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

AIEEE Solved Paper-2003 Total Questions - 75

1) 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

View Answer
2) 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)

                      \[{{a}^{2}}=4b\]

View Answer
3) 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)

\[-i\]

View Answer
4) 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

View Answer
5) 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

View Answer
6) 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

View Answer
7) 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

View Answer
8) 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

View Answer
9) 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

View Answer
10) 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)

\[\alpha =2ab,\,\,\beta ={{a}^{2}}+{{b}^{2}}\]

View Answer
11) 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

View Answer
12) 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)

      \[7!\,\,\times 5!\]

View Answer
13) 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)

      \[{{\omega }^{2}}\]

View Answer
14) 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)

\[^{n+1}{{C}_{r+1}}\]

View Answer
15) 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

View Answer
16) 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

View Answer
17) 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)

      \[{{\log }_{e}}\,\,\left( \frac{4}{e} \right)\]

View Answer
18) 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

View Answer
19) 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

View Answer
20) 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)

\[\frac{a}{4}\cot \left( \frac{\pi }{2n} \right)\]

View Answer
21) 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

View Answer
22) 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

View Answer
23) 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)

      \[\left| a \right|\ge \frac{1}{\sqrt{2}}\]

View Answer
24) 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

View Answer
25) 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

View Answer
26) 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)

\[\frac{7n\,(n+1)}{2}\]

View Answer
27) 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

View Answer
28) 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)

      \[(-1,0)\cup (1,2,)\cup (2,\infty )\]

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

A)

                                        \[\frac{1}{8}\]

B)

      0

C)

\[\frac{1}{32}\]

D)

      \[\infty \]

View Answer
30) 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

View Answer
31) 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

View Answer
32) 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

View Answer
33) 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

View Answer
34) 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

View Answer
35) 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)

\[F(t)=t{{e}^{-t}}\]

View Answer
36) 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)

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

View Answer
37) 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

View Answer
38) 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)

      \[\frac{1}{n+1}+\frac{1}{n+2}\]

View Answer
39) \[\underset{x\to \infty }{\mathop{\lim }}\,\frac{1+{{2}^{4}}+{{3}^{4}}+....+{{n}^{4}}}{{{n}^{5}}}\]                                                                             \[-\underset{x\to \infty }{\mathop{\lim }}\,\frac{1+{{2}^{3}}+{{3}^{3}}+....+{{n}^{3}}}{{{n}^{5}}}\] is

A)

        1/30

B)

0

C)

1/4

D)

      1/5

View Answer
40) 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

View Answer
41) 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

View Answer
42) 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)

\[e+\frac{{{e}^{2}}}{2}+\frac{5}{2}\]

View Answer
43) 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

View Answer
44) 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)

\[x{{e}^{2\,\,{{\tan }^{-1}}y}}={{e}^{{{\tan }^{-1}}y}}+k\]

View Answer
45) 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)

      \[\sqrt{a_{1}^{2}+b_{1}^{2}-a_{2}^{2}-b_{2}^{2}}\]

View Answer
46) 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)

\[{{(3x+1)}^{2}}+{{(3y)}^{2}}={{a}^{2}}-{{b}^{2}}\]

View Answer
47) 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)

\[pq=-1\]

View Answer
48) 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)

\[y(\cos \alpha +\sin \alpha )+x(\cos \alpha -\sin \alpha )=a\]

View Answer
49) 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)

      \[r>2\]

View Answer
50) 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)

\[{{x}^{2}}+{{y}^{2}}-2x+2y=62\]

View Answer
51) 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)

      \[{{t}_{2}}={{t}_{1}}+\frac{2}{{{t}_{1}}}\]

View Answer
52) 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

View Answer
53) 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)

\[{{90}^{o}}\]

View Answer
54) 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

View Answer
55) 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

View Answer
56) 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)

\[aa'+cc'+1=0\]

View Answer
57) 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

View Answer
58) 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)

\[\frac{1}{{{a}^{2}}}+\frac{1}{{{b}^{2}}}+\frac{1}{{{c}^{2}}}-\frac{1}{a{{'}^{2}}}-\frac{1}{{{b}^{'2}}}-\frac{1}{c{{'}^{2}}}=0\]

View Answer
59) 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

View Answer
60) 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)

      \[3u\,.\,v\times w\]

View Answer
61) 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.

View Answer
62) 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)

      \[\sqrt{288}\]

View Answer
63) 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

View Answer
64) 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

View Answer
65) 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

View Answer
66) 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

View Answer
67) 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)

\[\frac{2}{5}\]

View Answer
68) 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)

\[\left[ 0,1 \right]\]

View Answer
69) 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)

      \[\frac{1}{4}\]

View Answer
70) 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

View Answer
71) 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

View Answer
72) 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)

      \[H\sin \alpha -G\cos \alpha \]

View Answer
73) 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)

\[\frac{u\cos \alpha }{f}\]

View Answer
74) 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)

\[u\,\sqrt{\frac{2}{gh}}\]

View Answer
75) 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)

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

View Answer

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AIEEE Solved Paper-2003 Total Questions - 75