Mathematics Grade

10

1. Real Numbers2. Polynomials3. Pair of Linear Equations in Two Variables4. Quadratic Equations5. Arithmetic Progressions6. Triangles7. Coordinate Geometry8. Introduction to Trigonometry9. Some Applications of Trigonometry10. Circles11. Constructions12. Areas Related to Circles13. Surface Areas and Volumes14. Statistics15. Probability

Real Numbers 1. Euclid’s division lemma:Given positive integers a and b, there exist whole numbers q and r satisfying a = bq + r,0 ≤ r < b.2. Euclid’s division algorithm: This is based on Euclid’s division lemma. According to this,the HCF of any two positive integers a and b, with a > b, is obtained as follows:Step 1: Apply the division lemma to find q and r where a = bq + r, 0 ≤ r < b.Step 2: If r = 0, the HCF is b. If r ≠ 0, apply Euclid’s lemma to b and r.Step 3: Continue the process till the remainder is zero. The divisor at this stage will beHCF (a, b). Also, HCF (a, b) = HCF (b, r).3. The Fundamental Theorem of Arithmetic:Every composite number can be expressed (factorized) as a product of primes, and thisfactorization is unique, apart from the order in which the prime factors occur.4. If p is a prime and p divides a2, then p divides a, where a is a positive integer.5. To prove that √2, √3 are irrationals.6. Let x be a rational number whose decimal expansion terminates. Then we can express x in the form p/q, where p and q are coprime, and the prime factorization of q is of the form2n5m, where n, m are non-negative integers.7. Let x = p/q be a rational number, such that the prime factorization of q is of the form 2n5m,where n, m are non-negative integers. Then x has a decimal expansion which terminates.

8. Let x = p/q be a rational number, such that the prime factorization of q is not of the form 2n5m, where n, m are non-negative integers. Then x has a decimal expansion which is non-terminating repeating (recurring).

Polynomials

1. If p(x) is a polynomial in x, the highest power of x in p(x) is called the degree of the polynomial p(x).

2. A polynomial of degree 1 is called a linear polynomial.

3. A polynomial of degree 2 is called a quadratic polynomial.

4. A polynomial of degree 3 is called a cubic polynomial.

5. If p(x) is a polynomial in x, and if k is any real number, then the value obtained by replacing x by k in p(x), is called the value of p(x) at x = k, and is denoted by p(k).

6. A real number k is said to be a zero of a polynomial p(x), if p(k) = 0.

7. In general, if k is a zero of p(x) = ax + b, then p(k) = ak + b = 0, i.e., k = -b/a.

8. So, the zero of the linear polynomial ax + b is -b/a = - (Constant term)/Coefficient of a

9. In general, for a linear polynomial ax + b, a ≠ 0, the graph of y = ax + b is a

10. straight line which intersects the x-axis at exactly one point, namely,(-b/a , 0). Therefore, the linear polynomial ax + b, a ≠ 0, has exactly one zero, namely, the x-coordinate of the point where the graph of y = ax + b intersects the x-axis.

11. The zeroes of a quadratic polynomial ax2 + bx + c, a ≠ 0, are precisely the x-coordinates of the points where the parabola representing y = ax2 + bx + c intersects the x-axis.

12. A quadratic polynomial can have either two distinct zeroes or two equal zeroes (i.e., one zero), or no zero. This also means that a polynomial of degree 2 has atmost two zeroes.

13. In general, given a polynomial p(x) of degree n, the graph of y = p(x) intersects the x-axis at atmost n points. Therefore, a polynomial p(x) of degree n has at most n zeroes.

14. In general, if α and β are the zeroes of the quadratic polynomial p(x) = ax2 + bx + c, a ≠ 0, then you know that x – α and x – β are the factors of p(x). Therefore,

ax2 + bx + c = k(x – α) (x – β), where k is a constant = k[x2 – (α + β)x + α β] = kx2 – k(α + β)x + k α β

15. Comparing the coefficients of x2, x and constant terms on both the sides, we get a = k, b = – k(α + β) and c = kαβ.

This gives α + β = -b/ aαβ = c/a

16. In general, it can be proved that if α, β, γ are the zeroes of the cubic polynomial ax3 + bx2 + cx + d, then

17. Quadratic polynomial : x2 –(α + β)x + αβ18. Cubic polynomial: x3 –(α + β + γ) x2 +(αβ + βγ + γα)x - αβγ

If p(x) and g(x) are any two polynomials with g(x) ≠ 0, then we can find polynomials q(x) and r(x) such that p(x) = g(x) × q(x) + r(x), where r(x) = 0 or degree of r(x) < degree of g(x). This result is known as the Division Algorithm for polynomials.

Pair of Linear Equations in Two Variables1. You also know that an equation which can be put in the form ax

+ by + c = 0, where a, b and c are real numbers, and a and b are not both zero, is called a linear equation in two variables x and y.

2. Every solution of the equation is a point on the line representing it.

3. In fact, this is true for any linear equation, that is, each solution (x, y) of a linear equation in two variables, ax + by + c = 0, corresponds to a point on the line representing the equation, and vice versa.

4. The general form for a pair of linear equations in two variables x and y is a1x + b1y + c1 =0 and a2x + b2y + c2 = 0, where a1, b1, c1, a2, b2, c2 are all real numbers and a1

2 + b12 ≠ 0, a2

2 + b22 ≠

0.5. A pair of linear equations which has no solution, is called an

inconsistent pair of linear equations. 6. A pair of linear equations in two variables, which has a solution,

is called a consistent pair of linear equations. 7. A pair of linear equations which are equivalent has infinitely

many distinct common solutions. Such a pair is called a dependent pair of linear equations in two variables.

8. (i) the lines may intersect in a single point. In this case, the pair of equations has a unique solution (consistent pair of equations).(ii) the lines may be parallel. In this case, the equations have no solution (inconsistent pair of equations). (iii) the lines may be coincident. In this case, the equations have infinitely many solutions [dependent (consistent) pair of equations].

9.

10.

Quadratic Equations1. ax2 + bx + c = 0, a ≠ 0 is called the standard form of a

quadratic equation.2. In general, a real number α is called a root of the quadratic

equation ax2 + bx + c = 0, a ≠ 0 if a α2 + bα + c = 0. We also say that x = α is a solution of the quadratic equation, or that α satisfies the quadratic equation. Note that the zeroes of the quadratic polynomial ax2 + bx + c and the roots of the quadratic equation ax2 + bx + c = 0 are the same.

3. Completing the square method:a. Quadratic equation – 3x2 + 5x + 2 = 0b. Convert the equation to the form where 3x2 becomes x2

and LHS = 0c. 3x2 + 5x + 2 = 0 becomes x2 + (5x/3) + 2/3 = 0d. x2 + (5x/3) = -2/3e. Add {(5/3)*(1/2)}2 on LHS and RHSf. {x + (5/6)}2 = -2/3 + (5/6)2

= -2/3 + 25/36= -24+25/36 = 1/36(x + 5/6) = √1/36(x + 5/6) = + or – 1/6

(x + 5/6) = 1/6 (x + 5/6) = -1/6x = -4/6 x = -6/6 = -1

= -2/34. Quadratic Formula:

a. Nature of rootsD – discriminantD = b2 – 4acIf D > 0, 2 distinct real rootsIf D = 0, 2 equal real rootsIf D < 0, No real roots

b. Root of ax2 + bx + c = 0 is given by:x = (-b (+ or -) √D)/2a

Arithmetic Progressions

Triangles1. Two figures having the same shape but not necessarily the same size are called similar figures. 2. All the congruent figures are similar but the converse is not true. 3. Two polygons of the same number of sides are similar, if (i) their corresponding angles are equal and

(ii) their corresponding sides are in the same ratio (i.e., proportion). 4. 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 are divided in the same ratio. 5. If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side. 6. If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio and hence the two triangles are similar (AAA similarity criterion). 7. If in two triangles, two angles of one triangle are respectively equal to the two angles of the other triangle, then the two triangles are similar (AA similarity criterion). 8. If in two triangles, corresponding sides are in the same ratio, then their corresponding angles are equal and hence the triangles are similar (SSS similarity criterion). 9. If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are in the same ratio (proportional), then the triangles are similar (SAS similarity criterion). 10. The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. 11. If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, then the triangles on both sides of the perpendicular are similar to the whole triangle and also to each other. 12. In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (Pythagoras Theorem). 13. If in a triangle, square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.

Coordinate GeometryThe distance of a point from the y-axis is called its x-coordinate, or abscissa. The distance of a point from the x-axis is called its 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).

Introduction to Trigonometry

Some Applications of Trigonometry1. The line of sight is the line drawn from the eye of an observer

to the point in the object viewed by the observer. 2. The angle of elevation of the point viewed is the angle formed

by the line of sight with the horizontal when the point being viewed is above the horizontal level, i.e., the case when we raise our head to look at the object.

3. The angle of depression of a point on the object being viewed is the angle formed by the line of sight with the horizontal when the point is below the horizontal level, i.e., the case when we lower our head to look at the point being viewed.

4.

Circles

1. A tangent to a circle is a line that intersects the circle at only one point.

2. There is only one tangent at a point of the circle.3. The tangent to a circle is a special case of the secant, when the

two end points of its corresponding chord coincide.4. The common point of the tangent and the circle is called the

point of contact and the tangent is said to touch the circle at the common point.

5. Theorem 10.1: The tangent at any point of a circle is perpendicular to the radius through the point of contact.

6. By theorem above, we can also conclude that at any point on a circle there can be one and only one tangent.

7. The line containing the radius through the point of contact is also sometimes called the ‘normal’ to the circle at the point.

8. Case 1: There is no tangent to a circle passing through a point lying inside the circle. Case 2: There is one and only one tangent to a circle passing through a point lying on the circle. Case 3: There are exactly two tangents to a circle through a point lying outside the circle.

9. The length of the segment of the tangent from the external point P and the point of contact with the circle is called the length of the tangent from the point P to the circle.

10. Theorem 10.2: The lengths of tangents drawn from an external point to a circle are equal.

Constructions

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4.

5.

Areas Related to Circles

Surface Areas and Volumes

Statistics1.2.Finding the mean:

1)Direct method

where x – mean and xi means mark Sigma fi – sum of all the frequencies

Sigma fixi – sum of the products of fi and xi2) Assumed Mean methodThe first step is to choose one among the xi as the assumed mean, and denote it by ‘a’- either of the middlemost xi.

The next step is to find the difference di between a and each of the xi ’s, that is, the deviation of ‘a’ from each of the xi. i.e., di = xi – a.The third step is to find the product of di with the corresponding fi, and take the sum of all the fidi’s.

3)Step Deviation method

h – class size ui – mark 0 at middlemost mean’s class

Mode

MedianIf n is odd, the median is the (n+1)/2th observation. And, if n is even, then the median will be the average of n/2th and (n+1)/2th observations.The median class is the class containing the value equal to or greater than n/2.

Empirical formula - 3 Median = Mode + 2 MeanLess than Ogive – Compute the <cf

Mark the <cf towards the upper limitsX axis mark upper limits and Y axis mark cfMore than Ogive – Compute the >cf

Mark the >cf towards the lower limitsX axis mark lower limits and Y axis mark cfMedian can be found either by:

Probability

0 ≤ P(E) ≤ 1A deck of cards consists of 52 cards which are divided into 4 suits of 13 cards each— spades (♠), hearts (♥), diamonds (♦) and clubs (♣). Clubs and spades are of black colour, while hearts and diamonds are of red colour. The cards in each suit are ace, king, queen, jack, 10, 9, 8, 7, 6, 5, 4, 3 and 2. Kings, queens and jacks are called face cards.

Probability for rolling 2 dice