A ball whose kinetic energy is E, is projected at an angle of 45° to the horizontal. The kinetic energy of the ball at the highest point of its flight will be

- 0
- E
- E/2
- E/√2

Energy required to move a body of mass m from an orbit of radius 2R to 3R is

- (GMm)/(3R
^{2}) - (GMm)/(6R)
- (GMm)/(8R)
- (GMm)/(12R
^{2})

A light string passing over a smooth light pulley connects two blocks of masses m_{1} and m_{2} (vertically). If the acceleration of the system is g/8, then the ratio of the masses is

- 5:3
- 4:3
- 8:1
- 9:7

A spring of force constant 800 N/m has an extension of 5 cm. The work done is extending it from 5 cm to 15 cm is

- 32 J
- 8 J
- 24 J
- 16 J

A tuning fork arrangement (pair) produces 4 beats/sec with one fork of frequency 288 cps. A little wax is placed on the unknown fork and it then produces 2 beats/sec. The frequency of the unknown fork is

- 286 cps
- 288 cps
- 292 cps
- 294 cps

Length of a string tied to two rigid supports is 40 cm. Maximum length (wavelength in cm) of a stationary wave produced on it is

- 20
- 40
- 80
- 120

A cylinder of height 20 m is completely filled with water. The velocity of efflux of water (in ms^{-1}) through a small hole on the side wall of the cylinder near its bottom is

- 20
- 25.5
- 10
- 5

Cooking gas containers are kept in a lorry moving with uniform speed. The temperature of the gas molecules inside will

- decrease for some, while increase for others
- decrease
- increase
- remain same

An astronomical telescope has a large aperture to

- reduce spherical aberration
- have low dispersion
- increase span of observation
- Have high resolution

An object 2.4 m in front of a lens forms a sharp image on a film 12 cm behind the lens. A glass plate 1 cm thick, of refractive index 1.50 is interposed between lens and film with its plane faces parallel to film. At what distance (from lens) should object be shifted to be in sharp focus on film?

- 5.6 m
- 7.2 m
- 3.2 m
- 2.4 m

Capacitance (in F) of a spherical conductor with radius 1 m is

- 10
^{-3} - 1.1 X 10
^{-10} - 9 X 10
^{-9} - 10
^{-6}

A parallel plate capacitor is made of two circular plates separated by a distance of 5 mm and with a dielectric of dielectric constant 2.2 between them. When the electric field in the dielectric is 3 × 10^{4} V/m, the charge density of the positive plate will be close to

- 6 × 10
^{–7}C/m^{2} - 6 × 10
^{4}C/m^{2} - 3 × 10
^{4}C/m^{2} - 3 × 10
^{–7}C/m^{2}

If in a circular coil A of radius R, current I is flowing and in another coil B of radius 2R a current 2I is flowing, then the ratio of the magnetic fields B_{A} and B_{B}, produced by them will be

- 2
- 1/2
- 1
- 1/4

A wire when connected to 220 V mains supply has power dissipation P_{1}. Now the wire is cut into two equal pieces which are connected in parallel to the same supply. Power dissipation in this case is P_{2}. Then P_{2}:P_{1} is

- 1
- 2
- 3
- 4

Infra red radiation is detected by

- photometer
- nanometer
- spectrometer
- pyrometer

According to Einstein’s photoelectric equation, the plot of the kinetic energy of the emitted photoelectrons from a metal vs the frequency of the incident radiation gives a straight line whose slope

- depends both on the intensity of the radiation and the metal used
- depends on the nature of the metal used
- depends on the intensity of the radiation
- is the same for all metals and independent of the intensity of the radiation

At a specific instant, emission of radioactive compound is deflected in a magnetic field. The compound can emit

- Electrons
- Protons
- He
^{2+} - Neutrons

The emission at instant can be

- 1, 2, 3, 4
- 1, 2, 3
- 2, 3
- 4

In a transformer, number of turns in the primary coil are 140 and that in the secondary coil are 280. If current in primary coil is 4 A, then that in the secondary coil is

- 10 A
- 6 A
- 4 A
- 2 A

A diode detector is used to detect an amplitude modulated wave of 60% modulation by using a condenser of capacity 250 pico farad in parallel with a load resistance 100 kilo ohm. Find the maximum modulated frequency which could be detected by it.

- 10.62 KHz
- 5.31 MHz
- 5.31 KHz
- 10.62 MHz

A block of mass m=0.1 kg is connected to a spring of unknown spring constant k. It is compressed to a distance x from its equilibrium position and released from rest. After approaching half the distance (x/2) from equilibrium position, it hits another block and comes to rest momentarily, while the other block moves with a velocity 3 ms^{-1}. The total initial energy of the spring is:

- 1.5 J
- 0.6 J
- 0.3 J
- 0.8 J

A uniform solid cylindrical roller of mass m is being pulled on a horizontal surface with force F parallel to the surface and applied at its centre. If the acceleration of the cylinder is a and it is rolling without slipping then the value of F is:

- ma
- 2 ma
- 3/2 ma
- 5/3 ma

A very long (length L) cylindrical galaxy is made of uniformly distributed mass and has radius R (R<<L). A star outside the galaxy is orbiting the galaxy in a plane perpendicular to the galaxy and passing through its centre. If the time period of star is T and its distance from the galaxy's axis is r, then:

- T
^{2}∝ r^{3} - T ∝ r
^{2} - T ∝ r
- T ∝ √r

The flow rate of water from a tap of diameter 1.25 cm is 0.48 L/min. The coefficient of viscosity of water is 10^{-3} Pa s. After sometime the flow rate is increased to 3 L/min. Characterize the flow for both the flow rates.

If it takes 5 minutes to fill a 15 litre bucket from a water tap of diameter 2/√π cm then the Reynolds number for the flow is (density of water = 10^{3} kg/m^{3} and viscosity of water = 10^{-3} Pa.s) close to:

- 5500
- 11,000
- 550
- 1100

When current in a coil changes from 5 A to 2 A in 0.1 s, an average voltage of 50 V is produced. The self-inductance of the coil is:

- 0.67 H
- 1.67 H
- 3 H
- 6 H

A person trying to lose weight by burning fat lifts a mass of 10 kg upto a height of 1 m 1000 times. Assume that the potential energy lost each time he lowers the mass is dissipated. How much fat will he use up considering the work done only when the weight is lifted up? Fat supplies 3.8×10^{7} J of energy per kg which is converted to mechanical energy with a 20% efficiency rate. Take g = 9.8 ms^{-2}

- 6.45 × 10
^{-3}kg - 9.89 × 10
^{-3}kg - 12.89 × 10
^{-3}kg - 2.45 × 10
^{-3}kg

A point particle of mass m, moves along the uniformly rough track PQR as shown in the figure. The coefficient of friction, between the particle and the rough track equals μ. The particle is released, from rest, from the point P and it comes to rest at a point R. The energies, lost by the ball, over the parts, PQ and QR, of the track, are equal to each other, and no energy is lost when particle changes direction from PQ to QR.

The values of the coefficient of friction μ and the distance x(=QR), are, respectively close to:

- 0.2 and 3.5 m
- 0.29 and 3.5 m
- 0.29 and 6.5 m
- 0.2 and 6.5 m

A galvanometer having a coil resistance of 100 Ω gives a full scale deflection, when a current of 1 mA is passed through it. The value of the resistance, which can convert this galvanometer into ammeter giving a full scale deflection for a current of 10 A is:

- 2 Ω
- 0.1 Ω
- 3 Ω
- 0.01 Ω

An observer looks at a distant tree of height 10 m with a telescope of magnifying power of 20. To the observer the tree appears:

- 10 times nearer
- 20 times taller
- 20 times nearer
- 10 times taller

Half-lives of two radioactive elements A and B are 20 minutes and 40 minutes, respectively. Initially, the samples have equal number of nuclei. After 80 minutes, the ratio of decayed numbers of A and B nuclei will be:

- 4 : 1
- 1 : 4
- 5 : 4
- 1 : 16

An arc lamp requires a direct current of 10 A and 80 V to function. If it is connected to a 220 V (rms), 50 Hz AC supply, the series inductor needed for it to work is close to:

- 0.08 H
- 0.044 H
- 0.065 H
- 80 H

A pipe open at both ends has a fundamental frequency f in air. The pipe is dipped vertically in water so that half of it is in water. The fundamental frequency of the air column is now:

- 3f/4
- 2f
- f
- f/2

Arrange the following electromagnetic radiations per quantum in the order of increasing energy:

A: Blue light; B: Yellow light; C: X-ray; D: Radio wave

- A, B, D, C
- C, A, B, D
- B, A, D, C
- D, B, A, C

An electric dipole is placed along the x-axis at the origin P. A point P is at a distance of 20 cm from this origin such that OP makes an angle 60° with x axis. If electric field at P makes an angle θ with x-axis, the value of θ is?

A function f from the set of natural numbers to integers defined by

f(n) = (n-1)/2, when n is odd

f(n) = -n/2, when n is even

- one-one and onto both
- one-one and but not onto
- neither one-one nor onto
- onto but not one-one

For real x, let f(x) = x^{3} + 5x + 1, then

- f is one-one and onto R
- f is onto R but not one-one
- f is neither one-one nor onto R
- f is one-one but not onto R

If ((1 + i)/(1 - i))^{x} = 1, then

- x = 4n, where n is any positive integer.
- x = 2n, where n is any positive integer.
- x = 4n + 1, where n is any positive integer.
- x = 2n + 1, where n is any positive integer.

If |z - 4| < |z - 2|, its solution is given by

- Re(z) > 3
- Re(z) > 0
- Re(z) < 0
- Re(z) > 2

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

- 0, -1
- 1, 1
- 0, 1
- 2, 1

If a, b, c are distinct +ve real numbers and a^{2} + b^{2} + c^{2} = 1, then ab + bc + ca is

- greater than 1
- equal to 1
- less than 1
- any real number

If A^{2} – A + I = 0, then the inverse of A is

- A - I
- A
- I + A
- I - A

Number greater than 1000 but less than 4000 is formed using the digits 0, 2, 3, 4 when repetition allowed is

- 105
- 125
- 128
- 625

Let S(k) = 1 + 3 + 5 + .. + (2k – 1) = 3 + k^{2} . Then which of the following is true?

- principle of mathematical induction can be used to prove the formula
- S(k) implies S(k + 1)
- S(k) implies S(k - 1)
- S(1) is correct

Let f(x) = 4 and f′(x) = 4. Then Lim_{x→2} (x f(2) − 2 f(x)) / (x − 2) is given by

- -4
- 2
- 3
- -2

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 α be the angle between their directions of motion. The relative velocity of the second particle w.r.t. the first is least after a time

- t = (u sin α)/f
- t = (f cos α)/u
- t = (u sin α)
- t = (u cos α)/f

The area of the region bounded by the parabola (y – 2)^{2} = x – 1, the tangent to the parabola at the point (2, 3) and the x-axis is

- 3
- 6
- 9
- 12

The order and degree of the differential equation (1 + 3dy/dx)^{2/3} = 4d^{3}y/dx^{3} are

- (3, 3)
- (3, 1)
- (1, 2)
- (1, 2/3)

In a class of 100 students there are 70 boys whose average marks in a subject are 75. If the average marks of the complete class is 72, then what is the average of the girls?

- 74
- 65
- 68
- 73

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 median of the new set

- is two times the original median
- is increased by 2
- remains the same as that of the original set
- is decreased by 2