AIEEE 2003 (Maths) Mock Test

A function $f$ from the set of natural numbers to integers defined by$f(n)=\{\begin{array}{ll}\frac{n-1}{2},& \text{when }n\text{ is odd}\\ -\frac{n}{2},& \text{when }n\text{ is even}\end{array}$ is:

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:

If $z$ and $\omega$ are two non-zero complex numbers such that $|z\omega |=1$, and $\text{Arg}(z)-\text{Arg}(\omega )=\frac{\pi }{2}$, then $\bar{z}\omega$ is equal to:

If ${\left(\frac{1+i}{1-i}\right)}^x=1$, then:

If $\left|\begin{array}{ccc}a& a^2& 1+a^3\\ b& b^2& 1+b^3\\ c& c^2& 1+c^3\end{array}\right|=0$ and vectors $(1,a,a^2)$, $(1,b,b^2)$ and $(1,c,c^2)$ are non-coplanar, then the product $abc$ equals:

If the system of linear equations$x+2ay+az=0$$x+3by+bz=0$$x+4cy+cz=0$ has a non-zero solution, then $a,b,c$:

If the sum of the roots of the quadratic equation $a{x}^2+bx+c=0$ is equal to the sum of the squares of their reciprocals, then $\frac{a}{b}$, $\frac{c}{a}$ and $\frac{b}{c}$ are in:

The number of real solutions of the equation $x^2-3|x|+2=0$ is:

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:

If $A=\left[\begin{array}{cc}a& b\\ b& a\end{array}\right]$ and $A^2=\left[\begin{array}{cc}\alpha & \beta \\ \beta & \alpha \end{array}\right]$, then:

A student is to answer 10 out of 13 questions in an examination such that he must choose at least 4 from the first five questions. The number of choices available to him is:

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:

If $1,\omega ,{\omega }^2$ are the cube roots of unity, then$\Delta =\left|\begin{array}{ccc}1& {\omega }^n& {\omega }^{2n}\\ {\omega }^n& {\omega }^{2n}& 1\\ {\omega }^{2n}& 1& {\omega }^n\end{array}\right|$ is equal to:

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:

The number of integral terms in the expansion of $(\sqrt{3}+\sqrt[8]{5}{)}^{256}$ is:

If $x$ is positive, the first negative term in the expansion of $(1+x{)}^{27/5}$ is:

The sum of the series $\frac{1}{1\cdot 2}-\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}-\dots$ upto $\infty$ is equal to:

Let $f(x)$ be a polynomial function of second degree. If $f(1)=f(-1)$ and $a,b,c$ are in A.P., then $f^{\prime }(a)$, $f^{\prime }(b)$ and $f^{\prime }(c)$ are in:

If $x_1,x_2,x_3$ and $y_1,y_2,y_3$ are both in G.P. with the same common ratio, then the points $(x_1,y_1)$, $(x_2,y_2)$ and $(x_3,y_3)$:

The sum of the radii of inscribed and circumscribed circles for an $n$ -sided regular polygon of side $a$, is:

If in a triangle $ABC$, $a{\cos }^2\left(\frac{C}{2}\right)+c{\cos }^2\left(\frac{A}{2}\right)=\frac{3b}{2}$, then the sides $a,b$ and $c$:

In a triangle $ABC$, 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:

The trigonometric equation ${\sin }^{-1}x=2{\sin }^{-1}a$, has a solution for:

The upper $\frac{3}{4}$ th portion of a vertical pole subtends an angle ${\tan }^{-1}\frac{3}{5}$ at 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:

The real number $x$ when added to its inverse gives the minimum value of the sum at $x$ equal to:

If $f:R\to R$ satisfies $f(x+y)=f(x)+f(y)$, for all $x,y\in R$ and $f(1)=7$, then ${\sum }_{r=1}^{n}f(r)$ is:

If $f(x)=x^n$, then the value of $f(1)-\frac{f^{\prime }(1)}{1!}+\frac{f^{\prime \prime }(1)}{2!}-\frac{f^{\prime \prime \prime }(1)}{3!}+\cdots +(-1{)}^n\frac{f^n(1)}{n!}$ is:

Domain of definition of the function $f(x)=\frac{3}{4-x^2}+{\log }_{10}(x^3-x)$, is:

${\lim }_{x\to \pi }\frac{\tan [1-\sin (x/2)]}{[\pi -2x{]}^2}$ is:

If ${\lim }_{x\to 0}\frac{\log (3+x)-\log (3-x)}{x}=k$, the value of $k$ is:

Let $f(a)=g(a)=k$ and their $n$ th derivatives $f^n(a),g^n(a)$ exist and are not equal for some $n$. Further if ${\lim }_{x\to a}\frac{f(a)g(x)-f(a)g(a)-g(a)f(x)+g(a)f(a)}{f(x)-g(x)}=4$, then the value of $k$ is:

The function $f(x)=\log (x+\sqrt{x^2+1})$ is:

If $f(x)=\{\begin{array}{ll}x{e}^{-\left(\frac{1}{|x|}+\frac{1}{x}\right)},& x\ne 0\\ 0,& x=0\end{array}$ then $f(x)$ is:

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

If $f(y)=e^y$, $g(y)=y$ for $y>0$ and $F(t)={\int }_0^{t}f(t-y)g(y)dy$, then:

If $f(a+b-x)=f(x)$, then ${\int }_a^{b}xf(x)dx$ is equal to:

The value of ${\lim }_{x\to 0}\frac{{\int }_0^{x^2}{\sec }^{2}tdt}{x\sin x}$ is:

The value of the integral $I={\int }_0^{1}x(1-x{)}^{n}dx$ is:

${\lim }_{n\to \infty }\left[\frac{1^3+2^3+3^3+\cdots +n^3}{n^4}\right]-{\lim }_{n\to \infty }\left[\frac{1^4+2^4+3^4+\cdots +n^4}{n^5}\right]$ is:

Let $\frac{d}{dx}F(x)=\frac{e^{\sin x}}{x}$, $x>0$. If ${\int }_1^4\frac{3e^{\sin x^3}}{x}dx=F(k)-F(1)$, then one of the possible values of $k$, is:

The area of the region bounded by the curves $y=|x-1|$ and $y=3-|x|$ is:

Let $f(x)$ be a function satisfying $f^{\prime }(x)=f(x)$ 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:

The degree and order of the differential equation of the family of all parabolas whose axis is the $x$ -axis, are respectively:

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

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:

Locus of centroid of the triangle whose vertices are $(a\cos t,a\sin t)$, $(b\sin t,-b\cos t)$ and $(1,0)$, where $t$ is a parameter, is:

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:

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 $\alpha$ ($0<\alpha <\pi /4$) with the positive direction of $x$ -axis. The equation of its diagonal not passing through the origin is:

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:

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:

The normal at the point $(b{t}_1^2,2b{t}_1)$ on a parabola meets the parabola again in the point $(b{t}_2^2,2b{t}_2)$, then:

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:

A tetrahedron has vertices at $O(0,0,0)$, $A(1,2,1)$, $B(2,1,3)$ and $C(-1,1,2)$. Then the angle between the faces $OAB$ and $ABC$ will be:

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:

The lines $\frac{x-1}{2}=\frac{y-3}{1}=\frac{z-4}{k}$ and $\frac{x-1}{k}=\frac{y-4}{2}=\frac{z-5}{1}$ are coplanar if:

The two lines $x=ay+b$, $z=cy+d$ and $x=a^{\prime }y+b^{\prime }$, $z=c^{\prime }y+d^{\prime }$ will be perpendicular, if and only if:

The shortest distance from the plane $12x+4y+3z=327$ to the sphere $x^2+y^2+z^2+4x-2y-6z=155$ is:

Two systems of rectangular axes have the same origin. If a plane cuts them at distances $a,b,c$ and $a^{\prime },b^{\prime },c^{\prime }$ from the origin, then:

$\vec{a},\vec{b},\vec{c}$ are 3 vectors, such that $\vec{a}+\vec{b}+\vec{c}=0$, $|\vec{a}|=1,|\vec{b}|=2,|\vec{c}|=3$, then $\vec{a}\cdot \vec{b}+\vec{b}\cdot \vec{c}+\vec{c}\cdot \vec{a}$ is equal to:

If $\vec{u},\vec{v}$ and $\vec{w}$ are three non-coplanar vectors, then $(\vec{u}+\vec{v}-\vec{w})\cdot [(\vec{u}-\vec{v})\times (\vec{v}-\vec{w})]$ equals:

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:

The vectors $\overrightarrow{AB}=3\hat{i}+4\hat{k}$ and $\overrightarrow{AC}=5\hat{i}-2\hat{j}+4\hat{k}$ are the sides of a triangle $ABC$. The length of the median through $A$ is:

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:

Let $\vec{u}=\hat{i}+\hat{j}$, $\vec{v}=\hat{i}-\hat{j}$ and $\vec{w}=2\hat{i}+3\hat{j}+\hat{k}$. If $\hat{n}$ is a unit vector such that $\vec{u}\cdot \hat{n}=0$ and $\vec{v}\cdot \hat{n}=0$, then $|\vec{w}\cdot \hat{n}|$ is equal to:

The median of a set of 9 distinct observations is 20.5. If each of the largest 4 observations of the set is increased by 2, then the median of the new set:

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:

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:

Events $A,B,C$ are mutually exclusive 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:

The mean and variance of a random variable having a binomial distribution are 4 and 2 respectively, then $P(X=1)$ is:

The resultant of forces $\vec{P}$ and $\vec{Q}$ is $\vec{R}$. If $\vec{Q}$ is doubled then $\vec{R}$ is doubled. If the direction of $\vec{Q}$ is reversed, then $\vec{R}$ is again doubled. Then $P^2:Q^2:R^2$ is:

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:

A couple is of moment $\vec{G}$ and the force forming the couple is $\vec{P}$. If $\vec{P}$ is turned through a right angle, the moment of the couple thus formed is $\vec{H}$. If instead, the forces $\vec{P}$ are turned through an angle $\alpha$, then the moment of couple becomes:

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 with respect to the first is least after a time:

Two stones are projected from the top of a cliff $h$ meters 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:

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: