𝐌𝐀𝐓𝐇𝐄𝐌𝐀𝐓𝐈𝐂𝐀𝐋 𝐏𝐇𝐘𝐒𝐈𝐂𝐒

ɪᴛ ɪs ᴏᴜʀ ʜᴇᴀʀᴛs
ᴛʜᴀᴛ ᴍᴀᴋᴇ's ᴜs
ᴛʜᴇ ʙᴇsᴛ ᴡᴇ ᴄᴀɴ ʙᴇ.

ᴡᴇ ᴀʀᴇ ᴡᴏɴᴅᴇʀғᴜʟ
ғᴏʀ ᴡʜᴏ ᴡᴇ ᴀʀᴇ
ᴀɴᴅ ɴᴏᴛ ғᴏʀ ᴡʜᴀᴛ ᴡᴇ ʜᴀᴠᴇ•

sᴄᴀᴛᴛᴇʀ ᴛʜᴇ sᴇᴇᴅs
ᴏғ ʟᴏᴠᴇ ᴛᴏ ᴇᴠᴇʀʏᴏɴᴇ
ᴡɪᴛʜ ᴋɪɴᴅ ᴡᴏʀᴅs
ᴀɴᴅ ɢᴇɴᴛʟᴇ ᴡᴀʏs•

ɪᴍᴘᴀʀᴛ ᴛʀᴜᴇ ʟᴏᴠᴇ ᴀɴᴅ
ɢɪᴠᴇ ᴛʜᴀɴᴋs ᴛᴏ
ᴇᴠᴇʀʏᴛʜɪɴɢ.

ᴘʀɪᴅᴇ ᴍᴀᴋᴇs
ᴜs ᴀʀᴛɪғɪᴄɪᴀʟ
ʙᴜᴛ ʜᴜᴍɪʟɪᴛʏ
ᴍᴀᴋᴇs ᴜs ʀᴇᴀʟ!

ɢᴏᴅ ʙʟᴇss ᴜs ᴀʟʟ
ᴀɴᴅ ɢᴏᴏᴅ ᴍᴏʀɴɪɴɢ
ᴇᴠᴇʀʏᴏɴᴇ •

𝙌𝙪𝙖𝙙𝙧𝙖𝙩𝙞𝙘 𝙀𝙦𝙪𝙖𝙩𝙞𝙤𝙣𝙨

In order to solve a quadratic equation of the form ax2 + bx + c, we first need to calculate the discriminant with the help of the formula D = b2 – 4ac.

The solution of the quadratic equation ax2 + bx + c= 0 is given by x = [-b ± √ b2 – 4ac] / 2a

If α and β are the roots of the quadratic equation ax2 + bx + c = 0, then we have the following results for the sum and product of roots:

α + β = -b/a

α.β = c/a

α – β = √D/a

It is not possible for a quadratic equation to have three different roots and if in any case it happens, then the equation becomes an identity.

Nature of Roots:

Consider an equation ax2 + bx + c = 0, where a, b and c ∈ R and a ≠ 0, then we have the following cases:

D > 0 iff the roots are real and distinct i.e. the roots are unequal

D = 0 iff the roots are real and coincident i.e. equal

D < 0 iffthe roots are imaginary

The imaginary roots always occur in pairs i.e. if a+ib is one root of a quadratic equation, then the other root must be the conjugate i.e. a-ib, where a, b ∈ R and i = √-1.

Consider an equation ax2 + bx + c = 0, where a, b and c ∈Q and a ≠ 0, then

If D > 0 and is also a perfect square then the roots are rational and unequal.

If α = p + √q is a root of the equation, where ‘p’ is rational and √q is a surd, then the other root must be the conjugate of it i.e. β = p - √q and vice versa.

If the roots of the quadratic equation are known, then the quadratic equation may be constructed with the help of the formula
x2 – (Sum of roots)x + (Product of roots) = 0.

So if α and β are the roots of equation then the quadratic equation is

x2 – (α + β)x + α β = 0

For the quadratic expressiony = ax2 + bx + c, where a, b, c ∈ R and a ≠ 0, then the graph between x and y is always a parabola.

If a > 0, then the shape of the parabola is concave upwards

If a < 0, then the shape of the parabola is concave upwards

Inequalities of the form P(x)/ Q(x) > 0 can be easily solved by the method of intervals of number line rule.

The maximum and minimum values of the expression y = ax2 + bx + c occur at the point x = -b/2a depending on whether a > 0 or a< 0.

y ∈[(4ac-b2) / 4a, ∞] if a > 0

If a < 0, then y ∈ [-∞, (4ac-b2) / 4a]

The quadratic function of the form f(x, y) = ax2+by2 + 2hxy + 2gx + 2fy + c = 0 can be resolved into two linear factors provided it satisfies the following condition: abc + 2fgh –af2 – bg2 – ch2 = 0

In general, if α1,α2, α3, …… ,αn are the roots of the equation

f(x) = a0xn +a1xn-1 + a2xn-2 + ……. + an-1x + an, then

1.Σα1 = - a1/a0

2.Σ α1α2 = a2/a0

3.Σ α1α2α3 = - a3/a0

……… ……….

Σ α1α2α3 ……αn= (-1)n an/a0

Every equation of nth degree has exactly n roots (n ≥1) and if it has more than n roots then the equation becomes an identity.

If there are two real numbers ‘a’ and ‘b’ such that f(a) and f(b) are of opposite signs, then f(x) = 0 must have at least one real root between ‘a’ and ‘b’.

Every equation f(x) = 0 of odd degree has at least one real root of a sign opposite to that of its last term.

📚𝐓𝐫𝐢𝐠𝐨𝐧𝐨𝐦𝐞𝐭𝐫𝐢𝐜 𝐄𝐪𝐮𝐚𝐭𝐢𝐨𝐧𝐬 𝐚𝐧𝐝 𝐈𝐝𝐞𝐧𝐭𝐢𝐭𝐢𝐞𝐬📚

A function f(x) is said to be periodic if there exists some T > 0 such that f(x+T) = f(x) for all x in the domain of f(x).

In case, the T in the definition of period of f(x) is the smallest positive real number then this ‘T’ is called the period of f(x).

Periods of various trigonometric functions are listed below:

1) sin x has period 2π

2) cos x has period 2π

3) tan x has period π

4) sin(ax+b), cos (ax+b), sec(ax+b), cosec (ax+b) all are of period 2π/a

5) tan (ax+b) and cot (ax+b) have π/a as their period

6) |sin (ax+b)|, |cos (ax+b)|, |sec(ax+b)|, |cosec (ax+b)| all are of period π/a

7) |tan (ax+b)| and |cot (ax+b)| have π/2a as their period

➖Sum and Difference Formulae of Trigonometric Ratios

1) sin(a + ß) = sin(a)cos(ß) + cos(a)sin(ß)

2) sin(a – ß) = sin(a)cos(ß) – cos(a)sin(ß)

3) cos(a + ß) = cos(a)cos(ß) – sin(a)sin(ß)

4) cos(a – ß) = cos(a)cos(ß) + sin(a)sin(ß)

5) tan(a + ß) = [tan(a) + tan (ß)]/ [1 - tan(a)tan (ß)]

6)tan(a - ß) = [tan(a) - tan (ß)]/ [1 + tan (a) tan (ß)]

7) tan (π/4 + θ) = (1 + tan θ)/(1 - tan θ)

8) tan (π/4 - θ) = (1 - tan θ)/(1 + tan θ)

9) cot (a + ß) = [cot(a) . cot (ß) - 1]/ [cot (a) +cot (ß)]

10) cot (a - ß) = [cot(a) . cot (ß) + 1]/ [cot (ß) - cot (a)]

➖Double or Triple -Angle Identities

1) sin 2x = 2sin x cos x

2) cos2x = cos2x – sin2x = 1 – 2sin2x = 2cos2x – 1

3) tan 2x = 2 tan x / (1-tan 2x)

4) sin 3x = 3 sin x – 4 sin3x

5) cos3x = 4 cos3x – 3 cosx

6) tan 3x = (3 tan x - tan3x) / (1- 3tan 2x)

➖For angles A, B and C, we have

1) sin (A + B +C) = sinAcosBcosC + cosAsinBcosC + cosAcosBsinC - sinAsinBsinC

2) cos (A + B +C) = cosAcosBcosC- cosAsinBsinC - sinAcosBsinC - sinAsinBcosC

3) tan (A + B +C) = [tan A + tan B + tan C –tan A tan B tan C]/ [1- tan Atan B - tan B tan C –tan A tan C

4) cot (A + B +C) = [cot A cot B cot C – cotA - cot B - cot C]/ [cot A cot B + cot Bcot C + cot A cotC–1]


➖List of some other trigonometric formulas:

1) 2sinAcosB = sin(A + B) + sin (A - B)

2) 2cosAsinB = sin(A + B) - sin (A - B)

3) 2cosAcosB = cos(A + B) + cos(A - B)

4) 2sinAsinB = cos(A - B) - cos (A + B)

5) sin A + sin B = 2 sin [(A+B)/2] cos [(A-B)/2]

6) sin A - sin B = 2 sin [(A-B)/2] cos [(A+B)/2]

7) cosA + cos B = 2 cos [(A+B)/2] cos [(A-B)/2]

8) cosA - cos B = 2 sin [(A+B)/2] sin [(B-A)/2]

9) tanA ± tanB = sin (A ± B)/ cos A cos B

10)cot A ± cot B = sin (B ± A)/ sin A sin B

➖Method of solving a trigonometric equation:

1) If possible, reduce the equation in terms of any one variable, preferably x. Then solve the equation as you used to in case of a single variable.

2) Try to derive the linear/algebraic simultaneous equations from the given trigonometric equations and solve them as algebraic simultaneous equations.

3) At times, you might be required to make certain substitutions. It would be beneficial when the system has only two trigonometric functions.

➖Some results which are useful for solving trigonometric equations:
1) sin θ = sina and cosθ = cosa ⇒ θ = 2nπ + a

2) sin θ = 0 ⇒ θ = nπ

3) cosθ = 0 ⇒ θ = (2n + 1)π/2

4) tan θ = 0 ⇒ θ = nπ

5) sinθ = sina⇒ θ = nπ + (-1)na where a ∈ [–π/2, π/2]

6) cosθ= cos a ⇒ θ = 2nπ ± a, where a ∈[0,π]

7) tanθ = tana⇒ θ = nπ+ a, where a ∈[–π/2, π/2]

8) sinθ = 1 ⇒ θ= (4n + 1)π/2

9) sin θ = -1 ⇒ θ = (4n - 1) π /2

10) sin θ = -1 ⇒ θ = (2n +1) π /2

11) |sinθ| = 1⇒ θ =2nπ

12) cosθ = 1 ⇒ θ =(2n + 1)

13) |cosθ| = 1⇒ θ =nπ

📖Notes on Probability🎲

A negative binomial experiment is an experiment which consists of x repeated trials in which each trial can result in just two possible outcomes a success or a failure. In addition to this it has the following properties:

The probability of success, denoted by P, is the same on every trial.

The trials are independent, that is, the outcome on one trial does not affect the outcome on other trials.

The experiment continues until r successes are observed, where r is specified in advance.

Let A1, A2, .... An be a set of mutually exclusive and exhaustive events and E be some event which is associated with A1, A2, ...., An. Then probability that E occurs is given by
P (E) = ∑n(i=1) P(Ai)P(E/Ai).

If the set of n events related to a sample space are pair-wise independent, they must be mutually independent, but vice versa is not always true.

In probability problems which require the application of the total probability formula the events A1 and A2 must fulfill the following three conditions:

A1 ∩ A2 = Φ

A1 U A2 = S.

A1 ∈ S and A2 ∈ S.

Conditions for application of Bayes formula in probability questions include:

A1, A2, ......., An of the sample space are exhaustive and mutually exclusive i.e. A1 U A2 U ........... U An = S

and Ai ∩ Aj = ∅, where j, i = 1, 2, ........ n and i ≠ j

Priori events i.e. within the sample space there would exist an event B such that P (B) > 0.

The main aim is to compute a conditional probability of the form P (Ai /B).

We know that at least one of the two sets of the two probabilities are given below:

P(Ai ∩ B) for each Ai

P(Ai) and P(B/Ai) for each Ai

If in a problem some event has already happened and then the probability of another event is to be found, it is an application of Bayes Theorem. To recognize the question in which Bayes’ theorem is to be used, the key word is “is found to be".

📖𝙉𝙤𝙩𝙚𝙨 𝙤𝙣 𝙋𝙧𝙤𝙗𝙖𝙗𝙞𝙡𝙞𝙩𝙮🎲

The sum of all the probabilities in the sample space is 1.

The probability of an event which cannot occur is 0.

The probability of any event which is not in the sample space is zero.

The probability of an event which must occur is 1.

The probability of the sample space is 1.

The probability of an event not occurring is one minus the probability of it occurring.

The complement of an event E is denoted as E' and is written as P (E') = 1 - P (E)

P (A∪B) is written as P (A + B) and P (A ∩ B) is written as P (AB).

If A and B are mutually exclusive events, P(A or B) = P (A) + P (B)

When two events A and B are independent i.e. when event A has no effect on the probability of event B, the conditional probability of event B given event A is simply the probability of event B, that is P(B).

If events A and B are not independent, then the probability of the intersection of A and B (the probability that both events occur) is defined by P (A and B) = P (A) P (B|A).

A and B are independent if P (B/A) = P(B) and P(A/B) = P(A).

If E1, E2, ......... En are n independent events then P (E1 ∩ E2 ∩ ... ∩ En) = P (E1) P (E2) P (E3)...P (En).

Events E1, E2, E3, ......... En will be pairwise independent if P(Ai ∩ Aj) = P(Ai) P(Aj) i ≠ j.

P(Hi | A) = P(A | Hi) P(Hi) / ∑i P(A | Hi) P(Hi).

If A1, A2, ……An are exhaustive events and S is the sample space, then A1 U A2 U A3 U ............... U An = S

If E1, E2,….., En are mutually exclusive events, then P(E1 U E2 U ...... U En) = ∑P(Ei)

If the events are not mutually exclusive then P (A or B) = P (A) +P (B) – P (A and B)

Three events A, B and C are said to be mutually independent if P(A∩B) = P(A).P(B), P(B∩C) = P(B).P(C), P(A∩C) = P(A).P(C), P(A∩B∩C) = P(A).P(B).P(C)

The concept of mutually exclusive events is set theoretic in nature while the concept of independent events is probabilistic in nature.

If two events A and B are mutually exclusive,

P (A ∩ B) = 0 but P(A) P(B) ≠ 0 (In general)

⇒ P(A ∩ B) ≠ P(A) P(B)

⇒ Mutually exclusive events will not be independent.

The probability distribution of a count variable X is said to be the binomial distribution with parameters n and abbreviated B (n,p) if it satisfies the following conditions:

The total number of observations is fixed

The observations are independent.

Each outcome represents either a success or a failure.

The probability of success i.e. p is same for every outcome.

Some important facts related to binomial distribution:

(p + q)n = C0Pn + C1Pn-1q +...... Crpn-rqr +...+ Cnqn

The probability of getting at least k successes out of n trials is

P(x > k) = Σnx = k nCxpxqn-x

Σnx = k nCxqn-xpx = (q + p)n = 1

Mean of binomial distribution is np

Variance is npq

Standard deviation is given by (npq)1/2, where n

Sum of binomials is also binomial i.e. if X ~ B(n, p) and Y ~ B(m, p) are independent binomial variables with the same probability p, then X + Y is again a binomial variable with distribution X + Y ~ B(n + m, p).

If X ~ B(n, p) and, conditional on X, Y ~ B(X, q), then Y is a simple binomial variable with distributionY ~ B( n, pq).

The Bernoulli distribution is a special case of the binomial distribution, where n = 1. Symbolically, X ~ B (1, p) has the same meaning as X ~ Bern (p).

If an experiment has only two possible outcomes, then it is said to be a Bernoulli trial. The two outcomes are success and failure.

Any binomial distribution, B (n, p), is the distribution of the sum of n independent Bernoulli trials Bern (p), each with the same probability p.

The binomial distribution is a special case of the Poisson Binomial Distribution which is a sum of n independent non-identical Bernoulli trials Bern(pi). If X has the Poisson binomial distribution with p1 = … = pn = p then X ~ B(n, p).

A cumulative binomial probability refers to the probability that the binomial random variable falls within a specified range (e.g., is greater than or equal to a stated lower limit and less than or equal to a stated upper limit).

⬇️⬇️⬇️⬇️⬇️⬇️⬇️⬇️⬇️⬇️⬇️⬇️

Newton was relatively modest about his achievements, writing in a letter to Robert Hooke in February 1676 :

"If I have seen further it is by standing on the shoulders of giants."

⬇️⬇️⬇️⬇️⬇️⬇️⬇️⬇️⬇️⬇️⬇️⬇️

In a later memoir, Newton wrote:✍
➖➖➖➖➖➖➖➖➖➖➖➖
"I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me."
➖➖➖➖➖➖➖➖➖➖➖

𝘿𝙞𝙛𝙛𝙚𝙧𝙚𝙣𝙩𝙞𝙖𝙡 𝙀𝙦𝙪𝙖𝙩𝙞𝙤𝙣𝙨

The order of the differential equation is the order of the derivative of the highest order occurring in the differential equation.

The degree of a differential equation is the degree of the highest order differential coefficient appearing in it subject to the condition that it can be expressed as a polynomial equation in derivatives.

A solution in which the number of constants is equal to the order of the equation is called the general solution of a differential equation.

Particular solutions are derived from the general solution by assigning different values to the constants of general solution.

An ordinary differential equation (ODE) of order n is an equation of the formF(x, y, y',….., y(n) ) = 0, where y is a function of x and y' denotes the first derivative of y with respect to x.

An ODE of order n is said to be linear if it is of the form an(x)y(n) + an-1(x) y (n-1) + …. + a1(x) y' + a0 (x) y = Q(x)

If both m1 and m2 are constants, the expressions (D – m1) (D – m2) y and (D– m2) (D – m1) y are equivalent i.e. the expression is independent of the order of operational factors.

A differential equation of the form dy/ dx = f (ax+by+c) is solved by writing ax + by + c = t.

A differential equation, M dx + N dy = 0, is homogeneous if replacement of x and y by λx and λy results in the original function multiplied by some power of λ, where the power of λ is called the degree of the original function.

Homogeneous differential equations are solved by putting y = vx.

Linear equation are of the form of dy/dx + Py = Q, where P and Q are functions of x alone, or constants.

Linear equations are solved by substituting y =uv, where u and v are functions of x.

The general method for finding the particular integral of any function is 1/ (D-α)x = eαx∫Xe-αxdx

Various methods of finding the particular integrals:
1. When X = eaxin f(D) y = X, where a is a constant

Then 1/f(D) eax = 1/f(a) eax , if f(a) ≠ 0 and

1/f(D) eax = xr/fr(a) eax , if f(a) = 0, where f(D) = (D-a)rf(D)

2. To find P.I. when X = cos ax or sin ax

f (D) y = X

If f (– a2) ≠ 0 then 1/f(D2) sin ax = 1/f(-a2) sin ax

If f (– a2) = 0 then (D2 + a2) is at least one factor of f (D2)

3. To find the P.I.when X = xm where m ∈ N

f (D) y = xm

y = 1/ f(D) xm

4. To find the value of 1/f(D) eax V where ‘a’ is a constant and V is a function of x

1/f (D) .eax V = eax.1/f (D+a). V

5. To find 1/f (D). xV where V is a function of x

1/f (D).xV = [x- 1/f(D). f'(D)] 1/f(D) V

Some Results on Tangents and Normals:
1. The equation of the tangent at P(x, y) to the curve y= f(x) is Y – y = dy/dx .(X-x)

2. The equation of the normal at point P(x, y) to the curve y = f(x) isY – y = [-1/ (dy/dx) ].(X – x )

3. The length of the tangent = CP =y √[1+(dx/dy)2]

4. The length of the normal = PD = y √[1+(dy/dx)2]

5. The length of the Cartesian sub tangent = CA = y dy/dx

6. The length of the Cartesian subnormal = AD = y dy/dx

7. The initial ordinate of the tangent = OB = y – x.dy/dx

𝐃𝐢𝐟𝐟𝐞𝐫𝐞𝐧𝐭𝐢𝐚𝐛𝐢𝐥𝐢𝐭𝐲

The derivative of f, denoted by f'(x) is given by f'(x) = lim?x→0 (? y)/(? x) = dy/dx

The right hand derivative of f at x = a is denoted by f'(a+) and is given by f'(a+) = limh→0+(f(a+h)-f(a))/h

The left hand derivative of f at x = a is denoted by f'(a-) and is given by f'(a-) = limh→0- (f(a-h)-f(a))/-h
For a function to be differentiable at x=a, we should have f'(a-)=f'(a+) i.e. limh→0 (f (a-h)-f (a))/ (-h) = limh→0 (f (a+h)-f (a))/h.

limh→0 sin 1/h fluctuates between -1 and 1.

If at a particular point say x =a, we have f'(a+) = t1 (a finite number) and f'(a-) = t2 (a finite number) and if t1 ≠ t2, then f' (a) does not exist, but f(x) is a continuous function at x = a.

Continuity and differentiability are quite interrelated. Differentiability always implies continuity but the converse is not true. This means that a differentiable function is always continuous but if a function is continuous it may or may not be differentiable.
If a function is not derivable at a point, it need not imply that it is discontinuous at that point. But, however, discontinuity at a point necessarily implies non-derivability.

In case, a function is not differentiable but is continuous at a particular point say x = a, then it geometrically implies a sharp corner at x = a.

A function f is said to be derivable over a closed interval [a, b] if :

1.For the points a and b, f'(a+) and f'(b-) exist and

2.For ant point c such that a < c < b, f'(c+)and f'(c-) exist and are equal.

If y = f(u) and u = g(x), then dy/dx = dy/du.du/dx = f'(g(x)) g'(x). This method is also termed as the chain rule.

For composite functions, differentiation is carried out in this way:

If y = [f(x)]n, then we put u = f(x). So that y = un. Then by chain rule:

dy/dx = dy/du.du/dx = nu(n-1)f' (x) = [f(x)](n-1) f' (x)

Differential calculus problems involving parametric functions:

If x and y are functions of parameter t, first find dx/dt and dy/dt separately. Then dy/dx=(dy/dt)/(dx/dt).

If the functions f(x) and g(x) are derivable at x = a, then the following functions are also derivable:
1.f(x) + g(x)

2.f(x) - g(x)

3.f(x) . g(x)

4.f(x) / g(x), provided g(a) ≠ 0

If the function f(x) is differentiable at x =a while g(x) is not derivable at x = a, then the product function f(x). g(x) can still be differentiable at x = a.

Even if both the functions f(x) and g(x) are not differentiable at x = a, the product function f(x).g(x) can still be differentiable at x = a.

Even if both the functions f(x) and g(x) are not derivable at x = a, the sum function f(x) + g(x) can still be differentiable at x = a.

If function f(x) is derivable at x = a, this need not imply that f'(x) is continuous at x = a.

✍️ 𝙉𝙤𝙩𝙚𝙨 𝙤𝙣 𝙑𝙚𝙘𝙩𝙤𝙧𝙨✍️
➖➖➖➖➖➖➖➖➖➖➖➖

Some Basic Results of Vector Calculus:
1) Vectors in the same direction can be added by simply adding their magnitudes. But if the vectors to be added are in opposite directions, then their magnitudes are subtracted and not added.

2) Column vectors can be added by simply adding the values in each row.

3) You can find the magnitude of a vector in three dimensions by using the formula a2 = b2 + c2 + d2, where a is the magnitude of the vector, and b, c, and d are the components in each direction.

4) If l1a + m1b = l2a + m2b then l1 = l2 and m1 = m2

5) Collinear Vectors are also parallel vectors except that they lie on the same line.

6) When two vectors are parallel, the dot product of the vectors is 1 and their cross product is zero.

7)Two collinear vectors are always linearly dependent.

8) Two non-collinear non-zero vectors are always linearly independent

9) Three coplanar vectors are always linearly dependent.

10) Three non-coplanar non-zero vectors are always linearly independent

11) More than 3 vectors are always linearly dependent.

12) Three vectors are linearly dependent if they are coplanar that means any one of them can be represented as a linear combination of other two.

✍️ 𝙉𝙤𝙩𝙚𝙨 𝙤𝙣 𝙑𝙚𝙘𝙩𝙤𝙧𝙨✍️
➖➖➖➖➖➖➖➖➖➖➖➖
The length or the magnitude of the vector = (a, b, c) is defined by w = √a2+b2+c2

A vector may be divided by its own length to convert it into a unit vector, i.e. ? = u / |u|. (The vectors have been denoted by bold letters.)

If the coordinates of point A are xA, yA, zA and those of point B are xB, yB, zB then the vector connecting point A to point B is given by the vector r, where r = (xB - xA)i + (yB – yA) j + (zB – zA)k , here i, j and k denote the unit vectors along x, y and z axis respectively.

Some key points of vectors:
1) The magnitude of a vector is a scalar quantity

2) Vectors can be multiplied by a scalar. The result is another vector.

3) Suppose c is a scalar and v = (a, b) is a vector, then the scalar multiplication is defined by cv = c (a, b) = (ca, cb). Hence each component of vector is multiplied by the scalar.

4) If two vectors are of the same dimension then they can be added or subtracted from each other. The result is gain a vector.

If u, v and w are three vectors and c, d are scalars then the following results of vector addition hold true:
1) u + v = v + u (the commutative law of addition)

2) u + 0 = u

3) u + (-u) = 0 (existence of additive inverses)

4) c (du) = (cd)u

5) (c + d)u = cu + d u

6) c(u + v) = cu + cv

7) 1u = u

8) u + (v + w) = (u + v) + w (the associative law of addition)

Some Basic Rules of Algebra of Vectors:
1) a.a = |a|2 = a2

2) a.b = b.a

3) a.0 = 0

4) a.b = (a cos q)b = (projection of a on b)b = (projection of b on a) a

5) a.(b + c) = a.b + a.c (This is also termed as the distributive law)

6) (la).(mb) = lm (a.b)

7) (a ± b)2 = (a ± b) . (a ± b) = a2 + b2 ± 2a.b

8) If a and b are non-zero, then the angle between them is given by cos θ = a.b/|a||b|

9) a x a = 0

10) a x b = - (b x a)

11) a x (b + c) = a x b + a x c

Any vector perpendicular to the plane of a and b is l(a x b) where l is a real number.

Unit vector perpendicular to a and b is ± (a x b)/ |a x b|

The position of dot and cross can be interchanged without altering the product. Hence it is also represented by [a b c]

1) [a b c] = [b c a] = [c a b]

2) [a b c] = - [b a c]

3) [ka b c] = k[a b c]

4) [a+b c d] = [a c d] + [b c d]

5) a x (b x c) = (a x b) x c, if some or all of a, b and c are zero vectors or a and c are collinear.

Methods to prove collinearity of vectors:
1) Two vectors a and b are said to be collinear if there exists k ? R such that a = kb.

2) If p x q = 0, then p and q are collinear.

3) Three points A(a), B(b) and C(c) are collinear if there exists k ? R such that AB = kBC i.e. b-a = k (c-b).

4) If (b-a) x (c-b) = 0, then A, B and C are collinear.

5) A(a), B(b) and C(c) are collinear if there exists scalars l, m and n (not all zero) such that la + mb+ nc = 0, where l + m + n = o

Three vectors p, q and r are coplanar if there exists l, m ? R such that r = lp + mq i.e., one can be expressed as a linear combination of the other two.

If [p q r] = 0, then p, q and r are coplanar.

Four points A(a), B(b), C(c) and D(d) lie in the same plane if there exist l, m ? R such that b-a = l(c-b) + m(d-c).

If [b-a c-b d-c] = 0 then A, B, C, D are coplanar.

Two lines in space can be parallel, intersecting or neither (called skew lines). Let r = a1 + μb1 and r = a2 + μb2 be two lines.

They intersect if (b1 x b2)(a2 - a1) = 0

The two lines are parallel if b1 and b2 are collinear.

The angle between two planes is the angle between their normal unit vectors i.e. cos q = n1 . n2

If a, b and c are three coplanar vectors, then the system of vectors a', b' and c' is said to be the reciprocal system of vectors if aa' = bb' = cc' = 1 where a' = (b xc) /[a b c] , b' = (c xa)/ [a b c] and c' = (a x b)/[a b c] Also, [a' b' c'] = 1/ [a b c]

Dot Product of two vectors a and b defined by a = [a1, a2, ..., an] and b = [b1, b2, ..., bn] is given by a1b1 + a2b2 + ..., + anbn .

⭕️𝐀𝐥𝐠𝐞𝐛𝐫𝐚 - 𝐍𝐨𝐭𝐞𝐬 𝐨𝐧 𝐏𝐞𝐫𝐦𝐮𝐭𝐚𝐭𝐢𝐨𝐧𝐬⭕️


➖The concept of permutation is used for the arrangement of objects in a specific order i.e. whenever the order is important, permutation is used.

➖The total number of permutations on a set of n objects is given by n! and is denoted as nPn = n!

➖The total number of permutations on a set of n objects taken r at a time is given by nPr = n!/ (n-r)!

➖The number of ways of arranging n objects of which r are the same is given by n!/ r!

➖If we wish to arrange a total of n objects, out of which ‘p’ are of one type, q of second type are alike, and r of a third kind are same, then such a computation is done as n!/p!q!r!

➖Al most all permutation questions involve putting things in order from a line where the order matters. For example ABC is a different permutation to ACB.

➖The number of permutations of n distinct objects when a particular object is not to be considered in the arrangement is given by n-1Pr

➖The number of permutations of n distinct objects when a specific object is to be always included in the arrangement is given by r.n-1Pr-1.

➖If we need to compute the number of permutations of n different objects, out of which r have to be selected and each object has the probability of occurring once, twice or thrice… up to r times in any arrangement is given by (n)r.

➖Circular permutation is used when some arrangement is to be made in the form of a ring or circle.

➖When ‘n’ different or unlike objects are to be arranged in a ring in such a way that the clockwise and anticlockwise arrangements are different, then the number of such arrangements is given by (n – 1)!

➖If n persons are to be seated around a round table in such a way that no person has similar neighbor then it is given as ½ (n – 1)!

➖The number of necklaces formed with n beads of different colors = ½ (n – 1)!

➖nP0 =1

➖nP1 = n

➖nPn = n!/(n-n)! = n! /0! = n! /1= n!