AIEEE 2004 (Maths) Mock Test

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

The range of the function $f(x)={}^{7-x}{P}_{x-3}$ is

Let $z,w$ be complex numbers such that $\overline{z}+\overline{iw}=0$ and $\arg zw=\pi .$ Then $\arg z$ equals

If $z=x-iy$ and $z^{\frac{1}{3}}=p+iq,$ then $\frac{(\frac{x}{p}+\frac{y}{q})}{(p^2+q^2)}$ is equal to

If $|z^2-1|=|z{|}^2+1,$ then $z$ lies on

Let $A=\left(\begin{array}{ccc}0& 0& -1\\ 0& -1& 0\\ -1& 0& 0\end{array}\right).$ The only correct statement about the matrix $A$ is

Let $A=\left(\begin{array}{ccc}1& -1& 1\\ 2& 1& -3\\ 1& 1& 1\end{array}\right)$ and $10B=\left(\begin{array}{ccc}4& 2& 2\\ -5& 0& a\\ 1& -2& 3\end{array}\right).$ If $B$ is the inverse of matrix $A,$ then $a$ is

If $a_1,a_2,a_3,....,a_n,....$ are in G.P., then the value of the determinant
$$\left|\begin{array}{ccc}\log a_n& \log a_{n+1}& \log a_{n+2}\\ \log a_{n+3}& \log a_{n+4}& \log a_{n+5}\\ \log a_{n+6}& \log a_{n+7}& \log a_{n+8}\end{array}\right|$$
is

Let two numbers have arithmetic mean $9$ and geometric mean $4.$ Then these numbers are the roots of the quadratic equation

If $(1-p)$ is a root of quadratic equation $x^2+px+(1-p)=0,$ then its roots are

Let $S(K)=1+3+5+...+(2K-1)=3+K^2.$ Then which of the following is true?

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

The number of ways of distributing $8$ identical balls in $3$ distinct boxes so that none of the boxes is empty is

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

The coefficient of the middle term in the binomial expansion in powers of $x$ of $(1+\alpha x{)}^4$ and of $(1-\alpha x{)}^6$ is the same if $\alpha$ equals

The coefficient of $x^n$ in expansion of $(1+x)(1-x{)}^n$ is

If $S_n={\sum }_{r=0}^n\frac{1}{{}^n{C}_r}$ and $t_n={\sum }_{r=0}^n\frac{r}{{}^n{C}_r},$ then $\frac{t_n}{S_n}$ is equal to

Let $T_r$ be the $r$ th term of an A.P. 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

The sum of the first $n$ terms of the series $1^2+2\cdot 2^2+3^2+2\cdot 4^2+5^2+2\cdot 6^2+...$ is $\frac{n(n+1{)}^2}{2}$ when $n$ is even. When $n$ is odd the sum is

The sum of series $\frac{1}{2!}+\frac{1}{4!}+\frac{1}{6!}+...$ is

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 $\cos \frac{\alpha -\beta }{2}$ is

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

The sides of a triangle are $\sin \alpha ,\cos \alpha$ and $\sqrt{1+\sin \alpha \cos \alpha }$ for some $0<\alpha <\frac{\pi }{2}.$ Then the greatest angle of the triangle is

A 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 $40$ meter away from the tree the angle of elevation becomes $30^{\circ }.$ The breadth of the river is

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

The graph of the function $y=f(x)$ is symmetrical about the line $x=2$ then

The domain of the function $f(x)=\frac{{\sin }^{-1}(x-3)}{\sqrt{9-x^2}}$ is

If ${\lim }_{x\to \infty }(1+\frac{a}{x}+\frac{b}{x^2}{)}^{2x}=e^2,$ then the values of $a$ and $b,$ are

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

If $x=e^{y+e^{y+...to\cdot \infty }},$ $x>0$ then $\frac{dy}{dx}$ is

A point on the parabola $y^2=18x$ at which the ordinate increases at twice the rate of the abscissa is

A function $y=f(x)$ has a second order derivative $f^{\prime \prime }(x)=6(x-1).$ 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

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

If $2a+3b+6c=0,$ then at least one root of the equation $a{x}^2+bx+c=0$ lies in the interval

${\lim }_{n\to \infty }{\sum }_{r=1}^n\frac{1}{n}{e}^{\frac{r}{n}}$ is

If $\int \frac{\sin x}{\sin (x-\alpha )}dx=Ax+B\log \sin (x-\alpha )+C,$ then value of $(A,B)$ is

$\int \frac{dx}{\cos x-\sin x}$ is equal to

The value of ${\int }_{-2}^3|1-x^2|dx$ is

The value of $I={\int }_0^{\pi /2}\frac{(\sin x+\cos x{)}^2}{\sqrt{1+\sin 2x}}dx$ is

If ${\int }_0^{\pi }xf(\sin x)dx=A{\int }_0^{\pi /2}f(\sin x)dx,$ then $A$ is

If $f(x)=\frac{e^x}{1+e^x},I_1={\int }_{f(-a)}^{f(a)}xg\{x(1-x)\}dx$ and $I_2={\int }_{f(-a)}^{f(a)}g\{x(1-x)\}dx,$ then the value of $\frac{I_2}{I_1}$ is

The area of the region bounded by the curves $y=|x-2|,$ $x=1,x=3$ and the $x$ -axis is

The differential equation for the family of curves $x^2+y^2-2ay=0,$ where $a$ is an arbitrary constant is

The solution of the differential equation $ydx+(x+x^{2}y)dy=0$ is

Let $A(2,-3)$ and $B(-2,1)$ be vertices of a triangle $ABC.$ If the centroid of this triangle moves on the line $2x+3y=1,$ then the locus of the vertex $C$ is the line

The equation of the straight line passing through the point $(4,3)$ and making intercepts on the co-ordinate axes whose sum is $-1$ is

If the sum of the slopes of the lines given by $x^2-2cxy-7y^2=0$ is four times their product, then $c$ has the value

If one of the lines given by $6x^2-xy+4c{y}^2=0$ is $3x+4y=0,$ then $c$ equals

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 variable circle passes through the fixed point $A(p,q)$ and touches $x$ -axis. The locus of the other end of the diameter through $A$ is

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

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

If $a\ne 0$ and the line $2bx+3cy+4d=0$ passes through the points of intersection of the parabolas $y^2=4ax$ and $x^2=4ay,$ then

The eccentricity of an ellipse, with its centre at the origin, is $\frac{1}{2}.$ If one of the directrices is $x=4,$ then the equation of the ellipse is

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 ${\cos }^2\theta$ equals

Distance between two parallel planes $2x+y+2z=8$ and $4x+2y+4z+5=0$ is

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 co-ordinates of each of the point of intersection are given by

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 co-planar then $\lambda$ equals

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

Let $\vec{a},\vec{b}$ and $\vec{c}$ be three non-zero vectors such that no two of these are collinear. If the vector $\vec{a}+2\vec{b}$ is collinear with $\vec{c}$ and $\vec{b}+3\vec{c}$ is collinear with $\vec{a}$ then $\vec{a}+2\vec{b}+6\vec{c}$ equals

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

If $\vec{a},\vec{b},\vec{c}$ are non-coplanar vectors and $\lambda$ is a real number, then the vectors $\vec{a}+2\vec{b}+3\vec{c},\lambda \vec{b}+4\vec{c}$ and $(2\lambda -1)\vec{c}$ are non-coplanar for

Let $\vec{u},\vec{v},\vec{w}$ be such that $|\vec{u}|=1,|\vec{v}|=2,|\vec{w}|=3.$ If the projection $\vec{v}$ along $\vec{u}$ is equal to that of $\vec{w}$ along $\vec{u}$ and $\vec{v},\vec{w}$ are perpendicular to each other then $|\vec{u}-\vec{v}+\vec{w}|$ equals

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

Consider the following statements:
(a) Mode can be computed from histogram
(b) Median is not independent of change of scale
(c) Variance is independent of change of origin and scale.
Which of these is/are correct?

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

The probability that $A$ speaks truth is $\frac{4}{5},$ while this probability for $B$ is $\frac{3}{4}.$ The probability that they contradict each other when asked to speak on a fact is

A random variable $X$ has the probability distribution:

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

The mean and the variance of a binomial distribution are $4$ and $2$ respectively. Then the probability of $2$ successes is

With two forces acting at a point, the maximum effect is obtained when their resultant is $4\text{N}.$ If they act at right angles, then their resultant is $3\text{N}.$ Then the forces are

In a right angle $△ABC,$ $\angle A=90^{\circ }$ and sides $a,b,c$ are respectively, $5\text{cm},4\text{cm}$ and $3\text{cm}.$ If a force $\vec{F}$ has moments $0,9$ and $16$ in $\text{N cm}$ units respectively about vertices $A,B$ and $C,$ then magnitude of $\vec{F}$ is

Three forces $\vec{P},\vec{Q}$ and $\vec{R}$ acting along $IA,IB$ and $IC,$ where $I$ is the incentre of a $△ABC,$ are in equilibrium. Then $\vec{P}:\vec{Q}:\vec{R}$ is

A particle moves towards east from a point $A$ to a point $B$ at the rate of $4\text{km/h}$ and then towards north from $B$ to $C$ at the rate of $5\text{km/h}.$ If $AB=12\text{km}$ and $BC=5\text{km},$ then its average speed for its journey from $A$ to $C$ and resultant average velocity direct from $A$ to $C$ are respectively

A velocity $V=\frac{1}{4}\text{m/s}$ 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

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