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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
doneclear
B)
rhombus
doneclear
C)
rectangle
doneclear
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
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
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
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
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
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
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
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
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
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
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
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
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
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