Math Mastery Made Easy: A Quick Guide to CBSE Grade 10 Maths

Introduction

If one zero of the quadratic polynomial x2+ 3x + k is 2, then the value of k is

(A) 10 (B) –10 (C) 5 (D) –5

Geometrical Meaning of the Zeroes of a Polynomial

Look at the graphs in Figure given below. Each is the graph of y = p(x), where p(x) is a polynomial. For each of the graphs, find the number of zeroes of p(x).

Which of the following is not the graph of a quadratic polynomial?

In the adjoining figure, the graph of f(x) is drawn. Find the number of zeroes of f(x).

The graph of y = f(x) is given in the adjoining figure. Write the number of zeroes of f(x).

Relationship between Zeroes and Coefficients of a Quadratic Polynomial & Formation of Quadratic Polynomial.

Find the zeroes of the quadratic polynomial x2 + 7x + 10, and verify the relationship between the zeroes and the coefficients.

Find the zeroes of the polynomial x2 – 3 and verify the relationship between the zeroes and the coefficients.

If m and n are the zeroes of the polynomial 3x2 + 11x – 4, find the value of m/n + n/m.

Find a quadratic polynomial, the sum and product of whose zeroes are –3 and 2, respectively.

If one zero of 2x2 – 3x + k is reciprocal to the other, then find the value of k

If zeroes α and β of a polynomial  x2 – 7x + k are such that α – β = 1, then find the value of k.

If α and β are the zeroes of the polynomial 2y2 + 7y + 5, then find the value of α + β + αβ.

Relationship between Zeroes and Coefficients of a Cubic Polynomial & Formation of Cubic Polynomial:

Verify that 3, –1, –  1/3 are the zeroes of the cubic polynomial p(x) = 3x3 – 5x2 – 11x – 3, and then verify the relationship between the zeroes and the coefficients.

Find the cubic polynomial whose three zeroes are 3, -1 and –  1/3.

If α, β and γ are zeroes of the polynomial 6x3 + 3x2 - 5x + 1, then find the value of α−1 + β−1 + γ−1.

Given that the zeroes of the cubic polynomial x3 – 6x2 + 3x + 10 are of the form a, a + b, a + 2b for some real numbers a and b, find the values of a and b as well as the zeroes of the given polynomial.

Division Algorithm for Polynomials:

Divide2x2 + 3x + 1 by x + 2.

Divide3x3 + x2 + 2x + 5 by 1 + 2x + x2.

Divide 3x2 – x3 – 3x + 5 by x – 1 – x2, and verify the division algorithm.

On dividing x3 - 5x2 + 6x + 4 by a polynomial g(x), the quotient and the remainder were x - 3 and 4 respectively. Find g(x).

Division Algorithm for Polynomials & Finding remaining zeroes

Find all zeroes of the polynomial f(x) = x3+ 13x2 + 32x + 20. If one of its zeroes is - 2.

Find all the zeroes of 2x4 – 3x3 – 3x2 + 6x – 2, if you know that two of its zeroes are √2 and −√2.

Obtain all other zeroes of the polynomial x4 + 5x3− 2x2− 40x − 48, if two of its zeroes are 2√2 and −2√2

Introduction to Pair of Linear Equations:

Find the value of ‘a’ so that the point (a, 5) lies on the line represented by 2x - 3y = 5  ?

If x = α and y = β is the solution of the equations x - 2y = 2 and x + 2y = 4, then find the values of α and β?

Given that 2x + 3y = 10, x - 3y = -4 and y = mx + 3, then find the value of m ?

Method to Write a Given Statement as a Pair of Linear Equations in Two Variables (Algebraic representation):-

Akhila went to a fair in her village. She wanted to enjoy rides on the Giant Wheel and play Hoopla (a game in which you throw a ring on the items kept in a stall, and if the ring covers any object completely, you get it). The number of times she played Hoopla is half the number of rides she had on the Giant Wheel. If each ride costs ₹3, and a game of Hoopla costs ₹4, how would you find out the number of rides she had and how many times she played Hoopla, provided she spent ₹20. Represent this situation algebraically .

Seven times a two digit number is equal to four times the number obtained by reversing the order of its digits. If the difference between the digits is 3.Represent this situation algebraically .

The ratio of incomes of two persons is 11 : 7 and the ratio of their expenditures is 9 : 5. If each of them manages to save ₹400 per month. Represent this situation algebraically.

Five years ago, sagar was twice as old as Tiru. After 10 years Sagar's age will be ten years more than Tiru's age. To find their present ages form system of Linear Equations.

Method to write a given statement as a pair of L.E.s in 2 variables (Algebraically) and representing graphically (Geometrically).

Graphical Method of Solution of a Pair of Linear Equations: 

Akhila went to a fair in her village. She wanted to enjoy rides on the Giant Wheel and play Hoopla. The number of times she played Hoopla is half the number of rides she had on the Giant Wheel. If each ride costs ₹3, and a game of Hoopla costs ₹4, how would you find out the number of rides she had and how many times she played Hoopla, provided she spent ₹20. Represent this situation algebraically and graphically (geometrically).

Romila went to a stationery shop and purchased 2 pencils and3 erasers for ₹9. Her friend Sonali saw the new variety of pencils and erasers with Romila, and she also bought 4 pencils and 6 erasers of the same kind for₹18. Represent this situation algebraically and graphically.

Two rails are represented by the equations x + 2y – 4 = 0 and 2x+ 4y – 12 = 0. Represent this situation geometrically.

Champa went to a ‘Sale’ to purchase some pants and skirts. When her friends asked her how many of each she had bought, she answered, “The number ofskirts is two less than twice the number of pants purchased. Also, the number of skirts is four less than four times the number of pants purchased”. Help her friends to find how many pants and skirts Champa bought.

Behaviour of lines representing a pair of linear equations (Checking consistency).

On comparing the ratios a_1/a_2 ,b_1/b_2  and c_1/c_2 find out whether the lines representing the following pairs of linear equations intersect at a point or parallel or coincide:

3x - y = 7

2x + 5y + 1 = 0

Check graphically whether the pair of equations x + 3y = 6 and 2x – 3y = 12 is consistent. If so, solve them graphically.

Graphically, find whether the following pair of equations has no solution, unique solution or infinitely many solutions:

5x – 8y + 1 = 0

3x - 24/5y + 3/5= 0

For which values of p does the pair of equations given below has unique solution?

4x + py + 8 = 0

2x + 2y + 2 = 0

For what values of k will the following pair of linear equations have infinitely many solutions?

kx + 3y – (k – 3) = 0

12x + ky – k = 0

Algebraic Methods of Solving a Pair of Linear Equations: Substitution Method:

Solve the following pair of equations by substitution method:

7x – 15y = 2

x + 2y = 3

Two rails are represented by the equations x + 2y – 4 = 0 and 2x + 4y – 12 = 0. Will the rails cross each other?

Aftab tells his daughter, “Seven years ago, I was seven times as old as you were then. Also, three years from now, I shall be three times as old as you will be." then find their present ages?

Romila went to a stationery shop and purchased 2 pencils and 3 erasers for ₹9. Her friend Sonali saw the new variety of pencils and erasers with Romila, and she also bought 4 pencils and 6 erasers of the same kind for ₹18. Find the cost of each pencil and each eraser.

Algebraic Methods of Solving a Pair of Linear Equations: Elimination Method

Solve the following pair of linear equations by the elimination method:  3x + 4y = 10 and 2x – 2y = 2.

Use elimination method to find all possible solutions of the following pair of linear equations:

2x + 3y = 8

4x + 6y = 7

The ratio of incomes of two persons is 9: 7 and the ratio of their expenditures is 4 : 3. If each of them manages to save₹2000 per month, find their monthly incomes.

The sum of a two-digit number and the number obtained by reversing the digits is 66. If the digits of the number differ by 2,find the number. How many such numbers are there?

Algebraic Methods of Solving a Pair of Linear Equations: Cross-Multiplication Method

Solve using cross multiplication method:

5x + 4y - 4 = 0

x - 12y - 20 = 0

Solve the following pair of linear equations by cross multiplication method:

x + 2y = 2

x - 3 y = 7

From a bus stand in Bangalore, if we buy 2 tickets to Malleswaram and 3 tickets to Yeshwanthpur, the total cost is ₹46; but if we buy 3 tickets to Malleswaram and 5 tickets to Yeshwanthpur the total cost is ₹74. Find the fares from the bus stand to Malleswaram and to Yeshwanthpur.

Equations Reducible to a Pair of Linear Equations in Two Variables

Solve the pair of equations:

2/x+ 3/y= 13;

5/x- 4/y= - 2

Solve the following pair of equations by reducing them to a pair of linear equations:

5/(x - 1) + 1/(y - 2) = 2;

6/(x - 1) - 3/(y - 2) = 1

A boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours, it can go 40 km upstream and 55 km down-stream. Determine the speed of the stream and that of the boat in still water.

Introduction

Determine the value of ‘k’ for which the given value is a solution of the equation.

x2 + 2ax + k = 0, x = – a

If one root of the equation x2 + ax + 3 = 0 is 1,

then its other root is

A. 2

B. -2

C. 3

D. -3

Check whether –3 is a solution of the equation 3x2 + 5x + 2 = 0.

Representing word problems as Quadratic Equations 

Represent the following situations mathematically:

John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and the product of the number of marbles they now have is 124. We would like to find out how many marbles they had to start with.

Represent the following situations mathematically:

A cottage industry produces a certain number of toys in a day. The cost of production of each toy (in rupees) was found to be 55 minus the number of toys produced in a day. On a particular day, the total cost of production was ₹ 750. We would like to find out the number of toys produced on that day.

The hypotenuse of a right angled triangle is 6 metres more than twice the shortest side. If the third side is 2 meters less than the hypotenuse, find the sides of the triangle.

A man travels a distance of 300 km at a uniform speed. If the speed of the train is increased by 5 km an hour, the journey would have taken two hours less. Find the original speed of the train.

Method to check whether a given equation is quadratic or not & method to determine unknown constant involved in a Q.E.

Check whether the following are quadratic equations:

(i) (x – 2)2 + 1 = 2x – 3

(ii) x(x + 1) + 8 = (x + 2) (x – 2)

(iii) x (2x + 3) = x2 + 1

(iv) (x + 2)3 = x3 – 4

Which one of the following is not a quadratic equation?

(A) (x + 2)2= 2(x + 3)       

(B)  x2 + 3x = (–1)(1 – 3x)2

(C) (x + 2) (x – 1) = x2– 2x – 3

(D)  x3– x2+ 2x + 1 = (x + 1)3

Check whether 16/x – 1 = 15/(x+1) is a quadratic equation?

Check whether 1/(x+1) + 2/(x+2) = 4/(x+4) is a quadratic equation?

Learning Objectives:

How to find Solutions or Roots of a Quadratic Equation.

Different methods to find Solution of a Quadratic Equation.

Find the roots of the equation 2x2 – 5x + 3 = 0, by factorization.

Find the roots of the quadratic equation 6x2 – x – 2 = 0.

Find the roots of the quadratic equation 3x2− 2√6x + 2 = 0.

A charity trust decides to build a prayer hall having a carpet area of 300 square metres with its length one metre more than twice its breadth. What should be the length and breadth of the hall?

Solution of a Quadratic Equation by Completing the Square method:

Find the roots of the equation 2x2 – 5x + 3 = 0, by the method of completing the square.

Find the roots of the equation 5x2 – 6x – 2 = 0 by the method of completing the square.

Find the roots of x2 – 4x – 8 = 0 by the method of completing square.

Find the roots of 4x2 + 3x + 5 = 0 by the method of completing the square.

Solution of a Quadratic Equation by Quadratic Formula:

Find the roots of the following quadratic equations,

if they exist, using the quadratic formula:

(i) 3x2 – 5x + 2 = 0

(ii) x2 + 4x + 5 = 0

(iii) 2x2 – 2√2x + 1 = 0

Find the roots of the following equations:

(i) x + 1/x = 3,  x ≠ 0

(ii) 1/x-1/(x-2) = 3,  x ≠ 0, 2

The area of rectangular plot is 528m2. The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot. Solve the situation by using the quadratic formula.

Solution of a Quadratic Equation - word problems

Find two consecutive odd positive integers, sum of whose squares is 290.

A rectangular park is to be designed whose breadth is 3 m less than its length. Its area is to be 4 square meters more than the area of a park that has already been made in the shape of an isosceles triangle with its base as the breadth of the rectangular park and of altitude 12 m (see figure). Find its length and breadth.

A motor boat whose speed is 18 km/h in still water takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream.

P & Q are centres of circles of radii 9 cm and 2 cm respectively. PQ = 17 cm. R is the centre of the circle of radius x cm which touches given circles externally. Given that angle PRQ is 90°. Write an equation in x and solve it.

A pole has to be erected at a point on the boundary of a circular park of diameter 13 meters in such a way that the differences of its distances from two diametrically opposite fixed gates A and B on the boundary is 7 meters. Is it possible to do so? If yes, at what distances from the two gates should the pole be erected?

Nature of Roots :

Find the discriminant of the quadratic equation 2x2 – 4x + 3 = 0, and hence find the nature of its roots.

Find the discriminant of the equation 3x2 – 2x + 1/3 = 0 and hence find the nature of its roots. Find them, if they are real.

If – 3 is a root of quadratic equation 2x2 + px – 15 = 0, while the quadratic equation x2 − 4px + k = 0 has equal roots , find the value of ‘k ’ .

If the equation (1 + m2) x2 + 2 mcx + (c2 − a2) = 0 has equal roots, prove that c2 = a2 (1 + m2).

Find the value of  ‘k ’ for the quadratic equation kx (x − 2) + 6 = 0, so that they have two equal roots.

Find the non-zero values of ‘ k ’ for which the roots of the quadratic equation 9x2 - 3kx + k = 0 are real and equal.

Introduction:

For the AP: 3/2, 1/2,-1/2,-3/2, . . ., write the first term ‘a’ and the common difference ‘d’.

Find the common difference of the AP: 1/p, (1-p)/p, (1-2p)/p,....

Which of the following list of numbers form an AP? If they form an AP, write the next two terms:

(i) 4, 10, 16, 22, . . .   

(ii) 1, – 1, – 3, – 5, . . .

(iii) – 2, 2, – 2, 2, – 2, . . .

(iv) 1, 1, 1, 2, 2, 2, 3, 3, 3, . . .

In which of the following situations, do the lists of numbers involved form an AP? Give reasons for your answers.

1) The fee charged from a student every month by a school for the whole session, when the monthly fee is Rs 400. 

2) The fee charged every month by a school from Classes I to XII, when the monthly fee for Class I is Rs 250, and it increases by Rs 50 for the next higher class.

3) The amount of money in the account of Varun at the end of every year when Rs 1000 is deposited at simple interest of 10% per annum.

4) The number of bacteria in a certain food item after each second, when they double in every second.

Method to write an AP, when first term and common difference are given:

The nth term of an AP is 7 – 4n, then its common difference is

A .  – 3

B .  3

C .     4

D .  – 4

An A.P is defined by an = 4n + 5 , then write the sequence.

Write the first three terms of the AP, when ‘a’ and ‘d’ are as given below:

a = 1/2, d = −1/6

a = −5, d = −3

a = √2, d = 1/√2

Find the first 5 terms of the sequence defined by an = (−1)n – 1 × 2n and check whether the sequence is in AP?

Which is the next term of the AP: √2, √8, √18, √32,....?

For what value of k will k + 9, 2k − 1, and 2k + 7 are consecutive terms of an AP.

nth Term of an AP:

Type I: problem based on finding nth term when sequence of an AP is given.

Type II: Problems based on finding n when nth term and AP are given.

Find the 10th term of the AP: 2, 7, 12,  . . .

Which term of the AP: 21, 18, 15, . . . is – 81? Also, is any term 0? Give reason for your answer.

Check whether 301 is a term of the list of numbers 5, 11, 17, 23, . . .

Which term of the AP: 121, 117, 113, is its first negative term?

[Hint: Find n for an < 0]

How many two-digit numbers are divisible by 3?

If m times the mth term of an A.P is equal to n times its nth term, show that the (m + n)th term of the AP is zero.

nth Term of an AP:

• nth Term of an AP from the End(Last term):

• Middle Term

• problems based on finding the AP and nth term when its two terms are given

• Word problem

Determine the AP whose 3rd term is 5 and the 7th term is 9.

Find the middle term of the A.P. 213, 205, 197,.... 37.

Find the 11th term from the last term (towards the first term)of the AP: 10, 7, 4, . . . ,– 62.

A sum of ₹1000 is invested at 8% simple interest per year. Calculate the interest at the end of each year. Do these interests form an AP? If so, find the interest at the end of 30 years making use of this fact.

In a flower bed, there are 23 rose plants in the first row, 21 in the second, 19 in the third, and so on. There are 5 rose plants in the last row. How many rows are there in the flower bed?

Sum of First n Terms of an AP and Arithmetic Mean:

Find the sum of the first 22 terms of the AP: 8, 3, –2 . . .

Find the sum of :

(i) the first 1000 positive integers 

(ii) the first n positive integers

Solve the equation: 1 + 4 + 7 + 10 + .....+ x = 287.

Find the sum of first 24 terms of the list of numbers whose nth term is given by an = 3 + 2n?

The sum of the first terms of an A.P. is given by Sn = 2n2 + 3n. Find the sixteenth term of the A.P.

Sum of First n Terms of an AP – Word problems:

If the sum of the first 14 terms of an AP is 1050and its first term is 10, find the 20th term.

How many terms of the AP: 24, 21, 18 . . . must be taken so that their sum is 78?

Find the sum of the first 15 multiples of 8.

A manufacturer of TV sets produced 600 sets in the third year

and 700 sets in the seventh year. Assuming that the production increases uniformly by a fixed number every year, find :

(i) the production in the 1st year

(ii) the production in the 10th year

(iii) the total production in first 7 years

A sum of ₹700 is to be used in give seven cash prizes to students of a school for their overall academic performance. If each prize is ₹20 less than its preceding prize, find the value of each of the prizes.

Introduction & Similar Figures

In Figure, if ∆ABC ~ ∆DEF and their sides are of lengths (in cm) as marked along them, then find the lengths of the sides of each triangle.

In the given figure, △ABC ~△PQR. Find the value of y + z.

If in two triangles DEF and PQR, ∠D = ∠Q and ∠R = ∠E, then which of the following is not true?

EF/PR = DF/PQ

DE/PQ = EF/RP

DE/QR = DF/PQ

EF/RP = DE/QR

It is given that ABC ~ DFE, ∠A = 30°, ∠C = 50°, AB = 5cm, AC = 8 cm and DF= 7.5 cm. Then, the following is true:

DE = 12 cm, ∠F = 50°

DE = 12 cm, ∠F = 100°

EF = 12 cm, ∠D = 100°

EF = 12 cm, ∠D = 30°

Basic Proportionality Theorem (BPT)& Its applications.

If a line intersects sides AB and AC of a ∆ABC at D and E respectively and is parallel to BC, prove that (AD )/AB = (AE )/AC.

ABCD is a trapezium with AB || DC. E and F are points on

non-parallel sides AD and BC respectively such that EF is

parallel to AB. Show that AE/ED = BF/FC

In Fig. DE∥BC and CD∥EF. Prove that AD2 = AB × AF.

In △ABC, D and E are points on the sides AB and AC respectively, such that DE || BC. If AD = 4x - 3, AE = 8x - 7, BD = 3x - 1 and CE = 5x -3 , find the value of x.

Converse of Basic Proportionality Theorem &  its applications

In figure, PS/SQ = PT/TR and ∠PST = ∠PRQ. Prove that PQR is an isosceles triangle.

If the diagonals of a quadrilateral divide each other proportionally, prove that it is a trapezium.

In the given figure, ∠A = ∠B and AD = BE. Show that DE ∥ AB.

If D and E are points on the respective sides AB and AC. △ABC such that, AD = 6 cm, BD = 9 cm, AE = 8 cm, EC = 12 cm. Prove that DE || BC.

Criteria for Similarity of Triangles & related proofs.

• AAA Similarity Criterion

• SSS Similarity Criterion

• SAS Similarity Criterion

Problems Based on Similarity of Triangles

1. AAA Similarity Criterion:

If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio(or  proportion) and hence the two triangles are similar.

2. SSS Similarity Criterion:

If in two triangles, sides of one triangle are proportional to (i.e. in the same ratio ) to the side of the other triangle, then their corresponding angles are equal and hence the two triangles are similar.

3. SAS Similarity Criterion

If one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional, then two triangles are similar.

In the adjoining figure, △AHK is similar to △ABC. If AK = 10 cm, BC = 3.5 cm and HK = 7 cm, find AC.

Observe the figure and then find ∠P.

A girl of height 90 cm is walking away from the base of a lamp-post at a speed of 1.2 m/s. If the lamp is 3.6 m above the ground, find the length of her shadow after 4 seconds.

Proofs Based on Similarity of Triangles:

In figure, if PQ || RS, prove that ∆POQ ~ ∆SOR.

In figure, OA. OB = OC. OD. Show that ∠A = ∠C and ∠B = ∠D.

In figure, CM and RN are respectively the medians of ∆ABC and ∆PQR. If ∆ABC ~ ∆PQR, prove that:

∆AMC ~ ∆PNR

(CM )/RN =(AB )/PQ

∆CMB ~ ∆RNQ

In the given figure, AB ∥ PQ ∥ CD, AB = x, CD = y, PQ = z

Prove that (1 )/(x ) + (1 )/(y ) = (1 )/(z )

Areas of Similar Triangles& Problems Based on Areas of Similar Triangles

In figure, the line segment XY is parallel to side AC of ∆ABC and it divides the triangle into two parts of equal areas. Find the ratio AX/AB

In the given figure, PA/AQ = PB/BR = 3. If the area △PQR is 32 cm2, then find the area of the quadrilateral AQRB.

ΔABC and ΔDEF are similar and AB = 1/3DE, then find ar(ΔABC): ar(ΔDEF)

In the given figure, if DE∥BC and AD : DB = 5 : 4, then find (ar(△DFE) )/(ar(△CFB) )

Areas of Similar Triangles &Proofs Based on Areas of Similar Triangles

If △ABC ∼ △PQR and AD and PS are bisectors of corresponding angles A and P, then prove that

(ar(ΔABC) )/(ar(ΔPQR)) = AD^2/PS^2 .

If the area of two similar triangles is equal, prove that they are congruent.

Diagonals of a trapezium PQRS intersect each other at the point O, PQ ∥ RS and PQ = 3RS. Find the ratio of the areas of triangles △POQ and △ROS.

Prove that the area of the equilateral triangle drawn on the hypotenuse of a right angled triangle is equal to the sum of the areas of the equilateral triangles drawn on the other two sides of the triangle.

Pythagoras Theorem and its Applications.

In figure, ∠ACB = 900 and CD⊥ AB. Prove that (BC^2)/(AC^2 ) = BD/AD

In figure, if AD⊥ BC, prove that AB2 + CD2 = BD2 + AC2.

BL and CM are medians of a triangle ABC right angled at A. Prove that 4(BL2 + CM2) = 5BC2.

A ladder is placed against a wall such that its foot is at a distance of 2.5 m from the wall and its top reaches a window 6 m above the ground. Find the length of the ladder.

Proofs Based on Converse of Pythagoras Theorem:

O is any point inside a rectangle ABCD. Prove thatOB2 + OD2 = OA2 + OC2.

ΔABC is right angled at C. If p is the length of the perpendicular from C to AB and a, b, c are the lengths of the sides opposite ∠A, ∠B and ∠C respectively, then prove that 1/P^2  = 1/a^2  + 1/b^2

In a ΔABC, AD⊥BC and AD2 = BD × CD. Prove that ΔABC is a right triangle.

In an equilateral triangle of side 3√3 cm, find the length of the altitude.

Introduction:

In this chapter, first of all, we will review the concepts of coordinate geometry that we learnt in class IX. Further, in this chapter, we will learn how to find the distance between two points whose coordinates are given and to find the area of the triangle formed by three given points. We will also study to find the coordinates of the point which divides a line segment joining two given points in a given ratio.

Find the distance of :

1) The point P (2, 3) from the x-axis.

2) The point (4, 7) from the y-axis.

Find the coordinates of the point, where the line x - y = 5 cuts Y - axis.

If P is a point on x-axis such that its distance from the origin is 3 units, then the coordinates of a point Q on OY such that OP = OQ, are ?

1. (0, 0)

2. (3, 0)

3. (0, -3)

4. (0, 3)

Find the coordinates of the vertices of an equilateral triangle of side 2a.

Distance Formula & its applications.

Find the distance between the points A (0, 6) and B (0, –2).

Find the distance of the point P(–6, 8) from the origin.

Find a relation between x and y such that the point (x, y) is equidistant from the points (7, 1) and (3, 5).

Find the points of X-axis which are at a distance of 2√5 from the point (7, −4). How many such points are there?

Find a point on the y−axis which is equidistant from the points A (6, 5) and B (– 4, 3).

Distance Formula & its applications

Problems based on collinearity and equidistant points.

Problems based on geometrical figure.

Do the points (3, 2), (–2, –3) and (2, 3) form a triangle? If so, name the type of triangle formed.

Show that the points (1, 7), (4, 2), (–1, –1) and (– 4, 4) are the vertices of a square.

Figure shows the arrangement of desks in a classroom. Ashima, Bharti and Camella are seated at A(3, 1), B(6, 4) and C(8, 6) respectively. Do you think they are seated in a line? Give reasons for your answer.

Prove that the points (a, b + c), (b, c + a) and (c, a + b) are collinear.

Section Formula & mid-point formula derivation of Section Formula and mid-point formula: And their applications.

Find the coordinates of the point which divides the line segment joining the points (4, – 3) and (8, 5) in the ratio 3: 1 internally.

In what ratio does the point (– 4, 6) divide the line segment joining the points A (– 6, 10) and B (3, – 8)?

Determine the ratio in which the point P (m, 6) divides the join of A(–4, 3) and B(2, 8). Also find the value of m.

Find the coordinates of a point A, where AB is the diameter of a circle whose centre is (2, -3) and B is (1, 4).

Section Formula

Mid-point

Centroid formula

points of trisection

Find the coordinates of the points of trisection (i.e., points dividing in three equal parts) of the line segment joining the points A(2, – 2) and B(– 7, 4).

Find the ratio in which the y-axis divides the line segment joining the points (5, – 6) and (–1, – 4). Also find the point of intersection.

If the points A (6, 1), B (8, 2), C (9, 4) and D (p, 3) are the vertices of a parallelogram, taken in order, find the value of p.

If the point C (–1, 2) divides internally the line segment joining the points A (2, 5) and B(x, y) in the ratio 3: 4, find the value of x2 + y2.

Area of a Triangle & its applications.

Find the area of a triangle whose vertices are(1, –1), (– 4, 6) and(–3, –5).

Find the area of a triangle formed by the points A (5, 2), B (4, 7) and C (7, – 4).

Find the area of the triangle formed by the points P(–1.5, 3), Q (6, –2)and R (–3, 4).

If A (–5, 7), B (– 4, –5), C (–1, –6) and D (4, 5) are the vertices of a quadrilateral, find the area of the quadrilateral ABCD.

Area of a Triangle  &Collinearity of Three points Method to Find the Unknown when Three points are Collinear

Show that the points A (0, 1), B (2, 3) and C (3, 4) are collinear.

Check whether the points (0, 5), (0, –9) and (3, 6) are collinear.

If the points A(1, 2), O(0, 0) and C(a, b) are collinear, then

(A) a = b          (B) a = 2b           (C) 2a = b            (D) a = –b

Find the value of k if the points A (2, 3), B (4, k) and C(6, –3) are collinear.

If (5, 2), (− 3, 4) and (x, y) are collinear, show that x + 4y − 13 = 0.

Introduction

Given tan A = 4/3, find the other trigonometric ratios of the angle A.

If ∠B and ∠Q are acute angles such that sin B = sin Q, then prove that ∠B = ∠Q.

Consider ∆ACB, right-angled at C, in which AB = 29 units, BC = 21 units and ∠ABC = θ (see Figure). Determine the values of

(i) cos2θ + sin2θ,

(ii) cos2θ- sin2θ.

In a right triangle ABC, right-angled at B, if tan A = 1, then verify that 2 sin A cos A = 1.

Relations between Trigonometric Ratios& Its applications.

If θ is an acute angle and 3 sin θ = 4 cos θ, then find the value of 4 sin2 θ – 3 cos2 θ + 2.

Prove that: cosA/(1+tanA) - sinA/(1+cotA)= cos A - sin A.

If tan A + cot A = 2, then find the value of tan2A + cot2A.

In ∆OPQ, right-angled at P, OP = 7 cm and OQ – PQ = 1 cm (see Figure). Determine the values of sin Q and cos Q.

Trigonometric Ratios of Some Specific Angles

The value of  tan⁡〖〖30〗^0 〗/cot⁡〖〖60〗^0 〗 is

(A) 1/√2    (B) 1/√3      (C) √3      (D) 1 

Evaluate: sin2 30° cos2 45° + 4 tan2 30° + 1/2sin2 90° - 2 cos2 90° + 1/24.

If tan (3x + 30°) = 1 then find the value of x.

If angles A, B, C of a ΔABC form an increasing AP, then find sin B.

Problems Based on Trigonometric Ratios of Some Specific Angles

Evaluate: 4(sin430° + cos460°) - 3(cos2 45°- sin290°)

Evaluate the following: (2cos^2 60^0+3sec^2 30^0-2tan^2 45^0)/(sin^2 30^0+cos^2 45^0 ).

In ∆ABC, right-angled at B, AB = 5 cm and ∠ACB = 30° (see Figure). Determine the lengths of the sides BC and AC.

In ∆PQR, right -angled at Q (see Figure), PQ = 3 cm and PR = 6 cm. Determine ∠QPR and ∠PRQ.

Trigonometric Ratios of Complementary Angles

Evaluate(tan〖65〗^0)/(cot〖25〗^0 ).

Evaluate (tan〖15〗^0)/(cot〖75〗^0 ) + (sin〖25〗^0)/(cos〖65〗^0 )

Evaluate (tan〖50〗^0+sec〖50〗^0)/(cot〖40〗^0+cosec〖40〗^0 )+ cos40ocosec50o

Express cot 85° + cos 75° in terms of trigonometric ratios of angles between 0° and 45°.

Find the value of (tan 10 tan 20 tan 30…tan 890).

Trigonometric Ratios of Complementary Angles - Continuation….

If sin (A – B) = 1/2, cos (A + B) = 1/2, 0° < A + B ≤ 90°, A > B, find A and B.

If sin 3A = cos (A – 26°), where 3A is an acute angle, find the value of A.

In a triangle ABC, write cos ((B+C)/2)in terms of angle A.

If A + B = 90° and sec A = 2/3, then find the value of cosec B

If tan 2A = cot (A + 60), find the value of A where 2A is an acute angle.

Show that: cosec2θ - tan2(90° -θ) = sin2θ + sin2(90° -θ)

Trigonometric Identities

Derivations

Conversion of Trigonometric Ratios in Terms of Other Trigonometric Ratios

Express the ratios cos A, tan A and sec A in terms of sin A.

Prove that sec A (1 – sin A)(sec A + tan A) = 1.

Prove that: (sin θ + 1 + cos θ) (sin θ – 1 + cos θ). sec θ cosec θ = 2

Prove that Sinθ/(1+cosθ ) + (1+cosθ )/(sinθ )= 2cosecθ

Trigonometric Identities &Problems Based on Trigonometric Identities

Prove that (cotA-cosA)/(cotA+cosA) = (cosecA-1)/(cosecA+1)

Prove that (sinθ-cosθ+1)/(sinθ+cosθ-1) = 1/(secθ-tanθ) using the identity sec2q = 1 + tan2q.

Prove: (tanA+secA-1 )/(tanA-secA+1) = (1+sinA )/cosA

Prove that tanθ/(1-tanθ)– cotθ/(1-cotθ)= (cosθ+sinθ )/(cosθ-sinθ).

Trigonometric Identities &Problems Based on Trigonometric Identities- eliminating theta

Prove the trigonometric identity √((cosecA-1)/(cosecA+1)) + √((cosecA+1)/(cosecA-1)) = 2 sec A.

If sin X + sin2 X = 1, prove that cos2 X + cos4 X = 1.

Evaluate sin A. cos A – (sinAcos(90^0-A) cosA )/(sec(90^0-A)) −  (cosAsin(90^0-A)sinA  )/(cosec(90^0-A))

If cosec θ + cot θ = p, then prove that cos θ = (p^2-1)/(p^2+1).

If a cos θ- b sin θ = x and a sin θ + b cos θ = y, then prove that a2 + b2 = x2 + y2.

Introduction:

Trigonometry is used in our day-to-day life. Here, we shall study the use of trigonometry in measuring the heights and distance of towers, buildings and other objects. Measuring heights and distances is an important application of trigonometry.

If the length of the shadow of a tower is equal to its height,

then the angle of elevation of the sun is:

A. 300

B. 450

C. 600

D. 750

If a tower 30m high casts a shadow 10√3m long on the ground, then what is the angle of elevation of the sun?

An observer 1.5 m tall is 28.5 m away from a tower 30 m high. Find the angle of elevation of the top of the tower from his eye.

The figure shows the observation of point C from point A. Find the angle of depression from A, if AB = 4 cm and BC = 4√(3 ) cm.

Problems based on finding one side of right angled triangle when an acute angle and one of the other two sides are known.

A tower stands vertically on the ground. From a point on the ground, which is 15 m away from the foot of the tower, the angle of elevation of the top of the tower is found to be 60°. Find the height of the tower.

An electrician has to repair an electric fault on a pole of height 5 m. She needs to reach a point 1.3m below the top of the pole to undertake the repair work (see Fig.). What should be the length of the ladder that she should use which, when inclined at an angle of 60° to the horizontal, would enable her to reach the required position? Also, how far from the foot of the pole should she place the foot of the ladder? (You may take √3 = 1.73)

A tree breaks due to the storm and the broken part bends so that the top of the tree touches the ground making an angle of 30o with the ground. The distance from the foot of the tree to the point where the top touches the ground is 10 metres. Find the height of the tree.

A ladder 15 m long leans against a wall making an angle of 60o with the wall. Find the height of the wall from the point the ladder touches the wall.

Problems based on two right angled triangle having common base (common horizontal line) or perpendicular (common vertical line)

An observer 1.5 m tall is 28.5 m away from a chimney. The angle of elevation of the top of the chimney from her eyes is 45°. What is the height of the chimney?

From a point P on the ground the angle of elevation of the top of a 10 m tall building is 30°. A flag is hoisted at the top of the building and the angle of elevation of the top of the flagstaff from P is 45°. Find the length of the flagstaff and the distance of the building from the point P. (You may take √3 = 1.732)

The shadow of a tower standing on a level ground is found to be 40 m longer when the Sun’s altitude is 30° than when it is 60°. Find the height of the tower.

A vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff of height 6m. At a point on the plane, the angle of elevation of the bottom of the flagstaff is 30∘ and that of the top of the flagstaff is 600. Find the height of the tower.

[Use √3 = 1.732.]

Problems based on use of two right angled triangles when length of one side of each triangle are equal or a relation between them is known:

The angles of depression of the top and the bottom of an 8 m tall building from the top of a multi-storeyed building are 30° and 45°, respectively. Find the height of the multi-storeyed building and the distance between the two buildings.

The angle of elevation of a jet plane from a point A on the ground is 60°. After a flight of 30 seconds, the angle of elevation changes to 30°. If the jet plane is flying at a constant height of 1500√3 m, find the speed of the jet plane.

The angle of elevation of the top of a vertical tower from a point on the ground is 600. From another point 10m vertically above the first, its angle of elevation is 300. Find the height of the tower.

Problems based on right angled triangle formed by the angle of depression

From a point on a bridge across a river, the angles of depression of the banks on opposite sides of the river are 30° and 45°, respectively. If the bridge is at a height of 3 m from the banks, find the width of the river.

A peacock is sitting on the top of a tree. It observes a serpent on the ground making an angle of depression of 300. The peacock catches the serpent in 12 s with the speed of 300 m/min. what is the height of the tree?

The angle of elevation of a cloud from a point 120 m above a lake is 300 and the angle of depression of its reflection in the lake is 600. Find the height of the cloud.

Two ships are approaching a light house from opposite directions. The angle of depression of two ships from top of the light house are 300 and 450. If the distance between two ships is 100 m. Find the height of light-house.

Introduction

The TANGENT at any point of a circle is perpendicular to the radius through the point of contact.

From a point P, 10 cm away from the centre of a circle, a tangent PT of length 8 cm is drawn. Find the radius of the circle.

In the given figure, point P is 26 cm away from the centre O of a circle and the length PT of the tangent drawn from P to the circle is 24 cm. Find the radius of the circle.

Number of Tangents from a Point on a Circle

The lengths of two tangents drawn from an external point to a circle are equal.

Prove that in two concentric circles, the chord of the larger circle, which touches the smaller circle, is bisected at the point of contact.

In two concentric circles, a chord of length 8 cm of the larger circle touches the smaller circle. If the radius of the larger circle is 5 cm then find the radius of the smaller circle.

Two concentric circles are of radii 7 cm and r cm respectively where r > 7. A chord of the larger circle of the length 48 cm, touches the smaller circle. Find the value of r.

PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at P and Q intersect at a point T (see Figure). Find the length TP.

Number of Tangents from a Point on a Circle :

Problems based on length of the tangent from an external point to a circle.

Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that ∠PTQ = 2 ∠OPQ.

If PA and PB are tangents from an outside point P. such that PA = 10 cm and ∠APB = 60o. Find the length of chord AB.

If two tangents inclined at an angle of 600 are drawn to a circle of a radius 3 cm, then find the length of each tangent.

In two concentric circles, prove that all chord of the outer circle which touch the inner circle are of equal length.

Miscellaneous problems related to circles

In fig. the radius of the in circle of △ABC of area 84 cm2 and the lengths of the segments AP and BP into which side AB is divided by the point of contact are 6 cm and 8 cm. Find the lengths of the sides AC and BC

If a, b and c are the sides of a right angled triangle, where c is hypotenuse, then prove that the radius of the circle which touches the sides of the triangle is given by r = (a+b-c)/2

A circle touches all the four sides of a quadrilateral ABCD. Prove that AB + CD = BC + DA.

In fig., a circle with centre O is inscribed in a quadrilateral ABCD such that, it touches the sides BC, AB, AD and CD at points P, Q, P and S respectively. If AB = 29 cm, AD = 23 cm, ZB = 90o and DS = 5 cm, then find the radius of the circle (in cm.).