)$, then $\int_{a}^{b}{x\,f(x)\,dx}$ is equal to     AIEEE  Solved  Paper-2003

A)
$\frac{a+b}{2}\int_{a}^{b}{f(b-x)\,dx}$

B)
$\frac{a+b}{2}\int_{a}^{b}{f(x)\,dx}$

C)
$\frac{b-a}{2}\int_{a}^{b}{f(x)\,dx}$

D)
$\frac{a+b}{2}\int_{a}^{b}{f(a+b+x)\,dx}$

• question_answer37) The value of $\underset{x\to 0}{\mathop{\lim }}\,\frac{\int_{0}^{{{x}^{2}}}{{{\sec }^{2}}t\,dt}}{x\sin x}$ is     AIEEE  Solved  Paper-2003

A)
3

B)
2

C)
1

D)
-1

• question_answer38) The    value    of    the    integral $I=\int_{0}^{1}{x{{(1-x)}^{n}}dx}$ is     AIEEE  Solved  Paper-2003

A)
$\frac{1}{n+1}$

B)
$\frac{1}{n+2}$

C)
$\frac{1}{n+1}-\frac{1}{n+2}$

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

• question_answer39) $\underset{x\to \infty }{\mathop{\lim }}\,\frac{1+{{2}^{4}}+{{3}^{4}}+....+{{n}^{4}}}{{{n}^{5}}}$                                                                             $-\underset{x\to \infty }{\mathop{\lim }}\,\frac{1+{{2}^{3}}+{{3}^{3}}+....+{{n}^{3}}}{{{n}^{5}}}$ is

A)
1/30

B)
0

C)
1/4

D)
1/5

• question_answer40) Let $\frac{d}{dx}F(x)=\left( \frac{{{e}^{\sin x}}}{x} \right),x>0$. If $\int_{1}^{4}{\,\,\,\frac{3}{x}{{e}^{\sin {{x}^{3}}}}dx=F(k)-F(1)}$. then one of the possible value of k, is     AIEEE  Solved  Paper-2003

A)
15

B)
16

C)
63

D)
64

• question_answer41) The area of the region bounded by the curves $y=\left| x-1 \right|$ and $y=3-\left| x \right|$ is     AIEEE  Solved  Paper-2003

A)
2 sq units

B)
3 sq units

C)
4 sq units

D)
6 sq units

• question_answer42) Let $f(x)$ be a function satisfying $f'(x)=f(x)$ with $f(0)=1$ and $g(x)$ be a function that satisfies $f(x)+g(x)={{x}^{2}}$. Then, the value of the integral $\int_{0}^{1}{f(x)\,g(x)\,dx}$, is     AIEEE  Solved  Paper-2003

A)
$e-\frac{{{e}^{2}}}{2}-\frac{5}{2}$

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

C)
$-\frac{{{e}^{2}}}{2}-\frac{3}{2}$

D)
$e+\frac{{{e}^{2}}}{2}+\frac{5}{2}$

• question_answer43) The degree and order of the differential equation of the family of all parabolas whose axis is X-axis, are respectively     AIEEE  Solved  Paper-2003

A)
2, 1

B)
1, 2

C)
3, 2

D)
2, 3

• question_answer44) The solution of the differential equation $(1+{{y}^{2}})+(x-{{e}^{{{\tan }^{-1}}}}y)\frac{dy}{dx}=0$, is     AIEEE  Solved  Paper-2003

A)
$(x-2)=k{{e}^{-{{\tan }^{-1}}y}}$

B)
$2\,x\,{{e}^{-{{\tan }^{-1}}y}}={{e}^{2\,{{\tan }^{-1}}y}}+k$

C)
$x{{e}^{{{\tan }^{-1}}}}={{\tan }^{-1}}\,y+k$

D)
$x{{e}^{2\,\,{{\tan }^{-1}}y}}={{e}^{{{\tan }^{-1}}y}}+k$

• question_answer45) If the equation of the locus of a point equidistant from the points $({{a}_{1}}-{{b}_{1}})$ and $({{a}_{2}}-{{b}_{2}})$ is $({{a}_{1}}-{{a}_{2}})x+({{b}_{1}}-{{b}_{2}})\,y+c=0$, then the value of 'c' is     AIEEE  Solved  Paper-2003

A)
$\frac{1}{2}(a_{2}^{2}+b_{2}^{2}-a_{1}^{2}-b_{1}^{2})$

B)
$a_{1}^{2}-a_{2}^{2}+b_{1}^{2}-b_{2}^{2}$

C)
$\frac{1}{2}\,\,(a_{1}^{2}+a_{2}^{2}+b_{1}^{2}+b_{2}^{2})$

D)
$\sqrt{a_{1}^{2}+b_{1}^{2}-a_{2}^{2}-b_{2}^{2}}$

• question_answer46) Locus of centroid of the triangle whose vertices are (b sin t, - b cost) and (1, 0), where t is a parameter, is     AIEEE  Solved  Paper-2003

A)
${{(3x-1)}^{2}}+{{(3y)}^{2}}={{a}^{2}}-{{b}^{2}}$

B)
${{(3x-1)}^{2}}+{{(3y)}^{2}}={{a}^{2}}+{{b}^{2}}$

C)
${{(3x+1)}^{2}}+{{(3y)}^{2}}={{a}^{2}}+{{b}^{2}}$

D)
${{(3x+1)}^{2}}+{{(3y)}^{2}}={{a}^{2}}-{{b}^{2}}$

• question_answer47) If   the   pair   of   straight   lines ${{x}^{2}}-2pxy-{{y}^{2}}=0$and ${{x}^{2}}-2qxy-{{y}^{2}}=0$ be such that each pair bisects the angle between the other pair, then     AIEEE  Solved  Paper-2003

A)
$p=q$

B)
$p=-q$

C)
$pq=1$

D)
$pq=-1$

• question_answer48) A square of side a lies above the X-axis and has one vertex at the origin. The side passing through the origin makes an angle $\left( 0<\alpha <\frac{\pi }{4} \right)$ with the positive direction of X-axis. The equation of its diagonal not passing through the origin is     AIEEE  Solved  Paper-2003

A)
$y(\cos \alpha -\sin \alpha )-x(\sin \alpha -\cos \alpha )=a$

B)
$y(\cos \alpha +\sin \alpha )+x(\sin \alpha -\cos \alpha )=a$

C)
$y(\cos \alpha +\sin \alpha )+x(\sin \alpha +\cos \alpha )=a$

D)
$y(\cos \alpha +\sin \alpha )+x(\cos \alpha -\sin \alpha )=a$

• question_answer49) If the two circles ${{(x-1)}^{2}}+{{(y-3)}^{2}}={{r}^{2}}$ and ${{x}^{2}}+{{y}^{2}}-8x+2y+8=0$ intersect in two distinct points, then     AIEEE  Solved  Paper-2003

A)
$2<r<8$

B)
$r<2$

C)
$r=2$

D)
$r>2$

• question_answer50) The lines $2x-3y=5$ and $3x-4y=7$ are diameters of a circle having area as 154 sq units. Then, the equation of the circle is     AIEEE  Solved  Paper-2003

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

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

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

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

• question_answer51) The normal at the point $(bt_{1}^{2},b{{t}_{1}})$ on a parabola meets the parabola again in the point$(bt_{2}^{2},2b{{t}_{2}})$, then     AIEEE  Solved  Paper-2003

A)
${{t}_{2}}=-{{t}_{1}}-\frac{2}{{{t}_{1}}}$

B)
${{t}_{2}}=-{{t}_{1}}+\frac{2}{{{t}_{1}}}$

C)
${{t}_{2}}={{t}_{1}}-\frac{2}{{{t}_{1}}}$

D)
${{t}_{2}}={{t}_{1}}+\frac{2}{{{t}_{1}}}$

• question_answer52) The foci of the ellipse $\frac{{{x}^{2}}}{16}+\frac{{{y}^{2}}}{{{b}^{2}}}=1$ and the hyperbola $\frac{{{x}^{2}}}{144}-\frac{{{y}^{2}}}{81}=\frac{1}{25}$ coincide. Then, the value of ${{b}^{2}}$ is     AIEEE  Solved  Paper-2003

A)
1

B)
5

C)
7

D)
9

• question_answer53) A tetrahedron has vertices at O(0, 0, 0), A(1, 2, 3), B(2, 1, 3) and C(-1, 1, 2). Then, the angle between the faces OAB and ABC will be     AIEEE  Solved  Paper-2003

A)
${{\cos }^{-1}}\left( \frac{19}{35} \right)$

B)
${{\cos }^{-1}}\left( \frac{17}{31} \right)$

C)
${{30}^{o}}$

D)
${{90}^{o}}$

• question_answer54) The radius of the circle in which the sphere ${{x}^{2}}+{{y}^{2}}+{{z}^{2}}+2x-2y-4z-19=0$ is cut by the plane $x+2y+2z+7=0$, is     AIEEE  Solved  Paper-2003

A)
1

B)
2

C)
3

D)
4

• question_answer55) The lines $\frac{x-2}{1}=\frac{y-3}{1}=\frac{z-4}{-k}$ and $\frac{x-1}{k}=\frac{y-4}{2}=\frac{z-5}{1}$ are coplanar, if     AIEEE  Solved  Paper-2003

A)
k = 0 or - 1

B)
k = 1 or ? 1

C)
k = 0 or -3

D)
k = 3 or -3

• question_answer56) The two lines $x=ay\,+b,\,\,z=cy+d$ and$x=a'\,y+b',\,z=c'\,y+d'$ will be perpendicular, if and only if     AIEEE  Solved  Paper-2003

A)
$aa'+bb'+cc'+1=0$

B)
$aa'+bb'+cc'=0$

C)
$(a+a')\,(b+b')+(c+c')=0$

D)
$aa'+cc'+1=0$

• question_answer57) The shortest distance from the plane $12x+4y+3z=327$ to the sphere ${{x}^{2}}+{{y}^{2}}+{{z}^{2}}+4x-2y-6z=155$ is     AIEEE  Solved  Paper-2003

A)
26

B)
$11\frac{4}{13}$

C)
13

D)
39

• question_answer58) Two systems of rectangular axes have the same origin. If a plane cuts them at distances a, b, c and a', b?, c from the origin, then     AIEEE  Solved  Paper-2003

A)
$\frac{1}{{{a}^{2}}}+\frac{1}{{{b}^{2}}}+\frac{1}{{{c}^{2}}}+\frac{1}{a{{'}^{2}}}+\frac{1}{{{b}^{'2}}}+\frac{1}{c{{'}^{2}}}=0$

B)
$\frac{1}{{{a}^{2}}}+\frac{1}{{{b}^{2}}}-\frac{1}{{{c}^{2}}}+\frac{1}{a{{'}^{2}}}+\frac{1}{{{b}^{'2}}}-\frac{1}{c{{'}^{2}}}=0$

C)
$\frac{1}{{{a}^{2}}}-\frac{1}{{{b}^{2}}}-\frac{1}{{{c}^{2}}}+\frac{1}{a{{'}^{2}}}-\frac{1}{{{b}^{'2}}}-\frac{1}{c{{'}^{2}}}=0$

D)
$\frac{1}{{{a}^{2}}}+\frac{1}{{{b}^{2}}}+\frac{1}{{{c}^{2}}}-\frac{1}{a{{'}^{2}}}-\frac{1}{{{b}^{'2}}}-\frac{1}{c{{'}^{2}}}=0$

• question_answer59) a, b, c   are three vectors, such that $a+b+c=0,\,\,\left| a \right|=1,\,\,\left| b \right|=2,\,\,\left| c \right|=3$, then $a\,.\,b+b\,.\,\,c+c\,.\,\,a$ is equal to     AIEEE  Solved  Paper-2003

A)
0

B)
-7

C)
7

D)
1

• question_answer60) If u, v and w are three non-coplanar vectors, then (u + v - w) . [(u - v) $\times$ (v - w)] equals     AIEEE  Solved  Paper-2003

A)
0

B)
$u\,.\,v\times w$

C)
$u\,.\,w\times v$

D)
$3u\,.\,v\times w$

• question_answer61) Consider points A, B, C and D with position vectors  $7\hat{i}-4\hat{j}+7\hat{k}$, $\hat{i}-6\hat{j}+10\hat{k}$, $-\hat{i}-3\hat{j}+4\,\hat{k}$and $5\,\hat{i}-\hat{j}+5\,\hat{k}$, respectively. Then, ABCD is a     AIEEE  Solved  Paper-2003

A)
square

B)
rhombus

C)
rectangle

D)
parallelogram but not a rhombus None of the given option is correct.

• question_answer62) The vectors $AB=3\hat{i}+4\hat{k}$ and $AC=5\,\hat{i}-2\,\hat{j}+4\,\hat{k}$ are the sides of a $\Delta ABC$. The length of the median through A is     AIEEE  Solved  Paper-2003

A)
$\sqrt{18}$

B)
$\sqrt{72}$

C)
$\sqrt{33}$

D)
$\sqrt{288}$

• question_answer63) A particle acted on by constant forces $4\,\hat{i}+\hat{j}-3\,\hat{k}$ and $3\,\hat{i}+\hat{j}-\,\hat{k}$ is displaced from the point $\hat{i}+2\hat{j}+3\,\hat{k}$ to the point $5\,\hat{i}+4\,\hat{j}+\,\hat{k}$. The total work done by the forces is     AIEEE  Solved  Paper-2003

A)
20 units

B)
30 units

C)
40 units

D)
50 units

• question_answer64) Let $u=\hat{i}+\hat{j},\,v=\hat{i}-\hat{j}$ and $w=\hat{i}+2\,\hat{j}+3\,\,\hat{k}$. If $\hat{n}$ is a unit vector such that $u\,.\,\,\hat{n}=0$ and $v\,.\,\,\hat{n}=0$, then $\left| w\,.\,\,\hat{n} \right|$ is equal to     AIEEE  Solved  Paper-2003

A)
0

B)
1

C)
2

D)
3

• question_answer65) The median of a set of 9 distinct observations is 205. If each of the largest 4 observations of the set is increased by 2, then the median of the new set     AIEEE  Solved  Paper-2003

A)
is increased by 2

B)
is decreased by 2

C)
is two times the original median

D)
remains the same as that of the original set

• question_answer66) In an experiment with 15 observations on x, the following results were available $\sum {{x}^{2}}=2830,\,\,\sum x=170$. One observation that was 20, was found to be wrong and was replaced by the correct value 30. Then, the corrected variance is     AIEEE  Solved  Paper-2003

A)
78.00

B)
188.66

C)
177.33

D)
8.33

• question_answer67) Five horses are in a race. Mr A selects two of the horses at random and bets on them. The probability that Mr A selected the winning horse, is     AIEEE  Solved  Paper-2003

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

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

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

D)
$\frac{2}{5}$

• question_answer68) Events A, B, C are mutually exclusive 3x +1 events   such that $P(A)=\frac{3x+1}{3},\,P(B)=\frac{1-x}{4}$ and $P(C)=\frac{1-2x}{2}$. The set of possible values of x are in the interval     AIEEE  Solved  Paper-2003

A)
$\left[ \frac{1}{3},\frac{1}{2} \right]$

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

C)
$\left[ \frac{1}{3},\frac{13}{3} \right]$

D)
$\left[ 0,1 \right]$

• question_answer69) The mean and variance of a random variable  X  having a binomial distribution are 4 and 2 respectively, then $P(X=1)$ is     AIEEE  Solved  Paper-2003

A)
$\frac{1}{32}$

B)
$\frac{1}{16}$

C)
$\frac{1}{8}$

D)
$\frac{1}{4}$

• question_answer70) The resultant of forces P and Q is R. If Q is doubled, then R is doubled. If the direction of Q is reversed, then R is again doubled, then? ${{P}^{2}}:{{Q}^{2}}:{{R}^{2}}$ is     AIEEE  Solved  Paper-2003

A)
3 : 1 : 1

B)
2 : 3 : 2

C)
1 : 2 : 3

D)
2 : 3 : 1

• question_answer71) Let ${{R}_{1}}$ and ${{R}_{2}}$ respectively be the maximum ranges up and down an inclined plane and R be the maximum range on the horizontal plane. Then. ${{R}_{1}},R,{{R}_{2}}$ are in     AIEEE  Solved  Paper-2003

A)
AGP

B)
AP

C)
GP

D)
HP

• question_answer72) A couple is of moment G and the force forming the couple is P. If P is turned through a right angle, the moment of the couple thus formed is H. If instead, the forces P is turned through an angle a, then the moment of couple becomes     AIEEE  Solved  Paper-2003

A)
$G\sin \alpha -H\cos \alpha$

B)
$H\cos \alpha +G\sin \alpha$

C)
$G\cos \alpha +H\sin \alpha$

D)
$H\sin \alpha -G\cos \alpha$

• question_answer73) Two particles start simultaneously from the same point and move along two straight lines, one with uniform velocity u and the other from rest with uniform acceleration f. Let $\alpha$ be the angle between their directions of motion. The relative velocity of the second particle w.r.t. the first is least after a time     AIEEE  Solved  Paper-2003

A)
$\frac{u\sin \alpha }{f}$

B)
$\frac{f\cos \alpha }{u}$

C)
$u\sin \alpha$

D)
$\frac{u\cos \alpha }{f}$

• question_answer74) Two stones are projected from the top of a cliff h metres high, with the same speed u, so as to hit the ground at the same spot. If one of the stones is projected horizontally and the other is projected at an angle $\theta$ to the horizontal, then tan $\theta$ equals       AIEEE  Solved  Paper-2003

A)
$\sqrt{\frac{2u}{gh}}$

B)
$2g\,\sqrt{\frac{u}{h}}$

C)
$2h\,\sqrt{\frac{u}{g}}$

D)
$u\,\sqrt{\frac{2}{gh}}$

• question_answer75) A body travels a distance s in t seconds. It starts from rest and ends at rest. In the first part of the journey, it moves with constant acceleration f and in the second part with constant retardation r. The value of t is given by     AIEEE  Solved  Paper-2003

A)
$2s\left( \frac{1}{f}+\frac{1}{r} \right)$

B)
$\frac{2s}{\frac{1}{f}+\frac{1}{r}}$

C)
$\sqrt{2s\,(f+r)}$

D)
$\sqrt{2s\,\left( \frac{1}{f}+\frac{1}{r} \right)}$