Rational numbers: Numbers which can be written in the form of\[\frac{p}{q}(q\ne 0)\]where p and q are integers, are called rational numbers.
Note: Every terminating decimal and non-terminating repeating decimal can be expressed as a rational number.
Irrational numbers: Numbers which cannot be written in the form of\[\frac{p}{q}\]where p and q are integers and\[q\ne 0\]are called irrational numbers. In other words, numbers which are not rational are called irrational numbers.
Real numbers: The rational numbers and the irrational numbers together are called real numbers.
Note: Any number that can be represented on a number line is called a real number.
Lemma: A proven statement which is used to prove another statement is called a lemma.
Euclid's division lemma: For any two positive integers 'a' and 'b; there exist whole numbers 'q' and 'r' such that\[\text{a}=\text{bq}+\text{r},0\le \text{r}<\text{b}\].
Note: Euclid's division algorithm is stated only for positive integers, but can be extended for all negative integers.
Algorithm: An' algorithm is a process of solving particular problems.
Euclid's division algorithm: is used to find the greatest common divisor (G.C.D.) or Highest Common Factor (H.C.F.) of two numbers.
Finding H.C.F. using Euclid's division algorithm: Suppose the two positive numbers are 'a' and 'b', such that a > b. Then the H.C.F. of 'a' and 'b' can be found by following the steps given:
Apply the division lemma to find 'q' and 'r' where\[a=bq+r,0\le r<b\].
lf\[\text{r}=0\],then H.C.F. is b. If\[\text{r}\ne \text{0}\], then apply Euclid's lemma to find 'b' and 'r'.
Continue steps (a) and (b) till \[r=0\].The divisor at this state will be H.C.F. (a, b). Also, H.C.F. (a, b)= H.C.F. (b, r).
Fundamental theorem of Arithmetic: Every composite number can be expressed as a unique product of prime numbers. This is also called the unique prime factorization theorem.
Note: (i) The order in which the prime factors occur may differ. In general, any composite number x, can be expressed as a product of prime numbers as shown below.
\[x={{p}_{1}}{{p}_{2}}{{p}_{3}}...........{{p}_{n}}\]where\[{{p}_{1}},{{p}_{2}},{{p}_{3}},...........{{p}_{n}}\]are primes in ascending order.
If 'p' is a prime, 'a' is a positive integer, and if 'p' divides\[{{a}^{2}}\], then 'p' divides 'a'. Also, if 'p' divides\[{{a}^{3}}\], then 'p' divides 'a'.
If 'a' is a terminating decimal, then 'a' can be expressed as \[\frac{p}{q}(q\ne 0)\], where 'p' and 'q' are co primes and the prime factorization of q is of the form \[{{2}^{m}}{{5}^{n}}\], (where m and n are whole numbers.).
If \[\frac{p}{q}\]is a rational number and q is not of form \[{{2}^{m}}{{5}^{n}}(m,n\in W)\], then \[\frac{p}{q}\]has a non-terminating repeating decimal expansion.
C.F. of two numbers is the product of the smallest power of each common prime factor in the numbers.
M. of two numbers is the product of the greatest power of each prime factor involved in the numbers.
Polynomial: A function p(x) of the form\[p(x)={{a}_{0}}+{{a}_{1}}x+......+{{a}_{n}}{{x}^{n}}\], where\[{{a}_{0}},{{a}_{1}},...{{a}_{n}}\]are real numbers and 'n' is a non-negative (positive) integer is called a polynomial.
Note: \[{{a}_{0}},{{a}_{1}},...{{a}_{n}}\]are called the coefficients of the polynomial.
If the coefficients of a polynomial are integers, then it is called a polynomial over integers.
If the coefficients of a polynomial are rational numbers, then it is called a polynomial over rational.
If the coefficients of a polynomial are real numbers, then it is called a polynomial over real numbers.
A function\[p(x)={{a}_{0}}+{{a}_{1}}x,...+{{a}_{n}}{{x}^{n}}\]is not a polynomial if the power of the variable is either negative or a fractional number.
Standard form: A polynomial is said to be in a standard form if it is written either in the ascending or descending powers of the variable.
Degree of a polynomial: If p(x) is a polynomial, then the highest power of x in p(x) is called the degree of the polynomial.
Types of polynomials:
(a) A polynomial of degree 1 is called a linear polynomial.
(b) A polynomial of degree 2 is called a quadratic polynomial.
(c) A polynomial of degree 3 is called a cubic polynomial.
(d) A polynomial of degree 4 is called a biquadratic polynomial or a quadratic polynomial.
Polynomial
General form
Coefficients
Linear polynomial
\[ax+b\]
\[a,b\in R,a\ne 0\]
Quadratic polynomial
\[a{{x}^{2}}+bx+c\]
\[a,b,c\in R.a\ne 0\]
Cubic polynomial
\[a{{x}^{3}}+b{{x}^{2}}+cx+d\]
\[a,b,c,d\in R,a\ne 0\]
Quadratic polynomial
\[a{{x}^{4}}+b{{x}^{3}}+c{{x}^{2}}+dx+e\]
\[a,b,c,d,e\in R,a\ne 0\]
Value of a polynomial: If p(x) is a polynomial in x. and if 'a' is any real number, then the value obtained by replacing 'x' by 'a' in p(x), denoted by p(a) is called the value of p(x) at x = a.
Zero of a polynomial: Areal number 'a' for which the value of the polynomial p(x) is zero, is called the zero of the polynomial.
In other words, a real number 'a' is called a zero of a polynomial p(x) if p(a) =0.
Note: The number zero is known as zero polynomial and its degree is not defined.
Geometric meaning of the zero of a polynomial:
(a) The graph of a linear equation of the form \[y=ax+b,a\ne 0\]is a straight line which intersects the X-axis at\[\left( \frac{-b}{a},0 \right)\].
Zero of the polynomial ax + b is the x-coordinate of the point of intersection of the graph with X-axis. Thus, the zero of\[y=ax+b\]is \[\left( \frac{-b}{a} \right)\]
Note: A linear polynomial ax + b, a more...
Linear equation in two variables: An equation of the form\[\text{ax}+\text{by}=\text{c}\], where \[a\ne 0,b\ne 0\] and a, b and c are real numbers is known as a linear equation in two variables x and y.
A pair of linear equations in two variables: Two linear equations in the same two variables are called a pair of linear equations in two variables.
General form of a pair of linear equations in two variables: The general form of a pair of linear equations in two variables is \[{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0\] and\[{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0\], where \[{{a}_{1}},{{a}_{2}},{{b}_{1}},{{b}_{2}},{{c}_{1}}\]and\[{{c}_{2}}\]are real numbers such that \[a_{1}^{2}+b_{1}^{2}\ne 0\]and\[a_{2}^{2}+b_{2}^{2}\ne 0\].
Note: A pair of linear equations in two variables is called a system of simultaneous linear equations.
Solution of a pair of linear equations in two variables: A pair of values (x, y) that satisfies both the linear equations is called a solution of the system of simultaneous equations.
Note: (i) A pair of values (x, y) that satisfies an equation is called its solution. (ii) Every linear equation in two variables has an infinite number of solutions.
(iii) Every solution of a linear equation is a point on the line representing it.
Methods of solving a pair of linear equations in two variables: A pair of linear equations in two variables can be solved by (i) Graphical method (ii) Algebraic method.
(i) Graphical method: The graph of a linear equation is a straight line. The graph of a pair of linear equations in two variables is represented by two lines.
(a) If the two lines intersect at a point, then the pair of linear equations has a unique solution (the point) and is said to be consistent.
(b) If the two lines coincide, then the pair of linear equations has infinitely many solutions (each point on the line being a solution), and is said to be dependent or consistent.
(c) If the two lines are parallel, then the pair of linear equations has no solution (no common point) and is said to be inconsistent.
In other words, there are three types of solutions of a pair of linear equations in two variables:
(a) Unique solution
(b) Infinitely many solutions
(c) No solution.
Note: Graphical method does not give an accurate answer as error is likely to occur in reading the coordinates of a point.
(ii) Algebraic methods: To obtain accurate result of solution of simultaneous linear equations, algebraic methods are used.
There are three algebraic methods to learn for this year.
(a) Method of elimination by substitution.
(b) Method of elimination by equating the coefficients.
(c) Method of cross multiplication.
Conditions for solvability of a pair of linear equations in two variables:
Consider the pair of linear equations \[{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0;\,{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0\]
Quadratic equation: An equation of the form \[a{{x}^{2}}+bx+c=0\] where a, b and\[c\in R\]and \[a\ne 0\]is called a quadratic equation.
If p(x) is a quadratic polynomial, then p(x) = 0 is called a quadratic equation.
Note: (i) An equation of degree 2 is called a quadratic equation.
(ii) The quadratic equation of the form \[a{{x}^{2}}+bx+c=0\].
Solution or roots of a quadratic equation: If p(x) = 0 is a quadratic equation, then the zeros of the polynomial p(x) are called the solutions or roots of the quadratic equation p(x)=0.
Note: (i) Since the degree of a quadratic equation is 2, it has 2 roots or solutions.
(ii) x = a is the root of p(x) = 0, if p(a) = 0.
(iii) Finding the roots of a quadratic equation is called solving the quadratic equation.
Methods of solving a quadratic equation: There are different methods of solving a quadratic equation.
(a) Factorization method
(i) Splitting the middle term
(ii) Completing the square
(b) Formula method
(a) (i) Splitting the middle term: Consider the quadratic equation\[a{{x}^{2}}+bx+c=0\].
Step 1: Find the product of the coefficient of\[{{x}^{2}}\]and the constant term i.e., ac.
Step 2: (a) If ac is positive, then choose two factors of ac, whose sum is equal to b (the coefficient of the middle term).
(b) If ac is negative, then choose two factors of ac, whose difference is equal to b (the coefficient of the middle term).
Step 3: Express the middle terms as the sum (or difference) of the two factors obtained in step 2. [Now the quadratic equation has 4 terms]
Step 4: Separate the term common to the first two terms and then write the first two terms as a product. Take the common term (binomial) out of the last two terms and get another factor so that the last two terms are written as a product.
Step 5: Express the given quadratic equation as a product of two binomials, and solve them.
Step 6: The two values obtained in step 5 are the roots of the given quadratic equation.
Note: An important property used in solving a quadratic equation by splitting the middle term.
"If ab = 0, then either a = 0, or b = 0 or both a and b are 0, where 'a' and 'b' are real numbers"
(ii) Completing the square: In some cases where the given quadratic equation can be solved by factorization, a suitable term is added and subtracted. Then terms are regrouped in such a manner that a square is completed by three of the terms. The equation is then solved using factorization method.
Note: Usually, the term added and subtracted is the square of half the coefficient of x.
Formula method: The roots of a quadratic equation\[a{{x}^{2}}+bx+c=0\]are given by \[\frac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\]provided\[{{b}^{2}}-4ac\ge 0\]. This formula for finding more...
Sequence: Numbers arranged in a definite order according to definite rule are said to be in a sequence.
Term: Each number of a sequence is called a term.
\[{{\mathbf{n}}^{\mathbf{th}}}\]term: The term occurring at the\[{{n}^{th}}\]place of a sequence is called its n"1 term, usually denoted by\[{{t}_{n}}\].
Progressions: Sequences that follow a definite pattern are called progressions.
Arithmetic progressions: sequence in which each term differs from its preceding term by a fixed number (constant) is called an arithmetic progression, denoted as A.P.
Common Difference: The fixed number by which any two successive terms of an A.P. differ is called the common difference of A.P. denoted by 'd'. So,\[\text{d}={{\text{t}}_{n}}-{{t}_{n-1}}\].
An A.P. of 'n' terms with first term 'a' and common difference 'd' is a, a +d,...a +(n- 1)d.
Arithmetic series: A series obtained by adding the terms of an A.P. is called an arithmetic series.
The general term (\[{{\mathbf{n}}^{\mathbf{th}}}\]term) of an A.P.: If the first term of an A.P. is 'a' and the common
difference is 'd', then its n111 term is given by\[{{t}_{n}}=a+(n-1)d\].
The general term from the end of an A.P.: If 'a' is the first term, 'd' the common difference and \['l'\]the last term of a given A.P., then its \[{{n}^{th}}\]term from the end is \[l-(n-1)d\].
Selection of term of an A.P.: Terms of an A.P. must be selected in such a way, that on taking the sum of the terms, one unknown is eliminated automatically.
(a)To select three terms of an A.P. with common difference 'd', choose a - d, a, a + d.
(b) To select four terms of an A.P. with common difference 2d, choose a - 3d, a - d, a + d, a + 3d.
(c) To select five terms of an A.P. with common difference d, choose a - 2d, a - d, a, a + d, a + 2d.
(d) To select six terms of an A.P. with common difference 2d, choose \[\text{a}-\text{5d},\text{ a}-\text{3d}\], \[\text{a}-\text{d},\text{ a},\text{ a}+\text{d},\text{ a}+\text{3d},\text{ a}+\text{5d}\]
The sum to 'n' terms of an A.P.: The sum of first 'n' terms of an A.P. is given by \[S=\frac{n}{2}[2a+(n-1)d]\], where 'a' is the first term and 'd' is the common difference.
Arithmetic Mean: If a, A and b are in A.P., then A is said to be the arithmetic mean (A.M.) between a and b. The arithmetic mean between two numbers 'a' and 'b' is given by\[(a+b)/2\].
Similar figures: Two figures having the same shape (not necessarily the same size) are called similar figures.
Congruent figures: Two figures having the same shape and the same size are called congruent figures.
Note: Congruent figures are similar but similar figures are not congruent.
Similar polygons: Two polygons of the same number of sides are similar; if their corresponding angles are equal and their corresponding sides are in the same ratio (or proportion)
Note: The same ratio of the corresponding sides is referred to as the scale factor (or the Representative Fraction) for the polygons.
Two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional.
The ratio of any two corresponding sides in two equiangular triangles is always the same.
Basic Proportionality Theorem; If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides divided in the same ratio.
Converse of basic proportionality theorem: If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.
Criteria for similarity of triangles: If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides divided in the same ratio.
Symbol for similarity: \[\sim \]
Symbol forcongruency: \[\cong \]
(a) A.A.A. Similarity: If in two triangles, the corresponding angles are equal and their corresponding sides are proportional, then the two triangles are similar.
Corollary: (AA similarity) If two angles of one triangle are respectively equal to two angles of another triangle, then two triangles are similar.
(b) S.S.S. Similarity: If in two triangles, the sides of one triangle are proportional to the sides of the other triangle, and their corresponding angles are equal, then the triangles are similar.
(c) S.A.S. Similarity: If one angle of and the sides including it of a triangle are proportional to an angle and the sides including it, of another triangle then the two triangles are similar.
If two triangles are equiangular, then the ratio of their corresponding sides is the same as the ratio of their corresponding (a) medians (b) altitudes and (c) angle bisector segments.
The ratio of the areas of two similar triangles is equal to the ratio of the squares of any two corresponding sides.
The areas of two similar triangles are in the ratio of the squares of the corresponding altitudes.
The areas of two similar triangles are in the ratio of the squares of the corresponding angle bisector segments.
If equilateral triangles are drawn on the sides of a right angled triangle, then the area of the triangle on the hypotenuse is more...
The branch of geometry that sets up a definite correspondence between the position of a point in a plane and a pair of algebraic numbers called coordinates is called Coordinate geometry.
The distance of a point from the Y-axis is called the X-coordinate or Abscissa.
The distance of a point from the X-axis is called the Y-coordinate or ordinate.
The coordinates of a point on the X-axis are of the form (x, 0) and of a point on the Y-axis are of the form (0, y).
The abscissa and ordinate of a point taken together is known as coordinates of a point.
The point of intersection of the axes of coordinates is called the origin.
The quarter plane that results from the division of the plane by the coordinate axes is called a quadrant.
Signs of x and y coordinates in the four quadrants:
Quadrant
X-coordinate
y-coordinate
Sign of the coordinates of the point
I
Positive
Positive
\[(+,+)\]
II
Negative
Positive
\[(-,+)\]
III
Negative
Negative
\[(-,-)\]
IV
Positive
Negative
\[(+,-)\]
Distance Formula:
The distance between two points \[A({{x}_{1}},{{y}_{1}})\] and \[B({{y}_{2}},{{y}_{2}})\] is given by
The distance of the point \[p(x,y)\]from the origin 0 (0,0) is given by \[OP=\sqrt{{{x}^{2}}+{{y}^{2}}}\].
Properties of various types of quadrilaterals: a quadrilateral is a
(a) Rectangle: If its opposite sides are equal and the diagonals are equal.
(b) Square: If all its sides are equal and the diagonals are equal.
(c) Parallelogram: If its opposite sides are equal.
(d) Parallelogram but not a rectangle: If its opposite sides are equal and the diagonals are not equal.
(e) Rhombus but not a square: If all its sides are equal and the diagonals are not equal.
Collinear points: Points are said to be collinear if they lie on the same straight line.
Test for co linearity of given points: Given three points A, B and C, find the distances AB, BC and AC. If the sum of any two of these distances is equal to the third distance, then the given points are collinear.
Section formula: The coordinates of the point \[p(x,y)\] which divides the line segment joining \[A({{x}_{1}},{{y}_{1}})\]and \[B({{x}_{2}},{{y}_{2}})\]internally in the ratio m : n are given by \[x=\frac{m{{x}_{2}}+n{{x}_{1}}}{m+n}\]and \[y=\frac{m{{y}_{2}}+n{{y}_{1}}}{m+n}\]
Note: The coordinates of a point \[p(x,y)\]which divides the line segment joining \[A({{x}_{1}},{{y}_{1}})\]and\[B({{x}_{2}},{{y}_{2}})\]externally in the ratio m: n are given more...
Full names of the trigonometric ratios: sin = sine; cos = cosine; tan = tangent; cosec = cosecant; sec = secant; cot = cotangent
Note: (a) \[\sin \theta \] is a single symbol. It does not mean the product of sine and\[\theta \].
(b) In short, t-ratios is used for trigonometric ratios.
(c) Trigonometric ratios are real numbers.
Reciprocal relations: The reciprocal relations for the trigonometric ratios are (a) \[\frac{1}{\sin \theta }=\cos ec\theta \] (b) \[\frac{1}{\cos \theta }=sec\theta \]
Trigonometric identities: An equation involving trigonometric ratios of angle\[\theta \]is said to be an identity if it is satisfied for all values of\[\theta \].
Line of sight: The imaginary line drawn from the eye of the observer to the object, when the observer is looking at it is called the line of sight,
Note: Line of sight is called the line of vision.
Angle of elevation: The angle formed by the line of sight with the horizontal when the object is above the horizontal level is called the angle of elevation,
Note: When the observer looks at an object raising his, head an angle of elevation is formed.
Angle of depression: The angle formed by the line of sight with the horizontal when the object is below the horizontal level is called the angle of depression.
Note: When the observer looks at an object lowering his head, an angle of depression is formed.
The height or length of an object or the distance between two distant objects can be determined with the help of trigonometric ratios.
Secant: A line which intersects a circle at two distinct points is called a secant of a circle.
Tangent: A line touching a circle at exactly one point only is called a tangent to the circle at that point.
Point of contact: The point P at which the tangent touches the circle is called the point of contact.
Number of tangents to a circle:
Position of the point w.r.t. the circle
Number of tangents
Inside
0
On
1
Outside
2
Length of a tangent: The length of the line segment of the tangent between a given point and the given point of contact with the circle is called the length of the tangent from the point to the circle.
The tangent at any point of a circle is perpendicular to the radius through the point of contact. In other words, the angle between a tangent and the radius through the point of contact is\[{{90}^{o}}\].
The lengths of tangents drawn from an external point to a circle are equal.
If AP and AQ are two tangents from an external point A to the circle, then AP = AQ.
Two tangents drawn from an external point subtend equal angles at the centre and are equally inclined to the line segment joining the centre to that point.
The tangents drawn at the ends of a diameter of a circle are parallel.
The line segments joining the point of contact of two parallel tangents to a circle is a diameter of the circle.
The angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segments joining the points of contact to the centre.
There is one and only one tangent at any point on the circumference of a circle.
A parallelogram circumscribing a circle is a rhombus.
(a) \[\sin \theta =\frac{side\,opposite\,to\,\theta }{Hypotenuse}=\frac{AB}{AC}\]
(b) \[\cos \theta =\frac{side\,adjacent\,to\,\theta }{Hypotenuse}=\frac{BC}{AC}\]
(c) \[\tan \theta =\frac{side\,opposite\,to\,\theta }{side\,adjacent\,to\,\theta }=\frac{AB}{BC}\]
(d) \[\cos ec\theta =\frac{Hypotenuse}{side\,opposite\,to\,\theta }=\frac{AC}{AB}\]
(e) \[sec\theta =\frac{Hypotenuse}{side\,adjacent\,to\,\theta }=\frac{AC}{BC}\]
(f) \[\cot \theta =\frac{side\,adjacent\,to\,\theta }{side\,opposite\,to\,\theta }=\frac{BC}{AB}\]
Note: Line of sight is called the line of vision.
Note: When the observer looks at an object raising his, head an angle of elevation is formed.
Note: When the observer looks at an object lowering his head, an angle of depression is formed.
