01-06-2025

1001CJA106216250078 JA

PART-1 : PHYSICS

SECTION-I

1) An isolated system consists of two concentric thin metallic spherical shells A and B having radii a
and 2a respectively. The shells are neutral initially. A point charge +q is placed at the centre of the

shells. The shell A is earthed :-

(A) There exists an electric field in the cavity of the shell A

(B) The charge density on the inner surface of the shell A is nonuniform
(C) There is no electric field in the region a < r < 2a.
(D) There exists an electric field at r > 2a.

2) A galvanometer is modified so that it can measure a physical quantity in the circuit. In the
modification instead of using permanent magnet, current coil is used on a ferromagnet and the
arrangement is such that magnetic field due to current coil always remains perpendicular to the
area of voltage coil. The remaining arrangement of the galvanometer remains same. The modified
circuit is shown below. Choose the CORRECT option(s) :

(A) If the marking of the scale is uniform, it will measure power dissipated in the resistor.
(B) If the marking of the scale is non-uniform, it will measure power dissipated in the resistor.
For increasing the sensitivity of the device, the frame on which the current coil is wrapped
(C)
should be of ferro-magnetic material.
(D) On increasing the value of current to twice deflection on the scale will also be doubled.

3) An ideal parallel plate capacitor of area A is filled with three dielectric slabs having dielectric
constants K1 = 3.0, K2 = 5.0 and K3 = 2.0 as shown in the figure. Choose the correct option(s) :-

(A)
the equivalent capacitance of the system is
(B) the electric field in K1 is more than the electric field in K2.
(C) the electric field on the left half of K3 is less than the electric field on the right half of K3.
(D) the electric field on the left half of K3 is more than the electric field on the right half of K3.

4) A light rod of length 2m is suspended from a ceiling by means of two vertical wires of equal length
tied to its ends. One of the wires is made of steel and is of cross-section 10–3 m2 and the other is of
brass of cross-section 2 × 10–3 m2. x is the distance from steel wire end, at which a weight may be
11 11
hung. Ysteel = 2 × 10 Pa and Ybrass = 10 Pa. Which of the following statements is/are correct
(Assuming the rod remains in horizontal position in final equilibrium position)?

(A) x = 1.2 m, if the strains of both the wires are to be equal.

(B) x = 1.42 m, if the stresses of both the wires are to be equal.
(C) x = 1 m, if the strains of both the wires are to be equal.
(D) x = 1.33 m, if the stresses of both the wires are to be equal.

5) A uniform rod of cross-sectional area 'a'(= 0.2 cm2)is bent to form a square of side ℓ = 10cm.
Three of the ends of square A, B and C are maintained at 100 °C, 40 °C and 0 °C respectively. If the
conductivity of the material of the rod be k = 385 W/m °C, then pick out the correct statement(s) of

the following :- (Given the lateral surfaces are insulated)

(A) The rate of heat supplied by the reservoir at A is 8.47 J/s.

(B) Rate at which the heat is absorbed by the reservoir at B is 1.54 J/s.
(C) The heat current from A to C is 3.85 J/s.
(D) The rate at which heat is absorbed by the reservoir at junction C is 6.93 J/s.

6) A conducting rod AB of mass m slides without friction over two long conducting horizontal rails
separated by a distance ℓ. At the left the rails are inter connected by a resistor R. The system is
located in uniform magnetic field perpendicular to the plane of the loop. At the moment t = 0, the
rod AB starts moving to the right with an initial velocity V0. Neglect resistance of rails and self
induction of the circuit.

(A)
The rod covers a distance before stopping.

(B)
The rod covers a distance before stopping.

(C)
The heat liberated in the resistor during motion of the rod is .

(D)
The heat liberated in the resistor during motion of the rod is .

7) A long thin wire is carrying current along positive Z-axis. Another current carrying loop made of
uniform wire is arranged with its vertices at A(a,0,a); B(a,0,0); C(0,a,0); D(0,a,a) as shown in figure.
BC and DA are in the shape of a quarter circle centered on Z-axis. The loop is constrained to rotate

about Z-axis without friction.

(A) Mutual inductance between the long wire and the loop is zero.
(B) Net magnetic force acting on the loop due to the long wire is zero.
(C) Net torque due to magnetic force on the loop about Z-axis is zero.
Consider an ampere loop in the shape of a circle of radius 'a'. Plane of the loop is parallel to XZ-
(D)
plane with its centre at (0,a /2,0). Let be the net magnetic field in space. Then
around the Ampere loop is zero.

8) The graph shows the force-extension curve for a steel wire that is stretched by a tensile force. In
the graph, PQ and QR are two idealized straight line sections. The unextended length of the wire is
2.00 m. Assume that Q is proportionality limit and R is yield point. Then mark the CORRECT
statement(s).

(A) The strain in the wire at R is 0.025.

(B) The work done in slowly stretching the wire from Q to R is equal to 50 J.
(C) If we release the wire at point Q, it won't have any permanent set.
(D) The work done in slowly stretching the wire from Q to R is equal to 55 J.

SECTION-II

1) The circuit shown in the figure contains three resistors R1 = 100 Ω, R2 = 50Ω & R3 = 20Ω and cells
of emf's E1 = 2V & E2. The ammeter indicates a current of 50mA. Find the emf (in V) of the second
cell. (The internal resistance of the ammeter and of the cell should be neglected.)

2) A conducting rod of length 1 m aligned along the unit vector is moving with a velocity

. The motion take place in a uniform magnetic field tesla. Find the
magnitude of induced emf across the rod in volt.

3) A short electric dipole is kept at (0, b). Dipole moment of the dipole is . A point charge –Q lies at

the origin (O). Electric field at a point A(a, a + b) is zero. Find the ratio
4) Infinite semi-circular wires are placed out from a single wire in a continuous fashion as shown in
the figure. The radii of the semicircles increases as Rn = αn – 1R1 where Rn is the radius of the nth
semicircle and α is constant greater than 1. The semicircles are concentric. The net magnetic field at

the centre is given by where β is the numerical constant. Find the value of β.

5) An electron accelerated by a potential difference V = 3.6 volt first enters a uniform electric field
of a parallel-plate capacitor whose plates extend over a length ℓ = 6 cm in the direction of initial
velocity. The electric field is normal to the direction of initial velocity and its strength varies with
time as E = a × t where a = 3200 Vm–1s–1. Then the electron enters a uniform magnetic field of
induction B = π × 10–9T. Direction of magnetic field is same as that of the electric field. Calculate
pitch (in mm) of helical path traced by the electron in the magnetic field. (Mass of electron, m = 9 ×
10–31kg). [Neglect the effect of induced magnetic field]. (Electron does not collide with the parallel
plates during the motion)

6) A small solid sphere of radius R made of material of bulk modulus K is surrounded by a liquid in a
cylindrical container. A massless tightly sealed piston of area A floats on the surface of the liquid.
When a mass m is placed on the piston to compress the liquid, the fractional change in the volume of

the sphere is find x.

7) Two identical calorimeters A and B contain equal quantity of water at 20 °C. A 5gm piece of metal
X of specific heat 0.2 cal/gC° is dropped into A and a 5gm piece of metal Y into B. The equilibrium
temperature in A is 22°C and in B is 23°C. The initial temperature of both the metals is 40°C. The
specific heat of metal y in cal/gC° is 9K/85, where K =

8) A thin walled cylindrical tank of radius R is filled with water of density D at 0°C. The upper and
lower flat surfaces are covered with non conducting material and tank is placed in surrounding
whose temperature is –θ0 in °C. As a result water starts freezing on the curved wall of the cylinder.
The time in which half the volume of water in the tank freezes is found out to be

, where L is the latent heat of fusion and K is the thermal conductivity of ice.

Find the value of .

9) One end of a steel rod of length ℓ, radius R and modulus of rigidity η is fixed and the other end
is twisted through an angle θ. If elastic deformation energy stored in the rod
is . The value of k is :-

10) A time varying magnetic field B = αr2t is existing in a circular region of radius 'R' as shown in
the figure. If the induced electric field at r = R/2 is E1 and at r = 2R is E2. The ratio of (E2/E1) is :-

PART-2 : CHEMISTRY

SECTION-I

1) Hydrogen atoms in a particular excited state ‘n’, when all returned to ground state, 6 different
photons are emitted. Which of the following is/are correct.

(A) Out of 6 different photons only 2 photons have wavelength equal to that of visible light.
If highest energy photon emitted from the above sample is incident on the metal plate having
(B)
work function 8 eV, then KE of liberated photo-electron may be equal to or less than 4.75 eV.
(C) Total number of radial nodes in all the orbitals of nth shell is 6.
(D) Total number of angular nodes in all the orbitals in (n – 1)th shell is 13.

2) Consider the isothermal reversible expansion of an ideal gas and a real (Van der Waals) gas, from
the same initial volume V1 to the same final volume V2, at the same temperature T. The absolute
value of the work done

by the real gas is always higher than the work done by the ideal gas, if the interparticle
(A)
interaction in the former is negligible
 by the real gas is always higher than the work done by the ideal gas, if the volume of the
(B)
particles in the former is negligible
in both the cases will always be the same, since the initial and final volumes in both the cases
(C)
are the same.
by the ideal gas is always higher than the work done by the real gas, since there is no hindrance
(D)
due to interparticle interaction and the volume of the particles

3) The decomposition of hydrogen peroxide which is catalyzed by iodide ion in an alkaline medium
undergoes a two-step mechanism given below:
Reaction : 2H2O2(ℓ) → 2H2O(ℓ) + O2(g)
Mechanism :
(slow reaction)

(fast reaction)
Among the following, the correct plot(s) for the above reaction is(are)
( [P]0 denotes the initial concentration of species P)

(A)

(B)

(C)
(D)

4) LiAlH4 can reduce :

(A)

(B)

(C) CH2 = CH – CH2 – N(CH3)2

(D)

5) Compounds which can evolve two mole of CO2 on oxidative ozonolysis followed by heating.

(A)

(B)

(C)

(D)

6) Which of the following is INCORRECT regarding Ellingham diagram.

(A) Entropy will increase if a metal oxide will change the phase from gas to liquid or liquid to solid.
 There is a point in a curve above which ΔG° is positive and metal oxide will NOT decompose on
(B)
its own above this point.
The sum of ΔrG° for the combined reaction involving oxidation of reducing agent and reduction
(C)
of the metal oxide will always be negative.
The reduction of metal oxide is easier if the metal is formed in liquid state at temperature of
(D)
reductions

7) Select the INCORRECT statement :-

(A) Aqueous solution of [Co(H2O)6]Cl2 produce pink solution with excess conc. HCl.
Potassium ferricyanide is more stable and also have more metal carbon bond length as
(B)
compared to potassium ferrocyanide.
(C) [Co(H2O)6]Cl3 is diamagnetic but [Co(H2O)6]Cl2 is paramagnetic
If Hund's rule is violated to fill electrons in splitted orbitals, then [Mn(CN)6]3– will be
(D)
paramagnetic.

8)

Number of valence Homonuclear

electrons diatomic species

8 E2

10 A2 ,

11

12 D2
All the elements given belongs to second period of periodic table.
Identify the CORRECT statement using above information.
(A) Bond length of D2 will smaller than bond length of E2
(B) Bond length of and will be almost equal.
Bond order of D2 will be more than
(C)

(D) During the formation the electron is removed from ungerade molecular orbital.

SECTION-II

1) For complete combustion of methanol

the amount of heat produced as measured by bomb calorimeter is 726 kJ mol–1 at 27°C. The enthalpy
of combustion for the reaction is –x kJ mol–1, where x is ______. (Nearest integer)
(Given : R = 8.3 JK–1 mol–1)

2) A solution was prepared by dissolving 2g of a non-volatile non electrolyte solid B in 100g of

solvent A at 354K. The vapour pressure of the solution became exactly equal to the atmospheric
pressure of 1.0 atm. If the molar mass of A is 80 g mol–1, the molar mass (in g mol–1) of solid B is
_______ (Nearest integer).
[Given: Gas constant, R = 8.3 J K– mol–1, Vapour pressure of A at 353K = 1.0 atm Standard enthalpy
of vapourisation of A at 353K = 85 × 353 J mol–1, e0.0289 = 1.03]

3)

Y is including stereoisomers value of (X + Y) will be

4)
How many reactions are possible for the above compound?
(i) Nucleophilic substitution
(ii) Free radical substituion
(iii) Electrophilic addition
(iv) Nucleophilic addition
(v) Electrophilic substitution

5) In the given reaction sequence, find number of possible 5 and 6 membered Lactone(s) having
different boiling points :

6) Number of compounds which can show Iodoform test with NaOH, I2

(a) (b) (c) (d)

(e) (f) (g)

(h) EtOH (i) (j) (k)

7) Number of reactions which will from benzaldehyde as major product

(i) (ii)

(iii) (iv)

(v) (vi)

(vii) (viii)

8) Write sum of the molecule weight of P, Q, R, S, T

(1) NaNO3 [P] + N2 + O2

(2) P4 + NaOH → Gas [Q] + other products

(3) Zn + dil. H2SO4 → gas [R] + other products

(4) Cu + dil. HNO3 → gas [S] + other products

(5)
Given atomic weight of O = 16, N = 14, H = 1, Na = 23, P = 31, S = 32, Cr = 52, Mn = 55, Cl = 35.5

9)
 Property of the gas/gases
Reagent Anion
evolved

(A) dil. H2SO4 (P) , (i) Paramagnetic

Colourless and having two pπ-

(B) conc. H2SO4 (Q) , , (ii)
pπ bonds

(R) , (iii) Central atom have lone pair

Maximum 3 atoms in one

(S) , (iv)
plane

(T) , (v) Contains non polar bond

(vi) Burning smell of sulphur

Using the above table identify the number of correct combination.
(1) B-Q-vi (2) A-S-ii
(3) B-P-i (4) B-T-iv
(5) B-R-i (6) B-P-iii
(7) B-R-v (8) A-Q-ii
(9) A-T-ii (10) A-S-iii

10) Consider the following arrangement of boxes having different numbers and containing a complex
molecule/ion.

Sum of the box numbers having diamagnetic complex/ion = X

Total number of boxes have a complex/ion which shows optical isomerism = Y

Find the value of .

PART-3 : MATHEMATICS

SECTION-I

1) Which of the following is/are incorrect?

(A)
let f : R → R, such that f (x) = 2x+ + sinx cosx then (where [.] denotes
the greatest integer function) f is one-one onto

(B)
let f : R → R, such that f (x) = then f is one-one onto
(C) let f : R → [1, ∞) such that f(x) = then f(x) is one-one into
(D) let f : R – {0} → R such that f(x) = |x|ℓn|x| is one-one onto.

2) f : R → R,
Which of the statements are incorrect?

(A) y = f(x) f(2x) is continuous at x = 0

(B) y = f(x) + f(2x) is continuous at x = 0
(C) y = f(x) is continuous at x = 2
(D) y = f(x) is continuous at x = 3

3) Let f(x) = x3 – 3x + 1, then

(A) f(f(x)) = 0 has 7 solutions

(B) f(f(x)) = 0 has 4 solutions
(C) f(f(x)) = –1 has 7 solutions
(D) f(f(x)) = –1 has 4 solutions

4) Circumcircle of triangle ABC is x2 + y2 = 1. Let AB = AC = and point , then which

of the following is/are True ?

(A) equation of side BC is

(B) area of triangle ABC is 1 sq. unit

(C)
incentre of triangle ABC is
(D) equation of side BC is

5) Consider a function f(x) = ax3 + x2 – (2a + 1)x + 2. Which of the following is/are True ?

The number of value(s) of 'a' for which Rolle's theorem can be applied in the interval [0,1] is/are
(A)
1 (where a ∈ [0, ∞]
The number of value(s) of 'a' for which Rolle's theorem can be applied in the interval [0,1] is/are
(B)
2
If k1 and k2 (k1 ≠ k2) are respectively the local maximum and local minimum values of f(x) then
(C)
the sum of those roots of the equation f(x) = k1 and f(x) = k2 at which f'(x) ≠ 0, is (where a ∈
(0, ∞))
If k1 and k2 (k1 ≠ k2) are respectively the local maximum and local minimum values of f(x) then
(D)
the sum of those roots of the equation f(x) = k1 and f(x) = k2 at which f'(x) ≠ 0, is
6) For the equation , which of the
following is/are correct ? (Here, the inverse trigonometric functions sin x, tan x and cos–1x assume
–1 –1

values in , and [0, π] respectively)

(A) The number of solutions is 2

(B) It has at least one irrational solution
(C) The sum of solutions is positive
(D) The sum of solutions is negative

7) The value(s) of the real parameter 'a' for which the inequality x6 – 6x5 + 12x4 + ax3 + 12x2 – 6x + 1
≥ 0 is satisfied for all real x

(A) [–12, 37]

(B) [–12, 38]
(C) [–17, 14]
(D) [–17, 35]

8) Let f (x) = x3 + 2x2 – x + 1, then which of the following statement is correct

(A) (where α is real root of f(x) = 0 and [.] denotes

the Greatest integer function)
(B) f (x) = 0 has one real root
(C) y = f (x) is an increasing function

(D)
(where α is real root of f(x) = 0 and [.] denotes the Greatest integer function)

SECTION-II

1) Let A, B, C and D be four distinct point on a line in that order. The circles with diameters AC is
x2 + y2 + ax + c = 0 and BD is x2 + y2 – by = 0 intersect at X and Y the line XY meets BC at Z. Let P
be a point on XY other than Z, the line CP intersects the circle with diameter AC at C and M, and line
BP intersects the circle with diameter BD at B and N and the equation of line AM and DN are bx +
cy + a = 0 and cx + ay + b = 0 respectively, the value of (a + b + c) is.......
(Given that a > b and c are three distinct real numbers)

2) If then is equal to (where y2(x) is )

3) Let f be a continuous and differentiable function in (a, b), if and

and .
Then the minimum value of [ a2 – b2] + 80 is([.] → represent greatest integer function) :

4) Let = ,
then the value of λ equals

5) Let f′ (x) > 0 x R+ where f : R+ R and f (x) + = f–1 and f–1 > 0. If sin–1 (f(2)) is

then 'λ' equals

6) If minimum value of , ∀ x1 ∈ [–6, –2] and x2 ∈ (0,

∞) is then 'a' equals

7) Let f(x) is a quadratic polynomial such that f(0) = 1 and is a rational function, then
the value of f'(0) is

8) Let A = {–3, –2, –1, 0, 1, 2, 3}, then number of functions from A to A which are odd and f(i) ≠ i (∀ i
= –3, –2, –1, 1, 2, 3) equals

9) Let f : (0,∞) → R be defined by f(x) = . Let x = c be the point where f(x) is

discontinuous. If f(c) = eλ then 10|λ| equals

10) Consider ƒ : (0,1) → R is given by . Point A(h,ƒ(h)), 0 < h

< 1 lies on curve and tangent at point A intersect the y-axis at point B(0,k). If distance between
points A and B is d, then 'd' equals
 ANSWER KEYS

PART-1 : PHYSICS

SECTION-I

Q. 1 2 3 4 5 6 7 8
A. A,C A,C B,C C,D A,B,C,D B,C A,C A,B,C

SECTION-II

Q. 9 10 11 12 13 14 15 16 17 18
A. 4 4 2 1 9 1 3 3 8 4

PART-2 : CHEMISTRY

SECTION-I

Q. 19 20 21 22 23 24 25 26
A. A,B,D A B,D A,B,D A,C,D A,B,C A,D A,B

SECTION-II

Q. 27 28 29 30 31 32 33 34 35 36
A. 727 53 8 5 5 7 6 260 6 16

PART-3 : MATHEMATICS

SECTION-I

Q. 37 38 39 40 41 42 43 44
A. A,B,C,D A,C A,D A,B,C A,C B,C B A,B

SECTION-II

Q. 45 46 47 48 49 50 51 52 53 54
A. 0 3 81 2 10 50 3 216 5 1
 SOLUTIONS

PART-1 : PHYSICS

1) VA = 0
qA = –q
qB = 0.

2) Since magnetic field is produced due to the current flowing in the current coil. Therefore
torque on the voltage coil will be proportional to i2
∴ Device will measure power in the load.

3)
Now, we can compare electric fields easily.

4) If stress in steel = stress in brass,

then,

System is in equilibrium. So, taking moments about P

TS . x = TB(2 – x)

...(ii)
From equation (i) and (ii), we get
x = 1.33 m

If strain in steel = strain in brass,

then,
 ...(iii)
From equation (ii) and (iii), we get
x = 1m

5) Heat current from A to B

Similarly,
Heat current from A to C, IAC = 3.85 J/s
Heat current from B to C, IAC = 3.08 J/s
Rate of heat supplied by the reservoir at A = IAB + IAC = 8.47 J/s
Rate of heat supplied by the reservoir at B = IBBC + IAB = – 1.54 J/s
Rate of heat supplied by the reservoir at C = – IBC + IAC = – 6.93 J/s
(Negative sign of heat supplied indicates heat being absorbed)

7) Flux through the loop due to current in the long wire is zero because field is tangential to
the surface containing the loop. Force on BC and DA is zero. Force on CD is attractive and on
AB is repulsive. Magnetic force on the loop passes through z-axis. The ampere loop mentioned
encircles the current in CD and so the curl is non-zero.

8)

+ 40 × 10–3 × 1000
= 5 + 10 + 40 = 55 J
Work done in stretching wire from Q to R
55 – 5 = 50 J

Strain = = 25 × 10–3 = 0.25

Q is a proportionality limit → at this point it will contain its elastic property and come back to
original shape.

9)
Applying KVL in loop ABCD
E1 = (I + 0.05) R1 + IR2
⇒ I = – 20 mA
∴ Current through R1 = 30 mA towards right
Current through R2 = 20 mA towards left
Applying KVL in loop BGFE
E2 = (I + 0.05) 100 + (0.05) 20 = 4 volts
Ans. E2 = 4V, I1 = 30 mA, towards right.

10)

11) The field at A due to the dipole and point charge Q much be equal and opposite. Field at A
due to the dipole makes an angle with the line CA, where

The field due to the dipole must be along OA. A negative charge (Q) at O can create an equal
field along AO at point A.

12) .........
 .......

13)

Since, electron is accelerated through a potential difference V, therefore, its initial velocity
v0 is given by

Or ....(i)

Since, initial velocity is parallel to plates or normal to the direction of electric field, therefore,
component of velocity parallel to plates remains constant as v0.

Hence, time taken by the electron to cross electric field is

Now consider motion of electron, normal to plates.
At some instant t, its acceleration

=
Let velocity component normal to plates be vy

or ....(ii)
If Q is angular deviation of electron from its initial direction of motion.
The pitch of its helical path.

or
p = 9mm

14)
15) 5(0.2) (40 – 22) = mS(22 – 20)
5S(40 – 23) = mS(23 – 20)

16) Let's say at any instant radius of water is r and in time dt it becomes r – dr.
2πr dr hρL = dQ = Heat flow out in time dt

H in
17)

The elastic deformation energy stored in the rod is

18)
When r ≤ R

When r > R

At , and at r = 2R,
 Hence,

PART-2 : CHEMISTRY

19) Value of n is 4
For n = 4 to n = 1 total 6 spectral lines are observed
(4 → 3, 3 → 2, 2 → 1, 4 → 2, 4 → 1, 3 → 1)
the highest energy photon along them is 4 → 1

For = 3 + 3(2) + 5(1) + 0 = 14

For = 0 + 3(1) + 5(2) = 13

27)

ΔU = –726 KJ/mol

Δng = 1–3/2 =
ΔH = ΔU + ΔngRT

= – 726 – ×
= – 727.245

32) a, b, c, d, g, h, i

33) i, iii, iv, v, vi, vii

PART-3 : MATHEMATICS

37) (A)

So,

So: f (x) = 2x + [x] + sin 2x ⇒ f′(x) = 2 + cos2x (Whenever differentiable)

⇒ increasing ⇒ oneone
but this fn is discontinuous at integers. (Jump discontinuity)
∴ into function
B) f (0) = f(–2) = 0
∴ Manyone
C) f(2) = f(0) = 2
∴ f(2) = f(0) = 2
∴ Manyone
D) f(–x) = f(x) R – {0}
⇒ even function ⇒ many one

38) is – 1 or 0 depending upon

but f(x) + f(2x) always tends towards zero

is 0,

39) f'(x) = 3(x – 1) (x + 1)

⇒ f is strictly non-decreasing in (–∞,–1] and [1,∞) and non- increasing [–1,1]
f(–1) = 3, f(1) = –1, f(2) = 3
f(x) = 0 has three distinct roots
x1,x2,x3 i.e –2 < x1 < –1 <x2 < 1< x3 < 2
So f(x) = x1 has one solution
f(x) = x2 = has 3 solution
f(x) = x3 has 3 solution

40) Co-ordiantes of

Co-ordiantes of
slope of BC, m = tan120º =
incentre of triangle ABC

Area of triangle
41) (A) f(0) = f(1)
2 = a + 1 – 2a – 1 + 2
a = 0 ⇒ f(x) = x2 – x + 2
(C) f(x) = k1 has roots α,α,α'
f(x) = k2 has roots β,β,β'

2α + α' =

f′(x) = 0 ⇒ 3x2a + 2x – (2a + 1) = 0

2(α + β) + (α′ + β′) =

α′ + β′ =

42)

no. of solution is 3.

43) ∀ x > 0 and

∀x<0

Let x +
Now
(t) (t2 – 3) – 6 (t2–2) + 12 (t) + a ≥ 0 for ≥ 2
& (t) (t2 – 3) – 6 (t2–2) + 12 t + a ≤ 0 for ≤ 2
Let f(t) = t3 – 6t2 + 9t + 12 + a
f'(t) = 3t2 – 12t + 9
= 3(t – 3) (t – 1)
Ronghly y = f(t)
for t ≥ 2, f(t) ≥ 0 ⇒ f(3) ≥ 0
⇒
f(-2) ≤ 0
–8 –6(4) – 18 + a + 12 ≤ 0
a – 38 ≤ 0

a ∈ [–12, 38]

44) f(x) = x3 + 2x2 –x + 1

f' (x) = 3x2 + 4x –1

f' (x) = 0 ⇒

f (–2) = 3 & f (–3) < 0

Hence
[α] = –3
Now sin–1 sin (–3) + cos–1cos(3)
⇒ sin–1sin (3) + cos–1 cos 3
⇒ – sin–1 sin (π–3) + cos–1 cos 3
⇒ –π+3+3 =6–π

45)

Since in x AM and ND are concurrent =0

2 2 2
a (bc – a ) – b (b – ac) + c(ab–c ) = 0
abc – a3 – b3 + abc + abc – c3
a3 + b3 + c3 = 3abc = (a + b + c) (a + bω + cω2) (a + bω + cω) = 0
46) Taking logarithm of both sides, we get log

= (i)

Since

so
Multiplying (i) by y and then differentiating, we get

So

47) Given

Let g(x) = sin–1(f(x))2 – x2 is non decreasing function

; [a2 – b2] = 1

48)

Here, R.H.S
=λ=2

49) Given : f (x) + = f–1

⇒ f (f(x) + ) = ............................... (1)

Let f(x) + = y
So eqn. (1) becomes -

f(y) =
⇒ f(x). f(y) = 1...............................(2)
From (1) -:

f(f(y)+ ) =

⇒ f(f(y)+ ) = f(x)..............from 2

⇒ f(y)+ = x..................... {∵ f is one-one

⇒ ..............from 2
2 2
⇒ x (f(x)) – x f(x) – 1 = 0

⇒ f(x) =

⇒ f(x) = ∴ f1(x) > 0

⇒ f(x) =
∴ sin–1 (f(2)) = sin–1 (–sinπ/10)

50) min = shortest distance between the

curve
(x + 4)2 + (y – 7)2 = 4 and y2 = 4x

51)

Let g(x) = .........(1)

=
= Aln x – + c ln (1 + x) –
Since g(x) is a rational function hence A = C = 0

g(x) = ........(2)
Comparing (1) and (2), we get f(x) = (B + D)x3 + (3B + D + E)x2 + 3Bx + B
⇒ f(x) is quadratic
⇒ B + D = 0, f(0) = 1 gives B = 1 ⇒ D = –1
∴ f(x) = (2 + E)x2 + 3x + 1
f'(0) = 3

52) For odd function

f(x) + f(x) = 0
∴ f(0) = 0 & f(i) ≠ i {given}
⇒ f(1) + f(–1) = 0
It is possible if
f(1) = 2 of f(–1) = – 2
f(1) = –2 of f(1) = 2
f(1) = 3 of f(–1) = – 3
f(1) = – 3 of f(–1) = 3
f(1) = –1 of f(–1) = 1
f(1) = 0 of f(–1) = 0
6 cases for each pair.
⇒ 6 × 6 × 6 = 216
{as their are 3 such pairs}

53)

Let

So, clearly and f(c) = e–1/2

54)

AB = d = 1