question_answer 1) If \[\alpha \ne \beta ,\,{{\alpha }^{2}}=5\alpha -3\] and \[{{\beta }^{2}}=5\beta -3\], then the equation having \[\alpha /\beta \] and \[\beta /\alpha \] as its roots, is
AIEEE Solved Paper-2002
A)
\[3{{x}^{2}}+19x+3=0\]
done
clear
B)
\[3{{x}^{2}}-19x+3=0\]
done
clear
C)
\[3{{x}^{2}}-19x-3=0\]
done
clear
D)
\[{{x}^{2}}-16x+1=0\]
done
clear
View Answer play_arrow
question_answer 2) If \[y={{(x+\sqrt{1+{{x}^{2}}})}^{n}}\] , then \[(1+{{x}^{2}})\frac{{{d}^{2}}y}{d{{x}^{2}}}+\frac{dy}{dx}\] is
AIEEE Solved Paper-2002
A)
\[{{n}^{2}}y\]
done
clear
B)
\[-{{n}^{2}}y\]
done
clear
C)
\[-y\]
done
clear
D)
\[2{{x}^{2}}y\]
done
clear
View Answer play_arrow
question_answer 3) If 1, \[{{\log }_{3}}\sqrt{({{3}^{1-x}}+2)},\,{{\log }_{3}}\,({{4.3}^{x}}-1)\] are in AP, then x equals
AIEEE Solved Paper-2002
A)
\[{{\log }_{3}}4\]
done
clear
B)
\[1-{{\log }_{3}}4\]
done
clear
C)
\[1-{{\log }_{4}}3\]
done
clear
D)
\[{{\log }_{4}}3\]
done
clear
View Answer play_arrow
question_answer 4) A problem in Mathematics is given to three students A, B, C and their respective probability of solving the problem is \[\frac{1}{2},\frac{1}{3}\] and \[\frac{1}{4}\]. Probability that the problem is solved, is
AIEEE Solved Paper-2002
A)
3/4
done
clear
B)
1/2
done
clear
C)
2/3
done
clear
D)
1/3
done
clear
View Answer play_arrow
question_answer 5) The period of \[{{\sin }^{2}}\theta \] is
AIEEE Solved Paper-2002
A)
\[{{\pi }^{2}}\]
done
clear
B)
\[\pi \]
done
clear
C)
\[2\pi \]
done
clear
D)
\[\pi /2\]
done
clear
View Answer play_arrow
question_answer 6) \[l,\,m,\,n\] are the pth, qth and rth terms of a GP and all positive, then \[\left| \begin{matrix} \log \,\,l & p & 1 \\ \log \,\,m & q & 1 \\ \log \,\,n & r & 1 \\ \end{matrix} \right|\] equals
AIEEE Solved Paper-2002
A)
3
done
clear
B)
2
done
clear
C)
1
done
clear
D)
zero
done
clear
View Answer play_arrow
question_answer 7) \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sqrt{1-\cos 2x}}{\sqrt{2}x}\] is
AIEEE Solved Paper-2002
A)
\[\lambda \]
done
clear
B)
\[-1\]
done
clear
C)
zero
done
clear
D)
does not exist
done
clear
View Answer play_arrow
question_answer 8) A triangle with vertices (4, 0), (-1, -1), (3, 5) is
A)
isosceles and right angled
done
clear
B)
isosceles but not right angled
done
clear
C)
right angled but not isosceles
done
clear
D)
neither right angled nor isosceles
done
clear
View Answer play_arrow
question_answer 9) In a class of 100 students, there are 70 boys whose average marks in a subject are 75. If the average marks of the complete class is 72, then what is the average marks of the girls?
AIEEE Solved Paper-2002
A)
73
done
clear
B)
65
done
clear
C)
68
done
clear
D)
74
done
clear
View Answer play_arrow
question_answer 10) If \[{{\cot }^{-1}}(\sqrt{\cos \alpha })-{{\tan }^{-1}}(\sqrt{\cos \alpha })=x\], then \[\sin x\] is equal to
AIEEE Solved Paper-2002
A)
\[{{\tan }^{2}}\left( \frac{\alpha }{2} \right)\]
done
clear
B)
\[{{\cot }^{2}}\left( \frac{\alpha }{2} \right)\]
done
clear
C)
\[\tan \alpha \]
done
clear
D)
\[\cot \left( \frac{\alpha }{2} \right)\]
done
clear
View Answer play_arrow
question_answer 11) The order and degree of the differential equation \[{{\left( 1+3\frac{dy}{dx} \right)}^{2/3}}\] are
AIEEE Solved Paper-2002
A)
\[\left( 1,\frac{2}{3} \right)\]
done
clear
B)
(3, 1)
done
clear
C)
(3, 3)
done
clear
D)
(1, 2)
done
clear
View Answer play_arrow
question_answer 12) A plane which passes through the point (3, 2, 0) and the line \[\frac{x-4}{1}=\frac{y-7}{5}\frac{z-4}{4}\] is
AIEEE Solved Paper-2002
A)
\[x-y+z=1\]
done
clear
B)
\[x+y+z=5\]
done
clear
C)
\[x+2y-z=1\]
done
clear
D)
\[2x-y+z=5\]
done
clear
View Answer play_arrow
question_answer 13) The solution of the equation \[\frac{{{d}^{2}}y}{d{{x}^{2}}}={{e}^{-2x}}\] is
AIEEE Solved Paper-2002
A)
\[\frac{{{e}^{-2x}}}{4}\]
done
clear
B)
\[\frac{{{e}^{-2x}}}{4}\,+cx+d\]
done
clear
C)
\[\frac{1}{4}{{e}^{-2x}}+c\,{{x}^{2}}+d\]
done
clear
D)
\[\frac{1}{4}{{e}^{-2x}}+c\,+d\]
done
clear
View Answer play_arrow
question_answer 14) \[\underset{x\to \infty }{\mathop{\lim }}\,{{\left( \frac{{{x}^{2}}+5x+3}{{{x}^{2}}+x+2} \right)}^{x}}\] is equal to
AIEEE Solved Paper-2002
A)
\[{{e}^{4}}\]
done
clear
B)
\[{{e}^{2}}\]
done
clear
C)
\[{{e}^{3}}\]
done
clear
D)
e
done
clear
View Answer play_arrow
question_answer 15) The domain of \[{{\sin }^{-1}}[{{\log }_{3}}(x/3)]\] is
AIEEE Solved Paper-2002
A)
[1, 9]
done
clear
B)
[-1, 9]
done
clear
C)
[-9, 1]
done
clear
D)
[-9,-1]
done
clear
View Answer play_arrow
question_answer 16) The value of \[{{2}^{1/4}}.\,{{4}^{1/8}}.\,\,{{8}^{1/16}}.....\,\,\infty \] is
AIEEE Solved Paper-2002
A)
1
done
clear
B)
2
done
clear
C)
3/2
done
clear
D)
4
done
clear
View Answer play_arrow
question_answer 17) Fifth term of a GP is 2, then the product of its 9 terms is
AIEEE Solved Paper-2002
A)
256
done
clear
B)
512
done
clear
C)
1024
done
clear
D)
None of these
done
clear
View Answer play_arrow
question_answer 18) \[\int{{{_{0}}^{10\pi }}}\left| \sin x \right|dx\] is
AIEEE Solved Paper-2002
A)
20
done
clear
B)
8
done
clear
C)
10
done
clear
D)
18
done
clear
View Answer play_arrow
question_answer 19) \[{{I}_{n}}=\int_{0}^{\pi /4}{{{\tan }^{n}}\,x\,dx,}\] , then \[\underset{x\to \infty }{\mathop{\lim }}\,n\,[{{I}_{n}}+{{I}_{n+2}}]\] equals
AIEEE Solved Paper-2002
A)
\[\frac{1}{2}\]
done
clear
B)
1
done
clear
C)
\[\infty \]
done
clear
D)
zero
done
clear
View Answer play_arrow
question_answer 20) \[\int_{0}^{2}{[{{x}^{2}}]}\,dx\] is
AIEEE Solved Paper-2002
A)
\[2-\sqrt{2}\]
done
clear
B)
\[2+\sqrt{2}\]
done
clear
C)
\[\sqrt{2}-1\]
done
clear
D)
\[-\sqrt{2}-\sqrt{3}+5\]
done
clear
View Answer play_arrow
question_answer 21) \[\int{_{-\pi }^{\pi }}\frac{2\pi (1+\sin x)}{1+{{\cos }^{2}}x}dx\] is
AIEEE Solved Paper-2002
A)
\[\frac{{{\pi }^{2}}}{4}\]
done
clear
B)
\[{{\pi }^{2}}\]
done
clear
C)
zero
done
clear
D)
\[\frac{\pi }{2}\]
done
clear
View Answer play_arrow
question_answer 22) The period of the function \[f(x)={{\sin }^{4}}x+{{\cos }^{4}}x\] is
AIEEE Solved Paper-2002
A)
\[\pi \]
done
clear
B)
\[\frac{\pi }{2}\]
done
clear
C)
\[2\pi \]
done
clear
D)
None of these
done
clear
View Answer play_arrow
question_answer 23) The domain of definition of the function\[f(x)=\sqrt{{{\log }_{10}}\left( \frac{5x-{{x}^{2}}}{4} \right)}\] is
AIEEE Solved Paper-2002
A)
[1, 4]
done
clear
B)
[1, 0]
done
clear
C)
[0, 5]
done
clear
D)
[5, 0]
done
clear
View Answer play_arrow
question_answer 24) If \[\sin y=x\sin (a+y)\], then-,-is
AIEEE Solved Paper-2002
A)
\[\frac{\sin a}{{{\sin }^{2}}(a+y)}\]
done
clear
B)
\[\frac{{{\sin }^{2}}\,(a+y)}{\sin \,\,a}\]
done
clear
C)
\[\sin \,a\,{{\sin }^{2}}(a+y)\]
done
clear
D)
\[\frac{{{\sin }^{2}}(a-y)}{\sin a}\]
done
clear
View Answer play_arrow
question_answer 25) If \[{{x}^{y}}={{e}^{x-y}}\], then \[\frac{dy}{dx}\] is
AIEEE Solved Paper-2002
A)
\[\frac{1+x}{1+\log x}\]
done
clear
B)
\[\frac{1-\log x}{{{(1+\log x)}^{2}}}\]
done
clear
C)
not defined
done
clear
D)
\[\frac{\log x}{{{(1+\log x)}^{2}}}\]
done
clear
View Answer play_arrow
question_answer 26) The two curves \[{{x}^{3}}-3x{{y}^{2}}+2=0\] and \[3\,{{x}^{2}}y-{{y}^{3}}-2=0\]
AIEEE Solved Paper-2002
A)
cut at right angle
done
clear
B)
touch each other
done
clear
C)
cut at an angle \[\frac{\pi }{3}\]
done
clear
D)
cut at an angle \[\frac{\pi }{4}\]
done
clear
View Answer play_arrow
question_answer 27) The function \[f(x)={{\cot }^{-1}}x+x\] increases in the interval
AIEEE Solved Paper-2002
A)
\[(1,\infty )\]
done
clear
B)
\[(-1,\infty )\]
done
clear
C)
\[(-\infty ,\infty )\]
done
clear
D)
\[(0,\infty )\]
done
clear
View Answer play_arrow
question_answer 28) The greatest value of \[f(x)={{(x+1)}^{1/3}}-{{(x-1)}^{1/3}}\] on [0, 1] is
AIEEE Solved Paper-2002
A)
1
done
clear
B)
2
done
clear
C)
3
done
clear
D)
1/3
done
clear
View Answer play_arrow
question_answer 29) Evaluate \[\int{{{_{0}}^{\pi /2}}\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx}\]
AIEEE Solved Paper-2002
A)
\[\frac{\pi }{4}\]
done
clear
B)
\[\frac{\pi }{2}\]
done
clear
C)
0
done
clear
D)
1
done
clear
View Answer play_arrow
question_answer 30) \[\int{\frac{dx}{x({{x}^{n}}+1)}}\] is equal to
AIEEE Solved Paper-2002
A)
\[\frac{1}{n}\log \left( \frac{{{x}^{n}}}{{{x}^{n}}+1} \right)+C\]
done
clear
B)
\[\frac{1}{n}\log \left( \frac{{{x}^{n}}+1}{{{x}^{n}}} \right)+C\]
done
clear
C)
\[\log \left( \frac{{{x}^{n}}}{{{x}^{n}}+1} \right)+C\]
done
clear
D)
None of these
done
clear
View Answer play_arrow
question_answer 31) The area bounded by the curve \[y=2x-{{x}^{2}}\] and the straight line \[y=-x\] is given by
AIEEE Solved Paper-2002
A)
\[\frac{9}{2}\] sq units
done
clear
B)
\[\frac{43}{6}\] sq units
done
clear
C)
\[\frac{35}{6}\] sq units
done
clear
D)
None of these
done
clear
View Answer play_arrow
question_answer 32) The differential equation of all non-vertical lines in a plane is
AIEEE Solved Paper-2002
A)
\[\frac{{{d}^{2}}y}{d{{x}^{2}}}=0\]
done
clear
B)
\[\frac{{{d}^{2}}x}{d{{y}^{2}}}=0\]
done
clear
C)
\[\frac{dy}{dx}=0\]
done
clear
D)
\[\frac{dx}{dy}=0\]
done
clear
View Answer play_arrow
question_answer 33) Given two vectors are \[\hat{i}-\hat{j}\] and \[\hat{i}+2\hat{j}\], the unit vector coplanar with the two vectors and perpendicular to first is
AIEEE Solved Paper-2002
A)
\[\frac{1}{\sqrt{2}}(\hat{i}+\hat{j})\]
done
clear
B)
\[\frac{1}{\sqrt{5}}(2\hat{i}-\hat{j})\]
done
clear
C)
\[\pm \frac{1}{\sqrt{2}}(\hat{i}-\hat{k})\]
done
clear
D)
None of these
done
clear
View Answer play_arrow
question_answer 34) The vector \[\hat{i}+x\hat{j}+3\hat{k}\] is rotated through an angle \[\theta \] and doubled in magnitude, then it becomes \[4\hat{i}+(4x-2)\hat{j}+2\hat{k}\]. The values of \[x\] are
AIEEE Solved Paper-2002
A)
\[\left\{ -\frac{2}{3},2 \right\}\]
done
clear
B)
\[\left\{ \frac{1}{3},2 \right\}\]
done
clear
C)
\[\left\{ \frac{2}{3},0 \right\}\]
done
clear
D)
\[\left\{ 2,7 \right\}\]
done
clear
View Answer play_arrow
question_answer 35) A parallelepiped is formed by planes drawn through the points (2, 3, 5) and (5, 9, 7), parallel to the coordinate planes. The length of a diagonal of the parallelopiped is
AIEEE Solved Paper-2002
A)
7 units
done
clear
B)
\[\sqrt{38}\] units
done
clear
C)
\[\sqrt{155}\] units
done
clear
D)
None of these
done
clear
View Answer play_arrow
question_answer 36) The equation of the plane containing the line \[\frac{x-{{x}_{1}}}{l}=\frac{y-{{y}_{1}}}{m}=\frac{z-{{z}_{1}}}{n}\] is \[a(x-{{x}_{1}})+b(y-{{y}_{1}})+c(z-{{z}_{1}})=0\] where
AIEEE Solved Paper-2002
A)
\[a{{x}_{1}}+b{{y}_{1}}+c{{z}_{1}}=0\]
done
clear
B)
\[al+bm+cn=0\]
done
clear
C)
\[\frac{a}{l}=\frac{b}{m}=\frac{c}{n}\]
done
clear
D)
\[l\,{{x}_{1}}+m{{y}_{1}}+n{{z}_{1}}=0\]
done
clear
View Answer play_arrow
question_answer 37) A and B play a game where each is asked to select a number from 1 to 25. If the two numbers match, both of them win a prize. The probability that they will not win a prize in a single trial, is
AIEEE Solved Paper-2002
A)
\[\frac{1}{25}\]
done
clear
B)
\[\frac{24}{25}\]
done
clear
C)
\[\frac{2}{25}\]
done
clear
D)
None of these
done
clear
View Answer play_arrow
question_answer 38) If A and B are two mutually exclusive events, then
AIEEE Solved Paper-2002
A)
\[P(A)<P(\overline{B})\]
done
clear
B)
\[P(A)>P(\overline{B})\]
done
clear
C)
\[P(A)<P(B)\]
done
clear
D)
None of these
done
clear
View Answer play_arrow
question_answer 39) The equation of the directrix of the parabola \[{{y}^{2}}+4y+4x+2=0\] is
AIEEE Solved Paper-2002
A)
\[x=-1\]
done
clear
B)
\[x=1\]
done
clear
C)
\[x=-3/2\]
done
clear
D)
\[x=3/2\]
done
clear
View Answer play_arrow
question_answer 40) Let \[{{T}_{n}}\] denotes the number of triangles which can be formed using the vertices of a regular polygon of n sides. If \[{{T}_{n\,+1}}-{{T}_{n}}=21\], then n equals
AIEEE Solved Paper-2002
A)
5
done
clear
B)
7
done
clear
C)
6
done
clear
D)
4
done
clear
View Answer play_arrow
question_answer 41) In a \[\Delta ABC,\,\,2ca\,\sin \,\left( \frac{A-B+C}{2} \right)\] is equal to
AIEEE Solved Paper-2002
A)
\[{{a}^{2}}+{{b}^{2}}-{{c}^{2}}\]
done
clear
B)
\[{{c}^{2}}+{{a}^{2}}-{{b}^{2}}\]
done
clear
C)
\[{{b}^{2}}-{{c}^{2}}-{{a}^{2}}\]
done
clear
D)
\[{{c}^{2}}-{{a}^{2}}-{{b}^{2}}\]
done
clear
View Answer play_arrow
question_answer 42) For \[x\in R\underset{x\to \infty }{\mathop{\lim }}\,{{\left( \frac{x-3}{x+2} \right)}^{x}}\] is equal to
AIEEE Solved Paper-2002
A)
e
done
clear
B)
\[{{e}^{-1}}\]
done
clear
C)
\[{{e}^{-5}}\]
done
clear
D)
\[{{e}^{5}}\]
done
clear
View Answer play_arrow
question_answer 43) The incentre of the triangle with vertices \[(1,\sqrt{3})\], (0, 0) and (2,0) is
AIEEE Solved Paper-2002
A)
\[\left( 1,\frac{\sqrt{3}}{2} \right)\]
done
clear
B)
\[\left( \frac{2}{3},\frac{1}{\sqrt{3}} \right)\]
done
clear
C)
\[\left( \frac{2}{3},\frac{\sqrt{3}}{2} \right)\]
done
clear
D)
\[\left( 1,\frac{1}{\sqrt{3}} \right)\]
done
clear
View Answer play_arrow
question_answer 44) If the vectors a, b and c from the sides BC, CA and AB respectively of a \[\Delta ABC\], then
AIEEE Solved Paper-2002
A)
\[a.\,b=b.\,c=c.\,b=0\]
done
clear
B)
\[a\times b=b\times c=c\times a\]
done
clear
C)
\[a.\,b=b.\,c=c.\,a=0\]
done
clear
D)
\[a\times a+a\times c+c\times a=0\]
done
clear
View Answer play_arrow
question_answer 45) If \[\omega \] is an imaginary cube root of unity, then \[{{(1+\omega -{{\omega }^{2}})}^{7}}\] equals
AIEEE Solved Paper-2002
A)
\[128\,\omega \]
done
clear
B)
\[-128\,\omega \]
done
clear
C)
\[128\,{{\omega }^{2}}\]
done
clear
D)
\[-128\,{{\omega }^{2}}\]
done
clear
View Answer play_arrow
question_answer 46) If \[\left| \begin{matrix} 6\,i & -3\,i & 1 \\ 4 & 3\,i & -1 \\ 20 & 3 & i \\ \end{matrix} \right|=x+iy\], then
AIEEE Solved Paper-2002
A)
\[x=3,\,y=1\]
done
clear
B)
\[x=1,\,y=3\]
done
clear
C)
\[x=0,\,y=3\]
done
clear
D)
\[x=0,\,y=0\]
done
clear
View Answer play_arrow
question_answer 47) \[{{\sin }^{2}}\theta =\frac{4xy}{{{(x+y)}^{2}}}\] is true if and only if
AIEEE Solved Paper-2002
A)
\[x-y\ne 0\]
done
clear
B)
\[x=-y\]
done
clear
C)
\[x=y\]
done
clear
D)
\[x\ne 0,y\ne 0\]
done
clear
View Answer play_arrow
question_answer 48) The radius of the circle passing through the foci of the ellipse \[\frac{{{x}^{2}}}{16}+\frac{{{y}^{2}}}{9}\] and having its centre at (0, 3), is
A)
4 units
done
clear
B)
3 units
done
clear
C)
\[\sqrt{12}\] units
done
clear
D)
\[\frac{7}{2}\] units
done
clear
View Answer play_arrow
question_answer 49) The probability of India winning a test match against West-Indies is \[1/2\] assuming independence from match to match. The probability that in a match series India's second win occurs at the third test is
A)
\[1/8\]
done
clear
B)
\[1/4\]
done
clear
C)
\[1/2\]
done
clear
D)
\[2/3\]
done
clear
View Answer play_arrow
question_answer 50) If \[(\omega \ne 1)\] is a cubic root of unity, then \[\left| \begin{matrix} 1 & 1+i+{{\omega }^{2}} & {{\omega }^{2}} \\ 1-i & -1 & {{\omega }^{2}}-1 \\ -i & -1+\omega -i & -1 \\ \end{matrix} \right|\] equals
AIEEE Solved Paper-2002
A)
0
done
clear
B)
1
done
clear
C)
\[i\]
done
clear
D)
\[\omega \]
done
clear
View Answer play_arrow
question_answer 51) A biased coin with probability \[p,0<p<1\], of heads is tossed until a head appears for the first time. If the probability that the number of tosses required is even, is \[2/5\], then p equals
AIEEE Solved Paper-2002
A)
\[1/3\]
done
clear
B)
\[2/3\]
done
clear
C)
\[2/5\]
done
clear
D)
\[3/5\]
done
clear
View Answer play_arrow
question_answer 52) A fair die is tossed eight times. The probability that a third six is observed on the eighth throw, is
AIEEE Solved Paper-2002
A)
\[\frac{^{7}{{C}_{2}}\times {{5}^{5}}}{{{6}^{7}}}\]
done
clear
B)
\[\frac{^{7}{{C}_{2}}\times {{5}^{5}}}{{{6}^{8}}}\]
done
clear
C)
\[\frac{^{7}{{C}_{2}}\times {{5}^{5}}}{{{6}^{6}}}\]
done
clear
D)
None of these
done
clear
View Answer play_arrow
question_answer 53) Let \[f(2)=4\] and \[f'\,(2)=4\]. Then,\[\underset{x\to 2}{\mathop{\lim }}\,\frac{xf(2)-2f(x)}{x-2}\] is given by
AIEEE Solved Paper-2002
A)
2
done
clear
B)
\[-2\]
done
clear
C)
\[-4\]
done
clear
D)
3
done
clear
View Answer play_arrow
question_answer 54) Three straight lines \[2x+11y-5=0\], \[24x+7y-20=0\] and \[4x-3y-2=0\]
AIEEE Solved Paper-2002
A)
form a triangle
done
clear
B)
are only concurrent
done
clear
C)
are concurrent with one line bisecting the angle between the other two
done
clear
D)
None of the above
done
clear
View Answer play_arrow
question_answer 55) A straight line through the point (2, 2) intersects the lines \[\sqrt{3}x+y=0\] and \[\sqrt{3}x-y=0\] at the points A and B. The equation to the line AB so that the \[\Delta OAB\] is equilateral, is
AIEEE Solved Paper-2002
A)
\[x-2=0\]
done
clear
B)
\[y-2=0\]
done
clear
C)
\[x+y-4=0\]
done
clear
D)
None of these
done
clear
View Answer play_arrow
question_answer 56) The greatest distance of the point P(10,7) from the circle \[{{x}^{2}}+{{y}^{2}}-4x-2y-20=0\] is
AIEEE Solved Paper-2002
A)
10 units
done
clear
B)
15 units
done
clear
C)
5 units
done
clear
D)
None of these
done
clear
View Answer play_arrow
question_answer 57) The equation of the tangent to the circle \[{{x}^{2}}+{{y}^{2}}+4x-4y+4=0\] which make equal intercepts on the positive coordinate axes, is
AIEEE Solved Paper-2002
A)
\[x+y=2\]
done
clear
B)
\[x+y=2\sqrt{2}\]
done
clear
C)
\[x+y=4\]
done
clear
D)
\[x+y=8\]
done
clear
View Answer play_arrow
question_answer 58) The equation of the ellipse whose foci are \[(\pm \,2,0)\] and eccentricity is \[1/2\], is
AIEEE Solved Paper-2002
A)
\[\frac{{{x}^{2}}}{12}+\frac{{{y}^{2}}}{16}=1\]
done
clear
B)
\[\frac{{{x}^{2}}}{16}+\frac{{{y}^{2}}}{12}=1\]
done
clear
C)
\[\frac{{{x}^{2}}}{16}+\frac{{{y}^{2}}}{8}=1\]
done
clear
D)
None of these
done
clear
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question_answer 59) The equation of the chord joining two points \[({{x}_{1}},{{y}_{1}})\] and \[({{x}_{2}},{{y}_{2}})\] on the rectangular hyperbola \[xy={{c}^{2}}\]is
AIEEE Solved Paper-2002
A)
\[\frac{x}{{{x}_{1}}+{{x}_{2}}}+\frac{y}{{{y}_{1}}+{{y}_{2}}}=1\]
done
clear
B)
\[\frac{x}{{{x}_{1}}-{{x}_{2}}}+\frac{y}{{{y}_{1}}-{{y}_{2}}}=1\]
done
clear
C)
\[\frac{x}{{{y}_{1}}+{{y}_{2}}}+\frac{y}{{{x}_{1}}+{{x}_{2}}}=1\]
done
clear
D)
\[\frac{x}{{{y}_{1}}-{{y}_{2}}}+\frac{y}{{{x}_{1}}-{{x}_{2}}}=1\]
done
clear
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question_answer 60) If the vectors \[c,a=x\,\hat{i}+y\hat{j}+z\hat{k}\] and \[b=\hat{j}\] are such that a, c and b form a right handed system, then c is
AIEEE Solved Paper-2002
A)
\[z\hat{i}-x\hat{k}\]
done
clear
B)
0
done
clear
C)
\[y\,\hat{i}\]
done
clear
D)
\[-z\,\hat{i}+x\,\hat{k}\]
done
clear
View Answer play_arrow
question_answer 61) The centre of the circle given by \[r.\,(\hat{i}+2\hat{j}+2\hat{k})=15\] and \[\left| r-(\hat{j}+2\hat{k} \right|=4\]is
A)
(0, 1, 2)
done
clear
B)
(1, 3, 4)
done
clear
C)
(-1, 3, 4)
done
clear
D)
None of these
done
clear
View Answer play_arrow
question_answer 62) The value of \[\frac{1-{{\tan }^{2}}{{15}^{o}}}{1+{{\tan }^{2}}{{15}^{o}}}\] is
AIEEE Solved Paper-2002
A)
1
done
clear
B)
\[\sqrt{3}\]
done
clear
C)
\[\frac{\sqrt{3}}{2}\]
done
clear
D)
2
done
clear
View Answer play_arrow
question_answer 63) If \[\tan \theta =-\frac{4}{3}\] then sine is
AIEEE Solved Paper-2002
A)
\[-\frac{4}{5}\] but not \[\frac{4}{5}\]
done
clear
B)
\[-\frac{4}{5}\] or \[\frac{4}{5}\]
done
clear
C)
\[\frac{4}{5}\] but not \[-\frac{4}{5}\]
done
clear
D)
None of these
done
clear
View Answer play_arrow
question_answer 64) If \[\sin (\alpha +\beta )=1,\,\,\sin (\alpha -\beta )=\frac{1}{2}\] then \[\tan \,(a+2\beta )\tan \,(2\alpha +\beta )\] is equal to
AIEEE Solved Paper-2002
A)
1
done
clear
B)
- 1
done
clear
C)
zero
done
clear
D)
None of these
done
clear
View Answer play_arrow
question_answer 65) If \[y={{\sin }^{2}}\theta +\cos e{{c}^{2}}\theta ,\,\,\theta \ne 0\], then
AIEEE Solved Paper-2002
A)
\[y=0\]
done
clear
B)
\[y\le 2\]
done
clear
C)
\[y\ge -2\]
done
clear
D)
\[y\ge 2\]
done
clear
View Answer play_arrow
question_answer 66) In a \[\Delta ABC\], \[a=4,\,b=3,\,\,\angle A={{60}^{o}}\], then c is the root of the equation
AIEEE Solved Paper-2002
A)
\[{{c}^{2}}-3c-7=0\]
done
clear
B)
\[{{c}^{2}}+3c+7=0\]
done
clear
C)
\[{{c}^{2}}-3c+7=0\]
done
clear
D)
\[{{c}^{2}}+3c-7=0\]
done
clear
View Answer play_arrow
question_answer 67) In a \[\Delta ABC,\,\tan \frac{A}{2}=\frac{5}{6},\tan \frac{C}{2}=\frac{2}{5}\], then
AIEEE Solved Paper-2002
A)
a, c, b are in AP
done
clear
B)
a, b, c are in AP
done
clear
C)
b, a, c are in AP
done
clear
D)
a, b, care in GP
done
clear
View Answer play_arrow
question_answer 68) The equation \[a\sin x+b\cos x=c\], where \[\left| c \right|>\sqrt{{{a}^{2}}+{{b}^{2}}}\] has
AIEEE Solved Paper-2002
A)
a unique solution
done
clear
B)
infinite number of solutions
done
clear
C)
no solution
done
clear
D)
None of the above
done
clear
View Answer play_arrow
question_answer 69) If \[\alpha \] is a root of \[25{{\cos }^{2}}\theta +5\cos \theta -12=0\frac{\pi }{2}<a<\pi \], then \[\sin 2\alpha \] is equal to
AIEEE Solved Paper-2002
A)
\[\frac{24}{25}\]
done
clear
B)
\[-\frac{24}{25}\]
done
clear
C)
\[\frac{13}{18}\]
done
clear
D)
\[-\frac{13}{18}\]
done
clear
View Answer play_arrow
question_answer 70) \[{{\tan }^{-1}}\left( \frac{1}{4} \right)+{{\tan }^{-1}}\left( \frac{2}{9} \right)\] is equal to
AIEEE Solved Paper-2002
A)
\[\frac{1}{2}{{\cos }^{-1}}\left( \frac{3}{5} \right)\]
done
clear
B)
\[\frac{1}{2}{{\sin }^{-1}}\left( \frac{3}{5} \right)\]
done
clear
C)
\[\frac{1}{2}{{\tan }^{-1}}\left( \frac{3}{5} \right)\]
done
clear
D)
\[{{\tan }^{-1}}\left( \frac{1}{2} \right)\]
done
clear
View Answer play_arrow
question_answer 71) \[\sum\limits_{n=0}^{\infty }{\frac{{{({{\log }_{e}}x)}^{n}}}{n!}}\] is equal to
AIEEE Solved Paper-2002
A)
\[{{\log }_{e}}x\]
done
clear
B)
\[x\]
done
clear
C)
\[{{\log }_{x}}e\]
done
clear
D)
None of these
done
clear
View Answer play_arrow
question_answer 72) \[{{x}^{(x-1)-\frac{1}{2}{{(x-1)}^{2}}+\frac{{{(x-1)}^{3}}}{3}-\frac{{{(x-1)}^{4}}}{4}+....}}\] is equal to
AIEEE Solved Paper-2002
A)
\[\log \,(x-1)\]
done
clear
B)
\[\log \,\,x\]
done
clear
C)
\[x\]
done
clear
D)
None of these
done
clear
View Answer play_arrow
question_answer 73) The coefficient of \[{{x}^{5}}\] in \[{{(1+2x+3{{x}^{2}}+....)}^{-3/2}}\] is
AIEEE Solved Paper-2002
A)
21
done
clear
B)
25
done
clear
C)
26
done
clear
D)
None of these
done
clear
View Answer play_arrow
question_answer 74) If \[\left| x \right|<1\], then the coefficient of \[{{x}^{n}}\] in expansion of \[{{(1+x+{{x}^{2}}+{{x}^{3}}+....)}^{2}}\] is
AIEEE Solved Paper-2002
A)
n
done
clear
B)
\[n-1\]
done
clear
C)
\[n+2\]
done
clear
D)
\[n+1\]
done
clear
View Answer play_arrow
question_answer 75) The number of real roots of \[{{3}^{2{{x}^{2}}-7x+7}}=9\] is
AIEEE Solved Paper-2002
A)
zero
done
clear
B)
2
done
clear
C)
1
done
clear
D)
4
done
clear
View Answer play_arrow