Important Formulae

PROFIT AND LOSS

1. Profit = Selling Price (S.P) – Cost Price (C.P)

When S.P. > C.P.,

1. Profit = S.P. – C.P.
2. S.P. = C.P. + Profit
3. C.P. = S.P. – Profit

2. Profit % = (Profit/C.P.) X 100

3. S.P. = C.P. X (100 + Profit%)/100)

4. When S.P. < C.P., Loss = Cost Price (C.P) – Selling Price (S.P)

5. Loss % = ( Loss /C.P) X 100

6. S.P. = C.P. X ( 100 - Loss%)/ X 100

7. When there are two articles having the same cost price and if one article is sold at a% profit and the other is sold at a% loss, profit % or loss % is zero.

8. If there are two articles having the same selling price and one is sold at x% profit and the other is sold at x% loss, effectively, a loss will always be made and the loss percent is(X/10)2 %

9. Discount = M.P. – S. P.

10. Discount % = ( Discount/M.P.) x 100

11. When M.P and Discount % are given,
S.P. = (M.P. X (100 - Diccount&))/100

12. When an article is sold after two successive discounts of p% and q%, then the final selling price = M.P.((100-p)(100-q))/100 x 100.

13. If cost price of x articles is equal to the selling price of y articles, then profit or loss percentage =(X - Y/Y) x 100 %

14. If the shopkeeper sells goods at cost price but gives lesser weight than true weight, then
Gain percentage =(True weight - False weight)/(False weight) x 100

15. If the shopkeeper sells his goods at a% loss on cost price but uses b gm instead of c gm, then his % profit or loss [100 - a] c/b - 100 is as the signis +ve or –ve.

SIMPLE INTEREST – COMPOUND INTEREST

1. Simple Interest = PTR/100.

2. Amount (A) = P(1 + TR/100)

3. Compound interest = P[(1 + R/100) - 1]

4. Amount under C.I = P(1 + R/100)

5. If a sum becomes x times in y years at CI then it will be (x)n times in ny years.

6. If a sum of money becomes ‘m’ times in ‘t’ years at SI, the rate of interest is given by
(a) (100 (m - 1))/t %
(b) Also to become n times, time taken = (n - 1) x t/(m - 1)

7. If amount under Compound Interest for n years and (n + 1) years is known, then the rate of interest is given by
(Difference of amount after n yrs and (n+1)/rs x 100)/Amount after n yrs

8. Difference between the compound interest and simple interest on a certain sum of money for 2years at r% rate is given by
Sum(r/100)2

9. Difference between CI and SI on a certain sum for 3 years at r% is given by
P(r/100)2 (r/100 + 3)

AVERAGES – MIXTURES – ALLIGATION

1. Average = Sum of all items in the group/Number of items in the group.

2. Quantity of Cheaper/Quantity of Dearer = Rate of deared - Average Rate/Average Rate - Rate of Cheaper

TIME AND WORK – PIPES AND CISTERNS

1. Work and men are directly proportional to each other, i.e.,

• Men and days are inversely proportional
• Men and hours are inversely proportional

Joint variation of the above can be written as

M1D1H1/W1 = M2D2H2/W2
2. If A can do a piece of work in x days and B can do it in y days, then A and B working together will do the same work in XY/(X + Y) days

3. If a pipe can fill a tank in x hours , and another pipe can empty the full tank in y hours, then the time taken to fill the tank, when both the pipes are opened = XY/(X - Y)

SETS AND RELATIONS

1. Union of Sets:

1. A ∪ B = {x │x∈A or x∈B}
2. If A ⊂ B, then A ∪ B = B
3. A ∪ φ = A
4. A ∪ μ = μ
2. Intersection of Sets:

1. A ∩ B = {x | x ∈ A and x ∈ B}.
2. A ⊂ B, then A ∩ B = A.
3. A ∩ φ = φ.
4. A ∩ μ = A.
3. Difference of Sets:

1. A - B = {x | x ∈ A and x ∈ B}
Similarly, B - A = {x ∈ B│ x ∉ A}
4. A Δ B = (A ∪ B) – (A ∩ B) = (A – B) ∪ (B – A)
5. Some Results:

1. A ∪ A = A; A ∩ A = A
2. A ∪ B = B ∪ A
3. A ∩ B = B ∩ A
4. A ∪ (B ∪ C) = (A ∪ B) ∪ C
5. A ∩ (B ∩ C) = (A ∩ B) ∩ C
6. A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
7. A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
8. C - (C - A) = C ∩ A
9. C - (A ∩ B) = (C - A) ∪ (C - B)
10. C - (A ∪ B) = (C - A) ∩ (C - B)
11. (Ac)c = A
12. (A ∪ B)c = Ac ∩ Bc
13. (A ∩ B)c = Ac ∪ Bc
14. A Δ B = (A – B) ∪ (B – A) = (A ∪ B) – (A ∩ B)
15. n(A ∪ B) = n(A) + n(B) –n(A ∩ B)
6. Cartesian Product of Two Sets:

Let A and B be any two sets. Then the cartesian product of A and B is the set of all ordered pairs of the form (a, b), where a ∈ A and b ∈ B.

The product is denoted by A ´ B

A x B = { (a, b)│ a ∈ A, b ∈ B}

1. A x B ≠ B x A
2. n(A x B) = n(B x A) = n(A) x n(B)
3. The number of relations defined from A to B = 2n(A) x n(B)
7. Reflexive Relation:

1. A relation R on a set A is said to be reflexive, if for every x ∈ A, (x, x) ∈ R.
2. If 'A' has n elements then a reflexive relation must have at least 'n' ordered pairs.
3. The number of reflexive relations defined from A to A = 2r2 - 1.
8. Symmetric Relation:

A relation R on a set is said to be symmetric if (x, y) ∈ R ⇒ (y, x) ∈ R
1. The necessary and sufficient condition for a relation R to be symmetric is R = R-1.
9. Transitive Relation:

A relation R on a set is said to be symmetric if (x, y) ∈ R ⇒ (x, z) ∈ R
10. Equivalence Relation:

1. A relation which is reflexive, symmetric and transitive is called an equivalence relation.
2. The smallest equivalence relation on a set A is the identity relation.
3. The largest equivalence relation on a set A is the cartesian product A x A.
11. Antisymmetric Relation:

R is antisymmetric if (a, b) ∈ R, (b, a) ∈ R ⇒ a = b

12. Partially Ordered Relation:

Let R be a relation on a set A, then R is said to be a partial order relation if it is reflexive, antisymmetric and transitive.

FUNCTIONS

1. Number of functions from a set A containing m elements to another set B containing n elements is nm.
2. One-to-One Function (Injection):

3. f: A → B is one-one if x1 ≠ x2 ⇒ f(x1) ≠ f(x2), Equivalently f(x1) = f(x2) ⇒ x1 = x2.
4. The number of one-one functions defined from set A to set B is n(B)p1(A).
5. Onto Function (Surjection):

1. If f is onto, range of f = co-domain of f.
2. If n(A) = m; n(B) = 2, then the number of onto functions defined from A to B is 2m – 2.

Bijection:

If a function is both one-to-one and onto, then it is called a bijective function or bijection. The number of bijective functions defined from A to B is m! where n(A) = n(B) = m

Composite Function or Product Function:

If f: A → B and g : B → C are two functions, then g o f is a function from A to C, such that g o f(a) = g[f(a)], for every a Î A and is called the composite mapping of f and g.
1. If f: A → B and g: B → C are two bijective functions, then (g o f)- 1 = f- 1 o g- 1.
2. If h: A → B, g: B → C and f: C → D be any three functions, then f o (g o h) = (f o g) o h.
3. (f o f - 1) (x) = x or f o f- 1 = I.

Real Function:

Given below are the domain and range of the various trigonometric functions.

Function Domain Range
sin x R [-1, 1]
cos x R [-1, 1]
tan x R-{(2n+1) φ/2 | n ∈ Z } R
cot x R - {nφ | n ∈ Z} R
sec x R-{(2n+1) φ/2 | n ∈ Z} (- ∞, -1] ∪ [1, ∞)
cosec x R-{nφ | n ∈ Z} (- ∞, -1] ∪ [1, ∞)

MATHEMATICAL LOGIC

1. If p = T and q = T then p ∧ q = T
2. If p = F, q = F then p ∨ q = F
3. If p = T, q = F then p ⇒ q = F
4. (p ⇒ q) = ∼ p ∨ q
5. Converse of p ⇒ q = q ⇒ p
6. Inverse of p ⇒ q = ∼p ⇒ ∼q
7. Contropositive of p ⇒ q = ∼q ⇒ ∼p
8. p ⇔ q = (p ⇒ q) ∧(q ⇒ p)
9. ∼(p ∨ q) = ∼p ∧ ∼q
10. ∼(p ∧ q) = ∼p ∨ ∼q
11. ∼(p ⇒ q) =p ∧ ∼q
12. ∼(p ⇔ q) = ∼p ⇔ q
= p ⇔ ∼q

Some Logical Equivalences:

1. Commutative Properties:
p ∨ q ≡ q ∨ p ; p ∧ q ≡ q ∧ p
2. Associative Properties:
p ∨ (q ∨ r) ≡ (p ∨ q) ∨ r ; p ∧ (q ∧ r) ≡ (p ∧ q) ∧ r
3. Distributive Properties:
p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r) ;
p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)
4. Idempotent Properties:
p ∨ p = p ; p ∧ p = p
5. Absorption :
p ∨ (p ∧ q) = p
p ∧ (p ∨ q) = q

List of Tautologies:

1. p ∧ q ⇒ p or p ∧ q ⇒ q
2. p ⇒ p ∨ q or q ⇒ p ∨ q
3. ∼p ⇒ (p ⇒ q) or ∼(p ⇒ q) ⇒ p
4. (p ∧(p ⇒ q)) ⇒ q
5. (∼p ∧ (p ∨ q)) ⇒ q
6. (∼q ∧ (p ⇒ q)) ⇒ ∼p
7. ((p ⇒ q) ∧ (q ⇒ r)) ⇒ (p ⇒ r)
hypothetical syllogism or transitive rule

Operations with F and T:
(T – Tautology and F – Contradiction)

1. p ∨ ∼p = T
2. p ∧ ∼p = F
3. T ∨ p = T
4. F ∧ p = F
5. F ∨ p = p
6. T ∧ p = p
7. ∼F =
8. ∼T = F

INEQUALITIES AND MODULUS

1. If a > b, then b < a
2. If a > b and b > c, then a > c
3. If a < b and b < c, then a < c
4. If a > b and c > 0 then a ± c > b ± c
5. If a > b and c > 0, then ac > bc
6. If a < b and c > 0, then ac < bc
7. If a > b and c < 0, then ac < bc
8. If a < b and c < 0, then ac > bc
9. If a > 0, then -a < 0 and if a > b, then -a < -b
10. If a and b are positive numbers and a > b, then 1/a < 1/b,
11. If A, G and H are the Arithmetic mean, Geometric mean and Harmonic mean of n positive real numbers, then A ≥ G ≥ H, the equality occurring only when the numbers are all equal.

Properties of modulus:

1. x = 0 ⇔ |x| = 0
2. |x| ≥ 0 and – |x| ≤ 0
3. |x + y| ≤ |x| + |y|
4. | |x| - |y| | ≤ |x - y|
5. -|x| ≤ x ≤ |x|
6. |xy| = |x| . |y|
7. |x/y| =|y/x| , y ≠ 0
8. |x|2 = x2

MATRICES AND DETERMINANTS

1. A = , then A2 –(a + d) A = [bc – ad]I.
2. If A = , then An =
3. If A = , then det A = ad – bc
4. If A is a n ´ n matrix, then det(KA) = Kn det A.
5. det(A-1) =
6. If A is a n ´ n matrix, then det(adjA) = (det A)n - 1.
7. If A = , then adj A =
8. If A = , then A–1 =
9. det(AB) = detA.detB

BINOMIAL THEOREM AND REMAINDER THEOREM

1. The nPr and nCr representation:
1. For 0 ≤ r ≤ n,
nPr = n!/(n - r)! and nCr = n!/r!(n - r)!
2. nCr = nCn - r
3. If nCr = nCs, then r = s or r + s = n
2. Binomial Theorem for a Positive Integral Index:
1. If 'n' is a positive integer, then
(x + y)n = nC0xn + nC1xn - 1y1 + nC2xn - 2y2+..... + nCrxn - ryr +…..+nCnyn,
2. Number of terms in the expansion is n + 1
3. General Term in the Expansion of (x + y)n is
Tr + 1 = nCr xn - r yr,
4. The general term in the expansion of (x - y)n is Tr + 1 = (-1)r nCr xn - r . yr
5. If 'n' is even, there exists only one middle term, which is (n/2 + 1)n term
6. If 'n' is odd, there will be two middle terms i.e.(n + 1)/2 nand(n + 3)/2n terms.
7. The Greatest Coefficient in the Expansion of (1 + x)n (where n is a positive integer): is
= n/2 (if n is even)
nc = (n + 1)/2 or nc (n - 1)/2 (if n is odd)
8. The coefficient of xk in the expansion(axp + b/xq)n is Tr + 1, then r =( np - k )/p + q.
9. The constant term in(axp + b/xq)n is Tr + 1, then r =( np )/p + q. .
10. The number of terms in (a + b + c)n is (n + 1)(n +2)/2
3. Numerically Greatest term in the Expansion of (1 + x)n:
The numerically greatest term in the expansion of (1 + x)n is found out using the following process. We calculate the value of ((n + 1)|x|)/|x| + 1:
1. If ((n + 1)|x|)/|x| + 1 = an integer, say 'k', then kth and (k + 1)th terms are numerically greatest terms
2. If ((n + 1)|x|)/|x| + 1 is not an integer, say k + a; where 0 < a < 1, (k + 1)th term is numerically greatest term.
4. Properties of Binomial Coefficients:
1. C0 + C1 + C2 + ....... + Cn = 2n
i.e., sum of all the binomial coefficients in the expansion of (1 + x)n is 2n
2. C1 + C3 + C5 + . . . . =2n/2 = 2n - 1
C0 + CC2 + C4 . . . . = 2n - 1
i.e. sum of coefficients of odd terms is equal to the sum of coefficients of even terms and each sum is 2n - 1
3. Let f(x) = (ax +b)n
The sum of all coefficients = f(1)
The sum of even coefficients =(f(1) + f(-1))/2
The sum of odd coefficients = (f(1) - f(-1))/2
4. The ratio of the two successive terms in (ax + by)n is Tr/Tr + 1 = nr/n- r + 1 (by/ax)
5. Expansions:
1. (1 - x)-1 = 1 + x + x2 + x3 + ........ + xr + .......
2. (1 + x)-1 = 1 - x + x2 - x3 + ....... + (-1)r xr +......
3. (1 - x)-2 = 1 + 2x + 3x2 + 4x3 + ........ + (r + 1)xr + .......
4. (1 + x)-2 = 1 - 2x + 3x2 - 4x3 + ....... + (-1)r(r + 1)xr + ......
6. Remainder Theorem:
1. If a polynomial, say f(x), is divided by (x - a), then the remainder is f(a).
2. If f(x) is a polynomial such that f(a) = 0, then (x - a) is a factor of f(x).
3. If the sum of the coefficients of all the terms in the polynomial f(x) is zero, then (x - 1) is a factor of f(x).
4. If the sum of the coefficients of odd powers of x is equal to the sum of the coefficients of even powers of x, then (x+1) is a factor.
5. If f(x) is divided by ax + b, then remainder is f (- b/a).
6. When a polynomial, say f(x) is divided by a polynomial p(x) to give a quotient q(x) and a remainder r(x), by division algorithm
Dividend = [Divisor x Quotient] + Remainder we have, f(x) = p(x) x q(x) + r(x)

STATISTICS

1. Arithmetic mean (A.M. or ):

1. Individual series: =
2. Discrete series: = where x1, x2, .… x n are n distinct values with frequencies f1, f2, f3, …., f n respectively.
3. Continuous series: = = where f1, f2, f3 ,…. fn are the frequencies of the classes whose mid-values are m1, m2, … mn respectively.
4. The algebraic sum of deviations taken about mean is zero.
5. Mean of first n natural numbers is (n+1)/2
6. Arithmetic mean of two numbers a and b is (a + b)/2
7. Combined Mean: If x1 and x2 are the arithmetic means of two series with n1 and n2 observations respectively, the combined mean,
2. Geometric Mean (G.M.) for Individual series
1. G.M. = (x1. x2. x3 …. xn)1/ n ,
2. Geometric mean of two numbers a and b is√ab
3. If b is G.M. of a and c then a, b and c are in a geometric progression.
3. Harmonic Mean (H.M.) for Individual series:
1. H =
where x1, x2, …. xn are n observations.
2. Harmonic mean of two numbers a and b is .
3. (GM)2 = AM x HM.
4. Median
1. Individual series:
If x1, x2, …. xn are arranged in ascending order of magnitude then the median is the size of item.
2. Discrete series:
Median is the value of the variable x for which the cumulative frequency just exceeds or equals , N being the total frequency.
3. Continuous series:
Median = l +
where l = lower boundary of the median class (or the class in which N/2th item lies)
m = cumulative frequency upto the median.
c = width of the class interval
f = frequency of median class
N = Total frequency i.e., N = ∑fi
4. The sum of absolute deviations taken about median is least.
5. Mode for continuous series:

Mode =
where l1 = lower boundary of the modal class (class, where the frequency is maximum)

c = width of the class Δ1 = f – f1 Δ2 = f – f2
f = frequency of the modal class f1 = frequency of the class which immediately precedes modal class

f2 = frequency of the class which immediately succeeds modal class

6. Empirical Formula:
1. For moderately symmetrical distribution,
Mode = 3 median – 2 mean
2. For a symmetric distribution,
Mode = Mean = Median.
7. Measures of Dispersion:
8. Quartile Deviation (Q.D.): Q.D. =
where Q1 ® size of (n+1)/4th item
Q3 ® size of 3(n+1)/4th item
9. Mean Deviation (M.D.) for Individual series:
1. M.D. =
where x1, x2 …. xn are the n observations and A is the mean or median or mode.
2. Mean deviation about the median is the least.
3. Mean deviation of two numbers a and b is .
10. Standard Deviation (S.D.):
Individual series:
1. S.D. (σ ) = where x1, x2, …. xn are n observations with mean as x
2. σ =
3. For a discrete series in the form a, a + d, a + 2d, ……(A.P.), the standard deviation is given by S.D. = d, where n is number of terms in the series.
11. Coefficient of variation (CV) is defined as, CV =
12. Correlation: We confine our study of correlation to Spearman's rank correlation co-efficient.
ρ = 1 – [di = (xi - yi); i = 1, 2, 3, . . . . , n]
13. Co-efficient of correlation:
1. Limits of correlation coefficient –1 ≤ r ≤ 1.
2. ρ = + 1, perfect positive correlation.
3. ρ = – 1, perfect negative correlation.
4. ρ = 0, we infer that there is no linear correlation.
5. ρ > 0, positive correlation.
6. ρ < 0, negative correlation.
7. Coefficient of skewness:
Sk =

PROBABILITY

1. P(E) =
2. P = 1 – P(E).
3. P(A ∩ B) = P(A).P(B) when A, B are independent events.
4. P(A ∪ B) = P(A) + P(B) –P(A ∩ B)
5. When n coins are tossed the probability of getting r heads =
6. The odd infavour of E is a : b, then P(E) = ,P = ,

LIMITS

1. k f(x) = k f(x)
2. [f(x) ± g(x)] = f(x) ± g(x)
3. [f(x) . g(x)] = f(x) . g(x)
4. =
5. = 1
6. = 1
7. = a
8. = a/b
9. = a/b
10. =
11. If k and ‘n’ are constants, |x| > 0 and n > 0 then and
12. = 1
13. (1 + x)1/x = e
14. (1 + 1/x)x = e
15. = 1
16. = loge a
17. L’ Hospitals Rule
(i) If is of the indeterminate form , then .
(ii) If is also of indeterminate form and so on

DIFFERENTIATION

1. = f '(x)
2. (k) = 0
3. k.f(x) = k f(x)
4. [f(x) ± g(x)] = f(x) ± g(x)
5. {f(x).g(x)}= f '(x).g(x)+f(x).g ' (x)
(Product rule or 'uv' rule)
6. If y = f(u) and u = g(x) be two functions, then dy/dx = (dy/du) ´ (du/dx)
7. (i) (xn)= n.xn–1
(ii) =
8. [ax + b]n = n.a(ax + b)n–1
9. [eax] = a.eax
10. [logx] =
11. [ax] = ax.loga
12. [sinx] = cosx
13. [cosx] = – sinx
14. [tanx] = sec²x
15. [cotx] = – cosec²x
16. [secx] = secx.tanx
17. [cosecx] = – cosecx.cotx
18. If y = then then