AIEEE 2005 Maths Question Paper Mock Test

If $A^2-A+I=0$, then the inverse of $A$ is:

If the cube roots of unity are $1,\omega ,{\omega }^2$, then the roots of the equation $(x-1{)}^3+8=0$ are:

Let $A=\{3,6,9,12\}$ and $R=\{(3,3),(6,6),(9,9),(12,12),(6,12),(3,9),(3,12),(3,6)\}$ be a relation on the set $A$. The relation is:

Area of the greatest rectangle that can be inscribed in the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is:

The differential equation representing the family of curves $y^2=2c(x+\sqrt{c})$, where $c>0$ is a parameter, is of order and degree respectively:

The limit ${\lim }_{n\to \infty }\frac{1}{n}\left[{\sec }^2\frac{1}{n^2}+{\sec }^2\frac{4}{n^2}+\cdots +{\sec }^{2}1\right]$ equals:

$ABC$ is a triangle. Forces $P,Q,R$ acting along $IA,IB$ and $IC$ respectively are in equilibrium, where $I$ is the incentre of $△ABC$. Then $P:Q:R$ is:

If in a frequency distribution, the mean and median are $21$ and $22$ respectively, then its mode is approximately:

Let $P$ be the point $(1,0)$ and $Q$ a point on the locus $y^2=8x$. The locus of midpoint of $PQ$ is:

If $C$ is the midpoint of $AB$ and $P$ is any point outside $AB$, then:

If the coefficients of $r^{\text{th}}$, $(r+1{)}^{\text{th}}$ and $(r+2{)}^{\text{th}}$ terms in the binomial expansion of $(1+y{)}^m$ are in A.P., then $m$ and $r$ satisfy the equation:

In a triangle $PQR$, $\angle R=\frac{\pi }{2}$. If $\tan \frac{P}{2}$ and $\tan \frac{Q}{2}$ are the roots of $a{x}^2+bx+c=0,a\ne 0$, then:

The system of equations:$\alpha x+y+z=\alpha -1$,$x+\alpha y+z=\alpha -1$,$x+y+\alpha z=\alpha -1$ has no solution, if $\alpha$ is:

The value of $a$ for which the sum of the squares of the roots of the equation $x^2-(a-2)x-a-1=0$ assumes the least value is:

If roots of the equation $x^2-bx+c=0$ are two consecutive integers, then $b^2-4c$ equals:

If the letters of the word SACHIN are arranged in all possible ways and these words are written out as in a dictionary, then the word SACHIN appears at serial number:

The value of ${}^{50}{C}_4+{\sum }_{r=1}^6{}^{56-r}{C}_3$ is:

If $A=\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]$ and $I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$, then which one of the following holds for all $n\ge 1$?

If the coefficient of $x^7$ in $[a{x}^2+(1/bx){]}^{11}$ equals the coefficient of $x^{-7}$ in $[ax-(1/b{x}^2){]}^{11}$, then $a$ and $b$ satisfy the relation:

Let $f:(-1,1)\to B$ be a function defined by $f(x)={\tan }^{-1}\frac{2x}{1-x^2}$, then $f$ is both one-one and onto when $B$ is the interval:

If $z_1$ and $z_2$ are two non-zero complex numbers such that $|z_1+z_2|=|z_1|+|z_2|$, then $\arg z_1-\arg z_2$ is equal to:

If $\omega =\frac{z}{z-(i/3)}$ and $|\omega |=1$, then $z$ lies on:

If $a^2+b^2+c^2=-2$ and $f(x)=\left|\begin{array}{ccc}1+a^{2}x& (1+b^2)x& (1+c^2)x\\ (1+a^2)x& 1+b^{2}x& (1+c^2)x\\ (1+a^2)x& (1+b^2)x& 1+c^{2}x\end{array}\right|$, then $f(x)$ is a polynomial of degree:

The normal to the curve $x=a(\cos \theta +\theta \sin \theta )$, $y=a(\sin \theta -\theta \cos \theta )$ at any point $\theta$ is such that:

A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched?

Let $\alpha$ and $\beta$ be the distinct roots of $a{x}^2+bx+c=0$, then ${\lim }_{x\to \alpha }\frac{1-\cos (a{x}^2+bx+c)}{(x-\alpha {)}^2}$ is equal to:

Suppose $f(x)$ is differentiable at $x=1$ and ${\lim }_{h\to 0}\frac{f(1+h)}{h}=5$, then $f^{\prime }(1)$ equals:

Let $f$ be differentiable for all $x$. If $f(1)=-2$ and $f^{\prime }(x)\ge 2$ for $x\in [1,6]$, then:

If $f$ is a real-valued differentiable function satisfying $|f(x)-f(y)|\le (x-y{)}^2$ for $x,y\in \mathbb{R}$ and $f(0)=0$, then $f(1)$ equals:

If $x$ is so small that $x^3$ and higher powers of $x$ may be neglected, then $\frac{(1+x{)}^{3/2}-(1+\frac{1}{2}x{)}^3}{(1-x{)}^{1/2}}$ may be approximated as:

If $x={\sum }_{n=0}^{\infty }{a}^n,y={\sum }_{n=0}^{\infty }{b}^n,z={\sum }_{n=0}^{\infty }{c}^n$ where $a,b,c$ are in A.P. and $|a|<1,|b|<1,|c|<1$, then $x,y,z$ are in:

In a triangle $ABC$, let $\angle C=\frac{\pi }{2}$. If $r$ is the inradius and $R$ is the circumradius of the triangle, then $2(r+R)$ equals:

If ${\cos }^{-1}x-{\cos }^{-1}\frac{y}{2}=\alpha$, then $4x^2-4xy\cos \alpha +y^2$ is equal to:

If in a triangle $ABC$, the altitudes from the vertices $A,B,C$ on opposite sides are in H.P., then $\sin A,\sin B,\sin C$ are in:

If $I_1={\int }_0^1{2}^{x^2}dx,I_2={\int }_0^1{2}^{x^3}dx,I_3={\int }_1^2{2}^{x^2}dx$ and $I_4={\int }_1^2{2}^{x^3}dx$, then:

The area enclosed between the curve $y={\log }_e(x+e)$ and the coordinate axes is:

The parabolas $y^2=4x$ and $x^2=4y$ divide the square region bounded by the lines $x=4,y=4$ and the coordinate axes into three parts. If $S_1,S_2,S_3$ are respectively the areas of these parts numbered from top to bottom, then $S_1:S_2:S_3$ is:

If $x\frac{dy}{dx}=y(\log y-\log x+1)$, then the solution of the equation is:

The line parallel to the x-axis and passing through the intersection of the lines $ax+2by+3b=0$ and $bx-2ay-3a=0$, where $(a,b)\ne (0,0)$ is:

A spherical iron ball $10\text{ cm}$ in radius is coated with a layer of ice of uniform thickness that melts at a rate of $50{\text{ cm}}^3\text{/min}$. When the thickness of ice is $5\text{ cm}$, then the rate at which the thickness of ice decreases, is:

$\int {\left\{\frac{(\log x-1)}{1+(\log x{)}^2}\right\}}^{2}dx$ is equal to:

Let $f:\mathbb{R}\to \mathbb{R}$ be a differentiable function having $f(2)=6,f^{\prime }(2)=\frac{1}{48}$. Then ${\lim }_{x\to 2}\frac{{\int }_6^{f(x)}4t^{3}dt}{x-2}$ equals:

Let $f(x)$ be a non-negative continuous function such that the area bounded by the curve $y=f(x)$, x-axis and the ordinates $x=\pi /4$ and $x=\beta >\pi /4$ is $\beta \sin \beta +\frac{\pi }{4}\cos \beta +\sqrt{2}\beta$. Then $f(\pi /2)$ is:

The locus of a point $P(\alpha ,\beta )$ moving under the condition that the line $y=\alpha x+\beta$ is a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is:

If the angle $\theta$ between the line $\frac{x+1}{1}=\frac{y-1}{2}=\frac{z-2}{2}$ and the plane $2x-y+\lambda z+4=0$ is such that $\sin \theta =1/3$, then the value of $\lambda$ is:

The angle between the lines $2x=3y=-z$ and $6x=-y=-4z$ is:

If the plane $2ax-3ay+4az+6=0$ passes through the midpoint of the line joining the centres of the spheres $x^2+y^2+z^2+6x-8y-2z=13$ and $x^2+y^2+z^2-10x+4y-2z=8$, then $a$ equals:

The distance between the line $\vec{r}=(2\hat{i}-2\hat{j}+3\hat{k})+\lambda (\hat{i}-\hat{j}+4\hat{k})$ and the plane $\vec{r}\cdot (\hat{i}+5\hat{j}+\hat{k})=5$ is:

For any vector $\vec{a}$, the value of $|\vec{a}\times \hat{i}{|}^2+|\vec{a}\times \hat{j}{|}^2+|\vec{a}\times \hat{k}{|}^2$ is equal to:

If non-zero numbers $a,b,c$ are in H.P., then the straight line $\frac{x}{a}+\frac{y}{b}+\frac{1}{c}=0$ always passes through a fixed point. That point is:

If a vertex of a triangle is $(1,1)$ and the mid-points of two sides through this vertex are $(-1,2)$ and $(3,2)$, then the centroid of the triangle is:

If the circles $x^2+y^2+2ax+cy+a=0$ and $x^2+y^2-3ax+dy-1=0$ intersect in two distinct points $P$ and $Q$, then the line $5x+by-a=0$ passes through $P$ and $Q$ for:

A circle touches the x-axis and also touches the circle with centre at $(0,3)$ and radius $2$. The locus of the centre of the circle is:

If a circle passes through the point $(a,b)$ and cuts the circle $x^2+y^2=p^2$ orthogonally, then the equation of the locus of its centre is:

An ellipse has $OB$ as semi minor axis, $F$ and $F^{\prime }$ its foci and the angle $\angle FB{F}^{\prime }$ is a right angle. Then the eccentricity of the ellipse is:

Let $a,b$ and $c$ be distinct non-negative numbers. If the vectors $a\hat{i}+a\hat{j}+c\hat{k}$, $\hat{i}+\hat{k}$ and $c\hat{i}+c\hat{j}+b\hat{k}$ lie in a plane, then $c$ is:

If $\vec{a},\vec{b},\vec{c}$ are non-coplanar vectors and $\lambda$ is a real number then $[\lambda (\vec{a}+\vec{b}){\lambda }^2\vec{b}\lambda \vec{c}]=[\vec{a}\vec{b}+\vec{c}\vec{b}]$ for:

Let $\vec{a}=\hat{i}-\hat{k}$, $\vec{b}=x\hat{i}+\hat{j}+(1-x)\hat{k}$ and $\vec{c}=y\hat{i}+x\hat{j}+(1+x-y)\hat{k}$. Then $[\vec{a}\vec{b}\vec{c}]$ depends on:

Three houses are available in a locality. Three persons apply for the houses. Each applies for one house without consulting others. The probability that all the three apply for the same house is:

A random variable $X$ has Poisson distribution with mean $2$. Then $P(X>1.5)$ equals:

Let $A$ and $B$ be two events such that $P(\bar{A}\cup \bar{B})=1/6$, $P(A\cap B)=1/4$ and $P(\bar{A})=1/4$, where $\bar{A}$ stands for complement of event $A$. Then events $A$ and $B$ are:

A lizard, at an initial distance of $21\text{ cm}$ behind an insect, moves from rest with an acceleration of $2{\text{ cm/s}}^2$ and pursues the insect which is crawling uniformly along a straight line at a speed of $20\text{ cm/s}$. Then the lizard will catch the insect after:

Two points $A$ and $B$ move from rest along a straight line with constant acceleration $f$ and $f^{\prime }$ respectively. If $A$ takes $m\text{ sec.}$ more than $B$ and describes $n$ units more than $B$ in acquiring the same speed then:

$A$ and $B$ are two like parallel forces. A couple of moment $H$ lies in the plane of $A$ and $B$ and is combined with them. The resultant of $A$ and $B$ after combining is displaced through a distance:

The resultant $R$ of two forces acting on a particle is at right angles to one of them and its magnitude is one third of the other force. The ratio of larger force to smaller one is:

The sum of the series $1+\frac{1}{4\cdot 2!}+\frac{1}{16\cdot 4!}+\frac{1}{64\cdot 6!}+\dots$ ad inf. is:

The value of ${\int }_{-\pi }^{\pi }\frac{2x(1+\sin x)}{1+{\cos }^{2}x}dx$ is:

The plane $x+2y-z=4$ cuts the sphere $x^2+y^2+z^2-x+z-2=0$ in a circle of radius:

If the pair of lines $a{x}^2+2(a+b)xy+b{y}^2=0$ lie along diameters of a circle and divide the circle into four sectors such that the area of one of the sectors is thrice the area of another sector then:

Let $x_1,x_2,\dots ,x_n$ be $n$ observations such that $\sum x_i^2=400$ and $\sum x_i=80$. Then a possible value of $n$ among the following is:

A particle is projected from a point $O$ with velocity $u$ at an angle of $60^{\circ }$ with the horizontal. When it is moving in a direction at right angles to its direction at $O$, its velocity then is given by:

If both the roots of the quadratic equation $x^2-2kx+k^2+k-5=0$ are less than $5$, then $k$ lies in the interval:

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

A real valued function $f(x)$ satisfies the functional equation $f(x-y)=f(x)f(y)-f(a-x)f(a+y)$ where $a$ is a given constant and $f(0)=1$. Then $f(2a-x)$ is equal to:

If the equation $a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x=0,a_1\ne 0,n\ge 2$, has a positive root $x=\alpha$, then the equation $n{a}_n{x}^{n-1}+(n-1)a_{n-1}{x}^{n-2}+\cdots +a_1=0$ has a positive root, which is: