CBSE NOTES CLASS 11 PHYSICS

Mathematics In Physics

Basic Concepts and Important Formulae

Mathematical Signs and Symbols

= equals

≅ equals approximately

~ is the order of magnitude of

≠ is not equal to

≡ is identical to, is defined as

> is greater than (>> is much greater than)

< is less than (<< is much less than)

≥ is greater than or equal to (or, is no less than)

≤ is less than or equal to (or, is no more than)

± plus or minus

∝ is proportional to

Σ the sum of

$\stackrel{̅}{\mathrm{x}}$ or < x > or xav - the average value of x

Some Useful Concepts

Componendo

If $\frac{\mathrm{a}}{\mathrm{b}}=\frac{\mathrm{c}}{\mathrm{d}}$, then $\frac{\mathrm{a}+\mathrm{b}}{\mathrm{b}}=\frac{\mathrm{c}+\mathrm{d}}{\mathrm{d}}$

Dividendo

If $\frac{\mathrm{a}}{\mathrm{b}}=\frac{\mathrm{c}}{\mathrm{d}}$ , then $\frac{\mathrm{a}-\mathrm{b}}{\mathrm{b}}=\frac{\mathrm{c}-\mathrm{d}}{\mathrm{d}}$

Componendo and Dividendo

If $\frac{\mathrm{a}}{\mathrm{b}}=\frac{\mathrm{c}}{\mathrm{d}}$, then $\frac{\mathrm{a}+\mathrm{b}}{\mathrm{a}-\mathrm{b}}=\frac{\mathrm{c}+\mathrm{d}}{\mathrm{c}-\mathrm{d}}$.

Invertendo

If $\frac{\mathrm{a}}{\mathrm{b}}=\frac{\mathrm{c}}{\mathrm{d}}$, then  $\frac{\mathrm{b}}{\mathrm{a}}=\frac{\mathrm{d}}{\mathrm{c}}$

Alternetendo

If $\frac{\mathrm{a}}{\mathrm{b}}=\frac{\mathrm{c}}{\mathrm{d}}$, then  $\frac{\mathrm{a}}{\mathrm{c}}=\frac{\mathrm{b}}{\mathrm{d}}$

Geometry

For a circle of radius r:

• circumference = 2πr;

• area = πr2

For a sphere of radius r:

• area = 4πr2;

• volume = $\frac{4}{3}$ πr3

For right circular cylinder of radius r and height h:

• area = 2π r2 +2π r h;

• volume = π r 2h ;

For a triangle of base b and altitude h.

• area = $\frac{1}{2}$ b h

Pythagoras theorem

For a right triangle with hypotenuse = c and perpendicular sides a and b,

c2 = a2 + b2

Algebraic Identities

 (x + y)2 = x2 + 2xy + y2 ⇒ x2 + y2 = (x + y)2 - 2xy (x – y)2 = x2 – 2xy + y2 ⇒ x2 + y2 = (x - y)2 + 2xy (x + y)2 = (x - y)2 + 4xy ⇒ (x - y)2 = (x + y)2 - 2xy x2 – y2 = (x + y) (x – y) (x + a) (x + b) = x2 + (a + b)x + ab (x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2zx (x + y)3 = x3 + y3 + 3xy(x + y) = x3 + 3x2y + 3xy2 + y3 (x – y)3 = x3 – y3 – 3xy(x – y) = x3 – 3x2y + 3xy2 – y3 x3 + y3 + z3 – 3xyz = (x + y + z) (x2 + y2 + z2 – xy – yz – zx) x3 + y3 = (x + y) (x2 + y2 - xy) x3 - y3 = (x - y) (x2 + y2 + xy) If x + y + z = 0 then x3 + y3 + z3 = 3xyz

Roots of equation ax2 + bx + c = 0 are given by

$\mathrm{x}=\frac{-\mathrm{b}±\sqrt{{\mathrm{b}}^{2}-4\mathrm{a}\mathrm{c}}}{2\mathrm{a}}$

Case I: Two distinct real roots exist, if,

$\mathrm{Descriminent D =}{\mathrm{b}}^{2}-4\mathrm{a}\mathrm{c}>0$

Case II: Two identical real roots exist, if,

$\mathrm{Descriminent D =}{\mathrm{b}}^{2}-4\mathrm{a}\mathrm{c}=0$

Case III: No real roots exist, if,

$\mathrm{Descriminent D =}{\mathrm{b}}^{2}-4\mathrm{a}\mathrm{c}<0$

Logarithmic Function

If a is a positive real number, other than unity, then,

is defined as logarithmic function. b is called the base.

For 0 < b < 1, the exponential function is a strictly decreasing function and is negative

For b > 1, the exponential function is a strictly increasing function and is positive.

Laws of logarithms

If

x = b y,

then

logb (x) = y

For example,

100 = 102 ⇒ log10 (100) = 2

Logarithm as inverse function of exponential function

The logarithmic function,

y = logb(x)

is the inverse function of the exponential function,

Also x = logb (bx)

Natural logarithm (ln)

ln (x) = loge (x)

Inverse logarithm (antilog) calculation

The inverse logarithm (or anti logarithm) is calculated by raising the base b to the logarithm y

x = log-1(y) = b y

Logarithm rules

${\mathrm{l}\mathrm{o}\mathrm{g}}_{\mathrm{b}}\left(\mathrm{x}\right)=\frac{{\mathrm{l}\mathrm{o}\mathrm{g}}_{\mathrm{c}}\left(\mathrm{x}\right)}{{\mathrm{l}\mathrm{o}\mathrm{g}}_{\mathrm{c}}\left(\mathrm{b}\right)}$

Binomial Theorem

For any real numbers n and x

For x << 1,we ignore higher powers of x. Thus, we can write,

(1 + x)n = (1 + nx) and

(1 – x)n = (1 – nx)

Exponential Expantion (or Taylor’s expansion)

Logarithmic expansion

Important Trigonometric Concepts

Measurement of Angle

Angle is a measure of rotation of a given ray about its initial point. The original ray is called the initial side and the final position of the ray after rotation is called the terminal side of the angle. The point of rotation is called the vertex.

If the direction of rotation is anticlockwise, the angle is said to be positive and if the direction of rotation is clockwise, then the angle is negative

Degree: If a rotation from the initial side to terminal side isth $\frac{1}{360}$th of a revolution, the angle is said to have a measure of one degree (1°).

One degree is divided into 60 minutes, and a minute is divided into 60 seconds. That is, one sixtieth of a degree is called a minute, written as 1′, and one sixtieth of a minute is called a second, written as 1″.

Radian: Angle subtended at the centre by an arc of length 1 unit in a unit circle (circle of radius 1 unit) is said to have a measure of 1 radian.

 One complete revolution of the initial side subtends an angle of 2π radian. In a circle of radius r, an arc of length l will subtend an angle θ radian at the centre, given by,

$\mathrm{Or \pi radians =}180°$

1° = 60′ and 1′= 60′′

Some Important Angle Measures in Degree and Radian

 $\mathrm{d}\mathrm{e}\mathrm{g}$ $0$ $30$ $45$ $60$ $90$ $180$ $270$ $360$ $\mathrm{r}\mathrm{a}\mathrm{d}$ $0$ $\frac{\mathrm{\pi }}{6}$ $\frac{\mathrm{\pi }}{4}$ $\frac{\mathrm{\pi }}{3}$ $\frac{\mathrm{\pi }}{2}$ $\mathrm{\pi }$ $\frac{3\mathrm{\pi }}{2}$

Trigonometric Functions

Trigonometric ratios for an angle are the ratio of sides of a right angled triangle.

 $\mathrm{sin}\mathrm{x}=\frac{\mathrm{p}}{\mathrm{h}}$ $\mathrm{cos}\mathrm{x}=\frac{\mathrm{b}}{\mathrm{h}}$ $\mathrm{cot}\mathrm{x}=\frac{\mathrm{b}}{\mathrm{p}}=\frac{\mathrm{cos}\mathrm{x}}{\mathrm{sin}\mathrm{x}}\mathrm{, x \ne n\pi }$

sin2 x + cos2 x = 1

1 + tan2 x = sec2 x

1 + cot2 x = cosec2 x

Values of trigonometric ratios of some common angles

 $0$ $\frac{\mathrm{\pi }}{6}$ $\frac{\mathrm{\pi }}{4}$ $\frac{\mathrm{\pi }}{3}$ $\frac{\mathrm{\pi }}{2}$ $\mathrm{\pi }$ $\frac{3\mathrm{\pi }}{2}$ $2\mathrm{\pi }$ $\mathrm{s}\mathrm{i}\mathrm{n}$ $0$ $\frac{1}{2}$ $\frac{1}{\sqrt{2}}$ $\frac{\sqrt{3}}{2}$ $1$ $0$ $-1$ $0$ $\mathrm{c}\mathrm{o}\mathrm{s}$ $1$ $\frac{\sqrt{3}}{2}$ $\frac{1}{\sqrt{2}}$ $\frac{1}{2}$ $0$ $-1$ $0$ $1$ $\mathrm{t}\mathrm{a}\mathrm{n}$ $0$ $\frac{1}{\sqrt{3}}$ $1$ $\sqrt{3}$ $\mathrm{n}\mathrm{d}$ $0$ $\mathrm{n}\mathrm{d}$ $0$ $\mathrm{c}\mathrm{o}\mathrm{t}$ $\mathrm{n}\mathrm{d}$ $3$ $1$ $\frac{1}{\sqrt{3}}$ $0$ $\mathrm{n}\mathrm{d}$ $0$ $\mathrm{n}\mathrm{d}$ $\mathrm{s}\mathrm{e}\mathrm{c}$ $1$ $\frac{2}{\sqrt{3}}$ $\sqrt{2}$ $2$ $\mathrm{n}\mathrm{d}$ $-1$ $\mathrm{n}\mathrm{d}$ $1$ $\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{c}$ $\mathrm{n}\mathrm{d}$ $2$ $\sqrt{2}$ $\frac{2}{\sqrt{3}}$ $1$ $\mathrm{n}\mathrm{d}$ $-1$ $\mathrm{n}\mathrm{d}$

Signs of Trigonometric Functions

Trends of Trigonometric Functions

 f Q I Q II Q III Q IV sin increases from 0 to 1 decreases from 1 to 0 decreases from 0 to -1 increases from -1 to 0 cos decreases from 1 to 0 decreases from 0 to -1 increases from -1 to 0 increases from 0 to 1 tan increases from 0 to ∞ increases from -∞ to 0 increases from 0 to ∞ increases from -∞ to 0 cot decreases from ∞ to 0 decreases from 0 to -∞ decreases from ∞ to 0 decreases from 0 to -∞ sec increases from 1 to ∞ increases from -∞ to -1 decreases from -1 to -∞ decreases from ∞ to 1 cosec decreases from ∞ to 1 increases from 1 to ∞ increases from -∞ to -1 decreases from -1 to-∞

Graphical Representation of Trigonometric Functions

Important Trigonometric Formulae

 $\mathrm{sec}\left(-\mathrm{x}\right)=\mathrm{sec}\mathrm{x}$ $\mathrm{sin}\left(\frac{\mathrm{\pi }}{2}–\mathrm{x}\right)=\mathrm{cos}\mathrm{x}$ $\mathrm{tan}\left(\frac{\mathrm{\pi }}{2}–\mathrm{x}\right)=\mathrm{cot}\mathrm{x}$ $\mathrm{cot}\left(\frac{\mathrm{\pi }}{2}–\mathrm{x}\right)=\mathrm{tan}\mathrm{x}$ $\mathrm{cot}\left(\frac{3\mathrm{\pi }}{2}+\mathrm{x}\right)=-\mathrm{tan}\mathrm{x}$ If none of the angles x, y and (x + y) is an odd multiple of $\frac{\mathrm{\pi }}{2}$, then If none of the angles x, y and (x + y) is a multiple of π, then cos 2x = cos2x – sin2 x = 2 cos2 x – 1 = 1 – 2 sin2 x sin 2x = 2 sin x cos x sin 3x = 3 sin x – 4 sin3 x cos 3x = 4 cos3 x – 3 cos x 2 cos x cos y = cos (x + y) + cos (x – y) –2 sin x sin y = cos (x + y) – cos (x – y) 2 sin x cos y = sin (x + y) + sin (x – y) 2 cos x sin y = sin (x + y) – sin (x – y)

Trigonometric expansions

Sine formulae

In any triangle, sides are proportional to the sines of the opposite angles. Let A, B and C be angles of a triangle and a, b and c be lengths of sides opposite to angles A, B and C respectively, then

$\frac{\mathrm{sin}\mathrm{A}}{\mathrm{a}}=\frac{\mathrm{sin}\mathrm{B}}{\mathrm{b}}=\frac{\mathrm{sin}\mathrm{C}}{\mathrm{c}}$

Cosine formulae

Let A, B and C be angles of a triangle and a, b and c be lengths of sides opposite to angles A, B and C respectively, then

a2 = b2 + c2 – 2bc cos A

b2 = c2 + a2 – 2ca cos B

c2 = a2 + b2 – 2ab cos C

That is,

Tips for use of sine and cosine formulae

$\left(\mathrm{i}\right)\frac{\mathrm{sin}\mathrm{A}}{\mathrm{a}}=\frac{\mathrm{sin}\mathrm{B}}{\mathrm{b}}=\frac{\mathrm{sin}\mathrm{C}}{\mathrm{c}}=\mathrm{k}$

Or

$\left(\mathrm{i}\mathrm{i}\right)\frac{\mathrm{a}}{\mathrm{sin}\mathrm{A}}=\frac{\mathrm{b}}{\mathrm{sin}\mathrm{B}}=\frac{\mathrm{c}}{\mathrm{sin}\mathrm{C}}=\mathrm{k}$

(iii) A + B + C = π

(v) Use formulae for π - x and $\frac{\mathrm{\pi }}{2}$ $–\mathrm{x}$

Differentials and Integrals

 Integrals (Anti derivatives) $\frac{\mathrm{d}\left(\mathrm{x}\right)}{\mathrm{d}\mathrm{x}}=1$ $\frac{\mathrm{d}\left(\mathrm{c}\right)}{\mathrm{d}\mathrm{x}}=0$ $\frac{\mathrm{d}\left(\mathrm{k}\mathrm{f}\left(\mathrm{x}\right)\right)}{\mathrm{d}\mathrm{x}}=\mathrm{k}\frac{\mathrm{d}\left(\mathrm{f}\left(\mathrm{x}\right)\right)}{\mathrm{d}\mathrm{x}}$ $\int \mathrm{cos}\mathrm{x}\mathrm{d}\mathrm{x}=\mathrm{sin}\mathrm{x}+\mathrm{C}$ $\int \mathrm{sin}\mathrm{x}\mathrm{d}\mathrm{x}=-\mathrm{cos}\mathrm{x}+\mathrm{C}$ $\frac{\mathrm{d}\left({\mathrm{e}}^{\mathrm{x}}\right)}{\mathrm{d}\mathrm{x}}={\mathrm{e}}^{\mathrm{x}}$ $\frac{\mathrm{d}\left(\mathrm{log}|\mathrm{x}|\right)}{\mathrm{d}\mathrm{x}}=\frac{1}{\mathrm{x}}$

Chain Rule

Chain rule is rule to differentiate composites of functions.

$\frac{\mathrm{d}\mathrm{y}}{\mathrm{d}\mathrm{t}}=\frac{\mathrm{d}\mathrm{y}}{\mathrm{d}\mathrm{x}}×\frac{\mathrm{d}\mathrm{x}}{\mathrm{d}\mathrm{t}}$

uv formulae