Arithmetic and Trig Problems

• The conditional equation (x-1)/3 = 2 holds if x is equal to 7.
• The equation (x - 1)² = x² - 2x + 1 is called an identity.
• There are two types of rational fractions.
• An improper fraction can be reduced to a proper fraction by division.
• 1/(1-x) is an improper fraction

Partial Fractions of 1/x(x+1)

• Partial fractions are in the form of 1/x - 1/(x+1)

Partial Fraction of 1/(x²-1)

• It will be of the form A/(x-1) + B/(x+1)

Partial Fraction of (x²+1)/(x-1)(x+1)

• It is of the form 1 + A/(x-1) + B/(x+1)

Partial Fraction of 4x²/(x²-1)(x+1)²

• It is of the form A/(x-1) + B/(x+1) + C/(x+1)² + D/(x+1)³
• (x+3)(x+4) = x² + 7x + 12 is an identity.

Partial fractions of (x+4)/(x+1)(x²+1)

• They are of the form A/(x+1) + (Bx+C)/(x²+1)

Partial fractions of 1/(x+1)(x²-x+1)

• They will be of the form A/(x+1) + (Bx+C)/(x²-x+1).
• If aₙ = (-1)ⁿ⁺¹, the 26th term of the sequence is -1.
• If the nth term of an A.P. is (3 - n), then the first three terms are 3, 2, 1.
• The next term of the sequence 1, 3, 6, 10 is 15.
• A sequence is denoted by {aₙ}.
• The common difference of the arithmetic sequence 3, 5, 7 is 2.
• If 1/a, 1/b, 1/c are in A.P., then b = 2ac/(a+c).
• The nth term of an A.P is a₁ + (n - 1)d.
• If a, A, b are in A.P., then 2A = a + b.
• The arithmetic mean between 1/a and 1/b is 2ab/(a+b).
• The arithmetic mean between 2 + √2 and 2 - √2 is 2.
• No term of a geometric sequence can be 0.5.
• The sequence 4, 12, 36 is a G.P.
• If a and r are the first term and common ratio respectively, then the (n + 1)th term of G.P is arⁿ.
• If a, G, b be the G.P where a, b are numbers then G.M. is equal to √ab.
• The G.M. between 1/ab and -1/ab is ±√-1/ab
• The G.M. between 1 and 16 is ±4.
• The geometric means between -21 and 81 are ±4.
• The sum of an infinite geometric series exists if |r| < 1.
• For an infinite geometric series, if |r| < 1, then S = a/(1-r).
• The sum of the convergent series a + ar + ar² + ... to infinity is a/(1-r).
• The reciprocal of the terms of H.P. is A.P.
• Let A, G, H be arithmetic, geometric, and harmonic means between a and b, then G² = AH.

If a=1, b=5, then A*H is equal to

• -5 If A, G, and H are arithmetic, geometric, and harmonic means, then A=G²/H

• If a and b are two distinct positive real numbers and G=√ab, then A >G > H

• Lim x->1 (xⁿ-1)/(x-1) is equal to n.

• Σk, from k=1 -> n, is equal to n(n+1)/2

• Σk², from k=1 -> n, is equal to [n(n+1)(2n+1)]/6

• If n = 1, then the value of lim n->1 (1/n-1) is zero.

• o! + 0! is equal to 2.

• If n is a negative integer, then n! is not defined.

• The factorial form of 6 x 5 x 4 is 6!/3!.

• The factorial form of n(n - 1)(n - 2) ... (n - r + 1) = n!/(n-r)!.

• The value of ⁴P₁ is 18.

• With usual notation ⁶P₂ equals 360.

• If ⁿP₂ = 30, then n = 6.

• The number of ways of arranging 5 keys in a circular ring is 12.

• The number of ways in which 5 people can be seated at a round table is 4!.

• ⁿCᵣ is equal to n! / [r! (n - r)!].

• r! × ⁿCᵣ = ⁿPᵣ.

• If r = n, ⁿCᵣ is equal to 1.

• If 'r=0, "Co is equal to 1.

• If "C, "C, then r + q = n

• If "C, "Cu, then n is equal to 8.

• If S is the sample space and E is an event of S, then P(E) is n(E)/n(S).

Probability of an Event E

• If P(E) is the probability of an event E, then 0≤P(E) ≤1
• The probability of a fair coin toss resulting in head is 1/2.
• Probability of sample space in tossing coin is 1
  • A coin is tossed once then P(S) is equals to 1
• If two dice are rolled once, then the number of elements in the sample space is 36.
• If Pakistan and India play a hockey match, the probability that Pakistan will win is 1/3.
• If "n(S) = 20 & n(B) = 2 then P (B)= 1/10
• Probability Event E Does Not Occur
• The probability that an event E does not occur: P(E) = 1 - P(E).
• Where A and B are disjoint events
• If A and B are disjoint events: P(A U B) = P (A) + P(B)

Mutually Exclusive and Independent Events

• Mutually Exclusive Events
  • If A and B are mutually exclusive: P(AUB) = P(A) + P(B)
• Independent Events: P(A∩B)= P(A).P(B)
• If A and B are independent events and P(A) = 0.8 & P(B) = 0.7 then P(A ∩ B) is: 0.56
• The expression n² - n + 41 represents a prime number for n ∈ N where n ≤ 40.
• n² > n + 3 is true for all integral values of n ≥ 3.
• If n is any positive integer, then 2ⁿ > 2(n + 1) is true for all n > 3.
• An algebraic expression consisting of two terms is called a binomial. The sum of binomial coefficients is 2ⁿ.
• The sum of odd coefficients in the binomial expansion of (1 + x)ⁿ is 2ⁿ⁻¹.
• The sum of odd coefficients in the expansion of (a + b)⁶ is 8.
• The sum of all coefficients in the binomial expansion of (1 + x)ⁿ is 2ⁿ.

Expansion of (r+1)th term of Binomial Expansion of $(a+x)^n$

• General term : ⁿCᵣ aⁿ⁻ˣ xʳ, n ∈ N

The rth term in the expansion of (a + x)ⁿ is

• T = nCr a^(n-r) x^(r-1)
• The coefficient of the 8th term in the expansion of (a + b)¹⁰ is ¹⁰C₈.
• In the expansion of (a + b)⁷ the 2nd term is 7a⁶b.
• In the expansion of (3x/2 - 1/3x)¹¹, the sixth term from the end is the 7th term.
• The co-efficient of the last term in the expansion of (a - b)ⁿ is 1.
• If n is even, the middle term in the expansion of (a + b)ⁿ is (n/2 + 1)th.
• The middle terms in the expansion of (x/2 + 2/x)¹¹ will be the 7th term.

Expansions of x and 1/3

• Middle terms in the expansion of (x + y)²¹ are the 11th and 12th terms.
• The 2nd term in (1+x)½ is x.
• The second term in the expansion of (1 + (2x/3))¹¹ is (11/3)x.
• 1 + 2x + 3x² + 4x³ + ... is the expansion of (1 - x)⁻².
• The expansion of (1 - 3x)⁻½ is valid if |x| < 1/3.
• The expansion of (1 + x)⁻¹ is valid only if |x| < 1.
• The equation 1 - x + x² - x³ + ... + (-1)ʳxʳ + ... is the expansion of (1+x)^(-1)
• O radian is measured in a circular system.

DMS vs Radians

• A rotation anti-clockwise equals 90°.
• 60th part of 1-degree is called one second

Angles measured in radians

• The area of the sector of the circular region of radius r is given by (1/2)r²θ.
• One radian is equal to 57.296°.
• 1° equals π/180 radian.

Angle Conversions and Approximations

• 1º is approximately equal to 0.175°

Calculations with π

• 5π/4 radians = 225°.
• • Value of π
  • In one hour, through minute hand 2π .

Clock problems

• Angular Displacement
  • Clock Turn for two hours: π/6
• The vertex of an angle in standard position is at (0,0).
• π/2 radians is a quadrantal angle.

Not quadrantal

• The values of angle which does not fall under quadrants are

Theta

• Given: Cosec and Sin
• Value lies in the third quad

Conversions between trigonometric signs and values

• If sin = √3/2, then θ is equal to 60°.
• sin 390° is equal to sin 30°.
• cos²30 + sin²30 = 1.
• cot²θ - cosec²θ is equal to -1.
• sec²θ - tan²θ is equal to 1.

Simplifications

Trigonometry Conversions

• (1/ 1+sine ) + ( 1/1 – sine equal to 2(sec®)^2 = 2/(cose)^2 . If sin(θ- π/2)
• Simplifications Cos and Sin
  • Then: is equals to cos 0
  • Where values lie on (Lahore 2016, D. G. Khan 2015) If sin(Θ-π/2) = is (A) equals, (B) cos ©, (C) sin O Ans(B) - cos Θ with reference to (Lahore 2016, D. G. Khan 2015) If value of Cos(+0) There: "value " lies here
  • (Multan 2017)

Then find:

• Tan(+0is)=, (equals)
• Where value is (Multan 2016, D. G. Khan 2015) "lies " Find the range where
• Tann(л - a = 0
• "value " is ( lies ) given, In the follow section for (Multan 2017, Lahore 2016, D. G. Khan 2015) Value and Equations Given : sin(Θ- л/2) Which: is equals to cos 0
• With range to equal to (Lahore 2016, D. G. Khan 2015) In given: value that finds = Cos(+0) There : what "value " , that lies in the equations = is found in range where equals the equation (Mirpur 2016, D. G. Khan 2015) Then: we follow the equation to get
• Tan(+0is)= - Tan and
• Tann(л - a = 0

Applying values following given formulas

• Now putting values in sin+sin is Follow Given- cos Q=2sin Applying to : 2Sin-Q is range

Then Cos q is what is left and ,

• Now putting values in cos - cos is

and that cos following equation where value follows Then : if the function , is applying - Now putting values is cos(л - a= 0

• which comes from the function if and follows that ranges of value can and will
• Domain of y = cos x is -∞ < x=""><>x < π/2] is -π/2 , π/2].

Functions and Ranges For Trigonometry

• Range of a function y = sin⁻¹ x
• With following given points, the is "equals" Range: y = cos x
• When one interval : [0, π] if, If: y = cos^-1 x then its domains is the follow equals Where is,
• Equal/Or follow When: If x = sin -1 , the formula equals as,

Value to Equal

• Equal value to sin cos, and which term can find
• Equals the value cos Value to find
  • Find/Find angles: sin (cos^-1 is - equal too and ranges that term is for find "domain " function

Value to Domain - function A) and where

• In and where if find to functions

14: Trigonometry Equation

If find any "Trigonometry Equation" is for has at a = Value The that is find find the Solution The, number to to equation can is

Values and Functions for Solving Trigonometry

• "if= find ", cos: for then to" function's

Angles for Solving

and, and or where : Values can equals: to

Follow: :Find values to get and what equals/Follow :Functions in equation

Range for Solving

• Then:

Range to follows : Find Value : can and what equals to functions

Trigonometric Function

• An: Equation that "follows trigonometry"" is function

Find Solutions to Trigonometry Equations given

• And: Then - range the in the: Is value find .