Coulomb’s Law

Coulomb’s Law

Coulomb’s law is a quantitative statement about the force between two point charges. When the linear size of charged bodies are much smaller than the distance separating them, the size may be ignored and the charged bodies are treated as point charges. Coulomb measured the force between two point charges and found that it varied inversely as the square of the distance between the charges and was directly proportional to the product of the magnitude of the two charges and acted along the line joining the two charges. Thus, if two point charges q1, q2 are separated by a distance r in vacuum, the magnitude of the force (F) between them is given by 

F =|q1×q2|/r²

How did Coulomb arrive at this law from his experiments? Coulomb used a torsion balance* for measuring the force between two charged metallic spheres. When the separation between two spheres is much larger than the radius of each sphere, the charged spheres may be regarded as point charges.

Mathematics MCQ Class 12

CBSE Class 12 2023 : Maths Important MCQs for CBSE Board Exam 2023

In this post, CBSE Board 12th Maths Exam 2023 Important MCQs Question Answers have been given.

The purpose of these practice questions and answers is to help the students prepare for their exams and perform well in them.

It is advised to read this resource before the actual paper as it can give students an idea about the type of questions they can expect in the exam.

1 The number of equivalence relations that can be defined in the set A= {1,2,3} which containing the elements (1,2) is

(a) 0

(b) 1

(c) 2

(d) 3

2 The number of one-to-one functions that can be defined from the set {1,2,3,4,5} to {a, b}

a) 5

b) 0

c) 2

d) 3

3 What is the simplified form of cos-1(4x3 − 3x)

(a) 3sin−1 x

(b) 3cos−1 x

(c) π − 3 sin−1 x

(d) π − 3 cos−1 x

4 The number of all possible matrices of order2x3 with entry 1 or 2

1)16

2) 64

3) 6

4) 24

5 If the order of matrix P is 2x3 and the order of matrix Q is 3x4 , find the order of PQ.

1) 2x4

2) 2x2

3) 4x2

4) 3x3

6 Let A is a non – singular matrix of order 3 x 3 then | A ( adj A )| is equal to

a)| A |

b)| A |2

c)| A |3

d)3| A |

7 The function f(x) =[x] is continuous at

a)4

b)-2

c)1

d)1.5

8 The bottom of a rectangular swimming tank is 25 m by 40 m water is pumped into the tank at the rate of 500 cubic meters per minute. Find the rate at which the level of water in the tank is rising?

A) 1⁄4 m/min

B) 2/3 m/min

C) 1/3 m/min

D) 1⁄2 m/min

9 Area of region bound by circle x2+y2=1

a)2 sq units

b) sq units

c)3 sq units

d)4 sq units

10 Integrating factor of the differential equation dy/dx + y tan x – sec x = 0 is:

(A) cosx 

(B) secx 

(C) ecosx

(D) esecx

11 If points A (60 î+ 3 ĵ), (40 î– 8 ĵ) and C ( aî- 52ĵ) are collinear, then ‘a’ is equal to

a) 40

b) -40

c) 20

d) -20

12 Write direction cosines of a line parallel to z-axis.

(a) 1,0,0

(b) 0,0,1

(c) 1,1,0

(d) -1,-1,-1

13 Find the foot of the perpendicular drawn from the point (2,-3,4) on the y-axis.

(a) (2,0,4)

(b) (0.3.0)

(c) (0,-3,0)

(d) (-2,0,-4)

14 If α,β, Υ are the angles that a line makes with the positive direction of x,y,z axis respectively then the direction cosines of the line are

(a) cosα,sinβ, cosΥ

(b) cosα,cosβ, cosΥ

(c) sinα,sinβ, sinΥ

(d) 1, 1, 1

15 The feasible region of the inequality x+y≤1 and x–y≤1 lies in......... quadrants.

(a)Only I and II 

(b)Only I and III

(c)Only II and III

(d)All four

16 Two dice are thrown. If it is known that the sum of numbers on the dice was less than 6, the probability of getting a sum 3 is

a)1/18

b)5/18

c)1/5

d)2

17 The solution set of the inequality 3x + 5y < 4 is

a)an open half-plane not containing the origin.

b)an open half-plane containing the origin.

c)the whole XY-plane not containing the line 3x + 5y = 4.

d)a closed half plane containing the origin.

18 If A is a square matrix of order 3 and |A| = 5, then |adjA| =

(a) 5

(b) 25

(c) 125

(d) ⅕

19 The area of a triangle with vertices (2, −6), (5,4) and (k, 4) is 35 square units then , k is

A.12

B. −2

C. −12, −2

D. 12, −2

20 The vector having initial and terminal points as (2,5,0) and (-3,7,4) respectively is

A. 5î+ 2ĵ− 4k̂

B. −î+ 12ĵ+ 4k̂

C. −5î+ 2ĵ+ 4k̂

D.−5î+ 12ĵ+ 4k̂

21. Let A = { 1 , 2, 3 } and consider the relation R = {(1 , 1), (2 , 2), ( 3 , 3), (1 , 2), (2 , 3), (1,3)} Then , R is

(a) Reflexive but not symmetric

(b) Reflexive but not transitive

(c) Symmetric and transitive

(d)Neither symmetric nor transitive

22. If A is a skew – symmetric matrix , then A² is

(a) Symmetric

(b) Skew – symmetric

(c) A² = A

(d) A² ≠ A

23. The number of arbitrary constants in the general solution of a differential equation of fourth order are :

(a) 0

(b) 2

(c) 3

(d) 4

24. If y = log x/ (1+x) , then dy/ dx is equal to

(a) x/ (1+x)²

(b) 1/ (1+x) ²

(c) 2x/ (1+x)²

(d) 1/ x(1+x)

25. Corner points of the region for an LPP are (0, 2), (3, 0), (6, 0),(6, 8), and (0, 5) . Let F = 4x + 6y be the objective function. The minimum value of F occurs at

(a) (0,2) only

(b)(3,0) only

(c) The midpoint of the line segment joining the points (0,2) and ( 3, 0) only

(d) Any point on the line segment joining the points (0,2) and (3, 0) only.

26. f(x) = xˣ has a stationary point at

(a) x = e

(b) x = 1/ e

(c) x = 1

(d) x = √e

27. If A and B are events such that P ( A/ B) = P (B/ A) , then ………

(a) P(A) = P(B)

(b) P(A) ≠ P(B)

(c) P(A) + P(B) = 1

(d) None of these

28. If R = {(x , y ): x + 2y = 8 } is a relation N , then the range of R is :

(a){3, 2 , 1}

(b) { 3 , 2}

(c) {2 , 8 , 1}

(d) {3}

29. Let f ∶ R → R be defined as f(x) = 3x. Choose the correct answer.

(a) f is one – one onto

(b) f is many-one onto

(c) f is one – one but not onto

(d) f is neither one-one nor onto

30. The principal value of [tan-1 √3 − cot-1, (−√3)] is :

(a)π

(b) − π/2

(c) 0

(d) 2√3

31. If cos (sin-1 2/ 5 + cos-1 x) = 0 , then x is equal to

(a) 1/ 5

(b)2/ 5

(c) 0

(d) 1

32. The value of ∫ (cos 2x)/ (sinx +cosx)² dx is

(a) log | cos x + sin x | + C

(b) log | cos x − sin x | + C

(c) log | cos x + sin x |² + C

(d) log | cos x + sin x |-2 + C

33. The derivative of 3ˣ  w.r.t x is :

(a) log 3

(b) x. 3ˣ-1

(c) 0

(d) 3ˣ log 3

34. A is a square matrix of order 2, then adj(adj A) is :

(a) O

(b) I

(c) A-1

(d) A

35. The number of arbitrary constants in the general solution of a differential equation of fourth order is :

(a) 1

(b) 2

(c) 3

(d) 4

 

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Electromagnetic Waves

Electromagnetic Spectrum

At the time Maxwell predicted the existence of electromagnetic waves, the only familiar electromagnetic waves were the visible light waves. The existence of ultraviolet and infrared waves was barely established. By the end of the nineteenth century, X-rays and gamma rays had also been discovered. We now know that, electromagnetic waves include visible light waves, X-rays, gamma rays, radio waves, microwaves, ultraviolet and infrared waves. The classification of em waves according to frequency is the electromagnetic spectrum. There is no sharp division between one kind of wave and the next. The classification is based roughly on how the waves are produced and/or detected.

(Figure The electromagnetic spectrum, with common names for various part of it. The various regions do not have sharply defined boundaries.)

We briefly describe these different types of electromagnetic waves, in order of decreasing wavelengths.

Radio waves

Radio waves are produced by the accelerated motion of charges in conducting wires. They are used in radio and television communication systems. They are generally in the frequency range from 500 kHz to about 1000 MHz. The AM (amplitude modulated) band is from 530 kHz to 1710 kHz. Higher frequencies upto 54 MHz are used for short wave bands. TV waves range from 54 MHz to 890 MHz. The FM (frequency modulated) radio band extends from 88 MHz to 108 MHz. Cellular phones use radio waves to transmit voice communication in the ultrahigh frequency (UHF) band. How these waves are transmitted and received is described in Chapter 15.

Microwaves

Microwaves (short-wavelength radio waves), with frequencies in the gigahertz (GHz) range, are produced by special vacuum tubes (called klystrons, magnetrons and Gunn diodes). Due to their short wavelengths, they are suitable for the radar systems used in aircraft navigation. Radar also provides the basis for the speed guns used to time fast balls, tennis- serves, and automobiles. Microwave ovens are an interesting domestic application of these waves. In such ovens, the frequency of the microwaves is selected to match the resonant frequency of water molecules so that energy from the waves is transferred efficiently to the kinetic energy of the molecules. This raises the temperature of any food containing water.

Microwave oven

The spectrum of electromagnetic radiation contains a part known as microwaves. These waves have frequency and energy smaller than visible light and wavelength larger than it. What is the principle of a microwave oven and how does it work?Our objective is to cook food or warm it up. All food items such as fruit, vegetables, meat, cereals, etc., contain water as a constituent. Now, what does it mean when we say that a certain object has become warmer? When the temperature of a body rises, the energy of the random motion of atoms and molecules increases and the molecules travel or vibrate or rotate with higher energies. The frequency of rotation of water molecules is about

2.45 gigahertz (GHz). If water receives microwaves of this frequency, its molecules absorb this radiation, which is equivalent to heating up water. These molecules share this energy with neighbouring food molecules, heating up the food.

One should use porcelain vessels and not metal containers in a microwave oven because of the danger of getting a shock from accumulated electric charges. Metals may also melt from heating. The porcelain container remains unaffected and cool, because its large molecules vibrate and rotate with much smaller frequencies, and thus cannot absorb microwaves. Hence, they do not get heated up.

Thus, the basic principle of a microwave oven is to generate microwave radiation of appropriate frequency in the working space of the oven where we keep food. This way energy is not wasted in heating up the vessel. In the conventional heating method, the vessel on the burner gets heated first, and then the food inside gets heated because of transfer of energy from the vessel. In the microwave oven, on the other hand, energy is directly delivered to water molecules which is shared by the entire food.

Infrared waves

Infrared waves are produced by hot bodies and molecules. This band lies adjacent to the low-frequency or long-wave length end of the visible spectrum. Infrared waves are sometimes referred to as heat waves. This is because water molecules present in most materials readily absorb infrared waves (many other molecules, for example, CO2, NH3, also absorb infrared waves). After absorption, their thermal motion increases, that is, they heat up and heat their surroundings. Infrared lamps are used in physical therapy. Infrared radiation also plays an important role in maintaining the earth’s warmth or average temperature through the greenhouse effect. Incoming visible light (which passes relatively easily through the atmosphere) is absorbed by the earth’s surface and re-radiated as infrared (longer wavelength) radiations. This radiation is trapped by greenhouse gases such as carbon dioxide and water vapour. Infrared detectors are used in Earth satellites, both for military purposes and to observe growth of crops. Electronic devices (for example semiconductor light emitting diodes) also emit infrared and are widely used in the remote switches of household electronic systems such as TV sets, video recorders and hi-fi system.

Visible rays

It is the most familiar form of electromagnetic waves. It is the part of the spectrum that is detected by the human eye. It runs from about 4 × 1014 Hz to about 7 × 1014 Hz or a wavelength range of about 700 – 400 nm. Visible light emitted or reflected from objects around us provides us information about the world. Our eyes are sensitive to this range of wavelengths. Different animals are sensitive to different range of wavelengths. For example, snakes can detect infrared waves, and the ‘visible’ range of many insects extends well into the utraviolet.

Ultraviolet rays

It covers wavelengths ranging from about 4 × 10‐⁷ m (400 nm) down to 6 × 10‐¹⁰m (0.6 nm). Ultraviolet (uv) radiation is produced by special lamps and very hot bodies. The sun is an important source of ultraviolet light. But fortunately, most of it is absorbed in the ozone layer in the atmosphere at an altitude of about 40 – 50 km. uv light in large quantities has harmful effects on humans. Exposure to UV radiation induces the production of more melanin, causing tanning of the skin. UV radiation is absorbed by ordinary glass. Hence, one cannot get tanned or sunburn through glass windows.

Welders wear special glass goggles or face masks with glass windows to protect their eyes from large amount of UV produced by welding arcs. Due to its shorter wavelengths, UV radiations can be focussed into very narrow beams for high precision applications such as LASIK (Laser-assisted in situ keratomileusis) eye surgery. UV lamps are used to kill germs in water purifiers. 

Ozone layer in the atmosphere plays a protective role, and hence its depletion by chlorofluorocarbons (CFCs) gas (such as freon) is a matter of international concern. 

X-rays

Beyond the uv region of the electromagnetic spectrum lies the x-ray region. We are familiar with x-rays because of its medical applications. It covers wavelengths from about 10‐⁸ m (10 nm) down to 10‐¹⁰ m 

(10‐⁴ nm). One common way to generate X-rays is to bombard a metal target by high energy electrons. X-rays are used as a diagnostic tool in medicine and as a treatment for certain forms of cancer. Because x-rays damage or destroy living tissues and organisms, care must be taken to avoid unnecessary or over exposure. 

Gamma rays

They lie in the upper frequency range of the electromagnetic spectrum and have wavelengths of from about 10‐¹⁰m to less than 10‐¹⁰m. This high frequency radiation is produced in nuclear reactions and 

also emitted by radioactive nuclei. They are used in medicine to destroy cancer cells. 

Table summarises different types of electromagnetic waves, their production and detections. As mentioned earlier, the demarcation between different regions is not sharp and there are overlaps.