Mathematics Sample Book

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1

Linear, Quadratic, Exponential & Logarithmic Function

1. Introduction 1Relationship between two variables (dependent variable & Independent variables) is called function. A and B be two non-empty sets. Then, a Rule or correspondence f

Which associates to each element x  A, a unique element, denoted by f(x) of B, is called a function from A to B, and expressed as f:A B The element f(x) of B is called the image of x, while x is called the pre-image of f(x) The set A is called the domain of f, and set B is called the Co-domain of f The set f (A) = {f(x) : x  A} is called the range of f. f(A) =

Set of all images of element x  A

Remarks, If x  A and y  B and if there is a rule 'f' which associates x to y, then we write y = f(x)

Ch - 1 : Page 3

Here x is called independent variable which can be assigned any value. And y is called dependent variable because its value depend up on value of x.

2. Types of Function 11. Polynomial Function: A function f defined for all values of x in the domain of f and is given by f(x) = a0 + a1x + a2x2 +......... anxn, an 0 Where a0, a1, a2, a3 .......... an are real numbers and n is a non-negative integer is called a polynomial function of degree n. 2. Constant Function: A polynomial function of degree zero is called a constant function. 3. Linear Function: A polynomial function of degree one is called the linear function or monomial function. For Example: f(x) = mx + c is a linear function. 4. Quadratic Function: A polynomial function of degree two is called a quadratic function. For Example: f(x) = ax2 + bx + c is a quadratic function. 5. Cubic Function: A polynomial function of degree three is called a cubic function. For Example: f(x) = ax3 + bx2 + cx + d is a cubic function 6. Biquadratic Function: A polynomial function of degree four is called a biquadratic function. For Example: f(x) = ax4 + bx3 + cx2 + dx + c is a biquadratic function. 7. Power Function: A function of the form f(x) = xn Where n is a constant, is called a power function 8. Identity Function: The function f defined by f(x) = x for all value of x is called the identity function, obviously, it is a particular type of linear function. 9. Rational Function: A function φ defined by f (x) 1φ (x) = g (x) 1Where f(x) and g(x), g(x) 0 are Polynomial functions is called a rational function.

3. Linear Function 1A function of the form f(x) = mx + c, where 'm' and c are real numbers and m 0 is called a linear function. It is generally denoted by y = mx + c  (1) Here m is the slope and c is the intercept made by the line with y-axis. Remarks: Its graph is a straight line with slope m and c as intercept on y-axis ⇒ Putting y = 0 in the equation (1), we get 0 = mx + c ⇒ x =  c/m m 0 This is the xintercept. ⇒ If (x1, y1) and (x2, y2) are two different points on a line, then the slope of the line is y2 − y1 V e r tic a l C h a n g e 1m = = x2 − x1 H o r iz a n ta l C h a n g e

Ch - 1 : Page 4

∆y ∆x

= Y (0, c)

A

(-c/m

B

, 0)

0

X

1Graph of y = mx + c

4. 1Laws of exponents? 11. ax, ay = ax+y

2.

13. (ab)x = axbx

4.

15. a0 = 1

6.

(ax)y = axy 1 ax = x a x a 1= axy y a

5. 1Rules of Logarithms? 11. x = am  x 13. Loga   y

logax = m

  1 = Log ax  Log ay  1 15. Loga n x = 1logax n 17. logab × logba = 1

2.

Loga(x. y) = Logax + Logay

4.

Logaxn = n logax

6.

logab = logax × logxb

2

Progression

1. 1Sequence 1Sequence is the succession of numbers formed and arranged in a definite order according to a certain definite rule. The number occuring at the nth place of a sequence is called its nth term or the general term, is denoted by tn.

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A sequence is said to be finite if its number of terms is finite and it is said to be infinite if its number of times is infinite. For Example: The numbers 3, 5, 7, 9, 11 ....... form a sequence by the rule tn = 2n + 1

2. Series? 1By adding the terms of a sequence, we obtain a series.

3. 1Progressions? 1Sequences following certain patterns are called progressions.

4. 1Arithmetic Progression 1It is a sequence in which each term, except the first one, differs from its preceding term by a constant, called the common difference. ⇒ In an A.P. we usually denote the first term by a, the common difference by d and the nth term by tn. Thus d = tn  tn1 Remarks: In general an A.P. whose first term is 'a' and whose common difference is 'd' then it is defined as a, a + d, a + 2d, a + 3d, a + 4d, ..................... The general term of an A.P.

Let 'a' be the first term and 'd' be the common difference of an A.P. Then its general term tn is given by. tn = a+(n1)d Remarks: It is always convenient to assume ⇒ ⇒ 3 numbers in A.P. as (a  d), a, (a + d) ⇒ 4 numbers in A.P. as (a  3d), (a  d), (a + d), (a + 3d) ⇒ 5 numbers in A.P. as (a  2d), (a  d), a, (a + d) (a + 2d) ⇒ 6 numbers in A.P. as (a  5d), (a  3d), (a  d), (a + d), (a + 3d) (a + 5d)

5. Arithmetic Mean?1 1If a, A, b are in A.P., We say that A is the arithmetic mean between a and b. a + b It is given by  A = 2 1Also, if a, x1, x2, ........... xn b are in A.P. we say that x1, x2, ........... xn are the 'n' arithmetic means between a and b (b − a )  2 (b − a )  n (b − a )     x1 =  a + 1; x2 =  a + 1......... xn =  a +   (n + 1)  (n + 1)  ( n + 1 )    

6. Properties of Arithmetic Progressions 11. If a fixed non-zero number is added to each term of an A.P., then the resulting progression is also an A.P. 2. If a fixed non-zero number is multiplied or divided to each term of a given A.P., then the resulting progression is also an A.P.

7. Sum of some important Series Ch - 1 : Page 6

11. The sum of first n natural numbers Sn = 1 + 2 + 3 + 4 + 5 + .................... + n =

∑

n

n i − 1

n

n (n + 1) n = 2 i − 1 12. The sum of the squares of first n natural numbers n 2 Sn = 12 + 22 + 32 + 42+ ........................... +n2 = ∑ n i − 1 n ( )( ) ∑ n2 = n n + 16 2n + 1 i − 1 13. The sum of the cubes of the first n natural numbers n 3 3 3 3 3 Sn = 1 + 2 + 3 + ........................ + n = ∑ n i − 1 n 2  n (n + 1)  3 n = ∑   2   i − 1

∑

8. Geometric Progression 1A sequence of numbers in which every term, except the first one, bears a constant ratio with its preceding term, is called a geometrical progression. ⇒ The constant ratio is called the common ratio of the G.P. t + 1 ⇒ Thus common ratio r = n tn 1For example 2, 6, 18, 54, 162 .................. is a G.P.

9. To find the nth term of a G.P. 1Let a be the first term and r be the common ratio of a G.P. then its nth term is given by tn = ar n1

10. nth term from the end of a G.P. 1Let a be the first term, r be the common ratio and l be the last term of a G.P. 1 Then nth term from the end = n − 1 r 1Remarks: For solving problems on G.P. it is always convenient to take. (i) three numbers in G.P. as (a/r), (a), (ar) (ii) four numbers in G.P. as (a/r3), (a/r), (ar), (ar3) (iii) five numbers in G.P. as (a/r2), (a/r), (a), (ar), (ar2) Ch - 1 : Page 7

(iv) the numbers as a, ar, ar2, ....... when their product is not given.

11. Sum of n terms of a G.P. 1Let a be the first term and 'r' be the common ratio of a G.P. And if l be the last term of the G.P. Then sum of first n terms of the G.P. is denoted by Sn. Which is given by a − 1r a (1 − r n ) Sn = = 1, when r < 1 1 − r 1 − r 1r − a a (r n − 1 ) 1Sn = = 1, when r > 1 r − 1 r − 1

12. Sum of an infinite G.P. when |r| <1

1Let a G.P. whose first term is 'a' and the common ratio is 'r' where r <1, then sum of its infinite terms is given by a S = 1 − r

13. Geometric Mean?

1If three numbers a, G, b, are in G.P., we say that G is the geometric mean between a and b G= ab 1Also if a, y1, y2, ................. yn, b are in G.P. we say that y1, y2, ................. yn are the n geometric means between a and b. y1 = a  b  a 

1 n +1

1; y2 = a.  b  a 

2 n +1

1 ........; yn = a  b  a 

n n +1

14. Properties of G.P. 11. If all the terms of a G.P. are multiplied by the same non-zero number, then the new numbers form a G.P. 2. The reciprocal of terms in G.P. are in G.P. 3.

If each term of a G.P. be raised to the same power, the new terms formed are also in G.P.

3

1

Permutations & Combinations

1. Factorial? 1The continued product of first n natural numbers is called factional n, to be denoted by n! or n where n is a positive integer. n! = n(n – 1) (n – 2) – 3.2.1. Remarks: ⇒ The value of 0! = 1

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⇒ When n is negative or a fraction n! is not defined

2. Fundamental principle of counting or multiplication or Association 1If an event can occur in 'm' different ways and if following it, a second event can occur is 'n' different ways, then the two events is succession can occur in m × n different ways.

3. Addition Principle 4. Permutations 1The different arrangements which can be made out of a given number of things by tacking some or all at a time, are called permutations. ⇒ Let 1 r  n. Then the number of all permutations of n dis-similar things taken r at a time is denoted by p(n, r) or nPr. which is given by n! p(n, r) = (n − r )!

5. Theorems on Permutations 1Theorem 1. The number of all permutations of n different things, taken all at a time is given by P(n, n) = n! Theorem 2. The number of all permutations of n different things taken 'r' at a time , when a particular things is to be always included in each arrangement is r.p(n - 1, r  1). Theorem 3. The number of permutations of n different things taken 'r' at a time, when a particular thing is never taken in each arrangement is P(n  1, r) Theorem 4. Let there be n objects, of which 'm' objects are alike of one kind, and the remaining (n  m) objects are alike of another kind. Then, the total number of mutually distinguishable permutations that can be formed from those object is n! m ! (n − m )! 1Remark: The above theorem can be extended further i,e, if there are 'n' objects, of which 'p1' are alike of one kind ; 'p2' are alike of another kind; 'p3' are alike of 3rd kind; pr are alike of rth kind such that p1 + p2 + ... pr = n then the number of permutations of these n objects is n! ( P 1 ! ) × ( P 2 ! ) × . . .× ( P r ! ) 1Theorem 5. The number of permutations of n different objects, taken 'r' at a time when each may be repeated any number of times in each arrangement is nr Theorem 6. The number of circular permutation of n different objects is (n  1)! Theorem 7. Number of ways in which 'n' different beads can be arranged in to form a necklace is 1 (n  1)! 2

6. Combinations 1Each of the different groups or selections which can be formed by taking some or all of a number of objects, irrespective of their arrangements, is called a combination. ⇒ The number of all combinations of 'n' distinct objects, taken 'r' at a time is denoted by C(n, r) or nCr Which is given by

Ch - 1 : Page 9

C(n1 r) =

n! r !× (n − r )!

7. Difference between a Permutation and a Combination? 1In a combination only a group is made and the order in which the objects are arranged is immaterial. On the other hand, in a permutation, not only a group is formed but also an arrangement in a definite order is considered. Remarks: 1. nCn = 1 2. nC0 = 1 3. nC1 = n 4. nCr = nCn  r n n 5. If Cx = Cy then either x = y or x + y = n Pascal's Law

If n and r are non-negative integers, such that 1 r  n, then nCr + nCr  1 = n + 1 Cr.

8. Types of Combinations?

1Type I To find the total numbers of combinations of n dissimiler things, taking any number of them at a time.

Case I When all the things are different. In this case the total number of ways in which one or more things are taken =2n  1 Case II When all the things are not different. Suppose that out of (m + n + p...) things, m are alike of one kind, n alike of second kind, p alike of third kind and the rest all are different, say q. The total numbers of selection of combination = q[(m + 1)(n + 1)(p + 1)2q] 1 Type II Division into groups

The number of ways in which (p + q) things can be divided into two groups containing p ( p + q )! and q things respectively. = p + qCp = p! q! 1Case I When p = q, the groups are equal. In this case the two groups can be interchanged without forming a new sub-division. (2 p )! The number of sub-division = 2 !( q !) 2 1But if 2p things are divided equally among two persons the number of divisions (2 p )! will be = ( q !) 2 1Case IIThe number of ways in which (p + q + r) things can be arranged into three ( p + q + r)! groups containing p, q and r things respectively is = p! q! r! 1Case III The number of ways in which 3p things can be divided equally into three (3 p !) distinct group is = ( p !) 3

Ch - 1 : Page 10

4

.1

Matrices 1. 1What is Matrix? 1Presentation of set of elements/data in the form of rectangular array of rows and columns → for keeping various records. For example – Marks obtained by two students Vijay and Kamlesh in Accounts, Maths, Management and Economics are as follows Accounts Maths Management Economics Vijay 60 79 65 76 Kamlesh 75 80 60 68 These marks may be represented by the following rectangular array enclosed by a pair of brackets. 6 0 7 9 6 5 7 6  60 79 65 66   7 5 8 0 6 0 6 8  or  7 5 8 0 6 0 6 8    1Such a rectangular array of 2 rows and 4 columns subject to certain rules of operation is called a 2×4 matrix. ⇒ A matrix of order m×n having 'm' rows and 'n' columns can be written as a 13 − a 1n  a 11 a 12 a a 22 a 23 − a 2n   21  a 31 a 32 a 33 − a 3n    − − − −   −  − − − − −    a m1 a m 2 a m 3 − a m n  m ×n ⇒ In short an m×n matrix is defined by [aij]m × n ⇒ 1Any element aij indicates element in the ith row and jth column.

2. 1Types of Matrices? 1Various types of matrices are: 1. Row Matrix: A matrix having only one row is called a row matrix e.g. [7 9 8]1×3 2. Column Matrix: A matrix having only one column is called a column matrix e.g. 4    6   7  3×1 13. Square Matrix: If in any matrix the number of rows is equal to no. of columns, then the matrix is called a square matrix. Ch - 1 : Page 11

4.

Null (or zero) Matrix: A matrix of any order whose all elements are zero is called a null matrix and is denoted by 0. 0 0  0 0 0  e.g.  ,   0 0  0 0 0  15. Diagonal Matrix: A square matrix in which every non diagonal element is zero is called a diagonal matrix. 9 0 0    e.g.  0 5 0   0 0 − 4  16. Scalar Matrix: A diagonal matrix whose all the diagonal elements are equal is called a scalar matrix. 6 0 0  4 0    e.g.  , 0 6 0   0 4    0 0 6  7. 1Unit (or Identity) Matrix: A square matrix in which every non-diagonal element is 0 and every diagonal element is 1, is called a unit matrix. 1 0 0  1 0    1e.g. I2 =  1I3 =  0 1 0   0 1   0 0 1 

3. 1Conditions for two matrices to be equal 1Two matrices A and B are said to be equal if and only if (i) A and B have the same order. (ii) Each element of A is equal to the corresponding element of B.

4. 1Transpose of a matrix: 1The transpose of a matrix A is obtained by interchanging the rows and columns of A and it is denoted by AT or A1 .  1 9 5 1 − 2   t 1e.g. A =  1A =  − 2 8  8 6 9  5 6 

5. 1Various rules of operation on Matrices 1There are 3 rules of operation on matrices (i) Rule of addition: Two matrices A and B can be added only when they are of same order The process of addition is done by adding the corresponding elements of the two matrices. 9 5 6  2 9 4  9 + 2 5 + 9 6 + 4  5 1 4 1 0  1e.g.  1+  =  =      7 9 8  9 8 5  7 + 3 9 + 8 8 + 5  1 0 1 7 1 3  1(ii)Rule of Scalar Multiplication: Multiplication of a matrix A by a scalar K is done by multiplying each element of A by K

Ch - 1 : Page 12

5 2 9  5 × 5 2 × 5 9 × 5  2 5 1 0 4 5  1e.g A =  1, then 5A =      3 7 4   3 × 5 7 × 5 4 × 5  1 5 3 5 2 0  1 ⇒ If scalar be (1), then (1) A = A is called negative of A (iii) Rule of Subtraction: Let A and B be two matrices of same order, Then AB is obtained by ↓ adding A to the negative of B thus AB = A+( B) (iv) Rule of Multiplication of Matrices: The product AB of two matrices A and B is defined only when the number of columns is A is equal to the number of rows in B. If A be a matrix of order m×n and B be a matrix of order n×p, then Product AB = C is a matrix of order m×p Its element cij, is the sum of the products obtained by multiplying the elements of the ith row of A by the corresponding elements of the jth column of B.

6. 1The Properties of Matrix Algebra 11. Properties of addition of Matrices (i) Matrix addition is commutative i.e. A+B = B + A for all comparable matrices A, and B. (ii) Matrix addition is associative i.e. (A + B) + C = A (B + C) for all comparable matrices A, B and C. (iii) If K be a scalar and A, B be two matrices of the same order m×n , then K (A+B) = KA+KB (iv) If A be m×n matrix and 0 be the null matrix of the same order, then (a) A+0 = 0+A = A (b) A + (A) = (A) + A = 0. (v) If A, B, C be any three matrix of the same order m × n, then A+C=B+C ⇒ A=B 2. Properties of Matrix Multiplication (i) The product of matrices is not, in general commutative. (ii) Matrix multiplication is associative. i.e. (A.B) C = A (BC) (iii) Multiplication is distributive over addition in matrices i.e. A(B+C) = AB + AC (iv) If A, B, C are three matrices such that AB = AC, then, in general B C (v) A. I = A = I. A, where A be a square matrix of order n and I be the unit matrix of the same order. (vi) If 0 is a null matrix then A0 = 0 and 0A = 0

7. 1Determinant? 1With every square matrix, there is associated an expression, called the determinant of A, denoted by det A or |A|

Ch - 1 : Page 13

a1 b1  nd For the square matrix A =   1of 2 order, a b  2 2  a1 b1  nd 1|A| =   1is called a determinant of 2 order and its value is defined a b  2 2  ⇒

a1 1by  a 2

b1  1= a1 b2  a2 b1 b 2  a1  1 ⇒ For the square matrix A =  a 2  a 3

b1 b2 b2

c1 c2 c2

   1of order 3, 

a1 b1 c1    1|A| =  a 2 b 2 c 2  1is called determinant of 3rd order  a 3 b 2 c 2  1and its value is defined by b 2 c 2  a 2 c 2  a 2 b 2  b a a c 3  c 3  b 3  1|A| = a1  3 1 b1  3 1+ c1  3

9. 1Adjoint of a Matrices 1Let A = [aij] be a square matrix of order n and let Aij denote the cofactor of aij in |A|. Then, the adjoint of A, denoted by adj A, is defined as adj A = [Aij]n × n To find out the adjoint of a square matrix A ⇒ First we find out the co-factors of aij in |A|. Let it be Aij. ⇒ then we find out the matrix [Aij]n×n ⇒ then we find out the transpose of [Aij]n×n a 11 a 12 a 13    1e.g. If A =  a 2 1 a 2 2 a 2 3  1then  a 3 1 a 3 2 a 3 3   A 11 A 12 A 13   A 11 A 21 A 32      1Adj A =  A 2 1 A 2 2 A 2 3  =  A 1 2 A 2 2 A 3 2   A 3 1 A 3 2 A 3 3   A 1 3 A 2 3 A 3 3 

10. 1Inverse of a Matrix 1Let A be a square matrix such that |A| 0, the if there exists a matrix B such that A.B=B.A=I In this case, we say that the inverse of A is B, which is denoted by A1=B 1 Inverse of a matrix is also expressed as A1 = a d j A |A|

11. 1Methods of Solving the Linear Equation 11. Solution by the method of Inversion of the coefficient matrix Ch - 1 : Page 14

Let us consider two linear equations a11x + a12 y = b1 a21x + a22 y = b2 ⇒ Express the coefficient in Matrix form  a 11 a 12  1i.e. A =   1B = a 21 a 22 

x  y  1and C =  

b1  b   2

⇒ 1Express the two equations in the matrix form as AX = B Now, If |A| 0 then A-1 exists. It follows that A-1 (AX) = A-1B (A-1.A) X = A-1B I X = A-1B ⇒ X = A-1B This gives the required Solution. 2. Solution by the method of Reduction ⇒ Given equations are expressed in the form of augmented matrix [Ab]. Which is obtained by expressing the coefficients of variables in left side of vertical line and constants in the right side of vertical line. ⇒ Reduce the augmented matrix of the given equations into the form 1 0 0 d1   d 2  by adding to all the elements of any row 0 1 0 0 0 1 d 3  

the same multiple of the corresponding elemens of any other row or by multiple all the elements of any row by the same constant. After reduction of the augmented matrix into the above form, the given equations reduce to 1.x + 0.y + 0.z = d1 x = d1 ⇒ 0.x + 1.y + 0.z = d2 y = d2 0.x + 0.y + 1.z = d3 z = d3

5

Graph of Linear Inequalities

1

1

1. Basics of Linear Inequalities? Consider that two real numbers a and b such that a is not equal to b (i.e. a ≠ b), then either a is greater than b or a is less than b. Now Ch - 1 : Page 15

⇒ ⇒

(i) a is said to be greater than b when a − b is positive. (ii) a is said to be less than b when a − b is negative

We can write these two statement as follows ⇒ (i) a > b when a − b > 0 ⇒ (ii) a < b when a − b < 0 If a 1 b (i.e. a is not less then b), then either a > b or a = b. These two relations can be written in combine as a ≥ b, where the symbol ≥ means 'greater than or equal to'. Similarly if a 1 b (i.e. a is not greater than b), then either a < b or a = b. These can be written in combine as a ≤ b, where the symbol '≤' means 'less than or equal to'. Thus linear function that involves an inequality sign (viz. > or < or ≥ or ≤) is a linear inequality. ⇒ The simplest example of Linear equation of one Variable are x ≥ 0 or x ≤ 0. They can be represented on a number line as 1 x  0 1 x0   3

 2

 1

0

1

2

 3

3

 2

 1

0

1

2

3

In general, a linear inequation in two variables x and y is of the form ⇒ ax + by + c < 0 or ax + by + c > 0 or ax + by + c ≤ 0 or ax + by + c ≥ 0

2. Graph of linear inequations? The following steps are required for the graph of an inequation ax + by + c ≤ 0 or ax + by + c ≥ 0 ⇒ ⇒ ⇒

(i) Consider the given inequation as the equation ax + by + c = 0 (ii) Form a table of values of the equation ax + by + c = 0 (iii) Draw the graph of this equation, which is a straight line

This line divides the plane into two parts If the inequality is ≤ or ≥ the end points on this line are included and the line drawn should be thick.

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If the inequality is < or >, the end points on this line are excluded and line drawn should be dotted. ⇒ (iv) Choose a point [for convenience choose (0 ,0)] not lying on this line. If this point satisfies the given inequation then shade the part of the plane containing this point, otherwise shade the other part. The shaded portion represents the solution set of the given inequation. The dotted line is not a part of the solution, while thick line is a part of it.

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