29-09-2024

7501CJA101001240050 JM

PHYSICS

SECTION-I

1) Two identical balls are interconnected with a light and inextensible thread having length l. The
system is on a smooth horizontal table with the thread just taut. Initial velocities of both as shown.

The kinetic energy of the system after the string gets taut.

(A) mv2
(B) 2mv2
(C) 3/4 mv2
(D) 3/2 mv2

2) Two masses 'm' and '2m' are placed in fixed horizontal circular smooth hollow tube as shown. The
mass 'm' is moving with speed 'u' and the mass '2m' is stationary. After their first collision, the time

elapsed for next collision. (coefficient of restitution e = 1/2)

(A)

(B)

(C)

(D)

3) Consider regular polygons with number of sides n = 3, 4, 5 ..... as shown in the figure. The centre
of mass of all the polygons is at height h from the ground. They roll on a horizontal surface about the
leading vertex without slipping and sliding as depicted. The maximum increase in height of the locus
of the centre of mass for each polygon is Δ. Then Δ depends on n and h as :
(A)

(B)

(C)

(D)

4) A large number (n) of identical beads, each of mass m and radius r are strung on a thin smooth
rigid horizontal rod of length L (L > > r) and are at rest at random positions. The rod is mounted
between two rigid supports (see figure). If one of the beads is now given a speed v, the average force
experienced by each support after a long time is (assume all collisions are elastic) :-

(A)

(B)

(C) Zero

(D)

5) A frictionless tube lies in the vertical plane and is in the shape of a function that has its endpoints
at the same height but is otherwise arbitrary. A chain with uniform mass per unit length lies in the
tube from end to end, as shown in figure. Choose the correct statement(s) :

(A) The chain will move in the direction having steep slope.
(B) The chain will move in the direction having longer length.
(C) The chain will not move irrespective of any shape of tube.
(D) The chain will not move only if the shape of tube is symmetric.
6) Track OABCD (as shown is figure) is smooth and fixed in vertical plane. What minimum speed has
to be given to a particle lying at point A, so that it can reach point C?

(A) 60 m/s
(B) 100 m/s
(C) 70 m/s
(D) 80 m/s

7) Two blocks A and B of equal masses are released on two sides of a fixed wedge C as shown in
figure. The acceleration of centre of mass of blocks A and B is (Neglect friction):

(A)

(B)

(C)

(D)

8) Two particles of mass m, constrained to move along the circumference of a smooth circular hoop
of equal mass m, are initially located at opposite ends of a diameter and given equal velocities v0
shown in the figure. The entire arrangement is located in gravity free space. Their velocity just

before collision is:

(A)
(B)

(C)

(D)

9) A plate in the form of a semicircle of radius R has a mass per unit area of kr where k is a constant
and r is the distance from the centre of the straight edge. By dividing the plate into semicircular
rings find the distance of the centre of mass of the plate from the centre of its straight edge.

(A)

(B)

(C)

(D)

10) A ball of mass m moving with a velocity v collides with ( ; l is the length of the string)
the bob of a pendulum of the same mass at rest which is hanging vertically with the string of length
. If the collision is perfectly inelastic, the height to which both will rise is given by :-

(A)

(B)

(C)

(D)

11)

A particle of mass 2m is projected from the ground at an angle of 45° with horizontal with a velocity
of . After one second explosion takes place and the particle is broken into two equal
pieces. As a result of this explosion, one part comes to rest. The maximum height from the ground
attained by the other part is (g = 10 ms-2)

(A) 25 m
(B) 35 m
(C) 40 m
(D) 50 m

12) Particles P and Q of masses 20g and 40g, respectively, are projected from positions A and B on
the ground. The initial velocities of P and Q make angles of 45º and 135º, respectively with the
horizontal as shown in the Fig. Each particle has an initial speed of 49m/s. The separation AB is
245m. Both particles travel in the same vertical plane and undergo a collision. After the collision P
retraces its path. The separation of Q from its initial position when it hits the ground is

(A) 245 m

(B)

(C)

(D)

13) A particle of mass m = 0.1 kg is released from rest from a point A of a wedge of mass M = 2.4
kg free to slide on a frictionless horizontal plane. The particle slides down the smooth face AB of the
wedge. When the velocity of the wedge is 0.2 m/s the velocity of the particle in m/s relative to the

wedge is

(A) 4.8
(B) 5
(C) 7.5
(D) 10

14) A particle is moving in a circular path of radius 'a' under the action of a potential energy U = –

. Its kinetic energy is:

(A)

(B)

(C)

(D) Zero

15) From what minimum height h must the system be released when spring is unstretched so that
after perfectly inelastic collision (e = 0) with ground, B may be lifted off the ground (Spring
constant = k).

(A) mg/(4k)
(B) 4mg/k
(C) mg/(2k)
(D) none

16) A square plate of edge d and a circular disc of diameter d are placed touching each other at the
midpoint of an edge of the plate as shown in the figure. The centre of mass of the combination,
assuming same mass per unit area for the two plates from the centre of the disk is

(A)

(B)

(C)

(D)

17) The centre of mass of a non uniform rod of length L whose mass per unit length λ varies as λ =

where k is a constant & x is the distance of any point on rod from its one end, is (from the
same end)

(A)
L

(B)
L

(C)
(D)

18) A small glass ball is pushed with a speed v from A. It moves on a smooth surface and collides
with wall at B. If it loses half of its speed during the collision, what will be the average speed of the

ball till it reaches at its initial position:

(A) Zero

(B)

(C)

(D)

19) A particle of mass m strikes the incline vertically with velocity u and just after strike it moves
horizontally. The movable wedge does not move despite collision due to friction. The magnitude of

impulse of friction on the wedge during collision is :

(A)
(B) mu

(C)

(D)

20) A smooth sphere is moving on a horizontal surface with a velocity vector m/s
immediately before it hit a vertical wall. The wall is parallel to vector and coefficient of restitution
between the sphere and the wall is e = 1/2. The velocity of the sphere after it hits the wall is

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

SECTION-II

1) We stick a inextensible adhesive tape on a tabletop. We grab one end of it and pull it back
horizontally and uniformly with a velocity of 4 cm/s. What is the velocity (in cm/s) of the middle of

the moving part of the tape?

2) Figure shows an isosceles triangular plate of mass M and base length l. The apex lies at the origin
and the angle at the apex is 90°. The base is parallel to x-axis. The moment of inertia of the plate

about the x-axis is . Where k is

3) A extremely light ball A is placed on a massive ball B and both the balls are released without
initial velocity. If all the collisions are elastic and sizes of the balls are negligible as compared to the
height h, the maximum height attained by ball A is xh then value of x is.

4) Three equal masses m are rigidly connected to each other by massless rods of length ℓ forming an
equilateral triangle, as shown in the figure. What is the ratio of the moment of inertia of the
assembly for an axis through B compared with that for an axis through A (centroid). Both the axis

are perpendicular to the plane of triangle.

5)

A particle is given a certain velocity v at point P as shown on a hemispherical smooth surface. Find
the value of v (in m/s), such that when particle reaches Q, the normal reaction of surface becomes
equal to particle's weight. [R = 1.6 m, g = 10 m/s2]

6) The moment of inertia of a uniform semi-circular wire of mass M and radius R about an axis

passing through its centre of mass and perpendicular to its plane is Then the value of x?

(Take π2 = 10)

7) A square plate has uniform mass distribution. It has mass M = 24/11 kg and edge L = 2m as
shown in the figure. Calculate the moment of inertia of the plate about the axis AB (as shown in

figure) in the plane of the square plate. (in SI unit)

8) A cube of mass m slides down the smooth track from height 4R. Then the force track will exert on

block at point 2 is ____ times the weight of block :-

9) A ring and a solid sphere rotating about an axis passing through their centers have same radii of
gyration. The axis of rotation is perpendicular to plane of ring. The ratio of radius of ring to that of
sphere is . The value of x is ___.

10) A small block of mass 20 kg rests on a bigger block of mass 30 kg, which lies on a smooth
horizontal plane. Initially the whole system is at rest. The coefficient of friction between the blocks is
0.5. The horizontal force, F = 50N, is applied on the lower block. The work done by frictional force
on upper block in t = 2s is n x 10 J. Find n ?

CHEMISTRY

SECTION-I

1) Al2(SO4)3 solution of 1 molal concentrations is present in 1 L solution of density 2.684 g/mL. How
many moles of BaSO4would be precipitated on adding excess of BaCl2 in it?
[Atomic wt. : Al = 27, S = 32, Ba = 137,Cl = 35.5]

(A) 2 moles
(B) 3 moles
(C) 6 moles
(D) 12 moles

2) The correct order of hydration enthalpies of alkali metal ions is -

(A) Li+ > Na+ > K+ > Rb+ > Cs+

(B) Li+ > Na| > K+ > Cs+ > Rb+
(C) Na+ > Li+ > K| > Rb+ > Cs+
(D) Na+ > Li+ > K+ > Cs+ > Rb+

3)

20 mL of a mixture of CO and H2 were mixed with excess of O2 and exploded & cooled. There was a
volume contraction of 23 mL. All volume measurements corresponds to room temperature (27°C)
and one atmospheric pressure. Determine the volume ratio V1 : V2 of CO and H2 in the original
mixture

(A) 6.5 : 13.5

(B) 5 : 15
(C) 9 : 11
(D) 7 : 13

4) Find molality of aqueous ammonia solution in which mole fraction of solvent is x -

(A)

(B)

(C)

(D)

5) 1120 ml of ozonised oxygen (O2 + O3) at 1 atm & 273K weighs 1.76 gm. The reduction in volume
on passing this through alkaline pyrogallol solution is :-

(A) 896 ml
(B) 224 ml
(C) 448 ml
(D) 672 ml

6) The electronic configuration of an element is 1s2 2s2 2p6 3s2 3p4. The atomic number and the group
number of the element ‘X’ which is just below the above element in the periodic table are
respectively.

(A) 24 & 6
(B) 24 & 15
(C) 34 & 16
(D) 34 & 8

7) The correct order of electron affinity is :-

(A) Cl > F > O

(B) F > O > Cl
(C) F > Cl > O
(D) O > F > Cl

8) The vapour density of a mixture containing NO2 and N2O4 is 27.6. The mole fraction of N2O4 in the
mixture is:

(A) 0.1
(B) 0.2
(C) 0.5
(D) 0.8

9) Increasing order of Electron affinity for following configuration.

(a) 1s2, 2s2 2p2 (b) 1s2, 2s2 2p4
(c) 1s2, 2s2 2p6 3s2 3p4 (d) 1s2, 2s2 2p3

(A) d < a < b < c

(B) d < a < c < b
(C) a < b < c < d
(D) a < b < d < c

10) In which of the following arrangements the order is NOT according to the property indicated
against it?

(A) Al3+ < Mg2+ < Na+ < F– - increasing ionic size
(B) B < C < N < O - increasing first ionization energy
(C) l < Br < F < CI - increasing electron affinity
(D) Li < Na < K < Rb - increasing metallic radius

11) Which represents alkali metals (i.e. group 1 metals) based on (IE1) and (IE2)

IE1 IE2

(A) X 100 150

(B) Y 95 120

(C) Z 195 500

(D) M 200 250

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

12) The pair of amphoteric hydroxide is :

(A) Al(OH)3, LiOH

(B) Be(OH)2, Mg(OH)2
(C) B(OH)3 , Be(OH)2
(D) Be(OH)2 , Zn(OH)2

13) The correct order of radii is :-

(A) N < Be < B

(B) F– < O2– < N3–
(C) Na < Li < K
(D) Fe3+ < Fe2+ < Fe4+

14) Which of the following is the correct order of second ionisation potential ?

(A) C < F < N < O

(B) N < C < O < F
(C) C < N < F < O
(D) F < O < N < C
15) The set of quantum numbers not applicable for an electron in an atom

(A)
n = 2, ℓ = 1, m = 1,

(B)
n = 1, ℓ = 0, m = 0,

(C)
n = 1, ℓ = 0, m = 1,

(D)
n = 2, ℓ = 0, m = 0,

16) Consider the following three moving objects:

(i) a golf ball with a mass of 45.9 g moving at a speed of 50.0 m/s.
(ii) An electron moving at a speed of 3.50 × 105 m/s.
(iii) A neutron moving at a speed of 2.3 × 102 m/s.
Mark the correct order of their de–Broglie wavelength.

(A) λi < λii < λiii

(B) λii < λiii < λi
(C) λiii < λii < λi
(D) λi < λiii < λii

17) In the emission spectrum below for hydrogen, which transition falls under visible region?

(A) a
(B) b
(C) c
(D) d

18) The normalised wave function of 1s orbital is and radial distribution function is

where N is normalisation constant . Which of the following graph is correct for

the radial distribution ( ) of 1s electron with respect to ‘r’ for H-like specie of atomic number
Z ? (Where r is radial distance from nucleus)
(A)

(B)

(C)

(D)

19)

Given
Reaction Energy Change (in kJ)
Li(s) → Li(g) 161
Li(g) → Li+(g) 520

F2(g) → F(g) 77
F(g) + e– → F–(g) (Electron gain enthalpy)
Li+(g) + F–(g) → LiF(s) –1047

Li(s) + F2(g) → Li F(s) –617

Based on data provided, the value of electron gain enthalpy of fluorine would be :
(A) –300 kJ mol–1
(B) –328 kJ mol–1
(C) –350 kJ mol–1
(D) –228 kJ mol–1

20) 300 gm, 30% (w/w) NaOH solution is mixed with 500 gm 40% (w/w) NaOH solution. What is %
(w/v) NaOH if density of final solution is 2 gm /mL :

(A) 72.5
(B) 65
(C) 62.5
(D) 60

SECTION-II

1) A certain dye absorbs 4000 Å and fluoresces at 5000 Å. Under given conditions 40% of the
absorbed energy is emitted. If the ratio of the number of quanta emitted to the number absorbed is x
: 1.00, the value of 'x' is.

2) Among the following species, how many have their ionic size greater than S–2?
O–2, P–3, Cl–, Na+ , Ca+2 , K+

3) Find out the group number in periodic table of an element which has electronic configuration
2 2
86[Rn]7s 6d ?

4)

A hydrocarbon CxHy on complete combustion gives CO2(g) & H2O(ℓ). If volume of O2(g) required for
complete combustion of hydrocarbon is double the volume of CO2(g) produced, then find the value of

5) 500 ml of each three sample of H2O2 labelled 10 V, 15V and 20 V are mixed and then diluted with
equal volume of water. The volume strength of resultant H2O2 solution is :

6) 1292.5 gm of aqueous solution of '5m' NaCI is kept in a large bucket. The bucket is placed under
a tap from which a '2m' aqueous solution of NaCI is flowing. Rate of flow of solution from tap is 0.5
gm/sec. The total amount of solution (in gm) finally present in bucket when solution present in
bucket have concentration of NaCI 4m.

7) Ratio of frequency of revolution of electron in the second excited state of He+ and second state of
hydrogen is

8) A light source emits radiations of Number of photons emitted by this source varies
with time, as N = 1018t2. (t in sec). If this light source is used to emit photoelectrons from a metal of
work function 3.1 eV. Find the time(in sec) during which the total kinetic energy of all the emitted

photoelectrons from the beginning is 1296 J. . Assume all photoelectrons are

emitted with highest possible K.E.

9) Calculate molality of pure methyl alcohol (CH3OH) if its density is 1.5 gm/ml.

10) Out of following nine pairs, how many number of pairs of species in which size of first species is
higher as compare to second species.
(O, O+), (Sc, Y), (Na+, Mg2+), (Fe2+, Fe3+), (I–, Cl–), (Na+, Mg2+)

MATHEMATICS

SECTION-I

1) For the pair of straight lines x2 – 4αxy + y2 = 0, if sum of slopes is four times product of slopes,
then α is

(A) –2
(B) 2
(C) 1
(D) –1

2) Consider points A(2,1) and B(–2,7). If P be a point on the line 3x + 4y = 12. Such that PA + PB is
minimum, then coordinates of P are

(A)

(B) (0,3)

(C)

(D) (0,4)

3) The value of the determinant is:

(A) Always positive

(B) Always negative
(C) Always zero
(D) Cannot say anything
4) If the line x + y + 2 = 0 is shifted parallel to itself towards positive direction of x-axis by a
distance units, then the equation of the line in new position is -

(A) x + y – 6 = 0
(B) x + y + 6 = 0
(C) x + y + =0
(D) x + y – =0

5) If A,B,C are the angles of a triangle, then system of equations

(sinA)x + y + z = cosA
x + (sinB)y + z = cosB
x + y + (sinC)z = 1 – cosC has

(A) No Solution
(B) Unique Solution
(C) Infinitely many solution
(D) None of these

6) Equation of a line which is parallel to the line common to the pair of lines given by 6x2 – xy – 12y2
= 0 and 15x2 + 14xy – 8y2 = 0 and at a distance 7 unit from it is

(A) 3x – 4y + 35 = 0
(B) 5x – 2y – 7 = 0
(C) 3x + 4y – 35 = 0
(D) 2x – 3y – 7 = 0

7) Distance of the origin from the line measured along the line
is

(A)
(B) 10
(C) 5
(D)

8) A man standing on a level plain observes the elevation of the top of a pole of height h to be . He

then walks a distance x towards the pole and finds that the elevation is now , if h = 33m, then x2 is
equal to :-

(A) 66
(B) 60
(C) 3600
(D) 4356
9) If a2 + b2 + c2 = –2 and then f(x) is a polynomial of degree
:

(A) 0
(B) 1
(C) 2
(D) 3

10) For given lines x + ay + 5 = 0 and 3x + 2y + 1 = 0. If represents its

obtuse angle bisector, then sum of all possible integral values of a is (where a > –4) -

(A) –6
(B) –5
(C) –2
(D) 0

11)

The person walking along a straight road observes that a two points 1 km apart, the angles of
elevation of a pole in front of him are 30° and 75°. The height of the pole is -

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

12) The lines of family cx – y = 3 + 2c (where c is a parameter) are concurrent at a point P whose
distance from the line 3x + 4y – 9 = 0 is equal to

(A) 5
(B) 3
(C) 4

(D)

13) A straight line is drawn through the point P(2, 3) and is inclinded at an angle of 30° with positive
x-axis. Then the co-ordinate of two points on it at a distance 4 from P on either side of P, are given
by

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

14)

A vertical lamp-post of height 9 metres stands at the corner of a rectangular field. The angle of
elevation of its top from the farthest corner is 30°, while from another corner it is 45°. Then,

(A) the length of the diagonal of the field is

area of the field is
(B)

(C)
perimeter of the field is
angle subtended by lamp-post at the centre of the field is
(D)

15)

If = ax4 + bx3 + cx2 + dx + e is true for all x ∈ R, then value

of e, is

(A) 1
(B) 0
(C) 2
(D) 3

16) The locus of the centroid of the triangle whose vertices are (3cost, 3sint), (4sint, –4cost) and (2,
0) where t is the parameter is

(A)

(B)

(C)

(D)

17) A is a point on either of two rays y + |x| = 2 at a distance of units from their point of
intersection. The co-ordinates of the foot of perpendicular from A on the bisector of the angle
between them are
(A)

(B) (0, 0)

(C)

(D) (0, 4)

18) If x ∈ [0, 2π] and a, b, c ∈ R and P > 0 such that , then

the value of

(A) depend on a,b,c

(B) depend upon x
(C) depend upon P
(D) independent of a, b, c, x and P

19) Two sides of a rhombus are along the lines, x – y + 1 = 0 and 7x – y – 5 = 0. If its diagonals
intersect at (–1, –2), then which one of the following is a vertex of this rhombus ?

(A)

(B) (–3, –9)

(C) (–3, –8)

(D)

20)

The value of determinant is equal to-

(A) 1
(B) 0
(C) –1
(D) 2

SECTION-II

1) T is the area of region enclosed by the locus of point (x, y) which moves such that 2|x| + |y| = 4,
then the number of prime number(s) less than T
2) ABCD in order be a rectangle. The co-ordinate of A and C are (1, 3) and (5, 1) respectively. If the

gradient of the diagonal BD is 2 and the area of the rectangle is S, then the value of is equal to

3) If a3 + b3 + c3 – 3abc = 3, then find the value of

4)

A flag staff on the top of the tower 80 meter high, subtends an angle at a point on the

ground 100 meters away from the foot of the tower. If the height of the flag-staff is h. Find .

5) Number of integral value of a for which point (–2,a) lies inside the triangle formed by the lines y =
x, y = –x and 2x + 3y = 6, is

6) Let D, E, F are the points dividing the sides AB, BC, CA of ΔABC in λ : 1 internally. A line family
concurrent at a fixed point is drawn such that length of perpendicular from point D is equal to sum
of length of perpendiculars from E & F also given that the points E and F are on same side but D is
on opposite side, then co-ordinates of the fixed point is (a,b), then (a + b) is equal to [vertices A,B,C
are (0,0), (–3,2), (3,7) respectively]

7) Area of the parallelogram having sides 14x – 21y + 15 = 0, 14x – 21y – 6 = 0, 2x – 6y + 9 = 0 & 2x
– 6y – 3 = 0, is equal to

8) If a point (–1,2) undergoes the following transformations consecutively

(i) Shifted by 2 units in positive x-direction.
(ii) Image is taken in y = x line.
(iii) Shifted by units parallel to y = x line in downward direction. then co-ordinates of the final
point are (a,b) then |a + b| is equal to

9)

The angles of elevation of the top of a tower at the top and the foot of a pole of height 10 m are 30°

and 60° respectively. If the height of the tower is h. Find .

10) If , then the maximum value of Δ is

 ANSWER KEYS

PHYSICS

SECTION-I

Q. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
A. C B C D C D D D C A B C D C B B A C B B

SECTION-II

Q. 21 22 23 24 25 26 27 28 29 30
A. 3.00 8.00 9.00 2.00 4.00 6.00 4.00 3.00 5.00 4.00

CHEMISTRY

SECTION-I

Q. 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
A. C A D C A C A B A B C D B C C D B A B A

SECTION-II

Q. 51 52 53 54 55 56 57 58 59 60
A. 0.50 1.00 3.00 4.00 7.50 1851.00 1.18 to 1.19 30.00 31.25 5.00

MATHEMATICS

SECTION-I

Q. 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80
A. C C A A B C C D C B A B B C B A B D D B

SECTION-II

Q. 81 82 83 84 85 86 87 88 89 90
A. 6.00 1.00 9.00 5.00 1.00 3.00 6.00 7.00 3.00 5.00
 SOLUTIONS

PHYSICS

1)

so in the direction of string mv = 2mv1

KE of the system

2)

Let the speeds of balls of mass m and 2m after collision be v1 and v2 as shown in figure.
Applying conservation of momentum

mv1 + 2mv2 = mu and –v1 + v2 =

solving we get v1 = 0 and v2 =

Hence the ball of mass m comes to rest and ball of mass 2m moves with speed .

t=

3)

OA = h
OB =
Initial height of COM = h

Final height of COM

4) Time interval between consecutive collision with one support

Change in momentum in each collision = 2mv

5)

Let the curve be described by the function f(x), and let it run from x = a to x = b.
Consider a little piece of the chain between x and x + dx (see Fig.). The length of this piece is
, so its mass is , where ρ is the mass per unit length. The component of the

gravitational acceleration along the curve is – g sin θ = (using tan θ = f'), with positive
corresponding to moving along the curve from a to b. The total force along the curve is therefore

7)
mA = mB
 8)

VP = particle velocity just before impact

Vh = hoop velocity just before impact from conservation of momentum
2 mV0 = 3mVn -----------(1)
conservation of energy just before impact

2× = --------(2)
following eqn (1) & (2)

VP =

9)

dm = 2πrk rdr = 2πkr2dr

distance of CM of differential ring from O,

r1 =

rcm = =

= =

10)

By conservation of momentum

By conservation of mechanical energy

 11) In 1s, the particle's velocity becomes

It ascends a height

= 15 m
After the explosion:

12)

P, Q collide at mid point

Before collision = 49 cos 45°

= 49 cos 45 (– )
After collision

= 49 cos 45° (– )

=?
From momentum conservation :-
20 (49 cos 45°) – 40 (49 cos 45°) = – 20 (49 cos 45°) +
=0
So it will hit ground at (245/2)m

13)

Mu = m (V cos 60 – u ) [from conservation of momentum]

2.4 × 0.2 = 0.1

V = 10 m/s

14)
For r = a,

15)

(A)
kx0 = mg

compression from MP = Elongation from MP = x

0
From W.E.T. ⇒ mg(x + x ) +

=0– = –mgh

⇒
⇒ (x + x0)

From (1) kx – kx0 = 2mg Þ kx – k.

⇒x=

0
∴x+x =
Now from W.E.T. ⇒ from NLP to max compression

0
mg (x + x ) +

⇒ mg .
⇒

⇒h=
Alternate Solution

...(i)
kx = 2mg ...(ii)
From (i) & (ii)

h=

16)

17)

∴ xcm =
 18)

19)

v cosθ = u sinθ
v = u tan θ
–mu cos θ + J = mv sin θ
J = mv sin θ + mu cos θ

20)

∴ smooth sphere ⇒ f = 0
⇒ vy = s uncylindrical
∴

21)
 22)

23)

Let

By definition of 'e' : e =

24) IA =
IB = 2mℓ2
 ∴ ratio = 2 : 1

25)

WET between P and Q

v=4

26)

Parallel axis theorem,

27)

Apply parallel axis theorem

28)

N = 3 mg

29)
For ring
∴ Radius of gyration K1 = R1
For solid sphere

∴ Its radius of gyration = K2 =

∴ K1 = K2

∴
∴x=5

30)
= 100 N or 98 N
let's say blocks move together

CHEMISTRY

31)

Lex X mol = Al2(SO4)3

x=2
2 Al2(SO4)3 —→ 6SO4–2
2 6 mol

32)

HYDRATION ENTHALPY OF CATION

For alkali metal cations charge is same and size increases on moving down the group hence
hydration enthalpy decreases.
HYDRATION ENTHALPY: Li+ > Na+ > K+ > Rb+ > Cs+

33)
x x/2 x
 Vol cont. =
x=7

34)

Xsolvent = x , Xsolute = 1 – x

Molality =

35) ...(1)
32x + 48y = 1.76 ...(2)
y = 0.01, x = 0.04
alkaline pyrogallol absorbs
0.04 mole O2 = (0.04 × 22400) mℓ = 896mℓ

36) Se → Z = 34 and group number = 16.

37)

On moving left to right across the period ,electron affinity increases because size decreases and
attraction of nucleus on outermost electron (upcoming e–) also increases. Thus electron affinity of F
> O.
Electron affinity of Cl > F (due to small size of F atom, electronic repulsions are high in F atom thus
addition of e– becomes difficult as compared to Cl)

38)
Vapour density of mixture = 27.2
Mavg = 2 × VD
= 2 × 27.2 = 55.2
Let NO2 N2O4
x% 100–x%

N2O4 = 100 – 80% = 20% = 0.20.

39)
N<C<O <S

40)

Option B is Wrong
B < C < O < N → Ist IE

41)

42) Conceptual

43) In isoelectronic species, as Z↑ size ↓

44)

Based on electronic configuration.

45)

46)

⇒ λi < λiii < λii

47)

Balmer series (n2 = 2) falls in visible region.

48)

above function has a maxima at (a0/z) and zero at infinite,so it follows 1st graph.

49)
 L.E. = –1047
161 + 520 + 77 + EGE – 1047 = –617
EGE = –328

50)

Ans. (1)

51)

Given λabs = 4000 Å

λemit = 5000 Å

Since

52)

Only P–3 has greater size.

53) Electronic configuration

2 2
86[Rn] 7s 6d ⇒ Thorium

f-block element
Group → 3rd

54)

55) × 500 = Mf × 3000

Mf
Vol. strength of final solution = M × 11.35

= 7.5 V

56)
 1292.5 solution of bucket contains
5 mole NaCI in 1 kg water
Let x moles of NaCI & y kg of water is added to bucket from tap

⇒ x = 1 mol

amount of solution finally present

= 1292.5 + x × 58.5 + y + 100
= 1292.5 + 58.5 + 500
1851

57) ν ∝

58)

KE = 0.9 eV per photoelectrom

Total no. of photoelectrons =

t = 30 sec

59)

60)

I— > Cl—

MATHEMATICS

61)

m1 + m2 = 4a & m1m2 = 1
m1 + m2 = 4m2m2
4α = 4
α=1
 62)

Ans. (C)
Sol. Equation of AB =

2y + 3x = 8 ...(1)
3x + 4y = 12 ...(2)
from (1) & (2)

⇒ y = 2 and

63) Determinant formed by the cofactors of is

64)

PQ is line in new position.

COC' =

if equation of PC'Q is x + y = k
then k = 6
⇒ equation is x + y – 6 = 0
 65)
R1 → R1 – R2
R2 → R2 – R3

D = (1 – sinA)(1 – sinB)sinC + (1 – sinC)

(1 – sinA) + (1 – sinB)(1 – sinC)
D > 0, Hence unique solution

66)

6x2 – xy – 12y2 = (2x – 3y)(3x + 4y)

15x2 + 14xy – 8y2 = (5x – 2y)(3x + 4y)
required line ≡ 3x + 4y + λ = 0

67)

where θ = 60°

Now :

r=5

68) ,

BC = hcotβ
AC = hcotα
x = AC – BC = h(cotα – cotβ)

x = 66, x2 = 4356

69) Operating C1 → C1 + C2 + C3, we get

 [Operating R1 → R2 – R1 and R3 → R3 – R1]
= (1)[(1 – x)2 – 0]
= (1 – x)2
which is a polynomial of degree.

70)

3 + 2a < 0

71)

72)
c(x – 2) – (y + 3) = 0
point at which lines are concurrent = (2, –3)

=3

73)
PA = PB = 4 unite (given)
we will draw perpendicular from point A & point B on x axis & perpendicular from point P on
y-axis.
Hence ,

(vertically opposite angless)

AD = PA sin 30°

=4 units
PD = PA cos 30°

=4 = units
EB = PB sin 30°

=
EP = PB cos 30°

=4 units
Hence,
A(x1, y1) = A(2 + PD, 3 + AD)
= A(2+ )
A(x1, y1) = A (2 + , 5)
& B(x2, y2) = B (2–|EP|, 3– |EB|)
= B(2 – , 3–2)
B(x2, y2) = B (2– , 1)

So, the answer is

Hence, option (B) is correct

74) Let AP be the lamp-post of 9m standing at corner A of the rectangular field ABCD.
In Δ's BAP and CAP, we have
 and
and

Hence, area of the field = AB × BC

Perimeter

angle subtended at the centre

75) Put x = 0
 76)

Ans. (1)

(3h – 2)2 + (3k)2 = 25

77) y + |x| = 2
⇒ if x ≥ 0
& if x < 0
if

if x<0

PA =
|AO| =|A'O|
Hence, ∠APO = ∠A'PO
Since PO is the angle bisector and AO is perpendicular to PO, so O is the intersection point
having co-ordinates (0, 0).
⇒ Option (B) is correct.

78)

1 + sin2x = 0 ⇒p=3
(sinx + cosx)2 = 0
sinx + cosx = 0
a+b+c=0
C1 → C1 – C2
Value of determinant = 0

79) Equation of angle bisector of the lines

x – y + 1 = 0 and 7x – y – 5 = 0 is given by

⇒ 5(x – y + 1) = 7x – y – 5 and
5(x – y + 1) = –7x + y + 5
∴ 2x + 4y – 10 = 0 ⇒ x + 2y – 5 = 0 and
 12x – 6y = 0 ⇒ 2x – y = 0
Now equation of diagonals are
(x + 1) + 2(y + 2) = 0 ⇒ x + 2y + 5 = 0 ...(1) and
2(x + 1) – (y + 2) = 0 ⇒ 2x – y = 0 ...(2)

Clearly lies on (1)

80) Apply C1 → C1 + C2 + C3
so value of determinant = 0

81)

Area of rhombus = 16

82)

Let the equation of

BD is y = 2x + c ...(1)
passing through (3,2)
∴ BD : y = 2x – 4
Now \(AC = 2\sqrt 5 \), MD, MB = \(\sqrt 5 \)
∴ \(\frac{{x - 3}}{{\cos \theta }} = \frac{{y - 2}}{{\sin \theta }} = \pm \sqrt 5 \)
\(x = \pm \sqrt 5 \cos \theta + 3,y = \pm \sqrt 5 + 2\)
∴ tanθ = 2
∴ x = 4, y = 4 or x = 2, y = 0
∴ Area = 10 cm2

83)

Ans. 9
Sol. ∵

= (3abc – a3 – b3 – c3)2 = 9

84)

85)

x = –2 y = –x x = –2 2x + 3y = 6

A(–2, 2)
a = 3 only

86)

(a, b) will be centroid of ΔABC.

87)

use formula

88)

(i) (–1, 2) (1, 2)

(ii) Image in y = x is (2, 1)

(iii)
co-ordinates of the final point are (–3, –4)

89)

From 1st and 2nd

h = 3h – 30
2h = 30
h = 15

90)
Apply R3 → R3 – R1

= (3cosθ – sinθ)2 ≤ 10
max. value of Δ = 5