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QUESTION: 1

Find magnetic field at O.

Solution:

QUESTION: 2

Position of particle as a function of time is given as . Choose correct statement about and whereare velocity and acceleration of particle at time t.

Solution:

= ω (–sinωt cosωt + cosωt sinωt) = 0

so

QUESTION: 3

A Carnot engine, having an efficiency of η = 1/10 as heat engine, is used as a refrigerator. If the work done on the system is 10 J, the amount of energy absorbed from the reservoir at lower temperature is

Solution:

For Carnot engine using as refrigerator

It is given η = 1/10

⇒

⇒

So, Q_{2} = 90 J (as W = 10 J)

QUESTION: 4

Two uniformly charged solid spheres are such that E_{1 }is electric field at surface of 1st sphere due to itself. E_{2} is electric field at surface of 2^{nd} sphere due to itself. r_{1}, r_{2} are radius of 1^{st} and 2^{nd} sphere respectively. If then ratio of potential at the surface of spheres 1^{st} and 2^{nd} due to their self charges is:

Solution:

QUESTION: 5

Output at terminal Y of given logic circuit.

Solution:

QUESTION: 6

Velocity of a wave in a wire is v when tension in it is 2.06 × 10^{4} N. Find value of tension in wire when velocity of wave become v/2

Solution:

= 0.515 × 10^{4} N

QUESTION: 7

n mole of He and 2n mole of O_{2} is mixed in a container. Then will be

Solution:

QUESTION: 8

A uniform solid sphere of radius R has a cavity of radius 1m cut from it if centre of mass of the system lies at the periphery of the cavity then

Solution:

_{
}

**Alternative: **

QUESTION: 9

A solid sphere of mass m= 500gm is rolling without slipping on a horizontal surface. Find kinetic energy of a sphere if velocity of centre of mass is 5 cm/sec.

Solution:

K.E. of the sphere = Translational K.E + Rotational K.E.

K = Radius of gyration

QUESTION: 10

Two liquid columns of same height 5m and densities ρ and 2ρ are filled in a container of uniform cross sectional area. Then ratio of force exerted by the liquid on upper half of the wall to lower half of the wall is.

Solution:

QUESTION: 11

Two square plates of side 'a' are arranged as shown in the figure. The minimum separation between plates is 'd' and one of the plate is inclined at small angle α with plane parallel to another plate. The capacitance of capacitor is (given α is very small)

Solution:

QUESTION: 12

In YDSE path difference at a point on screen is λ/8. Find ratio of intensity at this point with maximum

intensity.

Solution:

QUESTION: 13

In the given circuit switch is closed at t = 0. The charge flown in time t = T_{C} (where T_{C} is time constant).

Solution:

QUESTION: 14

A particle is dropped from height h = 100 m, from surface of a planet. If in last 1/2 sec of its journey it covers 19 m. Then value of acceleration due to gravity that planet is :

Solution:

**Area of shaded trapezium
........(i)
......... (ii)
g = 8 m/s**

QUESTION: 15

A charge particle of mass m and charge q is released from rest in uniform electric field. Its graph between velocity (v) and distance travelled (x) will be :

Solution:

QUESTION: 16

An object is moving away from concave mirror of focal length f starting from focus. The distance of an object from pole of mirror is x. The correct graph of magnitude of magnification(m) verses distance x is:

Solution:

At focus, magnification is ∞

QUESTION: 17

In full scale deflection current in galvanometer of 100 Ω resistance is 1 mA. Resistance required in series to convert it into voltmeter of range 10 V.

Solution:

QUESTION: 18

There are two identical particles A and B. One is projected vertically upward with speed √2gh from ground and other is dropped from height h along the same vertical line. Collision between them is perfectly inelastic. Find time taken by them to reach the ground after collision in terms of .

Solution:

time for collision

After t_{1}

and

at the time of collision

and height from ground

so time

QUESTION: 19

Length of a simple pendulum is 25.0 cm and time of 40 oscillation is 50 sec. If resolution of stop watch is 1 sec then accuracy is g is (in %)

Solution:

QUESTION: 20

An electron is moving initially with velocityin uniform electric field . If initial wavelength of electron is λ_{0} and mass of electron is m. Find wavelength of electron as a function of time.

Solution:

Initially m

Velocity as a function of time =

so wavelength

*Answer can only contain numeric values

QUESTION: 21

An asteroid of mass m (m << m_{E}) is approaching with a velocity 12 km/s when it is at distance of 10 R from the centre of earth (where R is radius of earth). When it reaches at the surface of Earth, its velocity is (Nearest Integer) in km/s.

Solution:

= 16.028 km/s

*Answer can only contain numeric values

QUESTION: 22

In H–spectrum wavelength of 1^{st} line of Balmer series is λ = 6561Å. Find out wavelength of 2^{nd} line of same series in nm.

Solution:

= 4860 Å = 486 nm

*Answer can only contain numeric values

QUESTION: 23

There are three containers C_{1}, C_{2} and C_{3} filled with same material at different constant temperature. When we mix then for different volume then we get some final temperature as shown in the below table. So find value of final temperature θ as shown in the table.

Solution:

............(i)

.......(ii)

.........(iii)

and θ_{1} + θ_{2} + θ_{3} = (1 + 1 + 1)θ....(iv)

from (1) + (2) + (3)

3θ_{1} + 3θ_{2} + 3θ_{3} = 450

⇒ θ_{1} + θ_{2} + θ_{3} = 150

from (4) equation 150 = 3θ

θ = 50ºC

*Answer can only contain numeric values

QUESTION: 24

Two batteries (connected in series) of same emf 10 V of internal resistances 20Ω and 5Ω are connected to a load resistance of 30Ω. Now an unknown resistance x is connected in parallel to the load resistance. Find value of x so that potential drop of battery having internal resistance 20Ω becomes zero.

Solution:

*Answer can only contain numeric values

QUESTION: 25

An EMW is travelling along z-axis.

& Frequency of wave is 25 Hz, then electric field in volt/m.

Solution:

E/B = c

E = B × c

= 15 N/c

QUESTION: 26

Correct bond energy order of following is

Solution:

QUESTION: 27

Determine Bohr's radius of Li^{2+} ion for n = 2. Given (Bohr's radius of H-atom = a_{0})

Solution:

For Li^{2+}

QUESTION: 28

Given the following reaction sequence

A & B are respectively

Solution:

QUESTION: 29

Correct order of magnetic moment (spin only) for the following complexes

(a) [Pd(PPh_{3})_{2}Cl_{2}]

(b) [Ni(CO)_{4}]

(c) [Ni(CN)_{4}]^{2– }

(d)[Ni(H_{2}O)_{6}]^{2+}

Solution:

QUESTION: 30

Determine total number of neutrons in three isotopes of hydrogen.

Solution:

Number of neutrons

QUESTION: 31

Compare E_{a} (activation energy) for a, b, c and d.

Solution:

log k = log A –

slope

⇒ E_{b} > E_{c} > E_{d} > E_{a }

QUESTION: 32

Which of the following exhibit both Frenkel & Schottky defect?

Solution:

Only AgBr can exhibit both Schottky and Frenkel defect.

QUESTION: 33

Given:

Basicity of B is:

Solution:

Basicity = 1

QUESTION: 34

Which reaction does not occurs in the blast furnace in the metallurgy of Fe

(A) CaO + SiO_{2} → CaSiO_{3}

(B) Fe_{2}O_{3} + CO → Fe_{3}O_{4} + CO_{2}

(C) FeO + SiO_{2} → FeSiO_{3}

(D)

Solution:

Theory based

QUESTION: 35

Correct order of radius of elements is:

C, O, F, Cl, Br

Solution:

QUESTION: 36

Amongs the following which will show geometrical isomerism.

(a) [Ni(NH_{3})_{5}Cl]^{+}

(b) [Ni(NH_{3})_{4}ClBr]

(c) [Ni(NH_{3})_{3}Cl]^{+}

(d) [Ni(NH_{3})_{2}(NO_{2})_{2}]

Solution:

Ma_{4}bc can show 2 G.I.

Ma_{2}b_{2} can show 2 G.I.

(Square planar)

QUESTION: 37

Assertion: pH of water increases on increasing temperature.

Reason: H_{2}O → H^{+} + OH^{–} is an exothermic process.

Solution:

QUESTION: 38

Assertion: It has been found that for hydrogenation reaction the catalytic activity increases from group-5 to group-11 metals with maximum activity being shown by groups 7-9 elements of the periodic table.

Reason: For 7-9 group elements adsorption rate is maximum.

Solution:

QUESTION: 39

The major product of the following reactions is

Solution:

(Aromatic)

QUESTION: 40

Find the final major product of the following reactions

Solution:

On prototaoon, OH dispatches as H_{2}O making a carbocation. The H+ will shift to 3° carbon for a more stable carbocation and then elimination will occur to form.

QUESTION: 41

There are two compounds A and B of molecular formula C_{9}H_{18}O_{3}. A has higher boiling point than B. What are the possible structures of A and B?

Solution:

In (A), extensive inter-molecular H-bonding is possible while in (B) there is no Inter-molecular H-bonding.

QUESTION: 42

Kjeldahl method cannot be used for :

Solution:

Kjeldahl method is not applicable to nitro or diazo groups present in the ring, as nitrogen atom can't be converted to ammonium sulfate under the reaction conditions.

QUESTION: 43

A compound X that adds 2 hydrogen molecules on hydrogenation. The compound X also gives 3 - oxohexanedioic acid on oxidative ozonolysis. The compound 'X' is:

Solution:

QUESTION: 44

Formation of Bakelite follows:

Solution:

Formation of Bakelite follows electrophilic substitution reaction of phenol with formaldehyde followed by condensation.

QUESTION: 45

Products formed by hydrolysis of maltose are

Solution:

Maltose on hydrolysis gives 2 moles of α-D-glucose.

*Answer can only contain numeric values

QUESTION: 46

Temperature of 4 moles of gas increases from 300 K to 500 K find 'Cv' if ΔU = 5000 J.

Solution:

*Answer can only contain numeric values

QUESTION: 47

Given :

Determine at equilibrium

For cell reaction Sn | Sn ^{2+} || Pb^{ 2+} | Pb

take = 0.06 V

Solution:

At Equilibrium state.

E_{cell }= 0 ; Eº_{cell} = 0.01 V

= 10^{1/3} = 2.1544

*Answer can only contain numeric values

QUESTION: 48

Given following reaction,

NaClO_{3} + Fe → O_{2 }+ FeO + NaCl

In the above reaction 492 L of O_{2} is obtained at 1 atm & 300 K temperature.

Determine mass of NaClO_{3} required (in kg).

(R = 0.082 L atm mol^{–1} K^{–1} )

Solution:

mol of NaClO_{3} = mol of O_{2}

mol of O_{2 }=

= 20 mL

mass of NaClO_{3}= 20 × 106.5 = 2130 g

*Answer can only contain numeric values

QUESTION: 49

Complex [ML_{5}] can exhibit trigonal bipyramidal and square pyramidal geometry. Determine total number of 180º, 90º & 120º L-M-L bond angles.

Solution:

∠120º = 3; ∠90º = 6; ∠180º = 1

→Total = 10

∠90º = 8; ∠180º = 2 ⇒ Total = 10

*Answer can only contain numeric values

QUESTION: 50

How many atoms lie in the same plane in the major product (C)?

(Where A is the alkyne of lowest molecular mass)

Solution:

Number of atoms in one plane = 13

QUESTION: 51

Let and is nonzero vector and find

Solution:

QUESTION: 52

Let coefficient of x^{4} and x^{2} in the expansion of is α and β then α -β is equal to

Solution:

α = – 96 and β = 36

∴ α - β = -132

QUESTION: 53

Differential equation of x^{2 }= 4b(y + b), where b is a parameter, is

Solution:

2x = 4by'

So. differential equation is

QUESTION: 54

Image of (1, 2, 3) w.r.t a plane is then which of the following points lie on the plane

Solution:

d.r of normal to the plane

10/3, 10/3, 10/3

1, 1, 1

midpoint of P and Q is

equation of plane x + y + z = 1

QUESTION: 55

is equal to

Solution:

Using L’Hospital

QUESTION: 56

Let P be the set of points (x, y) such that x^{2} ≤ y ≤ – 2x + 3. Then area of region bounded by points in set P is

Solution:

Point of intersection of y = x^{2} & y = – 2x + 3 is

obtained by x^{2} + 2x – 3 = 0

⇒ x = - 3, 1

So, Area =

= 12 + 8 – 28/3 = 32/3

QUESTION: 57

Let f(x) = → R then range of f(x) is (where [ . ] denotes greatest integer function)

Solution:

∴ f(x) is a decreasing function

∴

QUESTION: 58

Let A = and I = then value of 10 A^{–1} is –

Solution:

Characteristics equation of matrix ‘A’ is

x^{2} – 6x – 10 = 0

∴ A^{2} - 6A - 10I = 0

⇒ 10A^{–1} = A – 6I

QUESTION: 59

Solution set of 3^{x}(3^{x }–1) + 2 = |3^{x} –1| + |3^{x} – 2| contains

Solution:

Let 3^{x} = t

t(t –1) + 2 = |t –1| + |t –2|

t^{2} – t + 2 = |t –1| + |t –2|

are positive solution

t = a

3^{x} = a

x = log_{3} a so singleton set

QUESTION: 60

Mean and variance of 20 observation are 10 and 4. It was found, that in place of 11, 9 was taken by mistake find correct variance.

Solution:

................... (1)

................... (2)

= 104 × 20 = 2080

Actual mean =

Variance =

= = 106 – 102.01 = 3.99

QUESTION: 61

λx + 2y + 2z = 5

2λx + 3y + 5z = 8

4x + λy + 6z = 10

for the system of equation check the correct option.

Solution:

D = (λ + 8) ( 2 - λ)

for λ = 2

= 5[18 – 10] – 2 [48 – 50] + 2 (16 – 30]

= 40 + 4 – 28 ≠ 0

No solutions for λ = 2

QUESTION: 62

For an A.P. T_{10} = 1/20; T_{20 }= 1/10 Find sum of first 200 term.

Solution:

QUESTION: 63

Let α = and a = . If a and b are roots of quadratic equation then quadratic equation is

Solution:

α = ω, b = 1 + ω^{3} + ω^{6} + .......... = 101

a = (1 + ω) (1 + ω^{2} + ω^{4} + ......... ω^{198} + ω^{200})

=

Equation: x^{2} – (101 +1)x + (101) × 1 = 0

⇒ x^{2} - 102x + 101 = 0

QUESTION: 64

Let f(x) is a three degree polynomial for which f '(–1) = 0, f ''(1) = 0, f(–1) = 10, f(1) = 6 then local minima of f(x) exist at

Solution:

Let f(x) = ax^{3} + bx^{2} + cx + d

a = 1/4 d = 35/4

b = -3/4 c = -9/4

⇒ f(x) = a(x^{3} – 3x^{2} – 9x) + d

f ' (x) = 3/4 (x^{2 }– 2x – 3)

⇒ f ' (x) = 0 ⇒ x = 3, –1

local minima exist at x = 3

QUESTION: 65

Let A and B are two events such that P(exactly one) = 2/5, P(A ∪ B) = then P(A ∩ B) =

Solution:

P(exactly one) = 2/5

⇒ P(A) + P(B) - 2P(A ∩ B) = 2/5

P (A ∪ B) = 1/2

⇒ P(A) + P(B) – P(A ∩ B) = 1/2

∴ P(A ∩ B) = 1/2 - 2/5 = (5-4)/10 = 1/10

QUESTION: 66

Let I = then

Solution:

1/3 < I < 1/√8

QUESTION: 67

Normal at (2, 2) to curve x^{2} + 2xy – 3y^{2} = 0 is L. Then perpendicular distance from origin to line L is

Solution:

x^{2} + 2xy – 3y^{2} = 0

x^{2} + 3xy – xy – 3y^{2} = 0

(x – y) (x + 3y) = 0

x – y = 0 x + 3y = 0

(2, 2) satisfy x – y = 0

Normal : x + y = λ

λ = 4

Hence x + y =4

perpendicular distance from origin = = 2√2

QUESTION: 68

Which of the following is tautology-

Solution:

(~p ∧ q) → (p ν q)

~{(~p ∧ q) ∧ (~p ∧ ~q)}

~{~p ∧ f}

QUESTION: 69

If a hyperbola has vertices (±6, 0) and P(10, 16) lies on it, then the equation of normal at P is

Solution:

Vertex is at (±6, 0)

∴ a = 6

Let the hyperbola is

Putting point P(10, 16) on the hyperbola

⇒ b^{2 }= 144

∴ hyperbola is

∴ equation of normal is

∴ putting we get 2x + 5y = 100

QUESTION: 70

If y = mx + c is a tangent to the circle (x – 3)^{2} + y^{2} = 1 and also the perpendicular to the tangent to the circle x^{2} + y^{2} = 1 at , then

Solution:

Slope of tangent to x^{2} + y^{2} = 1 at

x^{2} + y^{2 }= 1

2x + 2yy' = 0

y = mx + c is tangent of x^{2} + y^{2} = 1

so m = 1

y = x + c

now distance of (3, 0) from y = x + c is

= 1

c^{2} + 6c + 9 = 2

c^{2} + 6c + 7 = 0

*Answer can only contain numeric values

QUESTION: 71

Let = 1/7 and = 1/√10 where α, β ∈ . Then tan (α + 2β) is equal to

Solution:

and = 1/√10

tan α = 1/7

sin β = 1/√10

tanβ = 1/3

tan2β =

tan (α + 2β) = =

*Answer can only contain numeric values

QUESTION: 72

The number of four letter words that can be made from the letters of word "EXAMINATION" is

Solution:

EXAMINATION

2N, 2A, 2I, E, X, M, T, O

**Case I** All are different so = 8.7.6.5 = 1680

**Case II** 2 same and 2 different so

= 3.21.12 = 756

**Case III **2 same and 2 same so = 3.6 = 18

Total = 1680 + 756 + 18 = 2454

*Answer can only contain numeric values

QUESTION: 73

Let the line y = mx intersects the curve y^{2} = x at P and tangent to y^{2} = x at P intersects x-axis at Q. If area (ΔOPQ) = 4, find m (m > 0)

Solution:

2ty = x + t^{2}

Q(–t^{2}, 0)

= 4

|t|^{3 }= 8

t = ± 2 (t > 0)

m = 2

*Answer can only contain numeric values

QUESTION: 74

is equal to

Solution:

1/4 [1568 + 420 + 28] = 504

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