Quizrr Advanced Full Test 3 (Paper 2) – with Video Solutions - 2025

Total Time: 180 minutes | Total Marks: 204 | Total Questions: 54

GENERAL INSTRUCTIONS

1. The Question Paper consists of three parts (Physics, Chemistry and Mathematics) with a total of 54 questions.
2. Each subject is divided into three sections:
◦ Section 1: Single Correct Type Questions (4 options, only one correct)
◦ Section 2: Multiple Correct Type Questions (4 options, one or more correct)
◦ Section 3: Numerical Value Type Questions (answer to be entered as numerical value)
3. Marking Scheme:
◦ Single Correct: +3 for correct, -1 for wrong, 0 if unattempted.
◦ Multiple Correct: +4 for all correct options, partial marking +1 otherwise, -2 for wrong options.
◦ Numerical Value: +3 for correct, 0 for wrong/unattempted, no negative marking.

By: C I P H Ξ
R

Syllabus: Full Syllabus

SECTION 1 - Physics [SINGLE CORRECT TYPE]

Q.1

A stone with weight w is thrown vertically upward into air from ground level with initial speed v 0 . If a constant force f due to air drag acts on the stone throughout
its flight, the speed of stone just before impact with the ground is :-

(A) v 0 ( w−f ) 2
1

w+f

(B) v 0 ( w−f ) 2
3

w+f

(C) v 0 ( w−f ) 2
1

(D) v 0 ( w+f ) − 2
1

Q.2

A radioactive sample with half life = T emits α-particle. Its total activity is A i at some time and A f at a later time. The number of α-particles emitted by the
sample between these two points of time is -

(A) A i − A f
(B) T
ℓn2 ( Ai − Af )
(C) ℓn2
T [ Ai − Af ]
(D) T
ℓn2 [ 1
Ai − 1
Af ]

Q.3

Two light rods of length 1 m each are hinged together as shown in figure. Rod AB makes an angle θ with vertical while rod BC makes an angle ϕ with horizontal.
End C of rod BC remains in contact with horizontal. Rod AB is rotated with constant angular velocity ω = 1rad/s in clockwise direction. At the instant when
θ = 30 ∘ and ϕ = 30 ∘ match the variables in list-I with values in list-II.
 List-I List-II
(3√3+1)
(P) Angular velocity of rod BC in rad/s (1)
3√3
(√3−1)
(Q) Velocity of block D in m/s (2) √6
1
(R) Magnitude of angular acceleration of rod BC in rad/s 2 (3) √3
2
(S) Acceleration of point B in m/s (4) 1

(A) P → 4; Q → 1; R → 2; S → 3
(B) P → 1; Q → 4; R → 3; S → 2
(C) P → 3; Q → 3; R → 1; S → 4
(D) P → 3; Q → 2; R → 4; S → 1

Q.4

Consider a system of five large conducting plates of area A. The charge on plate 1, 2, 3, 4, 5 are given as Q, 2Q, 3Q, 4Q and 5Q respectively. Initially the two
switches are opened. Area of each plate is A. The distance between every successive plate is very very small equal to d.

List-I List-II
Qd
(P) The charge on surface b is (1) − 92 ( Aε 0
)
(Q) The potential difference between plate 2 and 3 ( V 2 − V 3 ) is (2) −0.6Q
Now both the switches S 1 and S 2 are closed simultaneously.
Now Consider a new steady state.
(R) The charge on surface b is (3) −1.3Q
(S) The charge on surface f is (4) None of these

(A) P → 4; Q → 3; R → 1; S → 4
(B) P → 2; Q → 4; R → 3; S → 2
(C) P → 3; Q → 3; R → 1; S → 4
(D) P → 4; Q → 1; R → 3; S → 4

SECTION 2 - Physics [MULTIPLE CORRECT TYPE]

Q.5

Assume that the earth changes its shape and turns into an infinite cylinder whose radius and distance of moon from the axis of cylindrical earth remains unchanged
and also moon is remain spherical. Assume density of earth remains same and distance of moon from earth is much greater than radius of earth. Which of the
following statement(s) is/are correct :

(A) Gravitational field at the moon due to earth increases.

(B) Gravitational field at the moon due to earth decreases.
(C) Speed of the moon in its orbit around the cylindrical earth increases.
(D) Speed of the moon in its orbit around the cylindrical earth decreases.

Q.6

ABC is an equilateral triangle, of side length ℓ 0 and resistance R 0 at temperature 0 ∘ C, kept in uniform magnetic field B 0 as shown.
Temperature coefficient of linear expansion of wire is α t and temperature coefficient of resistance of wire is α R . Now the temperature of the wire is slowly
changed according to the equation T = T 0 sin ωt, where T 0 is in ∘ C. Given that α R = α ℓ = α and α R T 0 ≪ 1&α ℓ T 0 ≪ 1 ). Choose the correct option(s):

(A) Amount of charge flown through the wire in the first quarter cycle of temperature starting from t = 0 is ( √3 B0 αℓ0 )T 0
2

2R 0

(B) Amount of charge flown through the wire in the first quarter cycle of temperature starting from t = 0 is ( √3 B0 αℓ0 )T 0
2

4R 0

(C) Rate of heat production from wire at any time 't' is

2 2 4 2 2
3 B0 α ℓ0 ω T0
4 R0 e 3a T cos 2 ωt

(D) Rate of heat production from wire at any time 't' is

2 2 4 2 2
3 B0 α ℓ0 ω T0
2 R0 e 3aT cos 2 ωt

Q.7

The density of a solid ball is to be determined in an experiment. The diameter of the ball is measured with a screw gauge, whose pitch is 0.5 mm and there are 100
division on the circular scale. The reading on the main scale is 2.3 mm and that on the circular scale is 40 divisions. If the measured mass of the ball has relative
error of 1% the percentage error in density is :

(A) 0.5%
(B) 0.8%
(C) 1.6%
(D) 2%

Q.8

A particle of mass m strikes another particle A of same mass which is connected to a particle B, also of same mass, by means of a light rigid rod of length L,
placed on smooth horizontal surface. Striking particle comes to rest as soon as it strikes A and the line of motion of striking particle makes an angle of 30 ∘ with the
rod as shown in the figure.

(A) the angular velocity of the rod after impact is v0

2L

(B) the instantaneous axis of rotation of the rod AB at the instant after the impact passes through a point which lies on AB
(C) velocity of particle A after the impact would be v0
4
√7
(D) the collision between striking particle and particle A is elastic

Q.9

One fourth part of an equiconvex lens of focal length 100 cm is removed as shown in the figure. An object of height 1 cm is placed in front of the lens. It is
observed that all the images are of equal height. Then

(A) Object is at a distance of 400

3 cm from the lens.
(B) The magnitude of magnification produced by upper and lower part is equal.
(C) The no. of images formed is two.
(D) The product of magnification of both the lenses is negative.

Q.10

Four coherent sources of light of same intensity I 0 and in same phase are placed on a line Perpendicular to a large screen as shown in figure. The whole
arrangement is placed in air. Assuming the net intensity at any point on screen is average (mean) of resultant intensities from both sides and point O lies on the
screen and on the line joining the sources. D 1 measured from mid point of S 1 and S 2 and similarly D 2 measured from mid point of S 3 and S 4 . Choose the correct
option(s) assuming that d 1 and d 2 are very small in comparison to D 1 and D 2 .

(A) The net intensity at point O is 2I 0

(B) The shape of maxima on the screen is circular on both sides.
(C) The net intensity at a distance 12 m from O is 2I 0 .
(D) The separation between maxima nearest to O due to sources on left side only and second maxima nearest to O due to sources on right side only is 21
4 m.

Q.11

The displacement of a particle as a function of time is shown in the figure. It indicates :

(A) The particle starts with a certain velocity, but the motion is retarded and finally the particle stops
(B) The velocity of the particle decreases
(C) The acceleration of the particle is in opposite direction to the velocity
(D) The particle moves with a constant velocity, the motion is accelerated and finally the particle moves with another constant velocity

SECTION 3 - Physics [NUMERICAL VALUE TYPE]

Q.12

A car windshield wiper blade sweeps the wet windshield rotating at a constant angular speed of ω. R is the radius of innermost arc swept by the blade. Length and
width of the blade are ℓ and b respectively. Coefficient of viscosity of water is η. The torque delivered by the motor to rotate the blade assuming that there is a
uniform layer of water of thickness t on the glass surface is ) − 1]. Find the value of α.
ηbωR 3 L 3
αt [(1 + R
(D)
(B)
(C)
(A)
Q.13

Moseley's law for K α photon is given by √v = a(Z − b) where a is a universal constant and b is a screening constant. Moseley's logic helps us assume b = 1 for
K a photon. If there is a percentage error = 10 −2 in the measurement of 'b' due to actual orbital configuration of an atom, then the relative error in the measurement
of v if Z = 51 is p × 10 −q . Find value of p+q (p is a natural number less than 5)

(D)
(B)
(A)
(C)
Q.14

A certain quantity of ideal gas takes up 56 J of heat in the process AB and 360 J in the process AC. What is the number of degree of freedom of the gas? (P-V i.e
sties sure - volunne graph is given)

(D)
(B)
(A)
(C)
Q.15

A Physical quantity P is given as - P = where, x = (2 ± 0.01)SI unit; y = (4 ± 0.02)SI unit; t = (2 ± 0.01) SI unit. Find the percentage error in
x2 y
(4−t) 3
calculation of P .

(D)
(B)
(A)
(C)
Q.16

A uniform rod is placed on two spinning wheels as shown in figure. The axes of the wheels are separated by a distance ℓ = m, the coefficient of friction
50
π2
between the rod and the wheels is μ = 0.1. The rod performs harmonic oscillations. The period of these oscillations is 10x sec. Find the value of x.

(D)
(B)
(A)
(C)
Q.17

Consider the nuclear reaction X 200 → A 120 + B 80 . If the binding energy per nucleon for X, A and B are 7.4 MeV, 8.2 MeV and 8.1 MeV respectively, then the
energy released in the reaction is (19 × n) MeV, the value of k is

(D)
(B)
(A)
(C)
Q.18

Figure shows a particle of mass m attached with 4 identical springs each of spring constant K and each of which are initially in their natural length L. The
gravitational force is neglected. If the mass is slightly displaced by distance x along a line perpendicular to the plane of the figure and released then the force
acting on particle just when it is released is proportional to x n , find n.
(D)
(C)
(B)
(A)

SECTION 1 - Chemistry [SINGLE CORRECT TYPE]

Q.19

The olivine series of minerals consists of crystal in which Fe 2+ and Mg 2+ ions may substitute for each-other causing substitutional impurity defect without
changing the volume of the unit cell. In olivine series of mineral, oxide ion exist as fcc with Si 4+ occupying 14 th of octahedral void and divalent ion occupying 1
4
th of tetrahedral void. The density of forsterite (Magnesium silicate) is 3.21 g/cc and that of fayalite (Ferrous silicate) is 4.34 g/cc. If density of olivine is
3.88 g/cc, then which of the following statement is INCORRECT.

(A) Forsterite = Mg 2 SiO 4 ; Fayalite = Fe 2 SiO 4

(B) An olivine contains 40.71% Forsterite and 59.29% Fayalite
(C) Forsterite Mg 2 SiO 4 with 59.29% percentage
(D) (A) & (B)

Q.20

A mixture of chlorides of copper, cadmium, chromium, iron and aluminium was dissolved in water.

It was acidified with dilute HCl and then hydrogen sulphide gas was passed for a sufficient time.

It was filtered, boiled and a few drops of nitric acid were added, while boiling. To this solution, ammonium chloride and ammonium hydroxide were added. To
this, an excess of sodium hydroxide was added and then filtered. The filtrate shall give the test for:

(A) sodium and iron

(B) sodium, chromium and aluminium
(C) aluminium and iron
(D) sodium, iron, cadmium and aluminium

Q.21

Match the thermodynamic processes given list I with the expressions given under list II.
List-I List-II
(P) Melting of ice at 273 K and 1 atm (1) q = 0
(Q) Expansion of 1 mol
of an ideal gas into a vacuum under isolated conditions (2) w = 0
(R) Mixing of equal volumes of two ideal gases at constant
temperature and pressure in an isolated container (3) ΔS sys > 0
(S) Reversible heating of H 2 ( g) (ideal) at 1 atm from K
to 600 K, followed by reversible cooling to 300 K at 1 atm (4) ΔU = 0
(5) ΔG = 0

(A) P → 2, 4, 5 Q → 2, 3 R → 3, 5 S → 1, 5
(B) P → 1, 4, 5 Q → 2, 3, 4 R → 3, 4 S → 1, 5
(C) P → 1, 2, 3, 4, 5 Q → 1, 2, 3 R → 3, 4 S → 1, 5
(D) P → 3, 5 Q → 1, 2, 3, 4 R → 1, 2, 3, 4 S → 1, 2, 4, 5

Q.22

Complete the following reactions identify the major products and apply a chemical test to distinguish between following pairs :
Reactions are as follows

(A) P → 3; Q → 2; R → 1; S → 4
(B) P → 2; Q → 3; R → 4; S → 1
(C) P → 2; Q → 1; R → 3; S → 4
(D) P → 1; Q → 2; R → 4; S → 3

SECTION 2 - Chemistry [MULTIPLE CORRECT TYPE]

Q.23

Consider the following values of I.E.(eV) for elements W and X :

Element I. E. .1 I. E. ⋅2 I. E. ⋅3 I. E. 4
W 10.5 15.5 24.9 79.8
X 8 14.8 78.9 105.8
Other two element Y and Z have outer electronic configuration ns 2 np 4 and ns 2 np 5 respectively. Then according to given information which of the following
compound(s) is/are not possible.

(A) W 2 Y 3
(B) X 2 Y 3
(C) WZ 2
(D) XZ 2

Q.24

In the crystal field of the complex [Fe(Cl)(CN) 4 (O 2 )] 4− the electronic configuration of metal is found to be t 62g , e 0g then which of the following is true about this
complex ion :

(A) It is a paramagnetic complex

(B) O − O bond length will be less than found in O 2 molecule
(C) Its IUPAC name will be chlorotetracyanosuperoxidoferrate (II) ion
(D) It is a diamagnetic complex

Q.25
The standard reduction potential data at 25 ∘ C is given below.
E ∘ (Fe 3+ ⋅ Fe 2+ ) = +0.77 V;
E ∘ (Fe 2+ . Fe) = −0.44 V;
E ∘ (Cu 2+ ⋅ Cu) = +0.34 V;
E ∘ (Cu + . Cu) = +0.52 V;
E ∘ (O 2 ( g) + 4H + + 4e − → 2H 2 O) = +1.23 V;
E ∘ (O 2 ( g) + 2H 2 O + 4e − → 4OH − ) = +0.40 V
E ∘ (Cr 3+ ⋅ Cr) = −0.74 V;
E ∘ (Cr 2+ . Cr) = −0.91 V
Which of the following statements is/are correct on the basis of above data :

(A) O 2 is a better oxidising agent in acidic medium.

(B) Cr 2+ show disproportionation to Cr and Cr 3+ in water.
(C) O 2 oxidises Fe 2+ to Fe 3+ in acidic medium.
(D) Cu + oxidises Fe to Fe 2+ and H 2 O to O 2 in acidic medium.

Q.26

Which of the following is/are correct statement (s) ?

(A)

(B)

(C)

(D)

Q.27

The correct statement(s) about reaction of X 2 molecules of group 17 elements is(are) :

(A) Yellow coloured F 2 reacts with water to gives colourless solution and paramagnetic gas
(B) Greenish yellow coloured Cl 2 reacts with water to gives colourless acidic solution
(C) Reddish brown coloured Br 2 when dissolved in Na 2 CO 3 solution form colourless solution
(D) Violet coloured iodine when dissolved in water having KI in presence of starch form blue coloured solution

Q.28

Which of following reactions represent correct major product?

(A)
(B)

(C)

(D)

Q.29

If the radius of first Bohr's orbit of H-atom is x, which of the following is the correct conclusion?

(A) The de-Broglie wavelength in the third Bohr orbit of H-atom = 6πx
(B) The fourth Bohr's radius of He + ion = 8x
(C) The de-Broglie wavelength in third Bohr's orbit of Li 2+ = 2πx
(D) The second Bohr's radius of Be 3+ = x

SECTION 3 - Chemistry [NUMERICAL VALUE TYPE]

Q.30

An alcohol (A) on treatment with conc. H 2 SO 4 /Δ gave an alkene (B). The compound (B), on reacting with Br 2 /CCl 4 and subsequent didehydrobromination with
NaNH 2 (2eq.) produced a compound (C). The compound (C) with dil. H 2 SO 4 in presence of HgSO 4 gave a compound 'D'. The compound D can also be obtained
by oxidation of A by pcc or from distillation of calcium acetate. If molecular weight of ' C ' is Y × 10, then write the value of Y .

(D)
(B)
(A)
(C)
Q.31

Number of CH 3 - I consumed during the following reaction. In which Hoffmann exhaustive elimination take place.

(A)
(D)
(C)
(B)
Q.32

A pentapeptide produces Alanine, Glycine, Valine, Leucine, Isoleucine on complete hydrolysis. If it contains −COOH on alanine and −NH 2 on Glycine, then
how many primary structures are possible for pentapeptide.

(D)
(B)
(A)
(C)
Q.33

The existence of coordination compounds with the same formula but different arrangements (Isomers) is crucial in the development of coordination chemistry.
There are two main forms of isomerism in coordination compound structural isomerism and stereoisomerism (spatial isomerism). Find total possible isomeric
compound for complex ion [Pt(Br) (NO 2 ) (NO 3 )(SCN)] 2−
Q.34
(B)
(A)
(D)
(C)

2 molal aqueous sugar solution is heated to 105.2 ∘ C. If the fraction of water (by mass) present in the solution that will vapourise out at this temperature is X. Then
what is the value 10X ? (Take K b of H 2 O = 0.52 ∘ C kg mol −1 )

(A)
(D)
(C)
(B)
Q.35

A free movable piston inside a closed cylinder is initially locked so as to divide the cylinder into 2 compartments of length L 1 = 10 cm and L 2 = 4.4 cm, which
are filled with air at 10 ∘ C and P 1 = 76 cm Hg and P 2 = 4 cm Hg respectively. The piston is then released and allowed to move reversibly until the pressures are
equal. Conditions are adiabatic γ = 1.4. Calculate the final temperature of the air in the larger compartment. (round off to nearest integer & take 10 0.44
13.66 = 250
283 )

(A)
(D)
(C)
(B)
Q.36

The graph of compressibility factor (Z)v/sP for 1 mol of a real gas is shown in following diagram. The graph is plotted at 273 K temperature. If slope of graph at
very high pressure ( dP
dZ
) is ( 2.8
1
) atm −1 then calculate volume of 1 mol of real gas molecules in (L/mol).
[Given N A = 6 × 10 23 and R = 22.4
273 L atm K −1 mol −1 ]

(D)
(C)
(B)
(A)

SECTION 1 - Mathematics [SINGLE CORRECT TYPE]

Q.37

Let the curve y = f(x) passes through origin and satisfies the differential equation dx + ∫ 0 ydx = 27. If a and b are chosen randomly from the set
dy 5

S = {1, 2, 3, 4} with replacement. The probability that the above curve passes through (a, b) is :

(A) 1
2

(B) 1
6

(C) 1
8

(D) 1
12

Q.38

If α and β are roots of equation 27

4 sin ( 9θ ) = sin 3 θ + 3 sin 3 ( 3θ ) + 9 sin 3 ( 9θ ) + 1
for 0 < θ < π
2 , then tan α + tan β is equal to
4√2

(A) 2 + √3
(B) 3 + √3
(C) 3 − √3
(D) 2 − √3

Q.39

Let ϕ(x) = 3f ( x3 ) + f (3 − x 2 )∀x ∈ (−3, 4) where f ′′ (x) > 0∀x ∈ (−3, 4), g(x) = (a 2 − 3a + 2) (cos 2
2
x x
4 − sin 2 4 ) + (a − 1)x + sin 1
Match List-I with List-II and select the correct answer using the code given below the list.
(A) P → 2; Q → 3; R → 1; S → 4
(B) P → 2; Q → 3; R → 4; S → 1
(C) P → 3; Q → 2; R → 1; S → 4
(D) P → 3; Q → 2; R → 4; S → 1

Q.40

(i) Let harmonic mean, arithmetic mean and geometric mean of two positive numbers a and b respectively be 4, A and G. If 2 A + G 2 = 27 then a 2 + b 2 = α 1
(ii) If the value of x + y + z = 15 where a, x, y, z, b are in arithmetic progression while the value of x1 + y1 + 1z is 58 where a, x, y, z, b are in harmonic
progression, then the value of a 2 + b 2 is α 2
(iii) The value of ∑ 16
r=1 ( 1+3+…+(2r−1) ) is α 3
3 3 3
1 +2 +……+r

(iv) If the geometric mean and harmonic mean of two positive numbers x 1 and x 2 are 18 and 216
13 respectively then the value of |x 1 − x 2 | is α 4 .
Match List-I with List-II and select the correct answer using the code given below the list.

(A) P → 2; Q → 1; R → 2; S → 3
(B) P → 2; Q → 2; R → 1; S → 4
(C) P → 2; Q → 1; R → 3; S → 4
(D) P → 1; Q → 1; R → 4; S → 2

SECTION 2 - Mathematics [MULTIPLE CORRECT TYPE]

Q.41
A line l intersects the plane P at a point A, where l and P are
r→ = (1 + 2t)^i + (2 + 3t)^j + (3 + 4t)k,
^ t∈R
r→. (^i − ^j + 2k)
^ + 2 = 0 respectively.
The equation of the straight line passing through A lying in the plane P and at minimum inclination of with the given line is(are) :

(A) r→ = −(^i + ^j + k)
^ + s(^i + 5^j + 2k),
^ s∈R

(B) r→ = (^i + 9^j + 3k)

^ + s(^i + 5^j + 2k),
^ s∈R

(C) r→ = 4^j + k
^ + s(^i + 5^j + 2k),
^ s∈R

(D) r→ = 2^j + 3k
^ + s(^i + 5^j + 3k),
^ s∈R

Q.42

Let S be the set of all non-zero real numbers α such that the quadratic equation αx 2 − x + α = 0 has two distinct real roots x 1 & x 2 satisfying the inequality
|x 1 − x 2 | < 1. Which of the following intervals is(are) a subset(s) of S ?

(A) (− 12 , − 1
)
√5

(B) (− 1
, 0)
√5

(C) (0, 1
)
√5

(D) ( 1
, 12 )
√5

Q.43

A curve of the differential equation (x 2 + xy + 6x + 3y + 9) dx − y 2 = 0 passes through the point (1, 1) then the solution curve:
dy

(A) Intersects y = x + 3 exactly at one point

(B) Intersects y = x + 3 exactly at two points
(C) Intersect y = (x + 3) 2
(D) Does not intersect y = (x + 3) 2

Q.44

If ω is the imaginary cube root of unity such that (∑ nr=1 (r ∑ rp=1 (ω p−1 ))) − 155ω = (∑ nr=1 (r ∑ rp=1 (ω p−1 ))) − 155ω, then the value of n is equal to

(A) 29
(B) 30
(C) 31
(D) 32

Q.45

A function f is defined by f(x) = ∫ 0 cos t cos(x − t)dt, 0 ≤ x ≤ 2π then which of the following hold(s) good?
π

(A) f(x) is continuous and differentiable in 0 < x < 2π

(B) Range of f(x) is [− π2 , π
2 ]
(C) Number of solutions of f(x) = cos x in x ∈ (0, 2π) is 2
(D) f(x) = π
| cos x| holds good in x ∈ (0, π
) ∪ ( 3π

Q.46
2

(A) tan A, tan B, tan C are in A.P.

(B) tan A, tan B, tan C are in G.P.
2

∣
In △ABC, if ∠B = sec −1 ( 54 ) + cosec −1 √5, ∠C = cosec −1 ( 25
2 , 2π)

7 ) + cot
−1 9
( 13 ) and c = 3.
(All symbols used have their usual meaning in a triangle.) which of the following is/are correct

(C) The distance between orthocentre and centroid of triangle with sides a 2 , b 3 and c is equal to
4

(D) The distance between orthocentre and centroid of triangle with sides a 2 , b and c is equal to

Q.47
4
3
5
3
10
3

The curves y = ax 3 + 4x(a ≠ 0) & xy = 1 touch each other at points P and Q, where P is in first quadrant. A triangle is formed by drawing a tangent to
y = ax 3 + 4x at point Q and coordinate axes. Then which of the following is/are correct?

(A) The value of a is -4

(B) The area of the triangle formed is 2
(C) The value of a is -2
(D) The area of the triangle formed is 1

SECTION 3 - Mathematics [NUMERICAL VALUE TYPE]

Q.48

Let A = [a ij ]3 × 3 be a matrix such that AA ⊤ = 4I and 2a ij + c ij = 0 where c ij is the cofactor of a ij ∀ i & j, I is the unit matrix of order 3 and A ⊤ is the
transpose of the matrix A.
a 11 + 4 a 12 a 13 a 11 + 1 a 12 a 13
If a 21 a 22 + 4 a 23 + 5λ a 21 a 22 + 1 a 23 = 0, then λ = a
b where a and b are coprime positive integers then the value of a + b is
a 31 a 32 a 33 + 4 a 31 a 32 a 33 + 1
________.

(D)
(B)
(A)
(C)
Q.49

If the co-efficient of x 4 in the expansion of (1 − x + 2x 2 ) is n+2 C r+1 + r ⋅ n+1 C r + n C r , then the coefficient of x r in the expansion of (1 + x) n/2 is
12

(D)
(B)
(A)
(C)
Q.50

j 2 +j
Let < a n > be an arithmetic sequence such that ∑ 50
i=1 a 2i−1 = 50, then ∑ j=1 (−1)
50 2 a 2j−1 is equal to

((D)
(B)
(C)
A)
Q.51

Let f : (0, 1) → (0, 1) be a differentiable function such that f ′ (x) ≠ 0∀x ∈ (0, 1) and f ( 12 ) = .

∣
If f(x) =

(D)
(B)
(A)
(C)
Q.52
lim t→x 0
t x
∫ √1−f 2 (s)ds−∫ 0 √1−f 2 (s)ds
f(t)−f(x)

differentiable and ∑ 5i=1 |x i | = 8, then

(A)
(D)
(C)
(B)
Q.53
1
5
then the value of f ( 14 ) =

lim x→∞
x 2 −f(x)
x equals
√m
4 and ∫0
1
f(x)dx = π
n
√3
2

, m, n ∈ N . Find the value of m − n.

Let f(x) be a real quadratic polynomial with leading coefficient 1. If x 1 , x 2 , x 3 , x 4 , x 5 be ×− coordinates of 5 distinct points where g(x) = |f(|x|)| is non-

Let f is a real valued function defined from R to R such that f(x) + f(−x) = 5∀x ∈ R, then the value of ∫ 1−x f −1 (t)dt is equal to

(A)
(D)
(C)
(B)
Q.54
4+x

Let x 1 , x 2 , … … … … . x 100 are 100 observations such that ∑ x i = 0, ∑ 1≤i<j≤100 |x i x j | = 80000 & mean deviation from their mean is 5, then their standard
deviation is _______

(D)
(C)
(B)
(A)