IPU CET 2017 Physics, Chemistry, Biology Question Paper(Available) : Download Solution PDF with Answer Key

Question 1:

A candle C sits between two parallel mirrors at a distance \( 0.2d \) from mirror 1. Here \( d \) denotes the distance between the mirrors. Multiple images of the candle appear in both mirrors. How far behind mirror 1 are the nearest three images of the candle in that mirror?

• (a) 0.2d, 1.8d, 2.2d
• (b) 0.2d, 2.2d, 4.2d
• (c) 0.2d, 1.8d, 3.8d
• (d) 0.2d, 0.8d, 1.4d

A block of mass \( m \) is in contact with the cart C as shown in the figure. The coefficient of static friction between the block and the cart is \( \mu \). The acceleration \( a \) of the cart that will prevent the block from falling satisfies:

• (a) \( a > \frac{mg}{\mu} \)
• (b) \( a > \frac{g}{\mu} \)
• (c) \( a \geq \frac{g}{\mu} \)
• (d) \( a < \frac{g}{\mu} \)

If the escape speed of a projectile on Earth's surface is \( 11.2 \, km/s \) and a body is projected out with thrice this speed, then determine the speed of the body far away from the Earth.

• (a) 56.63 km/s
• (b) 33 km/s
• (c) 39 km/s
• (d) 31.7 km/s

Consider the analogy between an oscillating spring-body system and an oscillating L-C-R circuit. Then, the correspondence between the two systems that is NOT correct is:

• (a) charge \( q \) corresponds to displacement \( x \) of the body
• (b) inductance \( L \) corresponds to mass \( m \) of the body
• (c) capacitance \( C \) corresponds to spring constant \( k \)
• (d) magnetic energy corresponds to kinetic energy of the body

A tank is filled with a liquid upto a height \( H \). A small hole is made at the bottom of this tank. Consider \( t_1 \) be the time taken to empty the first half of the tank and \( t_2 \) be the time taken to empty the rest half of the tank. Then, determine the ratio \( \frac{t_1}{t_2} \).

• (a) 1.33
• (b) 1.5
• (c) 2
• (d) 0.414

In a cinema, a picture 2.5 cm wide on the film is projected to an image 3.0 m wide on a screen that is 18 m away. The focal length of the lens is about:

• (a) 7.5 cm
• (b) 10 cm
• (c) 12.5 cm
• (d) 15 cm

The coefficient of volume expansion of glycerine is \( 49 \times 10^{-5} \, K^{-1} \). What is the fractional change in its density for a \( 30^\circ C \) rise in temperature?

• (a) \( 1.5 \times 10^{-2} \)
• (b) \( 2 \times 10^{-4} \)
• (c) \( 3.5 \times 10^{-3} \)
• (d) \( 2.5 \times 10^{-2} \)

A capacitor of capacitance 5 µF is connected as shown in the figure. The internal resistance of the cell is 0.5 Ω. The amount of charge on the capacitor plate is:

• (a) Zero
• (b) 5 µC
• (c) 10 µC
• (d) 25 µC

Use the diagram below to answer the following questions. 40 spheres of equal mass make two rings of 20 spheres each. The ring on the right has a radius twice as large as the ring on the left. At what position could a mass be placed so that the gravitational force it would experience would be the same from both rings?

• (a) A
• (b) B
• (c) C
• (d) D

If pressure of CO\(_2\) (real gas) in a container is given by \( P = \frac{RT}{2v-b} - \frac{a}{4v^2} \), then mass of the gas in the container is:

• (a) 11 g
• (b) 22 g
• (c) 33 g
• (d) 44 g

Two litres of water kept in a container at \( 27^\circ C \) is heated with a coil of 1 kW. The lid of the container is open and energy dissipates at the rate of 160 J/s. If the specific heat of water is 4.2 kJ/kg, then the time taken by coil to raise the temperature of water from \( 27^\circ C \) to \( 77^\circ C \) is:

• (a) 840 s
• (b) 500 s
• (c) 420 s
• (d) 372 s

The resistance \( R \) of a conductor varies with temperature \( t \) as shown in the figure. If the variation is represented by \( R_t = R_0 \left( 1 + \alpha t + \beta t^2 \right) \), then:

• (a) \( \alpha \) is positive and \( \beta \) is negative
• (b) \( \alpha \) is negative and \( \beta \) is positive
• (c) \( \alpha \) and \( \beta \) are both positive
• (d) \( \alpha \) is negative and \( \beta \) is negative

The bob of a pendulum is released from a horizontal position A as shown in the figure. If the length of the pendulum is 1.5 m, what is the speed with which the bob arrives at the lower most point B, given that it dissipated 5% of its initial energy against air resistance?

• (a) 5 m/s
• (b) 5.5 m/s
• (c) 5.3 m/s
• (d) 4.4 m/s

A ring of radius \( R \) is first rotated with an angular velocity \( \omega \), and then carefully placed on a rough horizontal surface. The coefficient of friction between the surface and the ring is \( \mu \). Time after which its angular speed is reduced to half is:

• (a) \( \frac{2 \omega R}{\mu g} \)
• (b) \( \frac{\omega R}{\mu g} \)
• (c) \( \frac{2 \omega R}{g} \)
• (d) \( \frac{\omega R}{2 \mu g} \)

A ball of mass \( m \) moving at a speed \( v \) makes a head-on collision with an identical ball at rest. The kinetic energy at the balls after the collision is \( \frac{3}{4} \) of the original. What is the coefficient of restitution?

• (a) \( \frac{1}{\sqrt{3}} \)
• (b) \( \frac{1}{\sqrt{2}} \)
• (c) \( \sqrt{2} \)
• (d) \( \sqrt{3} \)

A wire of arbitrary shape carries a current \( I = 2A \). Consider the portion of wire between \( (0, 0, 0) \) and \( (4, 4, 4) \). A magnetic field given by \[ B = \left( 1.2 \times 10^{-4} + 2 \times 10^{-4} \right) \, \hat{k} \]
\text{exists in the region. The force acting on the given portion of the wire is:

• (a) \( x = 10.25 \sin(\omega t + \theta) \)
• (b) \( x = 10.25 \sin(\omega t - \theta) \)
• (c) \( X = 11.25 \sin(\omega t + \omega) \)
• (d) \( x = 11.25 \sin(\omega t - \omega) \)

The power of the combination of two lenses made by keeping the convex lens of focal length 40 cm in contact with the concave lens of focal length 25 cm, is:

• (a) -1.5 D
• (b) -6.5 D
• (c) +6.5 D
• (d) +6.67 D

Two coherent point sources \( S_1 \) and \( S_2 \) vibrating in phase emit light of wavelength \( \lambda \). The separation between them is \( 2\lambda \). The light is collected on a screen placed at a distance \( D \gg \lambda \) from the slit \( S_1 \) as shown. The minimum distance, so that intensity at \( P \) is equal to intensity at \( O \), is:

• (a) \( \sqrt{2}D \)
• (b) \( \sqrt{3}D \)
• (c) \( \sqrt{8}D \)
• (d) \( \sqrt{5}D \)

What will be the displacement equation of the simple harmonic motion obtained by combining the motions?
Given: \[ x_1 = 2 \sin(\omega t), \quad x_2 = 4 \sin(\omega t + \frac{\pi}{2}), \quad x_3 = 6 \sin(\omega t) \]

• (a) \( x = 10.25 \sin(\omega t + \phi) \)
• (b) \( x = 10.25 \sin(\omega t - \phi) \)
• (c) \( x = 11.25 \sin(\omega t + \phi) \)
• (d) \( x = 11.25 \sin(\omega t - \phi) \)

A man holds a rectangular card in front of and parallel to a plane mirror. In order for him to see the entire image of the card, the least mirror area needed is:

• (a) that of the whole mirror, regardless of its size
• (b) that of the pupil of his eye
• (c) one-half that of the card
• (d) one-fourth that of the card

A stream of water flowing horizontally with the speed of 15 m/s gushes out of a tube of cross-sectional area \( 10^{-2} \, m^2 \) and hits a vertical wall nearby. What is the force exerted on the wall by the impact of water? Assuming it does not rebound.

• (a) 2250 N
• (b) 2000 N
• (c) 1500 N
• (d) 1000 N

The North pole of the Earth's magnetic field is at the geographical South pole. A compass is a small magnet whose North pole end is drawn in the approximate direction of

• (a) the geographical South pole along the lines of the magnetic field
• (b) the geographical North pole along the lines of the magnetic field
• (c) the geographical South pole against the lines of the magnetic field
• (d) the geographical North pole against the lines of the magnetic field

A shell of mass 0.020 kg is fired by a gun of mass 100 kg. If the muzzle speed of the shell is 80 m/s, what is the recoil speed of the gun?

• (a) 1.6 cm/s
• (b) 0.5 cm/s
• (c) 2 cm/s
• (d) 3 cm/s

An electron of a stationary hydrogen atom passes from the fifth energy level to the ground level. The velocity that the atom acquired as a result of photon emission will be:

• (a) \( \frac{24hR}{25m} \)
• (b) \( \frac{25hR}{24m} \)
• (c) \( \frac{24m}{25hR} \)
• (d) \( \frac{25m}{24hR} \)

If torques of equal magnitudes are applied to a hollow cylinder and a solid sphere both having the same mass and radius. The cylinder is free to rotate about its standard axis of symmetry and the sphere is free to rotate about an axis passing through its center. Which of the two will acquire a greater angular speed after a given time?

• (a) \( \omega_1 > \omega_2 \)
• (b) \( \omega_1 = \omega_2 \)
• (c) \( \omega_2 > \omega_1 \)
• (d) None of these

Which of the following factors by itself will increase the frequency at which an observer hears a sound emanating from a source?

• (a) A wind blows from the source to the observer
• (b) The source and the observer move away from each other at the same speed
• (c) The source and the observer move in the same direction at the same speed
• (d) The source moves away from the observer more slowly than the observer moves toward the source

If two simple pendulums first of bob mass \( M_1 \) and length \( l_1 \), and \( M_2 \) and \( l_2 \), Given \( M_1 = M_2 \) and \( l_1 = 2l_2 \), If the vibrational energies of both are same, then which of the following is correct?

• (a) Amplitude of \( B \) is smaller than \( A \)
• (b) Amplitude of \( B \) is greater than \( A \)
• (c) Amplitude will be same
• (d) None of the above

If 10% of a radioactive substance decays in every 5 years, then the percentage of the substance that will have decayed in 20 years, will be:

• (a) 40%
• (b) 50%
• (c) 65.6%
• (d) 34.4%

Ball A moving at 12 m/s collides elastically with ball B, initially at rest as shown. If both balls have the same mass, then what is the final velocity of ball A?

• (a) 3 m/s
• (b) 6 m/s
• (c) 9 m/s
• (d) 12 m/s

A stretched string of length 1 m fixed at both ends, when vibrated in one loop has a frequency of 200 Hz. It is now plucked at a point situated at 25 cm from one end. The stretched string would vibrate with a frequency of:

• (a) 100 Hz
• (b) 200 Hz
• (c) 400 Hz
• (d) 800 Hz

A and B are at an angle of 60° with each other. Their resultant makes an angle of 45° with a. If \( b = 2 \) units, then \( a \) is:

• (a) \( \sqrt{3} \)
• (b) \( \sqrt{3} + 1 \)
• (c) \( \sqrt{2} \)
• (d) \( \sqrt{3} - 1 \)

A beam of light (\( \lambda = 600 \, nm \)) from a distant source, falls on a single slit 1 mm wide and the resulting diffraction pattern is observed on a screen 2 m away. The distance between the first dark fringes on either side of the central bright fringe is:

• (a) 1.2 cm
• (b) 1.2 mm
• (c) 2.4 cm
• (d) 2.4 mm

A liquid is poured into a vessel at rest with the hole in a wall closed by a valve. It is filled to height \( H \). The distance of the hole from the top surface is \( h \). What is the horizontal acceleration required to move the vessel so that the liquid does not come out when the valve is opened (given \( l = \) length of the base)?

• (a) \( 2gh \)
• (b) \( g \)
• (c) \( \frac{1}{gH} \)
• (d) \( \frac{2gh}{l} \)

A glass prism ABC (refractive index 1.5), immersed in water (refractive index \( \frac{4}{3} \)). A ray of light is incident normally on face AB. If it is totally reflected at face AC, then

• (a) 40%
• (b) 50%
• (c) 65.6%
• (d) 34.4%

Two trains A and B of length 400 m each are moving on two parallel tracks with a uniform speed of 72 km/h in the same direction, with A ahead of speed B. The driver of B decides to overtake A and accelerates by 1 m/s². If after 50s, the guard of B just brushes past the driver of A, what was the original distance between them?

• (a) 100 m
• (b) 1150 m
• (c) 1300 m
• (d) 1250 m

An ideal gas is taken through a cyclic thermodynamical process through four steps. The amounts of heat involved in these steps are \( Q_1 = 5960 \, J \), \( Q_2 = -5585 \, J \), \( Q_3 = -2980 \, J \) and \( Q_4 = 3645 \, J \), respectively. The value of \( W_4 \) is:

• (a) 1315 J
• (b) 275 J
• (c) 765 J
• (d) 675 J

A rocket is fired from the Earth towards the Sun. At what distance from the Earth's centre, the gravitational force on the rocket is zero? Mass of the Sun = \( 2 \times 10^{30} \, kg \) and mass of the Earth = \( 6 \times 10^{24} \, kg \).

• (a) \( 2.6 \times 10^8 \, m \)
• (b) \( 3.2 \times 10^8 \, m \)
• (c) \( 3.9 \times 10^9 \, m \)
• (d) \( 2.3 \times 10^9 \, m \)

A block of mass m slides down with uniform speed on an inclined plane having inclination \( \theta \). If the coefficient of friction between the inclined plane and the block is \( \mu \), then the contact force between them is:

• (a) \( mg \sin \theta \)
• (b) \( mg \)
• (c) \( \sqrt{(mg \sin \theta)^2 + (\mu mg \cos \theta)^2} \)
• (d) \( mg \cos \theta \sqrt{1 + \mu^2} \)

A solid cylinder of mass 20 kg rotates about its axis with angular speed 100 rad/s. The radius of the cylinder is 0.25 m. What is the kinetic energy associated with the rotation of the cylinder? What is the magnitude of angular momentum of the cylinder about its axis?

• (a) 62.5 T-s
• (b) 70.4 T-s
• (c) 79.6 T-s
• (d) 60.5 T-s

A body of density \( D \), and mass \( M \) is moving downward in glycerine of density \( D_z \). What is the viscous force acting on it?

• (a) \( M g D \)
• (b) \( M g D_z \)
• (c) \( \frac{g}{D_z} \)
• (d) \( \frac{2g h}{D_z} \)

A stone of mass 0.25 kg tied to the end of a string is whirled round in a circle of radius 1.5 m with speed 40 rev/min in a horizontal plane. What is the tension in the string and what is the maximum speed with which the stone can be whirled around, if the string can withstand a maximum tension of 200 N?

• (a) 6 N, 35 m/s
• (b) 6 N, 37 m/s
• (c) 7.5 N, 46 m/s
• (d) 8 N, 38 m/s

If \( N_0 \) be the number of nuclei present at time \( t = 0 \), then the number of undecayed nuclei, \( N \), present after \( n \) mean life is

• (a) \( N = \left( \frac{1}{2} \right)^n N_0 \)
• (b) \( N = \left( \frac{1}{2} \right)^{1/n} N_0 \)
• (c) \( N = \left( \frac{1}{4} \right)^n N_0 \)
• (d) \( N = \left( \frac{1}{4} \right)^{1/n} N_0 \)

A U-shaped wire is dipped in a soap solution and removed. The thin soap film formed between the wire and light slider supports a weight of \( 1.5 \times 10^{-2} \, N \). The length of the slider is 30 cm. What is the surface tension of the film?

• (a) \( 3 \times 10^{-3} \, N/m \)
• (b) \( 2 \times 10^{-5} \, N/m \)
• (c) \( 4 \times 10^{-4} \, N/m \)
• (d) \( 2.5 \times 10^{-2} \, N/m \)

An object stands 4 cm in front of a converging lens. If the lens has a focal distance of 1 cm, where is the image formed?

• (a) 0.75 cm in front of the lens
• (b) 0.75 cm behind the lens
• (c) 1 cm behind the lens
• (d) 1.33 cm behind the lens

The length of a magnet is large compared to its width and breadth. The time period of its oscillation in vibration magnetometer is 2 s. The magnet is cut along its length into three equal parts and three parts are then placed on each other with their like poles together. The time period of this combination will be:

• (a) \( \frac{2}{3} \, s \)
• (b) \( \frac{1}{\sqrt{3}} \, s \)
• (c) \( \frac{2}{\sqrt{3}} \, s \)
• (d) \( \frac{3}{2} \, s \)

The image seen in a flat bathroom mirror is a

• (a) real image that appears behind the mirror
• (b) real image that appears in front of the mirror
• (c) virtual image that appears behind the mirror
• (d) virtual image that appears in front of the mirror

If electrical force between two charges is 200 N and we increase 10% charge on one of the charges and decrease 10% charge on the other, then electrical force between them for the same distance becomes

• (a) 200 N
• (b) 202 N
• (c) 198 N
• (d) 19 N

Electrons are accelerated through a potential difference \( V_e \), and protons are accelerated through a potential difference of 4 V. The de-Broglie wavelengths are \( \lambda_e \) and \( \lambda_p \) for electrons and protons, respectively. The ratio of \( \frac{\lambda_e}{\lambda_p} \) is given by (given \( m_e \) is mass of electrons and \( m_p \) is mass of protons):

• (a) \( \frac{\lambda_e}{\lambda_p} = \sqrt{\frac{m_e}{m_p}} \)
• (b) \( \frac{\lambda_e}{\lambda_p} = \sqrt{\frac{m_p}{m_e}} \)
• (c) \( \frac{\lambda_e}{\lambda_p} = \frac{1}{\sqrt{\frac{m_p}{m_e}}} \)
• (d) \( \frac{\lambda_e}{\lambda_p} = 2 \sqrt{\frac{m_p}{m_e}} \)

In hydrogen atom spectrum, frequency of \( 2.7 \times 10^{15} \, Hz \) of EM wave is emitted when transmission takes place from 2 to 1, the frequency emitted will be

• (a) \( 3.2 \times 10^{15} \, Hz \)
• (b) \( 3.2 \times 10^{15} \, Hz \)
• (c) \( 1.6 \times 10^{15} \, Hz \)
• (d) \( 16 \times 10^{15} \, Hz \)

Given
I. Plane mirrors
II. Concave mirrors
III. Convex mirrors
Given the preceding choices, virtual images can be formed by:

• (a) I, II and III
• (b) I and II
• (c) I and III
• (d) II only

Consider the following reaction sequence of alkene. \[ CH_3CH = CH_3 \xrightarrow{O_3} A \xrightarrow{H_2O} B \]
What is the product B?

• (a) CH\(_3\)CH\(_2\)COCH\(_3\)
• (b) CH\(_3\)COCH\(_2\)
• (c) CH\(_3\)CHO
• (d) CH\(_3\)CH\(_2\)CHO

Which of the following transition of an electron in H-atom will emit maximum energy?

• (a) \( n_6 \to n_5 \)
• (b) \( n_1 \to n_2 \)
• (c) \( n_3 \to n_2 \)
• (d) \( n_4 \to n_3 \)

Which of the following compounds produces the most heat per mole of compound when reacted with oxygen?

• (a) CH\(_4\)
• (b) C\(_2\)H\(_6\)
• (c) Cyclohexane
• (d) Cyclopentane

The correct set of quantum numbers for an element (Z=17) for the unpaired electron will be

• (a) 3, 1, 1, 1/2
• (b) 2, 0, 0, 1/2
• (c) 3, 0, 0, 1/2
• (d) 2, 1, 1, 0

From the following which pairs give the faster SN2 reaction?

The combustion of carbon monoxide yields carbon dioxide. The volume of oxygen gas needed to produce 22 g of carbon dioxide at STP is

• (a) 4.0 L
• (b) 5.6 L
• (c) 11 L
• (d) 22 L

What is the bond angle between Cl-O-Cl in Cl\(_2\)O\(_7\)?

• (a) 109°
• (b) 119°
• (c) 108°25'
• (d) 120°

A metal rod is in thermal contact with the two heat reservoirs both at constant temperature, one at 100K and the other at 200K. The rod conducts 1000 J of heat from the warmer to the colder reservoir. If no energy is exchanged with the surroundings, what is the total change of entropy?

• (a) -5 J/K
• (b) 0 J/K
• (c) 5 J/K
• (d) 10 J/K

Which of the following relation is correct for gaseous and reversible reactions?

• (a) \( \frac{K_C}{K_P} = (RT)^{\Delta n_g} \)
• (b) \( K_P = (P)^{\Delta n_g} \)
• (c) \( \frac{K_C}{K_X} = (P)^{-\Delta n_g} \)
• (d) \( \frac{K_C}{K} = \left( \frac{P}{RT} \right)^{\Delta n_g} \)

What is the conjugate base of H\(_2\)SO\(_4\)?

• (a) H\(_2\)O
• (b) OH\(^-\)
• (c) HSO\(_4^-\)
• (d) SO\(_4^{2-}\)

If the dipole moment of HBr is \( 2.60 \times 10^{-30} \, Cm \) and the interatomic spacing is 1.41 Å, then the percent ionic character of HBr is:

• (a) 16.23%
• (b) 13.21%
• (c) 11.50%
• (d) 15.81%

In the redox reaction \[ MnO_4^- + C_2O_4^{2-} \to Mn^{2+} + CO_2 + H_2O \]
The correct coefficients of \( C_2O_4^{2-} \) and \( \text{H^+ \) \text{are

• (a) 5, 8
• (b) 5, 4
• (c) 2, 3
• (d) 2, 1

In a sodium-sulphur battery, what is the half reaction for sodium in the spontaneous direction?

• (a) \( Na^+ + e^- \to Na \)
• (b) \( Na \to Na^+ + e^- \)
• (c) \( Na \to Na^+ + 2e^- \)
• (d) \( Na^+ + OH^- \to NaOH \)

Which of the following is not a true statement about ozone?

• (a) Both the O--O bonds in O\(_3\) are of equal length.
• (b) Bond order lies in between 1 and 2.
• (c) O--O bond angle in O\(_3\) is approximately 120°.
• (d) Ozone reacts with acidified K\(_2\)Cr\(_2\)O\(_7\) and oxidizes it.

Benzene and toluene combine to form an ideal solution. At 80°C, vapour pressure of pure benzene is 800 mmHg and the vapour pressure of pure toluene is 300 mmHg. If the vapour pressure of the solution is 400 mmHg, what are the mole fractions of benzene and toluene?

• (a) 60% benzene and 40% toluene
• (b) 50% benzene and 50% toluene
• (c) 40% benzene and 60% toluene
• (d) 20% benzene and 80% toluene

Which of the following statements about halogens are correct?

• (a) HF is the strongest hydrohalic acid
• (b) F\(_2\) is the strongest reducing agent among all types of halogen
• (c) The order of -ve electron gain enthalpy for halogens is F > Cl > Br > I
• (d) F\(_2\) has lower bond dissociation energy than Cl\(_2\)

Compound A undergoes the following reaction to form B and C: \[ KNO_3 + KOH \to Black coloured solid (A) \xrightarrow{heated} Green coloured solid B + Pink solution + A (C decolorised by Fe^{2+}) \]
Identify A, B, and C respectively

• (a) \( MnO_2, KMnO_4 \) and \( K_2MnO_4 \)
• (b) \( MnO_2, K_2MnO_4 \) and \( KMnO_4 \)
• (c) \( KMnO_4, MnO_2 \) and \( K_2MnO_4 \)
• (d) \( KMnO_4, K_2MnO_4 \) and \( MnO_2 \)

Terylene is a condensation polymer of ethylene glycol and

• (a) benzoic acid
• (b) phthalic acid
• (c) salicylic acid
• (d) terephthalic acid

When a sample of aluminium of unknown mass was subjected to 1.8 kJ of heat, the temperature of the aluminium sample increased from 26°C to 31°C. What was the mass of the sample?
(specific heat of Al = 0.90 J/g°C)

• (a) 200 g
• (b) 400 g
• (c) 600 g
• (d) 800 g

When Z grams of calcium carbonate completely burnt in air gives 28g of a solid compound, the mass of calcium carbonate used will be:

• (a) 200g
• (b) 100g
• (c) 56g
• (d) 50g

The correct increasing order of solubility of sulphates in water is:

• (a) \( BeSO_4 > MgSO_4 > CaSO_4 > SrSO_4 > BaSO_4 \)
• (b) \( BaSO_4 > SrSO_4 > CaSO_4 > MgSO_4 > BeSO_4 \)
• (c) \( H_2SO_3, NO_2 \) and \( SO_2 \)
• (d) \( SO_2, NO_2 \) and \( H_2O \)

Sulphur(s) when react with HNO\(_3\) (conc.) Green mainly gives:

• (a) \( H_2SO_4, NO_2 \) and \( H_2O \)
• (b) \( H_2S, N_2O_5 \) and \( SO_2 \)
• (c) \( H_2SO_3, NO_2 \) and \( SO_2 \)
• (d) \( SO_2, NO_2 \) and \( H_2O \)

100 g of O2 and 100 g of He(g) are in separate containers of equal volume. Both gases are at 100°C. Which one of the following statements is true?

• (a) Both gases would have the same pressure
• (b) The average kinetic energy of the O\(_2\) molecules is greater than that of the He molecules
• (c) There are equal numbers of He molecules and O\(_2\) molecules
• (d) The pressure of the He (g) would be greater than that of the O\(_2\) (g)

NH_3 molecule attract H^+ ion towards itself to form ammonium ion (NH_4^+) through:

• (a) electrovalent bond
• (b) metallic bond
• (c) co-ordinate bond
• (d) hydrogen bonding

Consider the following reactions,
C(s) + O2(g) → CO2(g), \Delta H = -94\text{ kcal
2CO(g) + O2 → 2CO2(g), \Delta H = -135.2\text{ kcalThen, the heat of formation of CO (g) is:

• (a) 26.4 kcal
• (b) -26.4 kcal
• (c) 41.2 kcal
• (d) -41.2 kcal

2HC = CH_2\text{Cl_2 \xrightarrow{\text{A \text{NH_4\text{Cl \xrightarrow{\text{B \text{Identify the compounds A and B

Which one of the following is the correct statement?

• (a) B₃N₃H₆ is known as inorganic benzene
• (b) Chlorides of both beryllium and aluminium have bridged chloride structure in gas phase
• (c) Boric acid is a protonic acid
• (d) Beryllium exhibits coordination number of six

A work is done on the system such that one mole of an ideal gas at 400K is compressed isothermally and reversibly to 1/10th of its original volume. The amount of (use R=2 cal) work done will be

• (a) 2.303 k-cal
• (b) 0.184 k-cal
• (c) 1.84 k-cal
• (d) 4.60 k-cal

The atomic radius of Ne is

• (a) greater than Ar
• (b) less than Ar
• (c) same as Ar
• (d) cannot be determined

The process involves smelting is

• (a) Al₂O₃ · 2H₂O \(\rightarrow\) Al₂O₃ + 2H₂O
• (b) 2PbS + 3O₂ \(\rightarrow\) 2PbO + SO₂
• (c) 2ZnO₃ \(\rightarrow\) ZnO + CO₂
• (d) Fe₂O₃ + 3C \(\rightarrow\) 2Fe + 3CO

Consider the following mechanism, \[ Cl + O_3 \rightarrow ClO + O_2 \] \[ ClO + O \rightarrow Cl + O_2 \]
In this mechanism, what is the catalyst?

• (a) Cl
• (b) O₃
• (c) ClO
• (d) O₂

Which of the following is used as a food preservative?

• (a) Sodium benzoate
• (b) Potassium chloride
• (c) Sodium bicarbonate
• (d) b and c both

Silica is a network solid of silicon and oxygen atoms. The empirical formula for silica is SiO₂. In silica, to how many oxygen atoms is each silicon bonded?

• (a) 1
• (b) 2
• (c) 3
• (d) 4

Bakelite is obtained by reaction of phenol with

• (a) acetaldehyde
• (b) acetal
• (c) formaldehyde
• (d) chlorobenzene

Curve X on the graph below shows the volume of oxygen formed during the catalytic decomposition of a 1.0 mol dm\(^{-3}\) solution of hydrogen peroxide.

\includegraphics{ig13.jpeg % Include your image here

Which change would produce the curve Y?

• (a) Adding water
• (b) Adding some 0.1 mol dm\(^{-3}\) hydrogen peroxide solution
• (c) Using a different catalyst
• (d) Lowering the temperature

Rate of dehydration of alcohols follows the order

• (a) 2\(^\circ\) > 1\(^\circ\) < CH\(_3\)OH < 3\(^\circ\)
• (b) 3\(^\circ\) > 2\(^\circ\) > 1\(^\circ\) > CH\(_3\)OH
• (c) 2\(^\circ\) > 3\(^\circ\) > 1\(^\circ\) > CH\(_3\)OH
• (d) CH\(_3\)OH > 1\(^\circ\) > 2\(^\circ\) > 3\(^\circ\)

As temperature is increased, the equilibrium of a gaseous reaction will always

• (a) shift to the right
• (b) shift to the left
• (c) remains constant
• (d) the answer cannot be determined from the given information

The ozone layer forms naturally by

• (a) the interaction of CFC with oxygen
• (b) the interaction of UV radiations with oxygen
• (c) the interaction of IR radiations with oxygen
• (d) the interaction of water vapour and oxygen

Which of the following statement is applicable for the Tyndall effect?

• (a) The diameter of the dispersed particle is much smaller than the wavelength of the light used
• (b) The diameter of the dispersed phase is much smaller than the wavelength of the light used
• (c) The refractive indices of the dispersed phase is much smaller than the wavelength of the light used
• (d) The refractive indices of the dispersed phase and the dispersion medium must differ greatly in

A catalyst will change all of the following except

• (a) enthalpy
• (b) activation energy
• (c) rate of the forward reaction
• (d) rate of the reverse reaction

On reduction of glycolic acid with HI, the product formed is

• (a) acetic acid
• (b) iodo-acetic acid
• (c) formic acid
• (d) None of these

The metallic sodium dissolved in liquid ammonia to form a deep blue colour solution. The deep blue colour is due to the formation of

• (a) solvated electrons, e\(^{-(NH_3)_x}\)
• (b) solvated atomic sodium Na(NH_3)_4
• (c) (Na\(^{+}\) + Na)
• (d) (NaNH_2 + H_2)

Which of the following statements is not correct?

• (a) At constant volume, the pressure of a contain amount of gas increases with increasing temperature
• (b) At constant temperature, the pressure of a certain amount of gas increases with increasing volume
• (c) At constant pressure, the volume of a certain amount of gas increases with increasing volume
• (d) In dealing with gas laws, the most convenient scale of temperature to use is the kelvin temperature scale

Which of the following flux is used to remove acidic impurities in metallurgical processes?

• (a) Silica
• (b) Limestone
• (c) Sodium Chloride
• (d) HCl

A cell has been set up as shown in the following diagram and E\(^0\) has been measured as 1.00V at 25°C. Calculate \(\Delta\)G\(^0\) for the reaction.

• (a) -386 kJ
• (b) -193 kJ
• (c) 1.00 kJ
• (d) 193 kJ

Select the incorrect statement.

• (a) Physical adsorption is reversible, while chemical is irreversible.
• (b) High pressure favors physical adsorption while low pressure favors chemical adsorption.
• (c) Physical adsorption is not specific, while chemical adsorption is highly specific.
• (d) High activation energy is required in chemical adsorption.

Which of the following is the correct Lewis structure for the ionic compound Ca(ClO2)2?

For the reactions
C + O2 \rightarrow CO2; \, \Delta H = -393 \, \text{kJ

2Zn + O2 \rightarrow 2ZnO, \, \Delta H = -412 \, \text{kJ

\text{the correct statement is:

• (a) carbon can oxidize Zinc
• (b) oxidation of carbon does not take place
• (c) zinc oxidation is not possible
• (d) Zinc can oxidize carbon

The decreasing order of reactivity towards electrophilic addition of the following is:

I. CH = CH

II. CH_2 = CH_2

III. H_2C = CH - Cl

• (a) I > II > III > IV
• (b) II > I > III > IV
• (c) IV > III > I > II
• (d) IV > I > III > II

An aqueous solution of 2% (wt/wt) non-volatile solute exerts a pressure of 1.004 bar at the boiling point of the solvent. What is the molecular mass of the solute?

• (a) 0.3655
• (b) 36.55
• (c) 41.34
• (d) 40.16

Consider line segments of lengths 1, 2, 3, \dots, 10, what is the number of triangles that can be formed from them?

• (a) 20
• (b) 30
• (c) 40
• (d) 50

The value of \[ \lim_{x \to 0} \int_0^x \sec^2 t \, dt \]
is:

• (a) 0
• (b) 3
• (c) 2
• (d) 1

How many paths are there from the point A to the point B in the figure below, if no point in a path is to be traversed more than once?

• (a) \( 2^3 \)
• (b) \( 2^6 \)
• (c) \( 12C_2 \)
• (d) 7

The number of real roots of \[ (6 - x)^4 + (8 - x)^4 = 16 \]

• (a) 0
• (b) 2
• (c) 4
• (d) 6

The expression \[ \left| (a + 1)^2 \, (b + 1)^2 \, (c + 1)^2 \right| \]
is equal to

• (a) \( -4(a - b)(b - c)(c - a) \)
• (b) \( 4(a - b)(b - c)(c - a) \)
• (c) \( 2(a - b)(b - c)(c - a) \)
• (d) \( 2(a - b)(b - c)(c - a) \)

The function \( f:[0,3] \to [1,29] \) defined by \[ f(x) = 2x^3 - 15x^2 + 36x + 1 \]
is

• (a) one-one and onto
• (b) onto but not one-one
• (c) one-one but not onto
• (d) neither one-one nor onto

For all \( n \in \mathbb{N}, 72n - 48n - 1 \) is divisible by

• (a) 25
• (b) 26
• (c) 1234
• (d) 2304

The sum \[ \sum_{k=0}^5 \left( 5C_k \right)^2 \]
is equal to

• (a) \( 25C5 \)
• (b) \( 15C5 \)
• (c) \( 10C5 \)
• (d) 1

If \[ f(x) = \begin{cases} \frac{1 - \sin x}{(n - 2x)^2} & if \quad x \neq \frac{\pi}{2}
\log (\sin x) \cdot \log \left( 1 + \frac{\pi}{4x + x^2} \right) & if \quad x = \frac{\pi}{2} \end{cases} \]
is continuous at \( x = \frac{\pi}{2} \), then \( k \) is equal to

• (a) \( -\frac{1}{6} \)
• (b) \( -\frac{1}{2} \)
• (c) \( -\frac{1}{4} \)
• (d) \( -\frac{2}{8} \)

The length of the axes of the conic \[ 9x^2 + 4y^2 - 6x + 4y + 1 = 0 \]
are

• (a) \( \frac{1}{2}, 9 \)
• (b) 3, 2
• (c) 1, \( \frac{3}{2} \)
• (d) 3, 2

A differentiable function \( f(x) \) has a relative minimum at \( x = 0 \), then the function \( y = f(x) + ax + b \) has a relative minimum at \( x = 0 \) for

• (a) all \( a \) and all \( b \)
• (b) all \( b \), if \( a = 0 \)
• (c) all \( b > 0 \)
• (d) all \( a > 0 \)

The maximum number of points into which 4 circles and 4 straight lines intersect, is

• (a) 26
• (b) 50
• (c) 56
• (d) 72

The solution of the differential equation \[ (x^2 - yx^2) \frac{dy}{dx} + y^2 + xy^2 = 0 \]
is

• (a) \( \log \left( \frac{x}{y} \right) = \frac{1}{x} + \frac{1}{y} + C \)
• (b) \( \log \left( \frac{x}{y} \right) = \frac{1}{x} + \frac{1}{y} + C \)
• (c) \( \log(xy) = \frac{1}{x} + \frac{1}{y} + C \)
• (d) \( \log(xy) + \frac{1}{x} + \frac{1}{y} = C \)

Given that \( \alpha_1, \alpha_2, \alpha_3 \) are the roots of \( 3x^3 - x^2 - 10x + 8 = 0 \), then the value of \( \alpha_1^2 + \alpha_2^2 + \alpha_3^2 \) is

• (a) 9/61
• (b) 61/9
• (c) 16/9
• (d) 9/16

The area bounded by the curves \( y = \cos x \) and \( y = \sin x \) between the ordinates \( x = 0 \) and \( x = \frac{3\pi}{2} \) is

• (a) \( 4\sqrt{2} - 1 \)
• (b) \( 4\sqrt{2} + 1 \)
• (c) \( 4\sqrt{2} - 2 \)
• (d) \( 4\sqrt{2} + 2 \)

The integral \[ \int_0^{\frac{\pi}{2}} \frac{\sqrt{\sin x + \cos x}}{\sin x + \cos x} \, dx \]
is equal to

• (a) 1
• (b) \( \frac{\pi}{2} \)
• (c) \( \frac{\pi}{4} \)
• (d) \( \frac{\pi}{3} \)

The value of the integral \[ \int_0^0 9 [x - 2\lfloor x \rfloor] \, dx \]
where \( [.] \) denotes the greatest integer function, is

• (a) 0.9
• (b) 0
• (c) 1.8
• (d) -0.9

The value of \[ \int_0^{\frac{\pi}{2}} \sin^7 \theta \cos^4 \theta \, d\theta \]
is

• (a) \( \frac{16}{1155} \)
• (b) \( \frac{16}{115} \)
• (c) \( \frac{385}{1155} \)
• (d) \( \frac{385}{16\pi} \)

The number of solutions of the equation \[ 3 \sin^2 x - 7 \sin x + 2 = 0 \]
in the interval \( [0, 5\pi] \) is

• (a) 0
• (b) 5
• (c) 6
• (d) 7

The inverse of matrix \[ \begin{pmatrix} 0 & 1 & -1
4 & -3 & 4
3 & -3 & 4 \end{pmatrix} \]
is

• (a) \[ \begin{pmatrix} 4 & 1 & -1
  3 & -1 & 3
  4 & -3 & -3 \end{pmatrix} \]
• (b) \[ \begin{pmatrix} 4 & -3 & 4
  3 & -3 & 4
  0 & 1 & -1 \end{pmatrix} \]
• (c) \[ \begin{pmatrix} 3 & -4 & 4
  4 & 1 & -3
  0 & 1 & 1 \end{pmatrix} \]
• (d) \[ \begin{pmatrix} -4 & 3 & 4
  3 & -3 & 4 \end{pmatrix} \]

If \[ a = \gamma \hat{i} + \delta \hat{j} + 2\zeta \hat{k}, \, b = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k}, \, r \times a = b \times a \, r \times b = a \times b \]
then a unit vector in the direction of \( r \) is

• (a) \( \frac{1}{\gamma \zeta} \left( \hat{i} + 3\hat{j} - \hat{k} \right) \)
• (b) \( \frac{1}{\gamma \zeta} \left( \hat{i} + \hat{j} + \hat{k} \right) \)
• (c) \( \frac{1}{\sqrt{3}} \left( \hat{i} + \hat{j} + \hat{k} \right) \)
• (d) None of these

The solution of the differential equation \[ \frac{d^2y}{dx^2} + 3y = -2x \]
is

• (a) \( c_1 \cos \sqrt{3}x + c_2 \sin \sqrt{3}x - \frac{2}{3}x \)
• (b) \( c_1 \cos \sqrt{3}x + c_2 \sin \sqrt{3}x - \frac{4}{5}x \)
• (c) \( c_1 \cos \sqrt{3}x + c_2 \sin \sqrt{3}x - 2x^2 + \frac{4}{3} \)
• (d) \( c_1 \cos \sqrt{3}x + c_2 \sin \sqrt{3}x - 2x^2 + \frac{9}{4} \)

From the bottom of a pole of height h, the angle of elevation of the top of a tower is \( \alpha \). The pole subtends an angle \( \beta \) at the top of the tower. The height of the tower is

If the numbers \( a_1, a_2, \ldots, a_n \) are different from zero and form an arithmetic progression, then \[ \frac{1}{a_1} + \frac{1}{a_2} + \frac{1}{a_3} + \dots + \frac{1}{a_n} \]
is equal to

The equation of a straight line passing through the point of intersection of \( x - y + 1 = 0 \) and \( 3x + y - 5 = 0 \) and perpendicular to one of them, is:

• (a) \( x + y + 3 = 0 \)
• (b) \( x - y - 3 = 0 \)
• (c) \( x - 3y - 5 = 0 \)
• (d) \( x - 3y + 5 = 0 \)

The number of integral values of \( \lambda \) for which the equation \[ x^2 + y^2 - 2\lambda x + 2\lambda y + 14 = 0 \]
represents a circle whose radius cannot exceed 6 is:

• (a) 10
• (b) 11
• (c) 12
• (d) 9

Let \( x_1 \) and \( x_2 \) be the roots of the equation \[ ax^2 + bx + c = 0 \quad (ac \neq 0) \]
Find the value of \[ \frac{1}{x_1} + \frac{1}{x_2} \]

• (a) \( \frac{\sqrt{b^2 - 4ac}}{c} \)
• (b) \( \frac{c}{\sqrt{b^2 - 4ac}} \)
• (c) \( \frac{b^2 - 4ac}{c^2} \)
• (d) \( \frac{c}{b^2 - 4ac} \)

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

• (a) \(\frac{2}{5}\)
• (b) \(\frac{1}{5}\)
• (c) \(\frac{3}{5}\)
• (d) \(\frac{2}{3}\)

The mid-point of the chord \(2x + y - 4 = 0\) of the parabola \(y^2 = 4x\) is

• (a) \(\left( \frac{5}{2}, -1 \right)\)
• (b) \(\left( -\frac{1}{2}, 1 \right)\)
• (c) \(\left( \frac{3}{2}, -1 \right)\)
• (d) None of these

Gas is being pumped into a spherical balloon. Then, the rate at of \(30 \, ft^3/min\) which the radius increases when it reaches the value 15 ft, is

• (a) \(\frac{1}{5} \, \frac{ft}{min}\)
• (b) \(\frac{1}{15} \, \frac{ft}{min}\)
• (c) \(\frac{1}{20} \, \frac{ft}{min}\)
• (d) \(\frac{1}{25} \, \frac{ft}{min}\)

If \(n\) is an integer, compute the value of the fraction
\[ \frac{(1 + \theta)^n}{(1 - \theta)^{n-2}} \]

• (a) \(2i\)
• (b) \((2i)^n\)
• (c) \(2r^{n-1}\)
• (d) \(2r^{n-2}\)

If the tangent at (1,1) on \( y^2 = x(2 - x^2) \) meets the curve again at P, then P is:

• (a) (4,4)
• (b) (-1,2)
• (c) \(\left( \frac{9}{4}, \frac{3}{8} \right)\)
• (d) None of these

If \( |a| < 1 \) and \( |b| < 1 \), then the sum of the series \[ a(a + b) + a^2(a^2 + b^2) + a^3(a^3 + b^3) + \cdots \]

• (a) \( \frac{a}{1 - ab} \)
• (b) \( \frac{a^2}{1 - ab} \)
• (c) \( \frac{b^2}{1 - b^2} \)
• (d) \( \frac{a}{1 - b^2} \)

Three points are chosen randomly and independently on a circle. What is the probability that all three pairwise distances between the points are less than the radius of the circle?

• (a) \( \frac{1}{36} \)
• (b) \( \frac{1}{24} \)
• (c) \( \frac{1}{18} \)
• (d) \( \frac{1}{12} \)

Choose the most appropriate option.
If \( A \) is a square matrix such that \( A^2 = A \) and \( B = I \), then \( AB + BA + I - (I - A)^2 \) is equal to:

• (a) \( A \)
• (b) \( 2A \)
• (c) \( -A \)
• (d) \( I - A \)

Calculate: \[ \left| \begin{matrix} x & y & x + y
y & x + y & x
x + y & x & y
\end{matrix} \right| \]

• (a) \( x^3 + y^3 \)
• (b) \( x^3 + y^3 + 3x^2y + 3xy^2 + 1 \)
• (c) \( -2(x^3 + y^3) \)
• (d) \( 2(x^3 + y^3) \)

Let \( f(x) = (x^3 + 2)^{30} \). If \( f^{(n)}(x) \) is a polynomial of degree \( 20 \), where \( f^{(n)}(x) \) denotes the \( n + h \)-order derivative of \( f(x) \), then the value of \( n \) is:

• (a) 60
• (b) 40
• (c) 70
• (d) 50

The set of values of \( x \) satisfying the system of inequalities \( 5x + 2 < 3x + 8 \) and \( \frac{x^2}{x-1} < 4 \) is:

• (a) \( (-\infty, 1) \)
• (b) \( (3, \infty) \)
• (c) \( (-\infty, 1) \cup (2, 3) \)
• (d) \( (-\infty, 1) \cup (2, 3) \cup (5, \infty) \)

Choose the most appropriate option.
The value of \(\lim_{x \to a \frac{\log x - 1}{x - a}\) is equal to

• (a) \( \frac{1}{a} \)
• (b) \( a \)
• (c) \( \log_a e \)
• (d) \( \frac{1}{a} \log_a e \)

Choose the most appropriate options.
\text{If \( |z^2 - 1| = |z|^2 + 1 \), \text{then \( z \) lies on a

• (a) Circle
• (c) Ellipse
• (d) None of these

Choose the most appropriate options.
\text{If \( f(x) = [x \sin n\pi x] \), \text{then which of the following is incorrect?

• (a) \( f(x) is continuous at x = 0 \)
• (b) \( f(x) is continuous in (-1, 0) \)
• (c) \( f(x) is differentiable at x = 1 \)
• (d) \( f(x) is differentiable in (-1, 1) \)

Choose the most appropriate option.
\text{Find the distance from the point A (2, 3, -1) to the given straight line. \[ x = 3t + 5, \quad y = 2t, \quad z = -2t - 25 \]

• (a) 15
• (b) 17
• (c) 19
• (d) 21

Choose the most appropriate options.
The degree of the differential equation \[ x = 1 + \frac{dy}{dx} + \frac{1}{2!} \left( \frac{d^2y}{dx^2} \right) + \frac{1}{3!} \left( \frac{d^3y}{dx^3} \right) + \dots \]

• (a) 3
• (b) 1
• (c) not defined
• (d) None of these

Choose the most appropriate option.
On the sphere \( (x - 1)^2 + (y + 2)^2 + (z - 3)^2 = 25 \), find the point \( M_0 \) to the plane \( 3x - 4z + 19 \).

• (a) \( (7, -2, -2) \)
• (b) \( (2, -2, 7) \)
• (c) \( (-2, -2, 7) \)
• (d) \( (-2, 7, -2) \)

Choose the most appropriate options.
The limit \[ \lim_{x \to 0} \frac{1 - \cos 2x}{x \tan 4x} \]

• (a) 4
• (b) 3
• (c) 2
• (d) \( \frac{1}{2} \)

Choose the most appropriate options.
If \( A(2,3) \) and \( B(-2,1) \) are two vertices of a triangle and the third vertex moves on the line \( 2x + 3y = 9 \), then the locus of the centroid of the new set of observations will be the triangle is

• (a) \( 2x + 3y = 3 \)
• (b) \( 2x + y = 3 \)
• (c) \( 2x - 3y = 1 \)
• (d) \( x - y = 1 \)

Choose the most appropriate option. \[ \lim_{x \to 0} \frac{\sin 2x}{\sin x} \]

• (a) 0
• (b) 1
• (c) \( \frac{1}{2} \)
• (d) \( \infty \)

Choose the most appropriate options.
Let \( f(x) = ax^3 + 5x^2 - bx + 1 \). If when divided by \( x - 1 \) it leaves a remainder of 5, and \( f(x) \) is divisible by \( 3x - 1 \), then

• (a) \( a = 26, b = 10 \)
• (b) \( a = 24, b = 12 \)
• (c) \( a = 24, b = 10 \)
• (d) \( a = 26, b = 12 \)

Choose the most appropriate options.
If the SD of a set of observations is 8 and each observation is divided by -2, then the SD of the new set of observation will be:

• (a) -4
• (b) -8
• (c) 8
• (d) 4

Choose the most appropriate option. \[ \lim_{x \to 1} \sin(x - 1) \cdot \tan\left( \frac{\pi}{x} \right) \]

• (a) 0
• (b) \( \frac{1}{\pi} \)
• (c) \( \frac{2}{\pi} \)
• (d) \( -\frac{3}{\pi} \)