AIEEE 2008 PYQ Mock Test

$AB$ is a vertical pole with $B$ at the ground level and $A$ at the top. A man finds that the angle of elevation of the point $A$ from a certain point $C$ on the ground is $60^{\circ }$. He moves away from the pole along the line $BC$ to a point $D$ such that $CD=7\text{m}$. From $D$, the angle of elevation of the point $A$ is $45^{\circ }$. Then the height of the pole is:

It is given that the events $A$ and $B$ are such that $P(A)=\frac{1}{4}$, $P(B|A)=\frac{1}{2}$ and $P(A|B)=\frac{2}{3}$. Then $P(B)$ is:

A die is thrown. Let $A$ be the event that the number obtained is greater than $3$. Let $B$ be the event that the number obtained is less than $5$. Then $P(A\cup B)$ is:

A focus of an ellipse is at the origin. The directrix is the line $x=4$ and the eccentricity is $1/2$. Then the length of the semi-major axis is:

A parabola has the origin as its focus and the line $x=2$ as the directrix. Then the vertex of the parabola is at:

The point diametrically opposite to the point $P(1,0)$ on the circle $x^2+y^2+2x+4y-3=0$ is:

Let $f:\mathbb{N}\to Y$ be a function defined as $f(x)=4x+3$, where $Y=\{y\in \mathbb{N}:y=4x+3\text{ for some }x\in \mathbb{N}\}$. The inverse of the function is:

The conjugate of a complex number is $\frac{1}{i-1}$. Then the complex number is:

Let $\mathbb{R}$ be the real line. Consider the following subsets of the plane $\mathbb{R}\times \mathbb{R}$:$S=\{(x,y):y=x+1\text{ and }0<x<2\}$$T=\{(x,y):x-y\text{ is an integer}\}$ Which one of the following is true?

The perpendicular bisector of the line segment joining $P(1,4)$ and $Q(k,3)$ has $y$ -intercept $-4$. Then a possible value of $k$ is:

The solution of the differential equation $\frac{dy}{dx}=\frac{y}{x}+x$ satisfying the condition $y(1)=1$ is:

The mean of the numbers $a,b,8,5,10$ is $6$ and the variance is $6.80$. Then which one of the following gives possible values of $a$ and $b$?

The vector $\vec{a}=\alpha \hat{i}+2\hat{j}+\beta \hat{k}$ lies in the plane of the vectors $\vec{b}=\hat{i}+\hat{j}$ and $\vec{c}=\hat{j}+\hat{k}$ and bisects the angle between $\vec{b}$ and $\vec{c}$. Then which one of the following gives possible values of $\alpha$ and $\beta$?

The non-zero vectors $\vec{a},\vec{b}$ and $\vec{c}$ are related by $\vec{a}=8\vec{b}$ and $\vec{c}=-7\vec{b}$. Then the angle between $\vec{a}$ and $\vec{c}$ is:

The line passing through the points $(5,1,a)$ and $(3,b,1)$ crosses the $yz$ -plane at the point $\left(0,\frac{17}{2},-\frac{13}{2}\right)$. Then:

If the straight lines $\frac{x-1}{k}=\frac{y-2}{2}=\frac{z-3}{3}$ and $\frac{x-2}{3}=\frac{y-3}{k}=\frac{z-1}{2}$ intersect at a point, then the integer $k$ is equal to:

Statement - 1: For every natural number $n\ge 2$, $1+\frac{1}{\sqrt{2}}+\cdots +\frac{1}{\sqrt{n}}>\sqrt{n}$.Statement - 2: For every natural number $n\ge 2$, $\sqrt{n(n+1)}<n+1$.

Let $A$ be a $2\times 2$ matrix with real entries. Let $I$ be the $2\times 2$ identity matrix. Denote by $\text{tr}(A)$ the sum of diagonal entries of $A$. Assume $A^2=I$.Statement - 1: If $A\ne I$ and $A\ne -I$, then $\det A=-1$.Statement - 2: If $A\ne I$ and $A\ne -I$, then $\text{tr}(A)\ne 0$.

Statement - 1: ${\sum }_{r=0}^n(r+1{)}^n{C}_r=(n+2)2^{n-1}$.Statement - 2: ${\sum }_{r=0}^n(r+1{)}^n{C}_r{x}^r=(1+x{)}^n+nx(1+x{)}^{n-1}$.

Let $p$ be the statement "$x$ is an irrational number", $q$ be the statement "$y$ is a transcendental number", and $r$ be the statement "$x$ is a rational number iff $y$ is a transcendental number".Statement - 1: $r$ is equivalent to either $q$ or $p$.Statement - 2: $r$ is equivalent to $\sim (p\leftrightarrow \sim q)$.

In a shop there are five types of ice-creams available. A child buys six ice-creams.Statement - 1: The number of different ways the child can buy the six ice-creams is ${}^{10}{C}_5$.Statement - 2: The number of different ways the child can buy the six ice-creams is equal to the number of different ways of arranging $6$ A’s and $4$ B’s in a row.

Let $f(x)=\{\begin{array}{ll}(x-1)\sin \left(\frac{1}{x-1}\right)& \text{if }x\ne 1\\ 0& \text{if }x=1\end{array}$. Then which one of the following is true?

The first two terms of a geometric progression add up to $12$. The sum of the third and the fourth terms is $48$. If the terms of the geometric progression are alternately positive and negative, then the first term is:

Suppose the cube $x^3-px+q$ has three distinct real roots where $p>0$ and $q>0$. Then which one of the following holds?

How many real solutions does the equation $x^7+14x^5+16x^3+30x-560=0$ have?

The statement $p\to (q\to p)$ is equivalent to:

The value of $\cot \left({\tan }^{-1}\frac{3}{4}+{\text{cosec}}^{-1}\frac{5}{3}\right)$ is:

The differential equation of the family of circles with fixed radius $5$ units and centre on the line $y=2$ is:

Let $I={\int }_0^1\frac{\sin x}{\sqrt{x}}dx$ and $J={\int }_0^1\frac{\cos x}{\sqrt{x}}dx$. Then which one of the following is true?

The area of the plane region bounded by the curves $x+2y^2=0$ and $x+3y^2=1$ is equal to:

The value of $\int \frac{\sin x}{\sin (x-\pi /4)}dx$ is:

How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which no two S are adjacent?

Let $a,b,c$ be any real numbers. Suppose that there are real numbers $x,y,z$ not all zero such that $x=cy+bz$, $y=az+cx$ and $z=bx+ay$. Then $a^2+b^2+c^2+2abc$ is equal to:

Let $A$ be a square matrix all of whose entries are integers. Then which one of the following is true?

The quadratic equations $x^2-6x+a=0$ and $x^2-cx+6=0$ have one root in common. The other roots of the first and second equations are integers in the ratio $4:3$. Then the common root is:

The organic chloro compound which shows complete stereochemical inversion during a ${\text{S}}_{N}2$ reaction is:

Toluene is nitrated and the resulting product is reduced with tin and hydrochloric acid. The product so obtained is diazotized and then heated with cuprous bromide. The reaction mixture so formed contains:

The coordination number and the oxidation state of the element ‘E’ in the complex $[\text{E}(\text{en}{)}_2({\text{C}}_2{\text{O}}_4)]{\text{NO}}_2$ (where ‘en’ is ethylenediamine) are, respectively:

Identify the wrong statement in the following:

Phenol, when it first reacts with concentrated sulphuric acid and then with concentrated nitric acid, gives:

In the following sequence of reactions, the alkene affords the compound ‘B’:
$${\text{CH}}_3\text{CH}={\text{CHCH}}_3\stackrel{{\text{O}}_3}{\to }\text{A}\stackrel{{\text{H}}_2\text{O/Zn}}{\to }\text{B}$$
The compound B is:

A larger number of oxidation states are exhibited by the actinoids than those by the lanthanoids, the main reason being:

In which of the following octahedral complexes of $\text{Co}$ (at. no. 27) will the magnitude of ${\Delta }_o$ be the highest?

At $80^{\circ }\text{C}$, the vapour pressure of pure liquid ‘A’ is $520\text{mm Hg}$ and that of pure liquid ‘B’ is $1000\text{mm Hg}$. If a mixture solution of ‘A’ and ‘B’ boils at $80^{\circ }\text{C}$ and $1\text{atm}$ pressure, the amount of ‘A’ in the mixture is ($1\text{atm}=760\text{mm Hg}$):

For a reaction $\frac{1}{2}A\to 2B$, the rate of disappearance of ‘A’ is related to the rate of appearance of ‘B’ by the expression:

The equilibrium constants $K_{P1}$ and $K_{P2}$ for the reactions $X\rightleftharpoons 2Y$ and $Z\rightleftharpoons P+Q$, respectively, are in the ratio of $1:9$. If the degree of dissociation of $X$ and $Z$ be equal, then the ratio of total pressure at these equilibria is:

Oxidizing power of chlorine in aqueous solution can be determined by the parameters:$\frac{1}{2}{\text{Cl}}_2(g)\stackrel{\frac{1}{2}{\Delta }_{\text{diss}}H}{\to }\text{Cl}(g)\stackrel{{\Delta }_{\text{eg}}H}{\to }{\text{Cl}}^{-}(g)\stackrel{{\Delta }_{\text{hyd}}H}{\to }{\text{Cl}}^{-}(aq)$ Using the data:${\Delta }_{\text{diss}}{H}_{{\text{Cl}}_2}=240\text{kJ/mol}$${\Delta }_{\text{eg}}{H}_{\text{Cl}}=-349\text{kJ/mol}$${\Delta }_{\text{hyd}}{H}_{\text{Cl}}=-381\text{kJ/mol}$ The energy involved in the conversion of $\frac{1}{2}{\text{Cl}}_2(g)$ to ${\text{Cl}}^{-}(aq)$ will be:

Which of the following factors is of no significance for roasting sulphide ores to the oxides and not subjecting the sulphide ores to carbon reduction directly?

Bakelite is obtained from phenol by reacting it with:

For the following three reactions, equilibrium constants are given:(a) $\text{CO}(g)+{\text{H}}_2\text{O}(g)\rightleftharpoons {\text{CO}}_2(g)+{\text{H}}_2(g);K_1$ (b) ${\text{CH}}_4(g)+{\text{H}}_2\text{O}(g)\rightleftharpoons \text{CO}(g)+3{\text{H}}_2(g);K_2$ (c) ${\text{CH}}_4(g)+2{\text{H}}_2\text{O}(g)\rightleftharpoons {\text{CO}}_2(g)+4{\text{H}}_2(g);K_3$ Which of the following relations is correct?

The absolute configuration of the following molecule is:

The electrophile, ${\text{E}}^{+}$, attacks the benzene ring to generate the intermediate $\sigma$ -complex. Of the following, which $\sigma$ -complex is of lowest energy?

$\alpha \text{-D-(+)-glucose}$ and $\beta \text{-D-(+)-glucose}$ are:

Standard entropy of ${\text{X}}_2$, ${\text{Y}}_2$ and ${\text{XY}}_3$ are $60$, $40$ and $50{\text{J K}}^{-1}{\text{mol}}^{-1}$, respectively. For the reaction $\frac{1}{2}{\text{X}}_2+\frac{3}{2}{\text{Y}}_2\to {\text{XY}}_3$, $\Delta H=-30\text{kJ}$. For the reaction to be at equilibrium, the temperature will be:

Four species are listed below:(i) ${\text{HCO}}_3^{-}$ (ii) ${\text{H}}_3{\text{O}}^{+}$ (iii) ${\text{HSO}}_4^{-}$ (iv) ${\text{HSO}}_3\text{F}$ Which one of the following is the correct sequence of their acid strength?

Which one of the following constitutes a group of isoelectronic species?

Which one of the following pairs of species have the same bond order?

The ionization enthalpy of hydrogen atom is $1.312\times 10^6\text{J/mol}$. The energy required to excite the electron in the atom from $n=1$ to $n=2$ is:

Which one of the following is the correct statement?

Given $E_{{\text{Cr}}^{3+}/\text{Cr}}^{\circ }=-0.72\text{V}$ and $E_{{\text{Fe}}^{2+}/\text{Fe}}^{\circ }=-0.42\text{V}$. The potential for the cell $\text{Cr}|{\text{Cr}}^{3+}(0.1\text{M})||{\text{Fe}}^{2+}(0.01\text{M})|\text{Fe}$ is:

Amount of oxalic acid present in a solution can be determined by its titration with ${\text{KMnO}}_4$ solution in the presence of ${\text{H}}_2{\text{SO}}_4$. The titration gives unsatisfactory results when carried out in the presence of $\text{HCl}$ because $\text{HCl}$:

The vapour pressure of water at $20^{\circ }\text{C}$ is $17.5\text{mm Hg}$. If $18\text{g}$ of glucose (${\text{C}}_6{\text{H}}_{12}{\text{O}}_6$) is added to $178.2\text{g}$ of water at $20^{\circ }\text{C}$, the vapour pressure of the resulting solution will be:

Among the following substituted silanes, the one which will give rise to cross-linked silicone polymer on hydrolysis is:

In context with the industrial preparation of hydrogen from water gas ($\text{CO}+{\text{H}}_2$), which of the following is the correct statement?

In a compound, atoms of element $\text{Y}$ form ccp lattice and those of element $\text{X}$ occupy $2/3$ rd of tetrahedral voids. The formula of the compound will be:

Gold numbers of protective colloids A, B, C and D are $0.50$, $0.01$, $0.10$ and $0.005$, respectively. The correct order of their protective powers is:

The hydrocarbon which can react with sodium in liquid ammonia is:

The treatment of ${\text{CH}}_3\text{MgX}$ with ${\text{CH}}_3\text{C}\equiv \text{C}-\text{H}$ produces:

The correct decreasing order of priority for the functional groups of organic compounds in the IUPAC system of nomenclature is:

The $p{K}_a$ of a weak acid, $\text{HA}$, is $4.80$. The $p{K}_b$ of a weak base, $\text{BOH}$, is $4.78$. The pH of an aqueous solution of the corresponding salt, $\text{BA}$, will be:

Electrons accelerated by potential $V$ are diffracted from a crystal. If $d=1\text{A˚}$ and $i=30^{\circ }$, $V$ should be about ($h=6.6\times 10^{-34}\text{J s},m_e=9.1\times 10^{-31}\text{kg},e=1.6\times 10^{-19}\text{C}$):

If a strong diffraction peak is observed when electrons are incident at an angle ‘$i$ ’ from the normal to the crystal planes with distance ‘$d$ ’ between them, de Broglie wavelength ${\lambda }_{dB}$ of electrons can be calculated by the relationship ($n$ is an integer):

In an experiment, electrons pass through a narrow slit of width ‘$d$ ’. They are detected on a screen at a distance ‘$D$ ’. Which graph correctly represents the number of electrons $N$ detected as a function of the detector position $y$ ($y=0$ corresponds to the middle of the slit)?

A planet in a distant solar system is $10$ times more massive than the earth and its radius is $10$ times smaller. Given that the escape velocity from the earth is $11\text{km/s}$, the escape velocity from the surface of the planet would be:

A spherical solid ball of volume $V$ is made of a material of density ${\rho }_1$. It is falling through a liquid of density ${\rho }_2$ (${\rho }_2<{\rho }_1$). Assume that the liquid applies a viscous force $F_{\text{viscous}}=-k{v}^2$. The terminal speed of the ball is:

Shown in the figure is a meter-bridge setup with null deflection in the galvanometer. One resistor is $55\Omega$ and the balancing length from left is $20\text{cm}$. The value of the unknown resistor $R$ on the right side is:

A thin rod of length $L$ is lying along the $x$ -axis from $x=0$ to $x=L$. Its linear density $\lambda$ varies as $\lambda =k(x/L{)}^n$. Which graph best approximates the dependence of the position $x_{CM}$ of the centre of mass on $n$ ($n\ge 0$)?

While measuring the speed of sound by performing a resonance column experiment, a student gets the first resonance at a column length of $18\text{cm}$ during winter. Repeating the same experiment during summer, she measures the column length to be $x\text{cm}$ for the second resonance. Then:

The dimension of magnetic field $B$ in M, L, T and C (Coulomb) is given as:

Consider a uniform square plate of side ‘$a$ ’ and mass ‘$m$ ’. The moment of inertia of this plate about an axis perpendicular to its plane and passing through one of its corners is:

A body of mass $m=3.513\text{kg}$ is moving along the $x$ -axis with a speed of $5.00\text{m/s}$. The magnitude of its momentum is recorded as:

An athlete in the olympic games covers a distance of $100\text{m}$ in $10\text{s}$. His kinetic energy can be estimated to be in the range:

A parallel plate capacitor with air between the plates has a capacitance of $9\text{pF}$. The separation between its plates is ‘$d$ ’. The space between the plates is now filled with two dielectrics. One has $k_1=3$ and thickness $d/3$ while the other has $k_2=6$ and thickness $2d/3$. Capacitance of the capacitor is now:

Statement - 1: Energy is released when heavy nuclei undergo fission or light nuclei undergo fusion.Statement - 2: For heavy nuclei, binding energy per nucleon increases with increasing $Z$ while for light nuclei it decreases with increasing $Z$.

Statement - 1: For a mass $M$ kept at the centre of a cube of side ‘$a$ ’, the flux of gravitational field passing through its sides is $4\pi GM$.Statement - 2: If the direction of a field due to a point source is radial and its dependence on the distance ‘$r$ ’ for the source is given as $1/r^2$, its flux through a closed surface depends only on the strength of the source enclosed and not on the size or shape of the surface.

A jar is filled with two non-mixing liquids $1$ and $2$ having densities ${\rho }_1$ and ${\rho }_2$ respectively. A solid ball of density ${\rho }_3$ is dropped and comes to equilibrium such that it floats at the interface, partially in both liquids. Which of the following is true for ${\rho }_1$, ${\rho }_2$ and ${\rho }_3$?

A working transistor with legs marked P, Q and R is tested. No conduction is found between P and Q. Connecting the negative terminal of the multimeter to R and positive to P or Q shows some resistance. Which is true?

Current $I$ enters at point A and leaves at point D of a conducting block. To find the voltage $\Delta V$ between points B and C, we assume current spreads over a hemispherical surface. The electric field at distance $r$ is $E=\rho j$. The potential difference $V_B-V_C$ is:

A student measures the focal length of a convex lens by measuring object distance ‘$u$ ’ and image distance ‘$v$ ’. The graph between $u$ and $v$ should look like:

A block of mass $0.50\text{kg}$ moving at $2.00\text{m/s}$ on a smooth surface strikes another mass of $1.00\text{kg}$ and they move together. The energy loss during collision is:

A capillary tube (A) is dipped in water. Another identical tube (B) is dipped in soap water solution. Which of the following shows the relative nature of the liquid columns?

Suppose an electron is attracted towards the origin by a force $k/r$ where $k$ is a constant. By applying Bohr model, the radius $r_n$ and kinetic energy $T_n$ of the $n^{\text{th}}$ orbital are:

A wave is described by $y(x,t)=0.005\cos (\alpha x-\beta t)$. If the wavelength and time period are $0.08\text{m}$ and $2.0\text{s}$ respectively, then $\alpha$ and $\beta$ are:

Two coaxial solenoids are made on a pipe of area $A=10{\text{cm}}^2$ and length $L=20\text{cm}$. If solenoids have $N_1=300$ and $N_2=400$ turns, their mutual inductance is (${\mu }_0=4\pi \times 10^{-7}\text{Tm/A}$):