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Chapter 11 Triangle and its Angles Exercise Ex. 11.1

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Two angles of a triangle are equal and the third angle is greater than each of those angles by 30^o. Determine all the angles of the triangle.Solution 4

Question 5

Solution 5

Question 6

Can a triangle have:

(i) Two right angles?

(ii) Two obtuse angles?

(iii) Two acute angles?

(iv) All angles more than 60^o?

(v) All angles less than 60^o?

(vi) All angles equal to 60^o?

Justify your answer in each case.Solution 6

(i) No

As two right angles would sum up to 180^o, and we know that the sum of all three angles of a triangle is 180^o, so the third angle will become zero. This is not possible, so a triangle cannot have two right angles.

(ii) No

A triangle cannot have 2 obtuse angles, since then the sum of those two angles will be greater than 180^o which is not possible as the sum of all three angles of a triangle is 180^o.

(iii) Yes

A triangle can have 2 acute angles.

(iv) No

The sum of all the internal angles of a triangle is 180^o. Having all angles more than 60^o will make that sum more than 180^o, which is impossible.

(v) No

The sum of all the internal angles of a triangle is 180^o. Having all angles less than 60^o will make that sum less than 180^o, which is impossible.

(vi) Yes

The sum of all the internal angles of a triangle is 180^o.  So, a triangle can have all angles as 60^o. Such triangles are called equilateral triangles.Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Chapter 11 Triangle and its Angles Exercise Ex. 11.2

Question 1

The exterior angles, obtained on producing both the base of a triangle both ways are 104^o and 136^o. Find all the angles of the triangle.

Solution 1

Question 2

In fig., the sides BC, CA and AB of a ΔABC have been produced to D, E, and F respectively. If ∠ACD = 105° and ∠EAF = 45°, find all the angles of the ΔABC.

Solution 2

Question 3(i)

Solution 3(i)

Question 3(ii)

Solution 3(ii)

Question 3(iii)

Solution 3(iii)

Question 4

In fig., AC ⊥ CE and ∠A : ∠B : ∠C = 3 : 2 : 1, find the value of ∠ECD.

Solution 4

Question 5

In fig. AB ∥ DE. Find ∠ACD.

Solution 5

Question 6

Which of the following statements are true (T) and which are false (F):

Solution 6

Question 7

Fill in the blanks to make the following statements true:

(i) Sum of the angle of triangle is ______ .

(ii) An exterior angle of a triangle is equal to the two ______ opposite angles.

(iii) An exterior angle of a traingle is always _______ than either of the interior oppsite angles.

(iv) A traingle cannot have more than ______ right angles.

(v) A triangles cannot have more than ______ obtuse angles.Solution 7

(i) 180^o

(ii) interior

(iii) greater

(iv) one

(v) oneQuestion 8

Solution 8

Question 9

Solution 9

Question 10

In fig., AB divides ∠DAC in the ratio 1 : 3 and AB = DB. Determine the value of x.

Solution 10

Question 11

Solution 11

Question 12

In fig., AM ⊥ BC and AN is the bisector of ∠A. If ∠B = 65° and ∠C = 33°, find ∠MAN.

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

In fig. AE bisects ∠CAD and ∠B = ∠C. Prove that AE ∥ BC.

Solution 15

Chapter 10 – Lines and Angles Exercise Ex. 10.1

Question 1

Write the complement of each of the following angles:

(i) 20^o

(ii) 35^o

(iii) 90^o

(iv) 77^o

(v) 30^oSolution 1

Question 2

Write the supplement of each of the following angles:

(i) 54^o

(ii) 132^o

(iii) 138^oSolution 2

(i) 54°

Since, the sum of an angle and its supplement is 180°

∴Its supplement will be 180° – 54° = 126°.

(ii) 132°

Since, the sum of an angle and its supplement is 180°

∴Its supplement will be 180° – 132° = 48°.

(iii) 138°

Since, the sum of an angle and its supplement is 180°

∴Its supplement will be 180° – 138° = 42°.Question 3

If an angle is 28^o less than its complement, find its measure.Solution 3

Question 4

If an angle is 30^o more than one half of its complement, find the measure of the angle.Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

If the complement of an angle is equal to the supplement of the thrice of it. Find the measure of angle.Solution 13

Let the measure of the angle be x^o.

Its complement will be (90^o – x^o) and its supplement will be (180^o – x^o).

Supplement of thrice of the angle = (180^o – 3x^o)

According to the given information:

(90^o – x^o) = (180^o – 3x^o)

3x – x = 180 – 90

2x = 90

x = 45

Thus, the measure of the angle is 45^o.

The measure of the angle is 45^oQuestion 14

Solution 14

Chapter 10 – Lines and Angles Exercise Ex. 10.2

Question 1

In fig., OA and OB are opposite rays:

(i) If x = 25°, what is the value of y?

(ii) if y = 35°, what is the value of x?

Solution 1

Question 2

In fig., write all pairs of adjacent angles and all the linear pairs.

Solution 2

Question 3

In fig., find x. further find ∠BOC, ∠COD and ∠AOD

Solution 3

Question 4

In fig., rays OA, OB, OC, OD and OE have the common end point O. Show that  ∠AOB + ∠BOC + ∠COD + ∠DOE + ∠EOA = 360ºSolution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

How many pairs of adjacent angles, in all, can you name in fig.

Solution 7

Question 8

In fig., determine the value of x.

Solution 8

Question 9

In fig., AOC is a line, find x.

Solution 9

Question 10

In Fig., POS is a line, find x.

Solution 10

Question 11

In fig., ACB is a line such that ∠DCA = 5x and ∠DCB = 4x. Find the values of ∠DCA and∠DCB

Solution 11

Question 12

Give POR = 3x and QOR = 2x + 10, find the value of x for which POQ will be aline.

Solution 12

Question 13

What value of y would make AOB a line in fig., if ∠AOC = 4y and ∠BOC = (6y + 30)?

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

In Fig., Lines PQ and RS intersect each other at point O. If ∠POR : ∠ROQ = 5:7, find all the angles.

Solution 16

Question 17

In Fig. If a greater than b by one third of a right-angle. find the values of a and b.

Solution 17

Question 18

Solution 18

Question 19In the given figure, POQ is a line. Ray OR is perpendicular to line PQ. OS is another ray lying between rays OP and OR. Prove that

                                    Solution 19Given that OR ⊥ PQ   ∴ POR = 90^o        ⇒ ∠POS  + ∠SOR = 90^o ∠ROS = 90^o  – ∠POS                … (1)     ∠QOR = 90^o                     (As OR ⊥ PQ)     ∠QOS – ∠ROS = 90^o     ∠ROS = ∠QOS – 90^o             … (2)     On adding equations (1) and (2), we have     2 ∠ROS = ∠QOS – ∠POS

Chapter 10 – Lines and Angles Exercise Ex. 10.3

Question 1

In fig., lines l₁ aans l₂ intersect at O, forming angles as shown in the figure. If x = 45, find the values of y, z and u.

Solution 1

Question 2

In fig., three coplanar lines intersect at a point O, forming angles as shown in the figure. Find the values of x, y, z and u.

Solution 2

Question 3

In fig. , find the values of x, y and z.

Solution 3

Question 4

In Fig., find the value of X.

Solution 4

Question 5

Solution 5

Question 6

In fig., rays AB and CD intersect at O.

(i) Determine y when x = 60^o

(ii) Determine x when y = 40

Solution 6

Question 7

In fig., lines AB, CD and EF intersect at O. Find the measures of ∠AOC, ∠COF, ∠DOE and ∠BOF.

Solution 7

Question 8

Solution 8

Question 9

In fig., lines AB, and CD intersect at O. If ∠AOC + ∠BOE = 70° and ∠BOD = 40°, find ∠BOE and reflex ∠COE.

Solution 9

Question 10

Which of the following statements are true (T) and which are false (F)?

(i) Angles forming a linear pair are supplementary.

(ii) If two adjacent angles are equal, then each angle measures 90^o.

(iii) Angles forming a linear pair can both be acute angles.

(iv) If angles forming a linear pair are equal, then each of these angles is of measure 90^o.Solution 10

(i) True

(ii) False

(iii) False

(iv) TrueQuestion 11

Fill in the blanks so as to make the following statements true:

(i) If one angle of a linear pair is acute, then its other angle will be ________.

(ii) A ray stands on a line, then the sum of the two adjacent angles so formed is _______.

(iii) If the sum of two adjacent angles is 180^o, then the ______ arms of the two angles are opposite rays.Solution 11

(i) obtuse.

(ii) 180^o

(iii) uncommonQuestion 12

Solution 12

Question 13

Solution 13

Chapter 10 – Lines and Angles Exercise Ex. 10.4

Question 1

In fig., AB ∥ CD and ∠1 and ∠2 are in the ratio 3:2. determine all angles from 1 to 8.

Solution 1

Question 2

In fig., l, m and n are parallel lines intersected by transversal p at x, y and z respectively. find ∠1, ∠2, ∠3.

Solution 2

Question 3

In fig., if AB ∥ CD and CD ∥ EF, find ∠ACE.

Solution 3

Question 4

In fig., state which lines are parallel and why.

Solution 4

Question 5

In fig. if l ∥ m, n ∥ p and ∠1 = 85°, find ∠2.

Solution 5

Question 6

If two straight lines are perpendicular to the same line, prove that they are parallel to each other.Solution 6

Question 7

Solution 7

Question 8

Solution 8

Let AB and CD be perpendicuar to line MN.

Question 9

In fig., ∠1 = 60° and ∠2 = (2/3)^rd of a right angle. prove that l ∥ m

Solution 9

Question 10

In fig., if l ∥ m ∥ n and ∠1 = 60°, find ∠2.

Solution 10

Question 11

Solution 11

Let AB and CD be perpendicuar to line MN.

Question 12

Solution 12

Question 13

Solution 13

Question 14

In fig., p is transversal to lines m and n, ∠2 = 120° and ∠5 = 60°. Prove that m ∥ n. 

Solution 14

Question 15

In fig., transceral l intersects two lines m and n, ∠4 = 110° and ∠7 = 65°. is m ∥ n ?

Solution 15

Question 16

Which pair of lines in Fig., are parallel? give reasons.

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

In fig., AB ∥ CD ∥ EF and GH ∥ KL. Find ∠HKL

Solution 20

Question 21

In fig., show that AB ∥ EF.

Solution 21

Question 22

In Fig., PQ ∥ AB and PR BC. IF ∠QPR = 102º, determine ∠ABC. Give reasons.

Solution 22

Question 23

Prove that if the two arms of an angle are perpendicular to the two arms of another angle, then the angles are either equal or supplementary.Solution 23

Consider the angles AOB and ACB.

Question 24

In fig.,  lines AB and CD are parallel and p is any point as shown in the figure. Show that ∠ABP + ∠CDP = ∠DPB.

Solution 24

Question 25

In fig., AB ∥ CD and P is any point shown in the figure. Prove that:

∠ABP + ∠BPD + ∠CDP = 360°

Solution 25

Question 26

In fig., arms BA and BC of ∠ABC are respectively parallel to arms ED and EF of ∠DEF. Prove that ∠ABC = ∠DEF

 Solution 26

Question 27

 Solution 27

Chapter 9 Introduction to Euclid’s Geometry Exercise Ex. 9.1

Question 1

Solution 1

Question 2(i)

Solution 2(i)

Question 2(ii)

Solution 2(ii)

Question 3(i)

Solution 3(i)

Question 3(ii)

Solution 3(ii)

Question 4

Write the truth value (T/F) of each of the following statements:

(i) Two lines intersect in a point.

(ii) Two lines may intersect in two points.

(iii) A segment has no length.

(iv) Two distinct points always determine a line.

(v) Every ray has a finite length.

(vi) A ray has one end-point only.

(vii) A segment has one end-point only.

(viii) The ray AB is same as ray BA.

(ix) Only a single line may pass through a given point.

(x) Two lines are coincident if they have only one point in common.Solution 4

(i) False

(ii) False

(iii) False

(iv) True

(v) False

(vi) True

(vii) False

(viii) False

(ix) False

(x) FalseQuestion 5

In fig., name the following:

(i) Five line segments.

(ii) Five rays.

(iii) Four collinear points.

(iv) Two pairs of non-intersecting line segments.Solution 5

Question 6

Fill in the blanks so as to make the following statements true:

(i) Two distinct points in a plane determine a ______ line.

(ii) Two distinct ______ in a plane cannot have more than one point in common.

(iii) Given a line and a point, not on the line, there is one and only ______ line which passes through the given point and is ______ to the given line.

(iv) A line separates a plane into ______ parts namely the ______ and the _______ itself.Solution 6

(i) unique

(ii) lines

(iii) perpendicular, perpendicular

(iv) three, two half planes, line.

Chapter 8 Co-ordinate Geometry Exercise Ex. 8.1

Question 1

Plot the following points on the graph paper:

(i) (2,5)

(ii) (4,-3)

(iii) (-5,-7)

(iv) (7,-4)

(v) (-3,2)

(vi) (7,0)

(vii) (-4,0)

(viii) (0,7)

(ix) (0,-4)

(x) (0,0)Solution 1

The graph of the given points are:

Question 2

Write the coordinates of each of the following points marked in the graph paper:

Solution 2

The coordinates of the given points are A(3,1), B(6,0), C(0,6), D(-3,0), E(-4,3), F(-2,-4), G(0,-5), H(3,-6), P(7,-3) and Q(7,6)

Chapter 7 Linear Equations in Two Variables Exercise Ex. 7.1

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Chapter 7 Linear Equations in Two Variables Exercise Ex. 7.2

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

If x = 1 and y = 6 is solution of the equation 8x – ay + a²= 0, find the value of a.Solution 6

Question 7(i)

Write two solutions of the form x = 0, y = a and x = b, y = 0 for the follwoing equation: 5x – 2y = 10Solution 7(i)

Question 7(ii)

Write two solutions of the form x = 0, y = a and x = b, y = 0 for the following equation: -4x + 3y = 12Solution 7(ii)

Question 7(iii)

Solution 7(iii)

Chapter 7 Linear Equations in Two Variables Exercise Ex. 7.3

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 1(v)

Solution 1(v)

Question 1(vi)

Solution 1(vi)

Question 1(vii)

Solution 1(vii)

Question 1(viii)

Solution 1(viii)

Question 2

Solution 2

Question 3

Solution 3

Question 4

Plot the points (3,5) and (-1,3) on a graph paper and verify that the straight line passing through these points also passes through the point (1,4).Solution 4

The given points on the graph:



It is dear from the graph, the straight line passing through these points also passes through the point (1,4).Question 5

From the choices given below, choose the equation whose graph is given in fig.,

(i) y = x

(ii) x + y = 0

(iii) y = 2x

(iv) 2 + 3y = 7x

 Solution 5

Question 6

From the choices given below, choose the equation whose graph is given in fig.,

(i) y = x + 2

(ii) y = x – 2

(iii) y = -x + 2

(iv) x + 2y = 6

 Solution 6

Question 7

Solution 7

Question 8

Draw the graph of the equation 2x + 3y = 12. Find the graph, find the coordinates of the point.

(i) whose y-coordinate is 3.

(ii) whose x-coordinate is -3Solution 8

Question 9(i)

Solution 9(i)

Question 9(ii)

Solution 9(ii)

Question 9(iii)

Solution 9(iii)

Question 9(iv)

Solution 9(iv)

Question 10

Solution 10

Question 11

Solution 11

Question 12

The sum of a two digit number and the number obtained by reversing the order of its digits is 121. If units and ten’s digit of the number are x and y respectively, then write the linear equation representing the above statement.Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Draw the graph of y = |x|.Solution 15

We have,

y = |X|                               …(i)

Putting x = 0, we get y = 0

Putting x = 2, we get y = 2

Putting x = -2, we get y = 2

Thus, we have the following table for the points on graph of |x|.

x	0	2	-2
y	0	2	2

The graph of the equation y = |x|:

Question 16

Draw the graph of y = |x| + 2.Solution 16

We have,

y = |x| + 2                                                  …(i)

Putting x = 0, we get y = 2

Putting x = 1, we get y = 3

Putting x = -1, we get y = 3

Thus, we have the following table for the points on graph of |x| + 2:

x	0	1	-1
y	2	3	3

The graph of the equation y = |x| + 2:

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Ravish tells his daughter Aarushi, “Seven years ago, I was seven times as old as you were then. Also, three years form now, I shall be three times as old as you will be”. If present ages of Aarushi and Ravish are x and y years respectively, represent this situation algebraically as well as graphically.Solution 20

Question 21

Solution 21

Chapter 7 Linear Equations in Two Variables Exercise Ex. 7.4

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

y + 3 = 0

y = -3

Point A represents -3 on number line.

On Cartesian plane, equation represents all points on x axis for which y = -3Question 1(iii)

Solution 1(iii)

y = 3

Point A represents 3 on number line.

On Cartesian plane, equation represents all points on x axis for which y = 3Question 1(iv)

Solution 1(iv)

Question 1(v)

Solution 1(v)

Question 2(i)

Give the geometrical representation of 2x + 13 = 0 as an equation in

One variableSolution 2(i)

Question 2(ii)

Give the geometrical representation of 2x + 13 = 0 as an equation in

Two variablesSolution 2(ii)

Question 3(i)

Solve the equation 3x + 2 = x – 8, and represent the solution on (i) the number line.Solution 3(i)

Question 3(ii)

Solve the equation 3x + 2 = x – 8, and represent the solution on (ii) the Cartesian plane.Solution 3(ii)

On Cartesian plane, equation represents all points on y axis for which x = -5Question 4

Write the equation of the line that is parallel to x-axis and passing through the point

(i) (0,3)

(ii) (0,-4)

(iii) (2,-5)

(iv) (3,4)Solution 4

(i) The equation of the line that is parallel to x-axis and passing through the point (0,3) is y = 3

(ii) The equation of the line that is parallel to x-axis and passing through the point (0,-4) is y = -4

(iii) The equation of the line that is parallel to x-axis and passing through the point (2,-5) is y = -5

(iv) The equation of the line that is parallel to x-axis and passing through the point (3, 4) is y = 4Question 5

Solution 5

Chapter 6 Factorisation of Polynomials Exercise Ex. 6.1

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8Give one example each of a binomial of degree 35, and of a monomial of degree 100.

Solution 8Degree of a polynomial is the highest power of variable in the polynomial.
Binomial has two terms in it. So binomial of degree 35 can be written as x³⁵ + 7 .  Monomial has only one term in it. So monomial of degree 100 can be written as 7x¹⁰⁰. Concept Insight: Mono, bi and tri means one, two and three respectively. So, monomial is a polynomial having one term similarly for binomials and trinomials. Degree is the highest exponent of variable.  The answer is not unique in such problems . Remember that the terms are always separated by +ve or -ve sign and not with  .  

Chapter 6 Factorisation of Polynomials Exercise Ex. 6.2

Question 1If f(x) = 2x³ – 13x² + 17x + 12, find:

(i) f(2)

(ii) f(-3)

(iii) f(0)Solution 1(i)

f(x) = 2x³ – 13x² + 17x + 12

f(2) = 2(2)³ – 13(2)² + 17(2) + 12

      = 16 – 52 + 34 + 12

      = 10

(ii)

f(-3) = 2(-3)³ – 13(-3)² + 17(-3) + 12

       = -54 – 117 – 51 + 12

       = – 210

(iii)

f(0) = 2(0)³ – 13(0)² + 17(0) + 12

      = 0 – 0 + 0 + 12

      =12Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7Find rational roots of the polynomial f(x) = 2x³ + x² – 7x – 6.Solution 7

Chapter 6 Factorisation of Polynomials Exercise Ex. 6.3

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

If the polynomials ax³ + 3x² – 13 and 2x³ -5x + a, when divided by (x-2) leave the same remainder, find the value of a.Solution 10

Question 11Find the remainder when x³ + 3x² + 3x + 1 is divided by 

Solution 11

Question 12

Solution 12

Chapter 6 Factorisation of Polynomials Exercise Ex. 6.4

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

What must be added to 3x³ + x² – 22x + 9 so that the result is exactly divisible by 3x² + 7x – 6?Solution 25

Chapter 6 Factorisation of Polynomials Exercise Ex. 6.5

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Using factor theorem, factorize: x⁴ – 7x³ + 9x² + 7x -10Solution 4

Question 5

Using factor theorem, factorize: 3x³ – x² – 3x + 1Solution 5

Question 6

Using factor theorem, factorize each of the following polynomials:

x³ – 23x² + 142x – 120Solution 6

Question 7

Using factor theorem, factorize: y³ – 7y + 6Solution 7

Question 8

Using factor theorem, factorize: x³ – 10x² – 53x – 42Solution 8

Question 9

Using factor theorem, factorize: y³ – 2y² – 29y – 42Solution 9

Question 10

Using factor theorem, factorize: 2y³ – 5y² – 19y + 42Solution 10

Question 11

x³ + 13x² + 32x + 20Solution 11

         Let p(x) = x³ + 13x² + 32x + 20
         The factors of 20 are 1,  2,  4,  5 … …
         By hit and trial method
         p(- 1) = (- 1)³ + 13(- 1)² + 32(- 1) + 20
                   = – 1 + 13 – 32 + 20
                   = 33 – 33 = 0
         As p(-1) is zero, so x + 1 is a factor of this polynomial p(x).

         Let us find the quotient while dividing x³ + 13x² + 32x + 20 by (x + 1)
          By long division

                   We know that
         Dividend = Divisor  Quotient + Remainder
         x³ + 13x² + 32x + 20 = (x + 1) (x² + 12x + 20) + 0
                                         = (x + 1) (x² + 10x + 2x + 20)
                                         = (x + 1) [x (x + 10) + 2 (x + 10)]
                                         = (x + 1) (x + 10) (x + 2)
                                         = (x + 1) (x + 2) (x + 10)Question 12

Factorise:

x³ – 3x² – 9x – 5Solution 12

        Let p(x) = x³ – 3x² – 9x – 5
        Factors of 5 are 1,  5.
        By hit and trial method
        p(- 1) = (- 1)³ – 3(- 1)² – 9(- 1) – 5
           = – 1 – 3 + 9 – 5 = 0
        So x + 1 is a factor of this polynomial
        Let us find the quotient while dividing x³ + 3x² – 9x – 5 by x + 1
        By long division

                Now, Dividend = Divisor  Quotient + Remainder
 x³ – 3x² – 9x – 5 = (x + 1) (x² – 4 x – 5) + 0
                                     = (x + 1) (x² – 5 x + x – 5)
                                     = (x + 1) [(x (x – 5) +1 (x – 5)]
                                     = (x + 1) (x – 5) (x + 1)
                                     = (x – 5) (x + 1) (x + 1) Question 13

Factorise:

2y³ + y² – 2y – 1Solution 13

         Let p(y) = 2y³ + y² – 2y – 1         By hit and trial method
         p(1) = 2 ( 1)³ + (1)² – 2( 1) – 1
                = 2 + 1 – 2 – 1= 0
         So, y – 1 is a factor of this polynomial
         By long division method,

                    p(y) = 2y³ + y² – 2y – 1
                 = (y – 1) (2y² +3y + 1)
                 = (y – 1) (2y² +2y + y +1)
                 = (y – 1) [2y (y + 1) + 1 (y + 1)]
                 = (y – 1) (y + 1) (2y + 1)

 Question 14

Using factor theorem, factorize: x³ – 2x² – x + 2Solution 14

Question 15

Solution 15

Question 16

Using factor theorem, factroize : x⁴ – 2x³ – 7x² + 8x + 12Solution 16

Question 17

Using factor theorem, factroize : x⁴ + 10x³ + 35x² + 50x + 24Solution 17

Question 18

Using factor theorem, factorize : 2x⁴ – 7x³ – 13x² + 63x – 45Solution 18

Chapter 5 Factorisation of Algebraic Expressions Exercise Ex. 5.1

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Give possible expressions for the length and breadth of the rectangle having

35y² + 13y – 12 as its area.Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

Solution 35

Chapter 5 Factorisation of Algebraic Expressions Exercise Ex. 5.2

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Chapter 5 Factorisation of Algebraic Expressions Exercise Ex. 5.3

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Chapter 5 Factorisation of Algebraic Expressions Exercise Ex. 5.4

Question 1

Solution 1

Question 2

Solution 2

Question 3Factorise : 27x³ – y³ – z³ – 9xyzSolution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 17

Solution 17

Chapter 4 Algebraic Identities Exercise Ex. 4.1

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 1(v)

Solution 1(v)

Question 2(i)

Solution 2(i)

Question 2(ii)

Solution 2(ii)

Question 2(iii)

Solution 2(iii)

Question 2(iv)

Solution 2(iv)

Question 3(i)

Solution 3(i)

Question 3(ii)

Solution 3(ii)

Question 3(iii)

Solution 3(iii)

Question 3(iv)

Solution 3(iv)

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10(i)

Solution 10(i)

Question 10(ii)

Solution 10(ii)

Question 11

Solution 11

Question 12

Solution 12

Question 13(i)

Solution 13(i)

Question 13(ii)

Solution 13(ii)

Question 13(iii)

Solution 13(iii)

Question 13(iv)

Solution 13(iv)

Question 14

Solution 14

Chapter 4 Algebraic Identities Exercise Ex. 4.2

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 1(v)

Solution 1(v)

Question 1(vi)

Solution 1(vi)

Question 1(vii)

Solution 1(vii)

Question 1(viii)

Solution 1(viii)

Question 1(ix)

Solution 1(ix)

Question 1 (x)Write the following in the expanded form:

(x + 2y + 4z)²Solution 1 (x)

Question 1 (xi)Write the following in the expanded form:

(2x – y + z)²Solution 1 (xi)

Question 1 (xii)Write the following in the expanded form:

(-2x + 3y + 2z)²Solution 1 (xii)

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6(i)

Solution 6(i)

Question 6(ii)

Solution 6(ii)

Question 6(iii)

Solution 6(iii)

Question 6(iv)

Solution 6(iv)

Question 6(v)

Solution 6(v)

Question 7(i)

Solution 7(i)

Question 7(ii)

Solution 7(ii)

Question 7(iii)

Solution 7(iii)

Chapter 4 Algebraic Identities Exercise Ex. 4.3

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11(i)

Solution 11(i)

Question 11(ii)

Solution 11(ii)

Question 11(iii)

Solution 11(iii)

Question 11(iv)

Solution 11(iv)

Question 11(v)

Solution 11(v)

Question 11(vi)

Solution 11(vi)

Question 12(i)

Solution 12(i)

Question 12(ii)

Solution 12(ii)

Question 12(iii)

Solution 12(iii)

Question 12(iv)

Solution 12(iv)

Question 13

Solution 13

Question 14(i)

Solution 14(i)

Question 14(ii)

Solution 14(ii)

Question 15

Solution 15

Question 16

Solution 16

Question 17(i)

Solution 17(i)

Question 17(ii)

Solution 17(ii)

Question 17(iii)

Solution 17(iii)

Question 17(iv)

Solution 17(iv)

Question 18

Solution 18

Question 19

Solution 19

Chapter 4 Algebraic Identities Exercise Ex. 4.4

Question 1 (i)

Solution 1 (i)

Question 1 (ii)

Solution 1 (ii)

Question 1 (iii)

Solution 1 (iii)

Question 1 (iv)

Solution 1 (iv)

Question 1 (v)

Solution 1 (v)

Question 1 (vi)

Solution 1 (vi)

Question 1 (vii)

Solution 1 (vii)

Question 1 (viii)

Solution 1 (viii)

Question 1 (ix)

Solution 1 (ix)

Question 1 (x)

Solution 1 (x)

Question 1 (xi)

Solution 1 (xi)

Question 1 (xii)

Solution 1 (xii)

Question 2 (i)

Solution 2 (i)

Question 2 (ii)

Solution 2 (ii)

Question 2 (iii)

Solution 2 (iii)

Question 2 (iv)

Solution 2 (iv)

Question 2 (v)

Solution 2 (v)

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6 (i)

Solution 6 (i)

Question 6 (ii)

Solution 6 (ii)

Question 6 (iii)

Solution 6 (iii)

Chapter 4 Algebraic Identities Exercise Ex. 4.5

Question 1 (i)

Solution 1 (i)

Question 1 (ii)

Solution 1 (ii)

Question 1 (iii)

Solution 1 (iii)

Question 1 (iv)

Solution 1 (iv)

Question 2(i)

Solution 2(i)

Question 2(ii)

Solution 2(ii)

Question 2(iii)

Solution 2(iii)

Question 2(iv)

Solution 2(iv)

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Chapter 3 Rationalisation Exercise Ex. 3.1

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Chapter 3 Rationalisation Exercise Ex. 3.2

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 1(v)

Solution 1(v)

Question 1(vi)

Solution 1(vi)

Question 1(vii)

Solution 1(vii)

Question 2(i)

Solution 2(i)

Question 2(ii)

Solution 2(ii)

Question 2(iii)

Solution 2(iii)

Question 2(iv)

Solution 2(iv)

Question 2(v)

Solution 2(v)

Question 2(vi)

Solution 2(vi)

Question 3(i)

Solution 3(i)

Question 3(ii)

Solution 3(ii)

Question 3(iii)

Solution 3(iii)

Question 3(iv)

Solution 3(iv)

Question 3(v)

Solution 3(v)

Question 3(vi)

Solution 3(vi)

Question 3(vii)

Solution 3(vii)

Question 3(viii)

Solution 3(viii)

Question 3(ix)

Solution 3(ix)

Question 4(i)

Solution 4(i)

Question 4(ii)

Solution 4(ii)

Question 4(iii)

Solution 4(iii)

Question 4(iv)

Solution 4(iv)

Question 4(v)

Solution 4(v)

Question 4(vi)

Solution 4(vi)

Question 5(i)

Solution 5(i)

Question 5(ii)

Solution 5(ii)

Question 5(iii)

Solution 5(iii)

Question 6(i)

Solution 6(i)

Question 6(ii)

Solution 6(ii)

Question 6(iii)

Solution 6(iii)

Question 6(iv)

Solution 6(iv)

Question 6(v)

Solution 6(v)

Question 6(vi)

Solution 6(vi)

Question 7

Solution 7

Question 8(i)

Solution 8(i)

Question 8(ii)

Solution 8(ii)

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 9(i)

Simplify:Solution 9(i)

Question 9(ii)

Simplify:Solution 9(ii)

Chapter 2 Exponents of Real Numbers Exercise Ex. 2.1

Question 1(i)

Simplify:

3(a⁴b³)¹⁰ × 5(a²b²)³Solution 1(i)

Question 1(ii)

Simplify:

(2x^-2y³)³Solution 1(ii)

Question 1(iii)

Simplify:

Solution 1(iii)

Question 1(iv)

Simplify:

Solution 1(iv)

Question 1(v)

Simplify:

Solution 1(v)

Question 1(vi)

Simplify:

Solution 1(vi)

Question 2(i)

If a = 3 and b = -2, find the value of:

a^a + b^bSolution 2(i)

Question 2(ii)

If a = 3 and b = -2, find the value of:

a^b + b^aSolution 2(ii)

Question 2(iii)

If a = 3 and b = -2, find the value of:

(a + b)^abSolution 2(iii)

Question 3(i)

Prove that:

Solution 3(i)

Question 3(ii)

Prove that:

Solution 3(ii)

Question 4(i)

Prove that:

Solution 4(i)

Question 4(ii)

Prove that:

Solution 4(ii)

Question 5(i)

Prove that:

Solution 5(i)

Question 5(ii)

Prove that:

Solution 5(ii)

Question 6

Solution 6

Question 7(i)

Simplify:

Solution 7(i)

Question 7(ii)

Simplify:

Solution 7(ii)

Question 7(iii)

Simplify:

Solution 7(iii)

Question 7(iv)

Simplify:

Solution 7(iv)

Question 8(i)

Solve the equation for x:

7^{2x + 3} = 1Solution 8(i)

Question 8(ii)

Solve the equation for x:

2^x+1 = 4^x-3Solution 8(ii)

Question 8(iii)

Solve the equation for x:

2^{5x + 3} = 8^{x + 3}Solution 8(iii)

Question 8(iv)

Solve the equation for x:

Solution 8(iv)

Question 8(v)

Solve the equation for x:

Solution 8(v)

Question 8(vi)

Solve the equation for x:

2^{3x – 7} = 256Solution 8(vi)

Question 9(i)

Solve the equation for x:

2^2x – 2^x+3 + 2⁴ = 0Solution 9(i)

Question 9(ii)

Solve the equation for x:

3^{2x + 4} + 1 = 2.3^{x + 2}Solution 9(ii)

Question 10

If 49392 = a⁴b²c³, find the values of a, b and c, where a, b and c are different positive primes.Solution 10

Question 11

If 1176 = 2^a × 3^b × 7^c, find a, b and c.Solution 11

Question 12

Given 4725 = 3^a 5^b 7^c , find

1. the integral values of a, b and c
2. the values of 2^-a3^b7^c

Solution 12

Question 13

If a = xy^{p – 1}, b = xy^q-1 and c = xy^r-1, prove that a^q-rb^r-p c^p-q = 1.Solution 13

Chapter 2 Exponents of Real Numbers Exercise Ex. 2.2

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 1(v)

Solution 1(v)

Question 1(vi)

Solution 1(vi)

Question 1(vii)

Assuming that x, y, z are positive real numbers, simplify each of the following:

Solution 1(vii)

Question 2(i)

Solution 2(i)

Question 2(ii)

Simplify:

Solution 2(ii)

Question 2(iii)

Solution 2(iii)

Question 2(iv)

Solution 2(iv)

Question 2(v)

Solution 2(v)

Question 2(vi)

Solution 2(vi)

Question 2(vii)

Solution 2(vii)

Question 3(i)

Solution 3(i)

Question 3(ii)

Solution 3(ii)

Question 3(iii)

Solution 3(iii)

Question 3(iv)

Solution 3(iv)

Question 3(v)

Solution 3(v)

Question 3(vi)

Solution 3(vi)

Question 3(vii)

Prove that:

Solution 3(vii)

Question 3(viii)

Solution 3(viii)

Question 3(ix)

Solution 3(ix)

Question 4(i)

Show that:

Solution 4(i)

Question 4(ii)

Show that:

Solution 4(ii)

Question 4(iii)

Show that:

Solution 4(iii)

Question 4(iv)

Show that:

Note: Question modifiedSolution 4(iv)

Note: Question modifiedQuestion 4(v)

Show that:

(x^a-b)^a+b(x^b-c)^b+c(x^c-a)^c+a = 1Solution 4(v)

Question 4(vi)

Show that:

Solution 4(vi)

Question 4(vii)

Show that:

Solution 4(vii)

Question 4(viii)

Show that:

Solution 4(viii)

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10(i)

Solution 10(i)

Question 10(ii)

Solution 10(ii)

Question 10(iii)

Solution 10(iii)

Question 10(iv)

Solution 10(iv)

Question 10(v)

Solution 10(v)

Question 10(vi)

Find the value of x if:

Solution 10(vi)

Question 10(vii)

Find the value of x if:

5^{2x + 3} = 1Solution 10(vii)

Question 10(viii)

Find the value of x if:

Solution 10(viii)

Question 10(ix)

Find the value of x if:

Solution 10(ix)

Question 11

If x = 2^1/3 + 2^2/3, show that x³ – 6x = 6.Solution 11

Question 12

Determine (8x)^x, if 9^x+2 = 240 + 9^x.Solution 12

Question 13

If 3^x+1 = 9^x-2, find the value of 2^1+x.Solution 13

Question 14

If 3^4x = (81)^-1 and 10^1/y = 0.0001, find the value of  2^-x+4y.Solution 14

Question 15

If 5^3x = 125 and 10^y = 0.001 find x and y.Solution 15

Question 16(i)

Solve the equation:

3^{x + 1} = 27 × 3⁴Solution 16(i)

Question 16(ii)

Solve the equation:

Solution 16(ii)

Question 16(iii)

Solve the equation

3^x-1 × 5^2y-3 = 225Solution 16(iii)

Question 16(iv)

Solve the equation:

Solution 16(iv)

Question 16(v)

Solve the equation:

Solution 16(v)

Question 16(vi)

Solve the equation:

Solution 16(vi)

Question 17

Solution 17

Question 18(i)

If a and b are different positive primes such that

Solution 18(i)

Question 18(ii)

If a and b are different positive primes such that

(a + b)^-1(a^-1 + b^-1) = a^xb^y, find x + y + 2.Solution 18(ii)

Question 19

If 2^x × 3^y × 5^z = 2160, find x, y and z. Hence, compute the value of 3^x × 2^-y × 5^-z.Solution 19

Question 20

If 1176 = 2^a × 3^b × 7^c, find the values of a, b and c. hence, compute the value of 2^a × 3^b × 7^-c as a fraction.Solution 20

Question 21(i)

Simplify :

Solution 21(i)

Question 21(ii)

Simplify:

Solution 21(ii)

Question 22

Show that:

Solution 22

Question 23(i)

If a = x^m+ny^l, b = x^n+ly^m and c = x^l+myⁿ, prove that a^m-nbn-lcl-m = 1.Solution 23(i)

Question 23(ii)

If x = a^m+n, y = a^n+l and z = a^l+m, prove that x^myⁿz^l = zⁿy^lz^m.Solution 23(ii)