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# JEE Main 2010 Paper -1 with Solutions

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## 84 Questions MCQ Test | JEE Main 2010 Paper -1 with Solutions

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JEE Main 2010 Paper -1 with Solutions - Question 1

### SECTION – I (Single Correct Choice Type) This Section contains 8 multiple choice questions. Each question has four choices A), B), C) and D) out of which ONLY ONE is correct. Q. Incandescent bulbs are designed by keeping in mind that the resistance of their filament increases with the increase in temperature. If at room temperature, 100 W, 60 W and 40 W bulbs have filament resistances R100, R60 and R40, respectively, the relation between these resistances is

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 1

JEE Main 2010 Paper -1 with Solutions - Question 2

### To Verify Ohm’s law, a student is provided with a test resistor RT, a high resistance R1, a small resistance R2, two identical galvanometers G1 and G2, and a variable voltage source V. The correct circuit to carry out the experiment is

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 2

G1 is acting as voltmeter and G2 is acting as ammeter.

JEE Main 2010 Paper -1 with Solutions - Question 3

### An AC voltage source of variable angular frequency ω and fixed amplitude V0 is connected in series with a capacitance C and an electric bulb of resistance R (inductance zero). When ω is increased

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 3

JEE Main 2010 Paper -1 with Solutions - Question 4

A thin flexible wire of length L is connected to two adjacent fixed points and carries a current I in the
clockwise direction, as shown in the figure. When the system is put in a uniform magnetic field of strength
B going into the plane of the paper, the wire takes the shape of a circle. The tension in the wire is

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 4

JEE Main 2010 Paper -1 with Solutions - Question 5

A block of mass m is on an inclined plane of angle θ. The coefficient of friction between the block and the plane is μ and tan θ > μ. The block is held stationary by applying a force P parallel to the plane. The direction of force pointing up the plane is taken to be positive. As P is varied from P1 = mg(sinθ − μ cosθ) to

P2 = mg(sinθ + μ cosθ), the frictional force f versus P graph will look like

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 5

Initially the frictional force is upwards as P increases frictional force decreases.

JEE Main 2010 Paper -1 with Solutions - Question 6

A thin uniform annular disc (see figure) of mass M has outer radius 4R and inner radius 3R. The work required to take a unit mass from point P on its axis to infinity is

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 6

JEE Main 2010 Paper -1 with Solutions - Question 7

Consider a thin square sheet of side L and thickness t, made of a material of resistivity ρ. The resistance
between two opposite faces, shown by the shaded areas in the figure is

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 7

JEE Main 2010 Paper -1 with Solutions - Question 8

A real gas behaves like an ideal gas if its

*Multiple options can be correct
JEE Main 2010 Paper -1 with Solutions - Question 9

SECTION – II (Multiple Correct Choice Type)

This section contains 5 multiple choice questions. Each question has four choices A), B), C) and D) out of which ONE OR MORE may be correct. Partially correct answers will be not be awarded any marks.

Q.  A point mass of 1 kg collides elastically with a stationary point mass of 5 kg. After their collision, the 1 kg
mass reverses its direction and moves with a speed of 2 ms−1. Which of the following statement(s) is (are)
correct for the system of these two masses?

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 9

*Multiple options can be correct
JEE Main 2010 Paper -1 with Solutions - Question 10

One mole of an ideal gas in initial state A undergoes a cyclic process ABCA, as shown in the figure. Its pressure at A is P0. Choose the correct option(s) from the following

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 10

Process AB is isothermal process

*Multiple options can be correct
JEE Main 2010 Paper -1 with Solutions - Question 11

A student uses a simple pendulum of exactly 1m length to determine g, the acceleration due to gravity. He
uses a stop watch with the least count of 1 sec for this and records 40 seconds for 20 oscillations. For this
observation, which of the following statement(s) is (are) true?

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 11

*Multiple options can be correct
JEE Main 2010 Paper -1 with Solutions - Question 12

A few electric field lines for a system of two charges Q1 and Q2 fixed at two different points on the x-axis are shown in the figure. These lines suggest that

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 12

No. of electric field lines of forces emerging from Q1 are larger than terminating at Q2

*Multiple options can be correct
JEE Main 2010 Paper -1 with Solutions - Question 13

A ray OP of monochromatic light is incident on the face AB of prism ABCD near vertex B at an incident angle of 60° (see figure). If the refractive index of the material of the prism is , which of the following is (are) correct?

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 13

JEE Main 2010 Paper -1 with Solutions - Question 14

SECTION –III (Paragraph Type)
This section contains 2 paragraphs. Based upon the first paragraph 2 multiple choice questions and based upon the second paragraph 3 multiple choice questions have to be answered. Each of these questions has four choices A), B), C) and D) out of WHICH ONLY ONE CORRECT.

Paragraph for Questions 14 to 15

Q. Electrical resistance of certain materials, known as superconductors, changes abruptly from a nonzero value to zero as their temperature is lowered below a critical temperature Tc(0). An interesting property of superconductors is that their critical temperature becomes smaller than Tc(0) if they are placed in a magnetic field, i.e., the critical temperature Tc(B) is a function of the magnetic field strength B. The dependence of Tc(B) on B is shown in the figure.

Q. In the graphs below, the resistance R of a superconductor is shown as a function of its temperature T for
two different magnetic fields B1 (solid line) and B2 (dashed line). If B2 is larger than B1 which of the following graphs shows the correct variation of R with T in these fields?

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 14

Larger the magnetic field smaller the critical temperature.

JEE Main 2010 Paper -1 with Solutions - Question 15

Paragraph for Questions 14 to 15

Q. Electrical resistance of certain materials, known as superconductors, changes abruptly from a nonzero value to zero as their temperature is lowered below a critical temperature Tc(0). An interesting property of superconductors is that their critical temperature becomes smaller than Tc(0) if they are placed in a magnetic field, i.e., the critical temperature Tc(B) is a function of the magnetic field strength B. The dependence of Tc(B) on B is shown in the figure.

Q. A superconductor has Tc (0) = 100 K. When a magnetic field of 7.5 Tesla is applied, its Tc decreases to
75 K. For this material one can definitely say that when

JEE Main 2010 Paper -1 with Solutions - Question 16

Paragraph for Questions 16 to 18

When a particle of mass m moves on the x-axis in a potential of the form V(x) = kx2 it performs simple harmonic motion. The corresponding time period is proportional to ,  as can be seen easily using dimensional analysis. However, the motion of a particle can be periodic even when its potential energy increases on both sides of x = 0 in a way different from kx2 and its total energy is such that the particle does not escape to infinity. Consider a particle of mass m moving on the x-axis. Its potential energy is V(x) = αx4 (α > 0) for |x| near the origin and becomes a constant equal to V0 for |x| ≥ X0 (see figure).

Q. If the total energy of the particle is E, it will perform periodic motion only if

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 16

Energy must be less than V0

JEE Main 2010 Paper -1 with Solutions - Question 17

Paragraph for Questions 16 to 18

When a particle of mass m moves on the x-axis in a potential of the form V(x) = kx2 it performs simple harmonic motion. The corresponding time period is proportional to ,  as can be seen easily using dimensional analysis. However, the motion of a particle can be periodic even when its potential energy increases on both sides of x = 0 in a way different from kx2 and its total energy is such that the particle does not escape to infinity. Consider a particle of mass m moving on the x-axis. Its potential energy is V(x) = αx4 (α > 0) for |x| near the origin and becomes a constant equal to V0 for |x| ≥ X0 (see figure).

Q. For periodic motion of small amplitude A, the time period T of this particle is proportional to

JEE Main 2010 Paper -1 with Solutions - Question 18

Paragraph for Questions 16 to 18

When a particle of mass m moves on the x-axis in a potential of the form V(x) = kx2 it performs simple harmonic motion. The corresponding time period is proportional to ,  as can be seen easily using dimensional analysis. However, the motion of a particle can be periodic even when its potential energy increases on both sides of x = 0 in a way different from kx2 and its total energy is such that the particle does not escape to infinity. Consider a particle of mass m moving on the x-axis. Its potential energy is V(x) = αx4 (α > 0) for |x| near the origin and becomes a constant equal to V0 for |x| ≥ X0 (see figure).

Q. The acceleration of this particle for |x| > X0 is

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 18

As potential energy is constant for |x| > X0, the force on the particle is zero hence acceleration is zero.

*Answer can only contain numeric values
JEE Main 2010 Paper -1 with Solutions - Question 19

SECTION –IV (Integer Type)

This section contains TEN questions. The answer to each question is a single digit integer ranging from 0 to 9.

Enter only the numerical value in the space provided below.

Q. Gravitational acceleration on the surface of a planet is  where g is the gravitational acceleration on
the surface of the earth. The average mass density of the planet is  times that of the earth. If the escape
speed on the surface of the earth is taken to be 11 kms-1, the escape speed on the surface of the planet in
kms-1 will be

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 19

*Answer can only contain numeric values
JEE Main 2010 Paper -1 with Solutions - Question 20

A piece of ice (heat capacity = 2100 J kg-1 °C-1 and latent heat = 3.36 ×105J kg-1) of mass m grams is at
-5°C at atmospheric pressure. It is given 420 J of heat so that the ice starts melting. Finally when the
ice-water mixture is in equilibrium, it is found that 1 gm of ice has melted. Assuming there is no other heat
exchange in the process, the value of m is

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 20

420 = (m × 2100 × 5 + 1 × 3.36 × 105 )×10−3

where m is in gm.

*Answer can only contain numeric values
JEE Main 2010 Paper -1 with Solutions - Question 21

A stationary source is emitting sound at a fixed frequency f0, which is reflected by two cars approaching the
source. The difference between the frequencies of sound reflected from the cars is 1.2% of f0. What is the
difference in the speeds of the cars (in km per hour) to the nearest integer ? The cars are moving at constant
speeds much smaller than the speed of sound which is 330 ms-1.

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 21

*Answer can only contain numeric values
JEE Main 2010 Paper -1 with Solutions - Question 22

The focal length of a thin biconvex lens is 20 cm. When an object is moved from a distance of 25 cm in
front of it to 50 cm, the magnification of its image changes from m25 to m50. The ratio

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 22

*Answer can only contain numeric values
JEE Main 2010 Paper -1 with Solutions - Question 23

An α- particle and a proton are accelerated from rest by a potential difference of 100 V. After this, their de
Broglie wavelengths are λα and λp respectively. The ratio , to the nearest integer, is

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 23

*Answer can only contain numeric values
JEE Main 2010 Paper -1 with Solutions - Question 24

When two identical batteries of internal resistance 1Ω each are connected in series across a resistor R, the
rate of heat produced in R is J1. When the same batteries are connected in parallel across R, the rate is J2.
If J1 = 2.25 J2 then the value of R in Ω is

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 24

*Answer can only contain numeric values
JEE Main 2010 Paper -1 with Solutions - Question 25

Two spherical bodies A (radius 6 cm ) and B(radius 18 cm ) are at temperature T1 and T2, respectively.
The maximum intensity in the emission spectrum of A is at 500 nm and in that of B is at 1500 nm.
Considering them to be black bodies, what will be the ratio of the rate of total energy radiated by A to that
of B?

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 25

*Answer can only contain numeric values
JEE Main 2010 Paper -1 with Solutions - Question 26

When two progressive waves y1 = 4 sin(2x - 6t) and  are superimposed, the
amplitude of the resultant wave is

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 26

Two waves have phase difference π/2.

*Answer can only contain numeric values
JEE Main 2010 Paper -1 with Solutions - Question 27

A 0.1 kg mass is suspended from a wire of negligible mass. The length of the wire is 1m and its crosssectional area is 4.9 × 10-7 m2. If the mass is pulled a little in the vertically downward direction and
released, it performs simple harmonic motion of angular frequency 140 rad s−1. If the Young’s modulus of
the material of the wire is n × 109 Nm-2, the value of n is

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 27

*Answer can only contain numeric values
JEE Main 2010 Paper -1 with Solutions - Question 28

A binary star consists of two stars A (mass 2.2Ms) and B (mass 11Ms), where Ms is the mass of the sun.
They are separated by distance d and are rotating about their centre of mass, which is stationary. The ratio
of the total angular momentum of the binary star to the angular momentum of star B about the centre of
mass is

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 28

JEE Main 2010 Paper -1 with Solutions - Question 29

SECTION – I (Single Correct Choice Type)

This Section contains 8 multiple choice questions. Each question has four choices A), B), C) and D) out of which ONLY ONE is correct.

Q. The synthesis of 3 – octyne is achieved by adding a bromoalkane into a mixture of sodium amide and an
alkyne. The bromoalkane and alkyne respectively are

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 29

JEE Main 2010 Paper -1 with Solutions - Question 30

The correct statement about the following disaccharide is

JEE Main 2010 Paper -1 with Solutions - Question 31

In the reaction  the products are

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 31

JEE Main 2010 Paper -1 with Solutions - Question 32

Plots showing the variation of the rate constant (k) with temperature (T) are given below. The plot that
follows Arrhenius equation is

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 32

JEE Main 2010 Paper -1 with Solutions - Question 33

The species which by definition has ZERO standard molar enthalpy of formation at 298 K is

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 33

Cl2 is gas at 298 K while Br2 is a liquid.

JEE Main 2010 Paper -1 with Solutions - Question 34

The bond energy (in kcal mol–1 ) of a C – C single bond is approximately

JEE Main 2010 Paper -1 with Solutions - Question 35

The correct structure of ethylenediaminetetraacetic acid (EDTA) is

JEE Main 2010 Paper -1 with Solutions - Question 36

The ionization isomer of is

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 36

Cl is replaced by 2NO in ionization sphere.

*Multiple options can be correct
JEE Main 2010 Paper -1 with Solutions - Question 37

SECTION – II (Multiple Correct Choice Type)

This section contains 5 multiple choice questions. Each question has four choices A), B), C) and D) out of which ONE OR MORE may be correct.

Q. In the Newman projection for 2,2-dimethylbutane

X and Y can respectively be

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 37

*Multiple options can be correct
JEE Main 2010 Paper -1 with Solutions - Question 38

Aqueous solutions of HNO3, KOH, CH3COOH, and CH3COONa of identical concentrations are provided.
The pair (s) of solutions which form a buffer upon mixing is(are)

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 38

In option (C), if HNO3 is present in limiting amount then this mixture will be a buffer. And the mixture given in option (D), contains a weak acid (CH3COOH) and its salt with strong base NaOH, i.e. CH3COONa.

*Multiple options can be correct
JEE Main 2010 Paper -1 with Solutions - Question 39

In the reaction  the intermediate(s) is(are)

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 39

Phenoxide ion is para* and ortho directing. (* preferably)

*Multiple options can be correct
JEE Main 2010 Paper -1 with Solutions - Question 40

The reagent(s) used for softening the temporary hardness of water is(are)

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 40

*Multiple options can be correct
JEE Main 2010 Paper -1 with Solutions - Question 41

Among the following, the intensive property is (properties are)

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 41

Resistance and heat capacity are mass dependent properties, hence extensive.

JEE Main 2010 Paper -1 with Solutions - Question 42

SECTION-III (Paragraph Type)
This section contains 2 paragraphs. Based upon the first paragraph 2 multiple choice questions and based upon the second paragraph 3 multiple choice questions have to be answered. Each of these questions has four choices A), B), C) and D) out of WHICH ONLY ONE CORRECT.

Paragraph for Question Nos. 42 to 43

The concentration of potassium ions inside a biological cell is at least twenty times higher than the outside.

The resulting potential difference across the cell is important in several processes such as transmission of nerve impulses and maintaining the ion balance. A simple model for such a concentration cell involving a metal M is:

For the above electrolytic cell the magnitude of the cell potential |Ecell|= 70 mV.

Q. For the above cell

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 42

JEE Main 2010 Paper -1 with Solutions - Question 43

Paragraph for Question Nos. 42 to 43

The concentration of potassium ions inside a biological cell is at least twenty times higher than the outside.

The resulting potential difference across the cell is important in several processes such as transmission of nerve impulses and maintaining the ion balance. A simple model for such a concentration cell involving a metal M is:

For the above electrolytic cell the magnitude of the cell potential |Ecell|= 70 mV.

Q. If the 0.05 molar solution of M+ is replaced by 0.0025 molar M+ solution, then the magnitude of the cell
potential would be

JEE Main 2010 Paper -1 with Solutions - Question 44

Paragraph for Question Nos. 44 to 46

Copper is the most noble of the first row transition metals and occurs in small deposits in several countries. Ores of copper include chalcanthite (CuSO4 ⋅5H2O) , atacamite (Cu2Cl(OH)3) , cuprite (Cu2O) , copper glance (Cu2S) and malachite (Cu2(OH)2CO3). However, 80% of the world copper production comes from the ore of chalcopyrite (CuFeS2) . The extraction of copper from chalcopyrite involves partial roasting, removal of iron and self-reduction.

Q. Partial roasting of chalcopyrite produces

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 44

JEE Main 2010 Paper -1 with Solutions - Question 45

Paragraph for Question Nos. 44 to 46

Copper is the most noble of the first row transition metals and occurs in small deposits in several countries. Ores of copper include chalcanthite (CuSO4 ⋅5H2O) , atacamite (Cu2Cl(OH)3) , cuprite (Cu2O) , copper glance (Cu2S) and malachite (Cu2(OH)2CO3). However, 80% of the world copper production comes from the ore of chalcopyrite (CuFeS2) . The extraction of copper from chalcopyrite involves partial roasting, removal of iron and self-reduction.

Q. Iron is removed from chalcopyrite as

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 45

JEE Main 2010 Paper -1 with Solutions - Question 46

Paragraph for Question Nos. 44 to 46

Copper is the most noble of the first row transition metals and occurs in small deposits in several countries. Ores of copper include chalcanthite (CuSO4 ⋅5H2O) , atacamite (Cu2Cl(OH)3) , cuprite (Cu2O) , copper glance (Cu2S) and malachite (Cu2(OH)2CO3). However, 80% of the world copper production comes from the ore of chalcopyrite (CuFeS2) . The extraction of copper from chalcopyrite involves partial roasting, removal of iron and self-reduction.

Q. In self-reduction, the reducing species is

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 46

*Answer can only contain numeric values
JEE Main 2010 Paper -1 with Solutions - Question 47

SECTION-IV (Integer Type)

This section contains TEN questions. The answer to each question is a single digit integer ranging from 0 to 9.

Enter only the numerical value in the space provided below.

Q. A student performs a titration with different burettes and finds titre values of 25.2 mL, 25.25 mL and 25.0
mL. The number of significant figures in the average titre value is

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 47
Average titre ( product of titration) is = (25.2 + 25.25 + 25.00) ÷3 = 25.15
significant figures = 2,5,1 total no. of significant figures are 3.
*Answer can only contain numeric values
JEE Main 2010 Paper -1 with Solutions - Question 48

The concentration of R in the reaction R → P was measured as a function of time and the following data is
obtained:

The order of the reaction is

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 48

*Answer can only contain numeric values
JEE Main 2010 Paper -1 with Solutions - Question 49

The number of neutrons emitted when undergoes controlled nuclear fission to  is

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 49

*Answer can only contain numeric values
JEE Main 2010 Paper -1 with Solutions - Question 50

The total number of basic groups in the following form of lysine is

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 50

*Answer can only contain numeric values
JEE Main 2010 Paper -1 with Solutions - Question 51

The total number of cyclic isomers possible for a hydrocarbon with the molecular formula C4H6 is

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 51

In C4H6, possible cyclic isomers are

*Answer can only contain numeric values
JEE Main 2010 Paper -1 with Solutions - Question 52

In the scheme given below, the total number of intra molecular aldol condensation products formed from ‘Y’ is

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 52

*Answer can only contain numeric values
JEE Main 2010 Paper -1 with Solutions - Question 53

Amongst the following, the total number of compound soluble in aqueous NaOH is

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 53

These four are soluble in aqueous NaOH.

*Answer can only contain numeric values
JEE Main 2010 Paper -1 with Solutions - Question 54

Amongst the following, the total number of compounds whose aqueous solution turns red litmus paper blue
is
KCN    K2SO4     (NH4)2C2O4       NaCl      Zn(NO3)2      FeCl3       K2CO3         NH4NO3       LiCN

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 54

KCN, K2CO3, LiCN are basic in nature and their aqueous solution turns red litmus paper blue.

*Answer can only contain numeric values
JEE Main 2010 Paper -1 with Solutions - Question 55

Based on VSEPR theory, the number of 90 degree F−Br−F angles in BrF5 is

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 55

*Answer can only contain numeric values
JEE Main 2010 Paper -1 with Solutions - Question 56

The value of n in the molecular formula BenAl2Si6O18 is

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 56

Be3Al2Si6O18 (Beryl)
(according to charge balance in a molecule)
2n + 6 + 24 − 36 = 0
n = 3

JEE Main 2010 Paper -1 with Solutions - Question 57

SECTION – I (Single Correct Choice Type)

This Section contains 8 multiple choice questions. Each question has four choices A), B), C) and D) out of which ONLY ONE is correct.

Q. Let ω be a complex cube root of unity with ω ≠ 1. A fair die is thrown three times. If r1, r2 and r3 are the
numbers obtained on the die, then the probability that ωr1 + ωr2 + ωr3 = 0 is

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 57

JEE Main 2010 Paper -1 with Solutions - Question 58

Let P, Q, R and S be the points on the plane with position vectors respectively. The quadrilateral PQRS must be a

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 58

Hence, PQRS is a parallelogram but not rhombus or rectangle.

JEE Main 2010 Paper -1 with Solutions - Question 59

The number of 3 × 3 matrices A whose entries are either 0 or 1 and for which the system  has
exactly two distinct solutions, is

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 59

Three planes cannot intersect at two distinct points.

JEE Main 2010 Paper -1 with Solutions - Question 60

The value of

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 60

JEE Main 2010 Paper -1 with Solutions - Question 61

Let p and q be real numbers such that p ≠ 0, p3 ≠ q and p3 ≠ − q. If α and β are nonzero complex numbers
satisfying α + β = − p and α3 + β3 = q, then a quadratic equation having  as its roots is

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 61

JEE Main 2010 Paper -1 with Solutions - Question 62

Let f, g and h be real-valued functions defined on the interval [0, 1] by f(x) = ,

g(x) = If a, b and c denote, respectively, the absolute maximum of f, g and h on [0, 1], then

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 62

JEE Main 2010 Paper -1 with Solutions - Question 63

If the angles A, B and C of a triangle are in an arithmetic progression and if a, b and c denote the lengths of
the sides opposite to A, B and C respectively, then the value of the expression  is

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 63

JEE Main 2010 Paper -1 with Solutions - Question 64

Equation of the plane containing the straight line  and perpendicular to the plane containing the
straight lines  is

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 64

*Multiple options can be correct
JEE Main 2010 Paper -1 with Solutions - Question 65

SECTION − II (Multiple Correct Choice Type)

This section contains 5 multiple choice questions. Each question has four choices A), B), C) and D) out of which ONE OR MORE may be correct.

Q. Let z1 and z2 be two distinct complex numbers and let z = (1 − t) z1 + tz2 for some real number t with 0 < t < 1. If Arg (w) denotes the principal argument of a non-zero complex number w, then

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 65

*Multiple options can be correct
JEE Main 2010 Paper -1 with Solutions - Question 66

The value(s) of

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 66

*Multiple options can be correct
JEE Main 2010 Paper -1 with Solutions - Question 67

Let ABC be a triangle such that ∠ACB = and let a, b and c denote the lengths of the sides opposite to A,
B and C respectively. The value(s) of x for which a = x2 + x + 1, b = x2 − 1 and c = 2x + 1 is (are)

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 67

*Multiple options can be correct
JEE Main 2010 Paper -1 with Solutions - Question 68

Let A and B be two distinct points on the parabola y2 = 4x. If the axis of the parabola touches a circle of radius r having AB as its diameter, then the slope of the line joining A and B can be

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 68

*Multiple options can be correct
JEE Main 2010 Paper -1 with Solutions - Question 69

Let f be a real-valued function defined on the interval Then which of the following statement(s) is (are) true?

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 69

*Multiple options can be correct
JEE Main 2010 Paper -1 with Solutions - Question 70

SECTION − III (Paragraph Type)

This section contains 2 paragraphs. Based upon the first paragraph 2 multiple choice questions and based upon the second paragraph 3 multiple choice questions have to be answered. Each of these questions has four choices A), B), C) and D) out of WHICH ONLY ONE CORRECT.

Paragraph for Questions 70 to 71

The circle x2 + y2 − 8x = 0 and hyperbola  intersect at the points A and B.

Q. Equation of a common tangent with positive slope to the circle as well as to the hyperbola is

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 70

*Multiple options can be correct
JEE Main 2010 Paper -1 with Solutions - Question 71

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The circle x2 + y2 − 8x = 0 and hyperbola  intersect at the points A and B.

Q. Equation of the circle with AB as its diameter is

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 71

JEE Main 2010 Paper -1 with Solutions - Question 72

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Let p be an odd prime number and Tp be the following set of 2 × 2 matrices :

Q. The number of A in Tp such that A is either symmetric or skew-symmetric or both, and det(A) divisible by
p is

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 72

JEE Main 2010 Paper -1 with Solutions - Question 73

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Let p be an odd prime number and Tp be the following set of 2 × 2 matrices :

Q. The number of A in Tp such that the trace of A is not divisible by p but det (A) is divisible by p is
[Note: The trace of a matrix is the sum of its diagonal entries.]

JEE Main 2010 Paper -1 with Solutions - Question 74

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Let p be an odd prime number and Tp be the following set of 2 × 2 matrices :

Q. The number of A in Tp such that det (A) is not divisible by p is

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JEE Main 2010 Paper -1 with Solutions - Question 75

SECTION − IV (Integer Type)

This section contains TEN questions. The answer to each question is a single digit integer ranging from 0 to 9.

Enter only the numerical value in the space provided below.

Q. Let Sk, k = 1, 2, ….. , 100, denote the sum of the infinite geometric series whose first term is  and the common ratio is . Then the value of  is

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 75

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JEE Main 2010 Paper -1 with Solutions - Question 76

The number of all possible values of θ, where 0 < θ < π, for which the system of equations

have a solution (x0, y0, z0) with y0z0 ≠ 0, is

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 76

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JEE Main 2010 Paper -1 with Solutions - Question 77

Let f be a real-valued differentiable function on R (the set of all real numbers) such that f(1) = 1. If the y intercept of the tangent at any point P(x, y) on the curve y = f(x) is equal to the cube of the abscissa of P,
then the value of f(−3) is equal to

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JEE Main 2010 Paper -1 with Solutions - Question 78

The number of values of θ in the interval , such that  for n = 0, ±1, ±2 and tanθ = cot 5θ as well as sin 2θ = cos 4θ is

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 78

JEE Main 2010 Paper -1 with Solutions - Question 79

The maximum value of the expression

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 79

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JEE Main 2010 Paper -1 with Solutions - Question 80

If are vectors in space given by  then the value of is

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JEE Main 2010 Paper -1 with Solutions - Question 81

The line 2x + y = 1 is tangent to the hyperbola . If this line passes through the point of intersection of the nearest directrix and the x-axis, then the eccentricity of the hyperbola is

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 81

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JEE Main 2010 Paper -1 with Solutions - Question 82

If the distance between the plane Ax − 2y + z = d and the plane containing the lines
and , then |d| is

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JEE Main 2010 Paper -1 with Solutions - Question 83

For any real number x, let |x| denote the largest integer less than or equal to x. Let f be a real valued
function defined on the interval [−10, 10] by

Then the value of

Detailed Solution for JEE Main 2010 Paper -1 with Solutions - Question 83

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JEE Main 2010 Paper -1 with Solutions - Question 84

Let ω be the complex number . Then the number of distinct complex numbers z satisfying  is equal to

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