Description

This mock test of JEE (Main) - 2010 Paper -1 for JEE helps you for every JEE entrance exam.
This contains 84 Multiple Choice Questions for JEE JEE (Main) - 2010 Paper -1 (mcq) to study with solutions a complete question bank.
The solved questions answers in this JEE (Main) - 2010 Paper -1 quiz give you a good mix of easy questions and tough questions. JEE
students definitely take this JEE (Main) - 2010 Paper -1 exercise for a better result in the exam. You can find other JEE (Main) - 2010 Paper -1 extra questions,
long questions & short questions for JEE on EduRev as well by searching above.

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

R_{100}, R_{60} and R_{40}, respectively, the relation between these resistances is

Solution:

QUESTION: 2

To Verify Ohm’s law, a student is provided with a test resistor R_{T}, a high resistance R_{1}, a small resistance

R_{2}, two identical galvanometers G_{1} and G_{2}, and a variable voltage source V. The correct circuit to carry out

the experiment is

Solution:

G_{1} is acting as voltmeter and G_{2} is acting as ammeter.

QUESTION: 3

An AC voltage source of variable angular frequency ω and fixed amplitude V_{0} is connected in series with a

capacitance C and an electric bulb of resistance R (inductance zero). When ω is increased

Solution:

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

Solution:

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 P_{1} = mg(sinθ − μ cosθ) to

P_{2} = mg(sinθ + μ cosθ), the frictional force f versus P graph will look like

Solution:

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

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

Solution:

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

Solution:

QUESTION: 8

A real gas behaves like an ideal gas if its

Solution:

*Multiple options can be correct

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?

Solution:

*Multiple options can be correct

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 P_{0}. Choose the correct option(s) from the following

Solution:

Process AB is isothermal process

*Multiple options can be correct

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?

Solution:

*Multiple options can be correct

QUESTION: 12

A few electric field lines for a system of two charges Q_{1} and Q_{2} fixed at two different points on the x-axis are shown in the figure. These lines suggest that

Solution:

No. of electric field lines of forces emerging from Q_{1} are larger than terminating at Q_{2}

*Multiple options can be correct

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?

Solution:

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 T_{c}(0). An interesting property of superconductors is that their critical temperature becomes smaller than T_{c}(0) if they are placed in a magnetic field, i.e., the critical temperature T_{c}(B) is a function of the magnetic field strength B. The dependence of T_{c}(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 B_{1 }(solid line) and B_{2} (dashed line). If B_{2} is larger than B_{1} which of the following graphs shows the correct variation of R with T in these fields?

Solution:

Larger the magnetic field smaller the critical temperature.

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 T_{c}(0). An interesting property of superconductors is that their critical temperature becomes smaller than T_{c}(0) if they are placed in a magnetic field, i.e., the critical temperature T_{c}(B) is a function of the magnetic field strength B. The dependence of T_{c}(B) on B is shown in the figure.

**Q.** A superconductor has T_{c} (0) = 100 K. When a magnetic field of 7.5 Tesla is applied, its T_{c} decreases to

75 K. For this material one can definitely say that when

Solution:

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) = kx^{2} 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 kx^{2} 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) = αx^{4} (α > 0) for |x| near the origin and becomes a constant equal to V_{0} for |x| ≥ X_{0} (see figure).

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

Solution:

Energy must be less than V_{0}

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) = kx^{2} 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 kx^{2} 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) = αx^{4} (α > 0) for |x| near the origin and becomes a constant equal to V_{0} for |x| ≥ X_{0} (see figure).

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

Solution:

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) = kx^{2} 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 kx^{2} 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) = αx^{4} (α > 0) for |x| near the origin and becomes a constant equal to V_{0} for |x| ≥ X_{0} (see figure).

Q. The acceleration of this particle for |x| > X_{0} is

Solution:

As potential energy is constant for |x| > X_{0}, the force on the particle is zero hence acceleration is zero.

*Answer can only contain numeric values

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

Solution:

*Answer can only contain numeric values

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

Solution:

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

where m is in gm.

*Answer can only contain numeric values

QUESTION: 21

A stationary source is emitting sound at a fixed frequency f_{0}, which is reflected by two cars approaching the

source. The difference between the frequencies of sound reflected from the cars is 1.2% of f_{0}. 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}.

Solution:

*Answer can only contain numeric values

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

Solution:

*Answer can only contain numeric values

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 λ_{α} and λ_{p} respectively. The ratio , to the nearest integer, is

Solution:

*Answer can only contain numeric values

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 J_{1}. When the same batteries are connected in parallel across R, the rate is J_{2}.

If J_{1} = 2.25 J_{2} then the value of R in Ω is

Solution:

*Answer can only contain numeric values

QUESTION: 25

Two spherical bodies A (radius 6 cm ) and B(radius 18 cm ) are at temperature T_{1} and T_{2}, 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?

Solution:

*Answer can only contain numeric values

QUESTION: 26

When two progressive waves y_{1} = 4 sin(2x - 6t) and are superimposed, the

amplitude of the resultant wave is

Solution:

Two waves have phase difference π/2.

*Answer can only contain numeric values

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} m^{2}. 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 × 10^{9} Nm^{-2}, the value of n is

Solution:

*Answer can only contain numeric values

QUESTION: 28

A binary star consists of two stars A (mass 2.2M_{s}) and B (mass 11M_{s}), where M_{s} 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

Solution:

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

Solution:

QUESTION: 30

The correct statement about the following disaccharide is

Solution:

QUESTION: 31

In the reaction the products are

Solution:

QUESTION: 32

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

follows Arrhenius equation is

Solution:

QUESTION: 33

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

Solution:

Cl_{2} is gas at 298 K while Br_{2} is a liquid.

QUESTION: 34

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

Solution:

QUESTION: 35

The correct structure of ethylenediaminetetraacetic acid (EDTA) is

Solution:

QUESTION: 36

The ionization isomer of is

Solution:

Cl^{–} is replaced by 2NO^{−} in ionization sphere.

*Multiple options can be correct

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

Solution:

*Multiple options can be correct

QUESTION: 38

Aqueous solutions of HNO_{3}, KOH, CH_{3}COOH, and CH_{3}COONa of identical concentrations are provided.

The pair (s) of solutions which form a buffer upon mixing is(are)

Solution:

In option (C), if HNO_{3} is present in limiting amount then this mixture will be a buffer. And the mixture given in option (D), contains a weak acid (CH_{3}COOH) and its salt with strong base NaOH, i.e. CH_{3}COONa.

*Multiple options can be correct

QUESTION: 39

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

Solution:

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

*Multiple options can be correct

QUESTION: 40

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

Solution:

*Multiple options can be correct

QUESTION: 41

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

Solution:

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

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 |E_{cell}|= 70 mV.

**Q.** For the above cell

Solution:

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 |E_{cell}|= 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

Solution:

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 (CuSO_{4} ⋅5H_{2}O) , atacamite (Cu_{2}Cl(OH)_{3}) , cuprite (Cu_{2}O) , copper glance (Cu_{2}S) and malachite (Cu_{2}(OH)_{2}CO_{3}). However, 80% of the world copper production comes from the ore of chalcopyrite (CuFeS_{2}) . The extraction of copper from chalcopyrite involves partial roasting, removal of iron and self-reduction.

**Q.** Partial roasting of chalcopyrite produces

Solution:

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 (CuSO_{4} ⋅5H_{2}O) , atacamite (Cu_{2}Cl(OH)_{3}) , cuprite (Cu_{2}O) , copper glance (Cu_{2}S) and malachite (Cu_{2}(OH)_{2}CO_{3}). However, 80% of the world copper production comes from the ore of chalcopyrite (CuFeS_{2}) . The extraction of copper from chalcopyrite involves partial roasting, removal of iron and self-reduction.

**Q. **Iron is removed from chalcopyrite as

Solution:

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 (CuSO_{4} ⋅5H_{2}O) , atacamite (Cu_{2}Cl(OH)_{3}) , cuprite (Cu_{2}O) , copper glance (Cu_{2}S) and malachite (Cu_{2}(OH)_{2}CO_{3}). However, 80% of the world copper production comes from the ore of chalcopyrite (CuFeS_{2}) . The extraction of copper from chalcopyrite involves partial roasting, removal of iron and self-reduction.

**Q.** In self-reduction, the reducing species is

Solution:

*Answer can only contain numeric values

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

Solution:

Average titre ( product of titration) is = (25.2 + 25.25 + 25.00) Ã·3 = 25.15

significant figures = 2,5,1 total no. of significant figures are 3.

*Answer can only contain numeric values

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

Solution:

*Answer can only contain numeric values

QUESTION: 49

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

Solution:

*Answer can only contain numeric values

QUESTION: 50

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

Solution:

*Answer can only contain numeric values

QUESTION: 51

The total number of cyclic isomers possible for a hydrocarbon with the molecular formula C_{4}H_{6} is

Solution:

In C_{4}H_{6}, possible cyclic isomers are

*Answer can only contain numeric values

QUESTION: 52

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

Solution:

*Answer can only contain numeric values

QUESTION: 53

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

Solution:

These four are soluble in aqueous NaOH.

*Answer can only contain numeric values

QUESTION: 54

Amongst the following, the total number of compounds whose aqueous solution turns red litmus paper blue

is

KCN K_{2}SO_{4} (NH_{4})_{2}C_{2}O_{4} NaCl Zn(NO_{3})_{2} FeCl_{3} K_{2}CO_{3} NH_{4}NO_{3} LiCN

Solution:

KCN, K_{2}CO_{3}, LiCN are basic in nature and their aqueous solution turns red litmus paper blue.

*Answer can only contain numeric values

QUESTION: 55

Based on VSEPR theory, the number of 90 degree F−Br−F angles in BrF_{5} is

Solution:

*Answer can only contain numeric values

QUESTION: 56

The value of n in the molecular formula Be_{n}Al_{2}Si_{6}O_{18} is

Solution:

Be_{3}Al_{2}Si_{6}O_{18} (Beryl)

(according to charge balance in a molecule)

2n + 6 + 24 − 36 = 0

n = 3

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 r_{1}, r_{2} and r_{3} are the

numbers obtained on the die, then the probability that ω^{r1} + ω^{r2} + ω^{r3 }= 0 is

Solution:

QUESTION: 58

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

Solution:

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

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

Solution:

Three planes cannot intersect at two distinct points.

QUESTION: 60

The value of

Solution:

QUESTION: 61

Let p and q be real numbers such that p ≠ 0, p^{3} ≠ q and p^{3} ≠ − q. If α and β are nonzero complex numbers

satisfying α + β = − p and α^{3} + β^{3} = q, then a quadratic equation having as its roots is

Solution:

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

Solution:

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

Solution:

QUESTION: 64

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

straight lines is

Solution:

*Multiple options can be correct

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 z_{1} and z_{2} be two distinct complex numbers and let z = (1 − t) z_{1} + tz_{2} for some real number t with 0 < t < 1. If Arg (w) denotes the principal argument of a non-zero complex number w, then

Solution:

*Multiple options can be correct

QUESTION: 66

The value(s) of

Solution:

*Multiple options can be correct

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 = x^{2} + x + 1, b = x^{2} − 1 and c = 2x + 1 is (are)

Solution:

*Multiple options can be correct

QUESTION: 68

Let A and B be two distinct points on the parabola y^{2} = 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

Solution:

*Multiple options can be correct

QUESTION: 69

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

Solution:

*Multiple options can be correct

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 x^{2} + y^{2} − 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

Solution:

*Multiple options can be correct

QUESTION: 71

**Paragraph for Questions 70 to 71**

The circle x^{2} + y^{2} − 8x = 0 and hyperbola intersect at the points A and B.

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

Solution:

QUESTION: 72

**Paragraph for Questions 72 to 74**

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

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

p is

Solution:

QUESTION: 73

**Paragraph for Questions 72 to 74**

Let p be an odd prime number and T_{p} 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.]

Solution:

QUESTION: 74

**Paragraph for Questions 72 to 74**

Let p be an odd prime number and T_{p} 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

Solution:

*Answer can only contain numeric values

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 S_{k}, 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

Solution:

*Answer can only contain numeric values

QUESTION: 76

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

have a solution (x_{0}, y_{0}, z_{0}) with y_{0}z_{0} ≠ 0, is

Solution:

*Answer can only contain numeric values

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

Solution:

*Answer can only contain numeric values

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

Solution:

QUESTION: 79

The maximum value of the expression

Solution:

*Answer can only contain numeric values

QUESTION: 80

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

Solution:

*Answer can only contain numeric values

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

Solution:

*Answer can only contain numeric values

QUESTION: 82

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

and , then |d| is

Solution:

*Answer can only contain numeric values

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

Solution:

*Answer can only contain numeric values

QUESTION: 84

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

Solution:

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