Solved papers for JEE Main & Advanced AIEEE Solved Paper-2004

AIEEE Solved Paper-2004 Total Questions - 75

1) Let\[R=\{(1,3),(4,2),(2,4),(2,3),(3,1)\}\]be a relation on the set\[A=\{1,2,3,4\}\]. The relation R is

A)

a function

B)

transitive

C)

not symmetric

D)

      reflexive

View Answer
2) The range of the function\[f(x){{=}^{7-x}}{{P}_{x-3}}\]is

A)

{1, 2, 3}

B)

     {1, 2, 3, 4, 5, 6}

C)

{1, 2, 3, 4}

D)

{1, 2, 3, 4, 5}

View Answer
3) Let\[z,w\]be complex numbers such that \[\overline{z}=i\overline{w}=0\] and \[arg(zw)=\pi \]. Then, \[\arg (z)\] equals

A)

\[\frac{\pi }{4}\]

B)

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

C)

\[\frac{3\pi }{4}\]

D)

                       \[\frac{5\pi }{4}\]

View Answer
4) If\[z=x-2y\]and\[{{z}^{1/3}}=p+iq,\]. Then \[{\left( \frac{x}{p}+\frac{y}{q} \right)}/{({{p}^{2}}+{{q}^{2}})}\;\] is equal to

A)

1

B)

                       \[-1\]

C)

2

D)

       \[-2\]

View Answer
5) If\[|{{z}^{2}}-1|=|z{{|}^{2}}+1,\]then z lies on

A)

the real axis

B)

the imaginary axis

C)

a circle

D)

an ellipse

View Answer
6) Let\[A=\left[ \begin{matrix} 0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0 \\ \end{matrix} \right].\]The only correct statement about the matrix A is

A)

A is a zero matrix

B)

\[A=(-\text{ }1)l,\]where\[l\]is a unit matrix

C)

\[{{A}^{-1}}\]does not exist

D)

\[{{A}^{2}}=l\]

View Answer
7) Let\[A=\left[ \begin{matrix}    1 & -1 & 1 \\    2 & 1 & -3 \\    1 & 1 & 1 \\ \end{matrix} \right]\]and\[10B=\left[ \begin{matrix}    4 & 2 & 2 \\    -5 & 0 & \alpha   \\    1 & -2 & 3 \\ \end{matrix} \right].\]If B is the inverse of matrix A, then a is

A)

-2

B)

1

C)

2

D)

5

View Answer
8) If\[{{a}_{1}},{{a}_{2}},{{a}_{3}}....,{{a}_{n}},...\]are in GP, then the value of the determinant \[\left| \begin{matrix}    \log {{a}_{n}} & \log {{a}_{n+1}} & \log {{a}_{n+2}} \\    \log \,\,{{a}_{n+3}} & \log {{a}_{n+4}} & \log {{a}_{n+5}} \\    \log {{a}_{n+6}} & \log {{a}_{n+7}} & \log {{a}_{n+8}} \\ \end{matrix} \right|,\] is

A)

0

B)

                                       1

C)

2

D)

\[-2\]

View Answer
9) Let two numbers have arithmetic mean 9 and geometric mean 4. Then, these numbers are the roots of the quadratic equation

A)

\[{{x}^{2}}+18x+16=0\]

B)

\[{{x}^{2}}-18x+16=0\]

C)

\[{{x}^{2}}+18x-16=0\]

D)

\[{{x}^{2}}+18x-16=0\]

View Answer
10) If\[(1-p)\]is a root of quadratic equation\[{{x}^{2}}+px\] \[+(1-p)=0,\]then its roots are

A)

\[0,1\]

B)

       \[-1,1\]

C)

       \[0,-1\]

D)

       \[-1,2\]

View Answer
11) Let\[S(K)=1+3+5+...+(2K-1)=3+{{K}^{2}}.\]Then, which of the following is true?

A)

S(1) is correct

B)

\[S(K)\Rightarrow S(K+1)\]

C)

\[S(K)S(K+1)\]

D)

Principle of mathematical induction can be used to prove the formula

View Answer
12) How many ways are there to arrange the letters in the word GARDEN with the vowels in alphabetical order?

A)

120

B)

                       240

C)

       360

D)

       480

View Answer
13) The number of ways of distributing 8 identical balls in 3 distinct boxes, so that none of the boxes is empty, is

A)

5

B)

                       21

C)

                       \[{{3}^{8}}\]

D)

       \[^{8}{{C}_{3}}\]

View Answer
14) If one root of the equation\[{{x}^{2}}+px+12=0\]is 4, while the equation\[{{x}^{2}}+px+q=0\]has equal roots, then the value of 'q' is

A)

\[\frac{49}{4}\]

B)

       12

C)

       3

D)

       4

View Answer
15) The coefficient of the middle term in the binomial expansion in powers of\[x\]of\[{{(1+\alpha x)}^{4}}\] and of\[{{(1-ax)}^{6}}\]is the same, if a equals

A)

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

B)

       \[\frac{10}{3}\]

C)

       \[-\frac{3}{10}\]

D)

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

View Answer
16) The coefficient of\[{{x}^{n}}\]in the expansion of \[(1+x){{(1-x)}^{n}}\]is

A)

\[(n-1)\]

B)

       \[{{(-1)}^{n}}(1-n)\]

C)

       \[{{(-1)}^{n-1}}{{(n-1)}^{2}}\]

D)

       \[{{(-1)}^{n-1}}n\]

View Answer
17) If\[{{S}_{n}}=\sum\limits_{r=0}^{n}{\frac{1}{^{n}{{C}_{r}}}}\]and\[{{t}_{n}}=\sum\limits_{r=0}^{n}{\frac{r}{^{n}{{C}_{r}}}}\]then\[\frac{{{t}_{n}}}{{{S}_{n}}}\]is equal to

A)

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

B)

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

C)

       \[n-1\]

D)

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

View Answer
18) Let\[{{T}_{r}}\]be the rth term of an AP whose first term is a and common difference is d. If for some positive integers \[m,n,m\ne n,{{T}_{m}}=\frac{1}{n}\]and\[{{T}_{n}}=\frac{1}{m},\]then\[a-d\]equals

A)

0

B)

                      1

C)

       \[\frac{1}{mn}\]

D)

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

View Answer
19) The sum of the first\[n\]terms of the series \[{{1}^{2}}+{{2.2}^{2}}+{{3}^{2}}+{{2.4}^{2}}+{{5}^{2}}+{{2.6}^{2}}+....\]is \[\frac{n{{(n+1)}^{2}}}{2},\] when n is even. When n is odd, the sum is

A)

\[\frac{3n(n+1)}{2}\]

B)

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

C)

       \[\frac{n{{(n+1)}^{2}}}{4}\]

D)

       \[{{\left[ \frac{n(n+1)}{4} \right]}^{2}}\]

View Answer
20) The sum of series\[\frac{1}{2!}+\frac{1}{4!}+\frac{1}{6!}+.....\]is

A)

\[\frac{({{e}^{2}}-1)}{2}\]

B)

                       \[\frac{{{(e-1)}^{2}}}{2e}\]

C)

       \[\frac{({{e}^{2}}-1)}{2e}\]

D)

       \[\frac{({{e}^{2}}-2)}{e}\]

View Answer
21) Let\[\alpha ,\beta \]be such that\[\pi <\alpha -\beta <3\pi .\].If\[\sin \alpha +\sin \beta =-\frac{21}{65}\]and\[\cos \alpha +\cos \beta =-\frac{27}{65},\]then the value of\[\cos \left( \frac{\alpha -\beta }{2} \right)\]is

A)

\[-\frac{3}{\sqrt{130}}\]

B)

       \[\frac{3}{\sqrt{130}}\]

C)

       \[\frac{6}{65}\]

D)

                       \[-\frac{6}{65}\]

View Answer
22) If\[u=\sqrt{{{a}^{2}}{{\cos }^{2}}\theta +{{b}^{2}}{{\sin }^{2}}\theta }\] \[+\sqrt{{{a}^{2}}{{\sin }^{2}}\theta +{{b}^{2}}{{\cos }^{2}}\theta },\] then the difference between the maximum and minimum values of\[{{u}^{2}}\]is given by

A)

\[2({{a}^{2}}+{{b}^{2}})\]

B)

       \[2\sqrt{{{a}^{2}}+{{b}^{2}}}\]

C)

       \[{{(a+b)}^{2}}\]

D)

       \[{{(a-b)}^{2}}\]

View Answer
23) The sides of a triangle are\[\sin \alpha ,\text{ }\cos \alpha \]and\[\sqrt{1+\sin \alpha \cos \alpha }\]for some\[0<\alpha <\frac{\pi }{2}\]. Then, the greatest angle of the triangle is

A)

\[60{}^\circ \]

B)

                                       \[90{}^\circ \]

C)

       \[120{}^\circ \]

D)

       \[150{}^\circ \]

View Answer
24) A person standing on the bank of a river observes that the angle of elevation of the top of a tree on the opposite bank of the river is \[60{}^\circ \]and when he retires 40 m away from the tree, the angle of elevation becomes\[30{}^\circ \]. The breadth of the river is

A)

20 m

B)

                       30 m

C)

       40 m

D)

       60 m

View Answer
25) If \[f:R\to S,\] defined by \[f(x)=\sin x-\sqrt{3}\cos x+1,\]is onto, then the interval of S is

A)

\[[0,\text{ }3]\]

B)

                       \[[-1,\text{ }1]\]

C)

       \[[0,\,1]\]

D)

       \[[-1,\text{ }3]\]

View Answer
26) The graph of the function\[y=f(x)\]is symmetrical about the line\[x=2,\]then

A)

\[f(x+2)=f(x-2)\]

B)

      \[f(2+x)=f(2-x)\]

C)

\[f(x)=f(-x)\]

D)

   \[f(x)=-f(-x)\]

View Answer
27) The domain of the function \[f(x)=\frac{si{{n}^{-1}}(x-3)}{\sqrt{9-{{x}^{2}}}}\]is

A)

[2, 3]

B)

                       [2, 3)

C)

       [1, 2]

D)

       [1, 2)

View Answer
28) If\[\underset{x\to \infty }{\mathop{\lim }}\,{{\left( 1+\frac{a}{x}+\frac{b}{{{x}^{2}}} \right)}^{2x}}={{e}^{2}},\]then the values of a and b are

A)

\[a\in R,b\in R\]

B)

     \[a=1,b\in R\]

C)

       \[a\in R,b=2\]

D)

       \[a=1,b=2\]

View Answer
29) Let\[f(x)=\frac{1-\tan x}{4x-\pi },x\ne \frac{\pi }{4},x\in \left[ 0,\frac{\pi }{2} \right]\]. If f(x) is continuous in\[\left[ 0,\frac{\pi }{2} \right]\],then\[f\left( \frac{\pi }{4} \right)\]is

A)

1

B)

                       1/2

C)

\[-1/2\]

D)

       \[-1\]

View Answer
30) If\[x={{e}^{y+{{e}^{y+....\infty }}}},x>0,\]then\[\frac{dy}{dx}\]is

A)

\[\frac{x}{1+x}\]

B)

                       \[\frac{1}{x}\]

C)

\[\frac{1-x}{x}\]

D)

       \[\frac{1+x}{x}\]

View Answer
31) A point on the parabola\[{{y}^{2}}=18x\]at which the ordinate increases at twice the rate of the abscissa, is

A)

(2, 4)

B)

(2, -4)

C)

       \[\left( -\frac{9}{8},\frac{9}{2} \right)\]

D)

       \[\left( \frac{9}{8},\frac{9}{2} \right)\]

View Answer
32) A function\[y=f(x)\]has a second order derivative\[f'\,'=6(x-1)\]. If its graph passes through the point (2, 1) and at that point, the tangent to the graph is\[y=3x-5,\]then the function is

A)

\[{{(x-1)}^{2}}\]

B)

                     \[{{(x-1)}^{3}}\]

C)

       \[{{(x+1)}^{3}}\]

D)

       \[{{(x+1)}^{2}}\]

View Answer
33) The normal to the curve\[x=a(1+\cos \theta ),\] \[y=a\sin \theta \]at\['\theta '\]always passes through the fixed point

A)

(a, 0)

B)

                       (0, a)

C)

       (0, 0)

D)

       (a, a)

View Answer
34) If\[2a+3b+6c=0,\] then atleast one root of the equation\[a{{x}^{2}}+bx+c=0\]lies in the interval

A)

(0, 1)

B)

(1, 2)

C)

       (2, 3)

D)

       (1, 3)

View Answer
35) \[\underset{n\to \infty }{\mathop{\lim }}\,\sum\limits_{r=1}^{n}{\frac{1}{n}{{e}^{r/n}}}\]is

A)

\[e\]

B)

                       \[e-1\]

C)

       \[1-e\]

D)

       \[e+1\]

View Answer
36) If \[\int{\frac{\sin x}{\sin (x-\alpha )}}dx\] \[=Ax+B\text{ }\log \text{ }\sin (x-\alpha )+C,\]then value of \[(A,\text{ }B)\]is

A)

\[(sin\text{ }\alpha ,\text{ }cos\text{ }\alpha )\]

B)

       (b)\[(cos\text{ }\alpha ,\text{ }sin\text{ }\alpha \text{)}\]

C)

       \[(-\sin \text{ }\alpha ,\text{ }\cos \text{ }\alpha )\]

D)

     \[(-\cos \text{ }\alpha ,\text{ }\sin \text{ }\alpha )\]

View Answer
37) \[\int{\frac{dx}{\cos x-\sin x}}\]is equal to

A)

\[\frac{1}{\sqrt{2}}\log \left| \tan \left( \frac{x}{2}-\frac{\pi }{8} \right) \right|+C\]

B)

\[\frac{1}{\sqrt{2}}\log \left| \cot \left( \frac{x}{2} \right) \right|+C\]

C)

\[\frac{1}{\sqrt{2}}\log \left| \tan \left( \frac{x}{2}-\frac{3\pi }{8} \right) \right|+C\]

D)

\[\frac{1}{\sqrt{2}}\log \left| \tan \left( \frac{x}{2}+\frac{3\pi }{8} \right) \right|+C\]

View Answer
38) The value of \[\int_{-2}^{3}{|1-{{x}^{2}}|}dx\]is

A)

\[\frac{28}{3}\]

B)

\[\frac{14}{3}\]

C)

       \[\frac{7}{3}\]

D)

       \[\frac{1}{3}\]

View Answer
39) The value of\[\int_{0}^{\pi /2}{\frac{{{(\sin x+\cos x)}^{2}}}{\sqrt{1+\sin 2x}}}dx\]is

A)

0

B)

                       1

C)

                       2

D)

       3

View Answer
40) If\[\int_{0}^{\pi }{x}f(\sin x)dx=A\int_{0}^{\pi /2}{f(\sin x)}dx,\]then A is equal to

A)

0

B)

                       \[\pi \]

C)

       \[\frac{\pi }{4}\]

D)

       \[2\,\pi \]

View Answer
41) If\[f(x)=\frac{{{e}^{x}}}{1+{{e}^{x}}},{{I}_{1}}=\int_{f(-a)}^{f(a)}{x}g\{x(1-x)\}dx\]and\[{{I}_{2}}=\int_{f(-a)}^{f(a)}{g\{x(1-x)\}dx},\] then the value of\[\frac{{{I}_{2}}}{{{I}_{1}}}\]is

A)

2

B)

                       \[-3\]

C)

       \[-1\]

D)

       1

View Answer
42) The area of the region bounded by the curves \[y=|x-2|,x=1,x=3\]and the X-axis is

A)

1

B)

                       2

C)

       3

D)

       4

View Answer
43) The differential equation for the family of curves\[{{x}^{2}}+{{y}^{2}}-2\text{ }ay=0,\]where a is an arbitrary constant, is

A)

\[2({{x}^{2}}-{{y}^{2}})y'=xy\]

B)

\[2({{x}^{2}}+{{y}^{2}})y'=xy\]

C)

       \[({{x}^{2}}-{{y}^{2}})y'=2xy\]

D)

     \[({{x}^{2}}+{{y}^{2}})y'=2xy\]

View Answer
44) The solution of the differential equation \[ydx+(x+{{x}^{2}}y)dy=0\]is

A)

\[-\frac{1}{xy}=C\]

B)

\[-\frac{1}{xy}+\log y=C\]

C)

       \[\frac{1}{xy}+\log y=C\]

D)

       \[\log y=Cx\]

View Answer
45) Let A (2, - 3) and B (- 2,1) be vertices of a\[\Delta ABC\] If the centroid of this triangle moves on the line \[2x+3y=1,\]then the locus of the vertex C is the line

A)

\[2x+3y=9\]

B)

\[2x-3y=7\]

C)

\[3x+2y=5\]

D)

       \[3x-2y=3\]

View Answer
46) The equation of the straight line passing through the point (4, 3) and making intercepts on the coordinate axes whose sum is -1, is

A)

\[\frac{x}{2}+\frac{y}{3}=-1\]and\[\frac{x}{-2}+\frac{y}{1}=-1\]

B)

\[\frac{x}{2}-\frac{y}{3}=-1\]and\[\frac{x}{-2}+\frac{y}{1}=-1\]

C)

\[\frac{x}{2}+\frac{y}{3}=1\]and\[\frac{x}{-2}+\frac{y}{1}=1\]

D)

\[\frac{x}{2}-\frac{y}{3}\,=1\] and \[\frac{x}{-2}\,+\frac{y}{1}\,=1\]

View Answer
47) If the sum of the slopes of the lines given by \[{{x}^{2}}-2cxy-7{{y}^{2}}=0\]is four times their product, then c has the value

A)

1

B)

       \[-1\]

C)

       2

D)

       \[-2\]

View Answer
48) If one of the lines   given by\[6{{x}^{2}}-xy+4c{{y}^{2}}=0\]is\[3x+4y=0,\]then c equals

A)

1

B)

       \[-1\]

C)

       3

D)

       \[-3\]

View Answer
49) If a circle passes through the point [a, b) and cuts the circle\[{{x}^{2}}+{{y}^{2}}=4\]orthogonally, then the locus of its centre is

A)

\[2ax+2by+({{a}^{2}}+{{b}^{2}}+4)=0\]

B)

\[2\,ax\,+2by-({{a}^{2}}+{{b}^{2}}\,+4)=0\]

C)

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

D)

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

View Answer
50) A variable circle passes through the fixed point \[A(p,\text{ }q)\]and touches x-axis. The locus of the other end of the diameter through A is

A)

\[{{(x-p)}^{2}}=4qy\]

B)

\[{{(x-q)}^{2}}=4qy\]

C)

\[{{(y-p)}^{2}}=4qx\]

D)

\[{{(y-q)}^{2}}=4px\]

View Answer
51) If the lines\[2x+3y+1=0\]and\[3x-y-4=0\]lie along diameters of a circle of circumference\[10\pi ,\]then the equation of the circle is

A)

\[{{x}^{2}}+{{y}^{2}}-2x+2y-23=0\]

B)

\[{{x}^{2}}+{{y}^{2}}-2x-2y-23=0\]

C)

\[{{x}^{2}}+{{y}^{2}}+2x+2y-23=0\]

D)

\[{{x}^{2}}+{{y}^{2}}+2x-2y-23=0\]

View Answer
52) The intercept on the line\[y=x\]by the circle\[{{x}^{2}}+{{y}^{2}}-2x=0\]is AB. Equation of the circle on AB as a diameter is

A)

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

B)

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

C)

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

D)

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

View Answer
53) If\[a\ne 0\]and the line\[2bx+3cy+4d=0\]passes through the points of intersection of the parabolas\[{{y}^{2}}=4\text{ }ax\]and\[{{x}^{2}}=4\text{ }ay,\]then

A)

\[{{d}^{2}}+{{(2b+3c)}^{2}}=0\]

B)

\[{{d}^{2}}+{{(3b+2c)}^{2}}=0\]

C)

\[{{d}^{2}}+{{(2b-3c)}^{2}}=0\]

D)

\[{{d}^{2}}+{{(3b-2c)}^{2}}=0\]

View Answer
54) The eccentricity of an ellipse with its centre at the origin, is\[\frac{1}{2}\]. If one of the directories is\[x=4,\]then the equation of the ellipse is

A)

\[3{{x}^{2}}+4{{y}^{2}}=1\]

B)

\[3{{x}^{2}}+4{{y}^{2}}=12\]

C)

       \[4{{x}^{2}}+3{{y}^{2}}=12\]

D)

       \[4{{x}^{2}}+3{{y}^{2}}=1\]

View Answer
55) A line makes the same angle\[\theta \]with each of the x and z-axes. If the angle\[\beta ,\]which it makes with y-axis, is such that\[{{\sin }^{2}}\beta =3{{\sin }^{2}}\theta ,\]then\[\cos \theta \]equals

A)

\[\frac{2}{3}\]

B)

                                       \[\frac{1}{5}\]

C)

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

D)

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

View Answer
56) Distance between two parallel planes \[2x+y+2z=8\]and\[4x+2y+4z+5=0\]is

A)

\[\frac{3}{2}\]

B)

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

C)

\[\frac{7}{2}\]

D)

       \[\frac{9}{2}\]

View Answer
57) A line with direction cosines proportional to 2,1,2 meets each of the lines\[x=y+a=z\]and \[x+a=2y=2z\]. The coordinates of each of the points of intersection are given by

A)

(3a, 3a, 3a), (a, a, a)

B)

(3a, 2a, 3a), (a, a, a)

C)

(3a, 2a, 3a), (a, a, 2a)

D)

(2a, 3a, 3a), (2a, a, a)

View Answer
58) If the straight lines \[x=1+s,y=-3-\lambda s,\] \[z=1+\lambda \,s\]and\[x=\frac{t}{2},y=1+t,z=2-t,\]with parameters s and t respectively, are coplanar, then\[\lambda ,\]equals

A)

\[-2\]

B)

       \[-1\]

C)

       \[-\frac{1}{2}\]

D)

       0

View Answer
59) The intersection of the spheres \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}+7x-2y-z=13\] and \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-3x+3y+4z=8\] is the same as the intersection of one of the sphere and the plane

A)

\[x-y-z=1\]

B)

\[x-2y-z=1\]

C)

                       \[x-y-2z=1\]

D)

       \[2x-y-z=1\]

View Answer
60) Let a, b and c be three non-zero vectors such that no two of these are collinear. If the vector \[a+2b\]is collinear with c and\[b+3c\]is collinear with a\[(\lambda \]being some non-zero scalar), then \[a+2b+6c\]equals

A)

\[\lambda a\]

B)

                       \[\lambda b\]

C)

       \[\lambda c\]

D)

       0

View Answer
61) A particle is acted upon by constant forces \[4\hat{i}+\hat{j}-3\hat{k}\]and\[3\hat{i}+\hat{j}-\hat{k}\]which displace it from a point\[\hat{i}+2\hat{j}-3\hat{k}\]to the point \[5\hat{i}+4\hat{j}+\hat{k}\]. The work done in standard units by the forces is given by

A)

40

B)

                       30

C)

       25

D)

       15

View Answer
62) If a, b, c are non-coplanar vectors and\[\lambda \]is a real number, then the vectors\[a+2b+3c,\lambda b+4c\]and\[(2\lambda -1)c\]are non-coplanar for

A)

all values of\[\lambda \]

B)

all except one value of \[\lambda \]

C)

all except two values of \[\lambda \]

D)

no value of\[\lambda \]

View Answer
63) Let\[u,v,w\]be such that\[|u|=1,|v|=2,\]\[|w|=3.\]If the projection v along u is equal to that of w along u and\[v,w\]are perpendicular to each other, then\[|u-v+w|\]equals

A)

2

B)

                       \[\sqrt{7}\]

C)

       \[\sqrt{14}\]

D)

       14

View Answer
64) Let a,b and c be non-zero vectors such that \[(a\times b)\times c=\frac{1}{3}|b||c|a.\]If\[\theta \]is the acute angle between the vectors b and c, then\[\sin \theta \]equals

A)

\[\frac{1}{3}\]

B)

                       \[\frac{\sqrt{2}}{3}\]

C)

       \[\frac{2}{3}\]

D)

       \[\frac{2\sqrt{2}}{3}\]

View Answer
65) Consider the following statements (1) Mode can be computed from histogram. (2) Median is not independent of change of scale. (3) Variance is independent of change of origin and scale. (4) Which of these is/are correct?

A)

Only (1)

B)

Only (2)

C)

       Only (1) and (2)

D)

(1), (2) and (3)

View Answer
66) In a series of\[2n\]observations, half of them equal\[a\]and remaining half equal\[-a\]. If the standard deviation of the observations is 2, then \[|a|\] equals

A)

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

B)

       \[\sqrt{2}\]

C)

\[2\]

D)

       \[\frac{\sqrt{2}}{n}\]

View Answer
67) The probability that A speaks truth is 4/5 while this probability for B is 3/4. The probability that they contradict each other when asked to speak on a fact, is

A)

\[\frac{3}{20}\]

B)

                       \[\frac{1}{5}\]

C)

       \[\frac{7}{20}\]

D)

       \[\frac{4}{5}\]

View Answer
68) A random variable X has the probability distribution

\[x\] 1 2 3 4 5 6 7 8

\[p(x)\] 0.15 0.23 0.12 0.10 0.20 0.08 0.07 0.05

For the events E = {X is a prime number} and\[F=\{X<4\},\]the probability\[P(E\cup F)\]is

A)

0.87

B)

                      0.77

C)

       0.35

D)

       0.50

View Answer
69) The mean and the variance of a binomial distribution are 4 and 2, respectively. Then, the probability of 2 successes is

A)

\[\frac{37}{256}\]

B)

       \[\frac{219}{256}\]

C)

\[\frac{128}{256}\]

D)

       \[\frac{28}{256}\]

View Answer
70) With two forces acting at a point, the maximum effect is obtained when their resultant is 4N. If they act at right angles, then their resultant is 3N. Then, the forces are

A)

\[(2+\sqrt{2})N\]and\[(2-\sqrt{2})N\]

B)

\[(2+\sqrt{3})N\]and\[(2-\sqrt{3})N\]

C)

\[\left( 2+\frac{1}{2}\sqrt{2} \right)N\]and \[\left( 2-\frac{1}{2}\sqrt{2} \right)N\]

D)

\[\left( 2+\frac{1}{2}\sqrt{3} \right)N\]and\[\left( 2-\frac{1}{2}\sqrt{3} \right)N\]

View Answer
71) In a right angled \[\Delta ABC,\,\,\angle A={{90}^{\text{o}}}\] and sides a,b,c are respectively, 5 cm, 4 cm and 3 cm. If a force F has moments 0, 9 and 16 (in N cm) units respectively about vertices A, B and C, the magnitude of F is

A)

3

B)

       4

C)

5

D)

       9

View Answer
72) Three forces P, Q and R acting along\[IA,IB\]and \[IC,\]where\[I\]is the incentre of a\[\Delta ABC,\]are in equilibrium. Then, P : Q : R is

A)

\[\cos \frac{A}{2}:\cos \frac{B}{2}:\cos \frac{C}{2}\]

B)

\[\sin \frac{A}{2}:\sin \frac{B}{2}:\sin \frac{C}{2}\]

C)

\[\sec \frac{A}{2}:\sec \frac{B}{2}:\sec \frac{C}{2}\]

D)

\[\cos ec\,\frac{A}{2}:\,\,\cos ec\,\frac{B}{2}\,:\,\cos ec\,\frac{C}{2}\]

View Answer
73) A particle moves towards East from a point A to a point B at the rate of 4 km/h and then towards North from B to C at rate of 5 km/h. If AB = 12 km and BC = 5 km, then its average speed for its journey from A to C and resultant average velocity direct from A to C are respectively

A)

\[\frac{17}{14}km/h\text{ }and\frac{13}{4}km/h\]

B)

\[\frac{13}{4}km/h\text{ }and\frac{17}{4}km/h\]

C)

\[\frac{17}{9}km/h\text{ }and\frac{13}{9}km/h\]

D)

\[\frac{13}{9}km/h\text{ }and\frac{17}{9}km/h\]

View Answer
74) A velocity 1/4 m/s is resolved into two components along OA and OB making angles \[30{}^\circ \]and\[45{}^\circ \]respectively with the given velocity, Then, the component along OB is

A)

\[\frac{1}{8}m/s\]

B)

       \[\frac{1}{4}(\sqrt{3}-1)m/s\]

C)

\[\frac{1}{4}m/s\]

D)

       \[\frac{1}{8}(\sqrt{6}-\sqrt{2})m/s\]

View Answer
75) If\[{{t}_{1}}\]and\[{{t}_{2}}\]are the times of flight of two particles having the same initial velocity u and range R on the horizontal, then\[t_{1}^{2}+t_{2}^{2}\]is equal to

A)

\[{{u}^{2}}/g\]

B)

\[4{{u}^{2}}/{{g}^{2}}\]

C)

\[{{u}^{2}}/2g\]

D)

       1

View Answer

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\[x\]	1	2	3	4	5	6	7	8
\[p(x)\]	0.15	0.23	0.12	0.10	0.20	0.08	0.07	0.05

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done AIEEE Solved Paper-2004 Total Questions - 75

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AIEEE Solved Paper-2004

AIEEE Solved Paper-2004 Total Questions - 75