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QUESTION: 1

A body is projected at t = 0 with a velocity 10 ms^{-1} at an angle of 60° with the horizontal. The radius of curvature of its trajectory at t = 1s is R. Neglecting air resistance and taking acceleration due to gravity g = 10 ms^{-2}, the value of R is :

Solution:

v_{x} = 10cos60° = 5 m/s

v_{y} = 10cos30° = 5 √3 m/s

velocity after t = 1 sec.

v_{x }= 5 m/s

QUESTION: 2

A particle is moving along a circular path with a constant speed of 10 ms^{-1}. What is the magnitude of the change in velocity of the particle, when it moves through an angle of 60° around the centre of the circle?

Solution:

QUESTION: 3

A hydrogen atom, initially in the ground state is excited by absorbing a photon of wavelength 980Å. The radius of the atom in the excited state, in terms of Bohr radius a_{0}, will be : (h_{c} = 12500 eV – Å)

Solution:

Energy of photon = 12500/980 = 12.75eV

∴ Electron will excite to n= 4

Since 'R' ∝ n^{2}

∴ Radius of atom will be 16a_{0}

QUESTION: 4

A liquid of density ρ is coming out of a hose pipe of radius a with horizontal speed v and hits a mesh. 50% of the liquid passes through the mesh unaffected. 25% looses all of its momentum and 25% comes back with the same speed. The resultant pressure on the mesh will be :

Solution:

Momentum per second carried by liquid per second is ρav^{2}

net force due to reflected liquid =

net force due to stopped liquid =

QUESTION: 5

An electromagnetic wave of intensity 50 Wm^{-2} enters in a medium of refractive index 'n' without any loss. The ratio of the magnitudes of electric fields, and the ratio of the magnitudes of magnetic fields of the wave before and after entering into the medium are respectively, given by :

Solution:

similarly

QUESTION: 6

An amplitude modulated signal is given by V(t) = 10[1 + 0.3cos(2.2 x 10^{4})]sin(5.5 x 10^{5}t). Here t is in seconds. The sideband frequencies (in kHz) are, [Given π = 22/7]

Solution:

Side band frequency are

QUESTION: 7

The force of interaction between two atoms is given by where x is the distance, k is the Boltzmann constant and T is temperature and α and β are two constants. The dimension of β is :

Solution:

QUESTION: 8

The charges Q + q and +q are placed at the vertices of a right-angle isosceles triangle as shown below. The net electrostatic energy of the configuration is zero, it the value of Q is:

Solution:

QUESTION: 9

In the circuit shown, the switch S_{1} is closed at time t = 0 and the switch S_{2} is kept open. At some later time (t_{0}), the switch S_{1} is opened and S_{2} is closed. The behaviour of the current I as a function of time 't' is given by :

Solution:

From time t = 0 to t = t_{0}, growth of current takes place and after that decay of current takes place.

most appropriate is (2)

QUESTION: 10

Equation of travelling wave on a stretched string of linear density 5 g/m is y = 0.03 sin(450 t – 9x) where distance and time are measured is SI units. The tension in the string is :

Solution:

y = 0.03 sin(450 t – 9x)

⇒ T = 2500 × 5 × 10^{–3}

= 12.5 N

QUESTION: 11

An equilateral triangle ABC is cut from a thin solid sheet of wood. (see figure) D, E and F are the mid-points of its sides as shown and G is the centre of the triangle. The moment of inertia of the triangle about an axis passing through G and perpendicular to the plane of the triangle is I_{0}. It the smaller triangle DEF is removed from ABC, the moment of inertia of the remaining figure about the same axis is I. Then:

Solution:

Suppose M is mass and a is side of larger

triangle, then M/4 and a/2 will be mass and side length of smaller triangle.

QUESTION: 12

There are two long co-axial solenoids of same length l. the inner and outer coils have radii r_{1} and r_{2} and number of turns per unit length n_{1} and n_{2} respectively. The ratio of mutual inductance to the self-inductance of the inner-coil is :

Solution:

QUESTION: 13

A rigid diatomic ideal gas undergoes an adiabatic process at room temperature,. The relation between temperature and volume of this process is TV^{x} = constant, then x is :

Solution:

For adiabatic process : TVg^{γ-1} = constant

For diatomic process :

QUESTION: 14

The gas mixture constists of 3 moles of oxygen and 5 moles of argon at temperature T. Considering only translational and rotational modes, the total inernal energy of the system is:

Solution:

QUESTION: 15

In a Young's double slit experiment, the path different, at a certain point on the screen, between two interfering waves is 1/8th of wavelength. The ratio of the intensity at this point to that at the centre of a bright fringe is close to :

Solution:

QUESTION: 16

If the deBronglie wavelength of an electron is equal to 10^{–3} times the wavelength of a photon of frequency 6 × 10^{14} Hz, then the speed of electron is equal to :

(Speed of light = 3 × 10^{8} m/s

Planck's constant = 6.63 × 10^{–34} J.s

Mass of electron = 9.1 × 10^{–31} kg)

Solution:

QUESTION: 17

A slab is subjected to two forces of same magnitude F as shown in the figure. Force is in XY-plane while force F_{1} acts along z-axisat the point . The moment of these forces about point O will be :

Solution:

Torque for F_{1} force

Torque for F_{2} force

QUESTION: 18

A satellite is revolving in a circular orbit at a height h from the earth surface, such that h << R where R is the radius of the earth.

Assuming that the effect of earth's atmosphere can be neglected the minimum increase in the speed requried so that the satellite could escape from the gravitational field of earth is :

Solution:

QUESTION: 19

In an experiment electrons are accelerated, from rest, by applying a voltage of 500 V. Calculate the radius of the path if a magnetic field 100 mT is then applied.

[Charge of the electron = 1.6 × 10^{–19} C Mass of the electron = 9.1 × 10^{–31} kg]

Solution:

QUESTION: 20

A particle undergoing simple harmonic motion has time dependent displacement given by The ratio of kinetic to potential energy of this particle at t = 210 s will be :

Solution:

Hence ratio is 3 (most appropriate)

QUESTION: 21

Ice at –20°C is added tp 50 g of water at 40°C. When the temperature of the mixture reaches 0°C, it is found that 20 g of ice is still unmelted. The amount of ice added to the water was close to

(Specific heat of water = 4.2 J/g/°C)

Specific heat of Ice = 2.1 J/g/°C

Heat of fusion of water at 0°C = 334 J/g)

Solution:

Let amount of ice is m gm.

According to principal of calorimeter

heat taken by ice = heat given by water

∴ 20 × 2.1 × m + (m - 20) × 334

= 50 × 4.2 × 40

376 m = 8400 + 6680

m = 40.1 g

QUESTION: 22

In the figure shown below, the charge on the left plate of the 10 μF capacitor is -30 μC. The charge on the right plate of the 6 μF capacitor is :

Solution:

6µF & 4µF are in parallel & total charge on this combination is 30 µC

∴ Charge on 6µF capacitor =

= 18 µC

Since charge is asked on right plate therefore is +18µC

Correct answer is (4)

QUESTION: 23

In the given circuit the current through Zener Diode is close to :

Solution:

Since voltage across zener diode must be less than 10V therefore it will not work in breakdown region, & its resistance will be infinite & current through it = 0

∴ correct answer is (4)

QUESTION: 24

The variation of refractive index of a crown glass thin prism with wavelength of the incident light is shown. Which of the following graphs is the correct one, if D_{m} is the angle of minimum deviation?

Solution:

Since D_{m} = (µ – 1)A

& on increasing the wavelength, µ decreases & hence D_{m} decreases. Therefore correct answer is (2)

QUESTION: 25

The resistance of the meter bridge AB in the given figure is 4Ω. With a cell of emf ε = 0.5 V and rheostat resistance R_{h} = 2Ω the null point is obtained at some point J. When the cell is replaced by another one of emf ε = ε_{2} the same null point J is found for R_{h} = 6 Ω. The emf ε_{2} is;

Solution:

Potential gradient with R_{h} = 2Ω

Let null point be at ℓ cm

Now with R_{h} = 6Ω new potential gradient is and at null point

dividing equation (1) by (2) we get

QUESTION: 26

The given graph shows variation (with distance r from centre) of :

Solution:

QUESTION: 27

Two equal resistance when connected in series to a battery, consume electric power of 60 W. If these resistances are now connected in parallel combination to the same battery, the electric power consumed will be :

Solution:

In series condition, equivalent resistance is 2R thus power consumed is

In parallel condition, equivalent resistance is R/ 2 thus new power is

o r P' = 4P = 240W

QUESTION: 28

An object is at a distance of 20 m from a convex lens of focal length 0.3 m. The lens forms an image of the object. If the object moves away from the lens at a speed of 5 m/s, the speed and direction of the image will be :

Solution:

From lens equation

velocity of image wrt. to lens is given by v_{I/L} = m^{2}v_{O/L}

direction of velocity of image is same as that of object

v_{O/L} = 5 m/s

= 1.16 × 10^{–3} m/s towards the lens

QUESTION: 29

A body of mass 1 kg falls freely from a height of 100 m on a platform of mass 3 kg which is mounted on a spring having spring constant k = 1.25 x 10^{6} N/m. The body sticks to the platform and the spring's maximum compression is found to be x. Given that g = 10 ms^{-2}, the value of x will be close to :

Solution:

Velocity of 1 kg block just before it collides with 3kg block =

Applying momentum conversation just before and just after collision.

initial compression of spring 1.25 × 10^{6} x_{0} = 30 ⇒ x_{0} ≈ 0

applying work energy theorem,W_{g} + W_{sp} = ΔKE

solving x ≈ 4 cm

QUESTION: 30

In a Wheatstone bridge (see fig.), Resistances P and Q are approximately equal. When R = 400 Ω, the bridge is equal. When R = 400 Ω, the bridge is balanced. On inter-changing P and Q, the value of R, for balance, is 405 Ω. The value of X is close to :

Solution:

After interchanging P and Q

From (i) and (ii)

QUESTION: 31

For the cell Zn(s) | Zn^{2+}(aq) || M^{x+} (aq) | M(s), different half cells and their standard electrode potentials are given below :

If , which cathode will give a mximum value of E^{0}_{cell }per electron transferred?

Solution:

QUESTION: 32

The correct match between items-I and II is :

Solution:

QUESTION: 33

If a reaction follows the Arrhenius equation, the plot lnk vs 1/(RT) gives straight line with agradient (–y) unit. The energy required to activate the reactant is :

Solution:

QUESTION: 34

The concentration of dissolved oxygen (DO) in cold water can go upto :

Solution:

In cold water, dissolved oxygen (DO) can reach a concentration upto 10 ppm

QUESTION: 35

The major product of the following reaction is:

Solution:

QUESTION: 36

Th correct statements among (a) to (d) regarding H_{2} as a fuel are :

(a) It produces less pollutant than petrol

(b) A cylinder of compressed dihydrogen weighs ~ 30times more than a petrol tank producing the same amount of energy

(c) Dihydrogen is stored in tanks of metal alloys like NaNi_{5}

(d) On combustion, values of energy released per gram of liquid dihydrogen and LPG are 50 and 142 kJ, respectively

Solution:

QUESTION: 37

The major poduct of the following reaction is:

Solution:

QUESTION: 38

The element that usually does not show variable oxidation states is :

Solution:

Usally Sc(Scandium) does not show variable oxidation states.

Most common oxidation states of :

(i) Sc : +3

(ii) V : +2, +3, +4, +5

(iii) Ti : +2, +3, +4

(iv) Cu : +1, +2

QUESTION: 39

An organic compound is estimated through Dumus method and was found to evolve 6 moles of CO_{2}. 4 moles of H_{2}O and 1 mole of nitrogen gas. The formula of the compound is :

Solution:

Hence, C_{6}H_{8}N_{2}

QUESTION: 40

The major product of the following reaction is :

Solution:

QUESTION: 41

Among the following compound which one is found in RNA?

Solution:

For the given structure 'uracil' is found in RNA

QUESTION: 42

Which compound(s) out of the following is/are not aromatic?

Solution:

out of the given options only is aromatic.

Hence (B), (C) and (D) are not aromatic

QUESTION: 43

The correct match between Item (I) and Item (II) is :

Solution:

(A) Norethindrone – Antifertility

(B) Ofloaxacin – Anti-Biotic

(C) Equanil – Hypertension (traiquilizer)

QUESTION: 44

Heat treatment of muscular pain involves radiation of wavelength of about 900 nm. Which spectral line of H-atom is suitable for this purpose?

[RH = 1 × 10^{5} cm^{–1}, h = 6.6 × 10^{–34} Js, c = 3 × 10^{8} ms^{–1}]

Solution:

QUESTION: 45

Consider the reaction,

N_{2}(g) + 3H_{2}(g) ⇔ 2NH_{3}(g)

The equilibrium constant of the above reaction is K_{P}. If pure ammonia is left to dissociate, the partial pressure of ammonia at equilibrium is given by

(Assume that P_{NH3} <<P_{total} at equilibrium)

Solution:

QUESTION: 46

Match the ores(Column A) with the metals (column B) :

Solution:

Siderite : FeCO_{3}

Kaolinite : Al_{2}(OH)_{4}Si_{2}O_{5}

Malachite : Cu(OH)_{2}.CuCO_{3}

Calamine : ZnCO_{3}

QUESTION: 47

The correct order of the atomic radii of C, Cs, Al and S is :

Solution:

Atomic radii order : C < S < Al < Cs

Atomic radius of C : 170 pm

Atomic radius of S : 180 pm

Atomic radius of Al : 184 pm

Atomic radius of Cs : 300 pm

QUESTION: 48

Match the metals (Column I) with the coordination compound(s) / enzyme(s) (Column II)

Solution:

(i) Wilkinson catalyst : RhCl(PPh_{3})_{3}

(ii) Chlorophyll : C_{55}H_{72}O_{5}N_{4}Mg

(iii) Vitamin B_{12}(also known as cyanocobalamin) contain cobalt.

(iv) Carbonic anhydrase contains a zinc ion.

QUESTION: 49

A 10 mg effervescent tablet contianing sodium bicarbonate and oxalic acid releases 0.25 ml of CO_{2} at T = 298.15 K and p = 1 bar. If molar volume of CO_{2} is 25.0 L under such condition, what is the percentage of sodium bicarbonate in each tablet ?

[Molar mass of NaHCO_{3} = 84 g mol^{-1}]

Solution:

QUESTION: 50

The major product of the following reaction is:

Solution:

QUESTION: 51

Two blocks of the same metal having same mass and at temperature T_{1} and T_{2}, respectively. are brought in contact with each other and allowed to attain thermal equilibrium at constant pressure. The change in entropy, ΔS, for this process is :

Solution:

QUESTION: 52

The chloride that CANNOT get hydrolysed is :

Solution:

CCl_{4} cannot get hydrolyzed due to the absence of vacant orbital at carbon atom.

QUESTION: 53

For the ch emical reaction X ⇔ Y, the standard reaction Gibbs energy depends on temperature T (in K) as :

Solution:

QUESTION: 54

The freezing point of a diluted milk sample is found to be –0.2°C, while it should have been –0.5°C for pure milk. How much water has been added to pure milk to make the diluted sample?

Solution:

QUESTION: 55

A solid having density of 9 × 10^{3} kg m^{–3} forms face centred cubic crystals of edge length 200√2 pm. What is the molar mass of the solid ?

(Avogadro constant ≌ 6 x 10^{23} mol^{-1}, π ≌ 3)

Solution:

QUESTION: 56

The polymer obtained from the following reactions is :

Solution:

QUESTION: 57

An example of solid sol is :

Solution:

QUESTION: 58

Peroxyacetyl nitrate (PAN), an eye irritant is produced by :

Solution:

Photochemical smog produce chemicals such as formaldehyde, acrolein and peroxyacetyl nitrate (PAN).

QUESTION: 59

NaH is an example of :

Solution:

NaH is an example of ionic hydride which is also known as saline hydride.

QUESTION: 60

The amphoteric hydroxide is :

Solution:

Be(OH)_{2} is amphoteric in nature while rest all alkaline earth metal hydroxide are basic in nature.

QUESTION: 61

Let It AA^{T} = I_{3}, then |p| is

Solution:

A is orthogonal matrix

QUESTION: 62

The area (in sq. units) of the region bounded by the curve x^{2} = 4y and the straight line x = 4y – 2 :-

Solution:

x = 4y – 2 & x^{2} = 4y

⇒ x^{2 }= x + 2 ⇒ x^{2} – x – 2 = 0

x = 2, – 1

QUESTION: 63

The outcome of each of 30 items was observed; 10 items gave an outcome 1/2 -d each, 10 items gave outcome 1/2 each and the remaining 10 items gave outcome 1/2 + d each. If the variance of this outcome data is 4/3 then |d| equals :-

Solution:

Variance is independent of origin. So we shift the given data by 1/2.

QUESTION: 64

The sum of an infinite geometric series with positive terms is 3 and the sum of the cubes of its terms is 27/19. Then the common ratio of this series is :

Solution:

QUESTION: 65

Let and be coplanar vectors. Then the non-zero vector is:

Solution:

QUESTION: 66

Let , where x and y are real numbers, then y – x equals :

Solution:

Hence, y – x = 198 – 107 = 91

QUESTION: 67

Let and g(x) = |f(x)| + f (|x|). Then, in the interval (–2, 2), g is :-

Solution:

and f(|x|) = x^{2} - 1, x ∈ [-2, 2]

It is not differentiable at x = 1

QUESTION: 68

Let f : R → R be defined by x ∈ R. Then the range of f is :

Solution:

f(0) = 0 & f(x) is odd.

Further, if x > 0 then

QUESTION: 69

The sum of the real values of x for which the middle term in the binomial expansion of equals 5670 is :

Solution:

⇒ 70x^{8} = 5670

QUESTION: 70

The value of r for which 20C_{r} 20C_{0} + 20C_{r–1} 20C_{1} + 20C_{r–2} 20C_{2} + .... 20C_{0} 20C_{r }is maximum, is

Solution:

Given sum = coefficient of x^{r} in the expansion of (1 + x)^{20}(1 + x)^{20},

which is equal to ^{40}C_{r}

It is maximum when r = 20

QUESTION: 71

Let a_{1}, a_{2}, ....., a_{10} be a G.P. If a_{3}/a_{1} = 25, then a_{9}/a_{5} equals.

Solution:

a_{1}, a_{2}, ....., a_{10} are in G.P.,

Let the common ratio be r

QUESTION: 72

If for a suitable chosen integer m and a function A(x), where C is a constant of integration then (A(x))^{m} equals :

Solution:

Case-II x ≤ 0

QUESTION: 73

In a triangle, the sum of lengths of two sides is x and the product of the lengths of the same two sides is y. If x^{2} - c^{2} = y, where c is the length of the third side of the triangle, then the circumradius of the triangle is :

Solution:

Given a + b = x and ab = y

If x^{2} – c^{2} = y ⇒ (a + b)^{2} – c^{2} = ab

⇒ a^{2} + b^{2} – c^{2} = –ab

QUESTION: 74

The value of the integral (where [x] denotes the greatest integer less than ^{20}Cr or equal to x) is :

Solution:

QUESTION: 75

If the system of linear equations

2x + 2y + 3z = a

3x - y + 5z = b

x - 3y + 2z = c

where a, b, c are non-zero real numbers, has more then one solution, then :

Solution:

P_{1} : 2x + 2y + 3z = a

P_{2} : 3x – y + 5z = b

P_{3} : x – 3y + 2z = c

We find

P_{1} + P_{3} = P_{2} ⇒ a + c = b

QUESTION: 76

A square is inscribed inthe circle x^{2} + y^{2} – 6x + 8y – 103 = 0 with its sides parallel to the corrdinate axes. Then the distance of the vertex of this square which is nearest to the origin is :-

Solution:

QUESTION: 77

Let f_{k}(x) = 1/k (sin^{k }x + cos^{k} x) for k = 1,2,3,....... Then for all x ∈ R, the value of f_{4}(x) – f_{6}(x) is equal to :

Solution:

QUESTION: 78

Let [x] denote the greatest integer less than or equal to x. Then :-

Solution:

R.H.L. ≠ L.H.L.

*Multiple options can be correct

QUESTION: 79

The direction ratios of normal to the plane through the points (0, –1, 0) and (0, 0, 1) and making an anlge π/4 with the plane y–z+5=0 are:

Solution:

Let the equation of plane be a(x – 0) + b(y + 1) + c(z – 0) = 0 It passes through (0,0,1) then

b + c = 0 ...(1)

⇒ a^{2} = –2bc and b = –c

we get a^{2} = 2c^{2}

⇒ direction ratio (a, b, c) = (√2, -1, 1) or (√2, 1, -1)

QUESTION: 80

If xlog_{e}(log_{e} x) – x^{2} + y^{2} = 4(y > 0), then dy/dx at x = e is equal to :

Solution:

Differentiating with respect to x,

at x = e we get

QUESTION: 81

The straight line x + 2y = 1 meets the coordinate axes at A and B. A circle is drawn through A, B and the origin. Then the sum of perpendicular distances from A and B on the tangent to the circle at the origin is :

Solution:

Equation of circle

Equation of tangent of origin is 2x + y = 0

QUESTION: 82

If q is false and p ∧ q⇔r is true, then which one of the following statements is a tautology?

Solution:

QUESTION: 83

If y(x) is the solution of the differential equation where, then :

Solution:

QUESTION: 84

The maximum value of the function f(x) = 3x^{3} – 18x^{2} + 27x – 40 on the set S = {x ∈ R : x^{2} + 30 ≤ 11x} is:

Solution:

S = {x ∈ R, x^{2} + 30 – 11x ≤ 0}

= {x ∈ R, 5 ≤ x ≤ 6}

Now f(x) = 3x^{3} – 18x^{2} + 27x – 40

⇒ f'(x) = 9(x – 1)(x – 3),

which is positive in [5, 6]

⇒ f(x) increasing in [5, 6]

Hence maximum value = f(6) = 122

QUESTION: 85

If one real root of the quadratic equation 81x^{2} + kx + 256 = 0 is cube of the other root, then a value of k is

Solution:

81x^{2} + kx + 256 = 0 ; x = α, α^{3}

QUESTION: 86

Two circles with equal radii are intersecting at the points (0, 1) and (0, –1). The tangent at the point (0, 1) to one of the circles passes through the centre of the other circle. Then the distance between the centres of these circles is :

Solution:

QUESTION: 87

Equation of a common tangent to the parabola y^{2} = 4x and the hyperbole xy = 2 is :

Solution:

Let the equation of tangent to parabola

y^{2} = 4x be y = mx + 1/m

It is also a tangent to hyperbola xy = 2

So tangent is 2y + x + 4 = 0

QUESTION: 88

The plane containing the line and also containing its projection on the plane 2x + 3y – z = 5, contains which one of the following points ?

Solution:

The normal vector of required plane

So, direction ratio of normal is (–1, 1, 1) So required plane is

–(x – 3) + (y + 2) + (z – 1) = 0

⇒ –x + y + z + 4 = 0

Which is satisfied by (2, 0, –2)

QUESTION: 89

If tangents are drawn to the ellipse x^{2} + 2y^{2} = 2 at all points on the ellipse other than its four vertices then the mid points of the tangents intercepted betwen the coordinate axes lie on the curve :

Solution:

Equation of general tangent on ellipse

Let the midpoint be (h, k)

QUESTION: 90

Two integers are selected at random from the set {1, 2,...., 11}. Given that the sum of selected numbers is even, the conditional probability that both the numbers are even is :

Solution:

Since sum of two numbers is even so either both are odd or both are even. Hence number of elements in reduced samples space

= ^{5}C_{2} + ^{6}C_{2}

so required probability =

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