Tamil Nadu Board of Secondary EducationSSLC (English Medium) Class 10th

# Samacheer Kalvi solutions for Mathematics Class 10 SSLC Tamil Nadu State Board chapter 3 - Algebra [Latest edition]

## Chapter 3: Algebra

Exercise 3.1Exercise 3.2Exercise 3.3Exercise 3.4Exercise 3.5Exercise 3.6Exercise 3.7Exercise 3.8Exercise 3.9Exercise 3.10Exercise 3.11Exercise 3.12Exercise 3.13Exercise 3.14Exercise 3.16Exercise 3.17Exercise 3.18Exercise 3.19Exercise 3.20Unit Exercise – 3
Exercise 3.1 [Pages 92 - 93]

### Samacheer Kalvi solutions for Mathematics Class 10 SSLC Tamil Nadu State Board Chapter 3 Algebra Exercise 3.1 [Pages 92 - 93]

Exercise 3.1 | Q 1. (i) | Page 92

Solve the following system of linear equations in three variables

x + y + z = 5; 2x – y + z = 9; x – 2y + 3z = 16

Exercise 3.1 | Q 1. (ii) | Page 92

Solve the following system of linear equations in three variables

1/x - 2/y + 4 = 0; 1/y - 1/z + 1 = 0; 2/z + 3/x = 14

Exercise 3.1 | Q 1. (iii) | Page 92

Solve the following system of linear equations in three variables

x + 20 = (3y)/2 + 10 = 2z + 5 = 110 – (y + z)

Exercise 3.1 | Q 2. (i) | Page 93

Discuss the nature of solutions of the following system of equations

x + 2y – z = 6; – 3x – 2y + 5z = – 12; x – 2z = 3

Exercise 3.1 | Q 2. (ii) | Page 93

Discuss the nature of solutions of the following system of equations

2y + z = 3(– x + 1); – x + 3y – z = – 4; 3x + 2y + z = -1/2

Exercise 3.1 | Q 2. (iii) | Page 93

Discuss the nature of solutions of the following system of equations

(y + z)/4 = (z + x)/3 = (x + y)/2; x + y + z = 27

Exercise 3.1 | Q 3 | Page 93

Vani, her father and her grandfather have an average age of 53. One-half of her grandfather’s age plus one-third of her father’s age plus one-fourth of Vani’s age is 65. Four years ago if Vani’s grandfather was four times as old as Vani then how old are they all now?

Exercise 3.1 | Q 4 | Page 93

The sum of the digits of a three-digit number is 11. If the digits are reversed, the new number is 46 more than five times the former number. If the hundreds digit plus twice the tens digit is equal to the units digit, then find the original three-digit number?

Exercise 3.1 | Q 5 | Page 93

There are 12 pieces of five, ten and twenty rupee currencies whose total value is ₹ 105. When first 2 sorts are interchanged in their numbers its value will be increased by ₹ 20. Find the number of currencies in each sort

Exercise 3.2 [Pages 96 - 97]

### Samacheer Kalvi solutions for Mathematics Class 10 SSLC Tamil Nadu State Board Chapter 3 Algebra Exercise 3.2 [Pages 96 - 97]

Exercise 3.2 | Q 1. (i) | Page 96

Find the G.C.D. of the given polynomials

x4 + 3x3 – x – 3, x3 + x2 – 5x + 3

Exercise 3.2 | Q 1. (ii) | Page 96

Find the G.C.D. of the given polynomials

x4 – 1, x3 – 11x2 + x – 11

Exercise 3.2 | Q 1. (iii) | Page 96

Find the G.C.D. of the given polynomials

3x4 + 6x3 – 12x2 – 24x, 4x4 + 14x3 + 8x2 – 8x

Exercise 3.2 | Q 1. (iv) | Page 96

Find the G.C.D. of the given polynomials

3x3 + 3x2 + 3x + 3, 6x3 + 12x2 + 6x + 12

Exercise 3.2 | Q 2. (i) | Page 97

Find the L.C.M. of the given expressions

4x2y, 8x3y2

Exercise 3.2 | Q 2. (ii) | Page 97

Find the L.C.M. of the given expressions

– 9a3b2, 12a2b2c

Exercise 3.2 | Q 2. (iii) | Page 97

Find the L.C.M. of the given expressions

16m, – 12m2n2, 8n2

Exercise 3.2 | Q 2. (iv) | Page 97

Find the L.C.M. of the given expressions

p2 – 3p + 2, p2 – 4

Exercise 3.2 | Q 2. (v) | Page 97

Find the L.C.M. of the given expressions

2x2 – 5x – 3, 4x2 – 36

Exercise 3.2 | Q 2. (vi) | Page 97

Find the L.C.M. of the given expressions

(2x2 – 3xy)2, (4x – 6y)3, (8x3 – 27y3)

Exercise 3.3 [Pages 97 - 98]

### Samacheer Kalvi solutions for Mathematics Class 10 SSLC Tamil Nadu State Board Chapter 3 Algebra Exercise 3.3 [Pages 97 - 98]

Exercise 3.3 | Q 1. (i) | Page 97

Find the LCM and GCD for the following and verify that f(x) × g(x) = LCM × GCD

21x2y, 35xy2

Exercise 3.3 | Q 1. (ii) | Page 97

Find the LCM and GCD for the following and verify that f(x) × g(x) = LCM × GCD

(x3 – 1) (x + 1), (x3 + 1)

Exercise 3.3 | Q 1. (iii) | Page 97

Find the LCM and GCD for the following and verify that f(x) × g(x) = LCM × GCD

(x2y + xy2), (x2 + xy)

Exercise 3.3 | Q 2. (i) | Page 97

Find the LCM pair of the following polynomials

a2 + 4a – 12, a2 – 5a + 6 whose GCD is a – 2

Exercise 3.3 | Q 2. (ii) | Page 97

Find the LCM pair of the following polynomials

x4 – 27a3x, (x – 3a)2 whose GCD is (x – 3a)

Exercise 3.3 | Q 3. (i) | Page 98

Find the GCD pair of the following polynomials

12(x4 – x3), 8(x4 – 3x3 + 2x2) whose LCM is 24x3 (x – 1)(x – 2)

Exercise 3.3 | Q 3. (ii) | Page 98

Find the GCD pair of the following polynomials

(x3 + y3), (x4 + x2y2 + y4) whose LCM is (x3 + y3) (x2 + xy + y2)

Exercise 3.3 | Q 4. (i) | Page 98

Given the LCM and GCD of the two polynomials p(x) and q(x) find the unknown polynomial in the following table

 LCM GCD p(x) q(x) a3 – 10a2 + 11a + 70 a – 7 a2 – 12a + 35
Exercise 3.3 | Q 4. (ii) | Page 98

Given the LCM and GCD of the two polynomials p(x) and q(x) find the unknown polynomial in the following table

 LCM GCD p(x) q(x) (x4 –  y4)(x4 + x2y2 + y2) (x2 –  y2) (x4 –  y4)(x2 + y2 –  xy)
Exercise 3.4 [Page 99]

### Samacheer Kalvi solutions for Mathematics Class 10 SSLC Tamil Nadu State Board Chapter 3 Algebra Exercise 3.4 [Page 99]

Exercise 3.4 | Q 1. (i) | Page 99

Reduce the following rational expression to its lowest form

(x^2 - 1)/(x^2 + x)

Exercise 3.4 | Q 1. (ii) | Page 99

Reduce the following rational expression to its lowest form

(x^2 - 11x + 18)/(x^2 - 4x + 4)

Exercise 3.4 | Q 1. (iii) | Page 99

Reduce the following rational expression to its lowest form

(9x^2 + 81x)/(x^3 + 8x^2 - 9x)

Exercise 3.4 | Q 1. (iv) | Page 99

Reduce the following rational expression to its lowest form

("p"^2 - 3"p" - 40)/(2"p"^3 - 24"p"^2 + 64"p")

Exercise 3.4 | Q 2. (i) | Page 99

Find the excluded values, of the following expression

y/(y^2 - 25)

Exercise 3.4 | Q 2. (ii) | Page 99

Find the excluded values, of the following expression

"t"/("t"^2 - 5"t" + 6)

Exercise 3.4 | Q 2. (iii) | Page 99

Find the excluded values, of the following expression

(x^2 + 6x + 8)/(x^2 + x - 2)

Exercise 3.4 | Q 2. (iv) | Page 99

Find the excluded values, of the following expression

(x^3 - 27)/(x^3 + x^2 - 6x)

Exercise 3.5 [Page 101]

### Samacheer Kalvi solutions for Mathematics Class 10 SSLC Tamil Nadu State Board Chapter 3 Algebra Exercise 3.5 [Page 101]

Exercise 3.5 | Q 1. (i) | Page 101

Simplify (4x^2y)/(2z^2) xx (6xz^3)/(20y^4)

Exercise 3.5 | Q 1. (ii) | Page 101

Simplify ("p"^2 - 10"p" + 21)/("p" - 7) xx ("p"^2 + "p" - 12)/("p" - 3)^2

Exercise 3.5 | Q 1. (iii) | Page 101

Simplify (5"t"^3)/(4"t" - 8) xx (6"t" - 12)/(10"t")

Exercise 3.5 | Q 2. (i) | Page 101

Simplify (x + 4)/(3x + 4y) xx (9x^2 - 16y^2)/(2x^2 + 3x - 20)

Exercise 3.5 | Q 2. (ii) | Page 101

Simplify (x^3 - y^3)/(3x^2 + 9xy + 6y^2) xx (x^2 + 2xy + y^2)/(x^2 - y^2)

Exercise 3.5 | Q 3. (i) | Page 101

Simplify (2"a"^2 + 5"a" + 3)/(2"a"^2 + 7"a" + 6) ÷ ("a"^2 + 6"a" + 5)/(-5"a"^2 - 35"a" - 50)

Exercise 3.5 | Q 3. (ii) | Page 101

Simplify ("b"^2 + 3"b" - 28)/("b"^2 + 4"b" + 4) ÷ ("b"^2 - 49)/("b"^2 - 5"b" - 14)

Exercise 3.5 | Q 3. (iii) | Page 101

Simplify (x + 2)/(4"y") ÷ (x^2 - x - 6)/(12y^2)

Exercise 3.5 | Q 3. (iv) | Page 101

Simplify (12"t"^2 - 22"t" + 8)/(3"t") ÷ (3"t"^2 + 2"t" - 8)/(2"t"^2 + 4"t")

Exercise 3.5 | Q 4 | Page 101

If x = ("a"^2 + 3"a" - 4)/(3"a"^2 - 3) and y = ("a"^2 + 2"a" - 8)/(2"a"^2 - 2"a" - 4) find the value of x2y–2

Exercise 3.5 | Q 5 | Page 101

If a polynomial p(x) = x2 – 5x – 14 is divided by another polynomial q(x) we get (x - 7)/(x + 2), find q(x)

Exercise 3.6 [Page 103]

### Samacheer Kalvi solutions for Mathematics Class 10 SSLC Tamil Nadu State Board Chapter 3 Algebra Exercise 3.6 [Page 103]

Exercise 3.6 | Q 1. (i) | Page 103

Simplify (x(x + 1))/(x - 2) + (x(1 - x))/(x - 2)

Exercise 3.6 | Q 1. (ii) | Page 103

Simplify (x + 2)/(x + 3) + (x - 1)/(x - 2)

Exercise 3.6 | Q 1. (iii) | Page 103

Simplify (x^3)/(x - y) + (y^3)/(y - x)

Exercise 3.6 | Q 2. (i) | Page 103

Simplify ((2x + 1)(x - 2))/(x - 4) - ((2x^2 - 5x + 2))/(x - 4)

Exercise 3.6 | Q 2. (ii) | Page 103

Simplify (4x)/(x^2 - 1) - (x + 1)/(x - 1)

Exercise 3.6 | Q 3 | Page 103

Subtract 1/(x^2 + 2) from (2x^3 + x^2 + 3)/(x^2 + 2)^2

Exercise 3.6 | Q 4 | Page 103

Identify rational expression should be subtracted from (x^2 + 6x + 8)/(x^3 + 8) to get 3/(x^2 - 2x + 4)

Exercise 3.6 | Q 5 | Page 103

If A = (2x + 1)/(2x - 1), B = (2x - 1)/(2x + 1) find 1/("A" - "B") - (2"B")/("A"^2 - "B"^2)

Exercise 3.6 | Q 6 | Page 103

If A = x/(x + 1) B = 1/(x + 1) prove that (("A" + "B")^2 + ("A" - "B")^2)/("A" + "B") = (2(x^2 + 1))/(x(x + 1)^2

Exercise 3.6 | Q 7 | Page 103

Pari needs 4 hours to complete the work. His friend Yuvan needs 6 hours to complete the same work. How long will it take to complete if they work together?

Exercise 3.6 | Q 8 | Page 103

Iniya bought 50 kg of fruits consisting of apples and bananas. She paid twice as much per kg for the apple as she did for the banana. If Iniya bought ₹ 1800 worth of apples and ₹ 600 worth bananas, then how many kgs of each fruit did she buy?

Exercise 3.7 [Page 105]

### Samacheer Kalvi solutions for Mathematics Class 10 SSLC Tamil Nadu State Board Chapter 3 Algebra Exercise 3.7 [Page 105]

Exercise 3.7 | Q 1. (i) | Page 105

Find the square root of the following rational expression

(400x^4y^12z^16)/(100x^8y^4z^4)

Exercise 3.7 | Q 1. (ii) | Page 105

Find the square root of the following rational expression

(7x^2 + 2sqrt(14)x + 2)/(x^2 - 1/2 x + 1/16)

Exercise 3.7 | Q 1. (iii) | Page 105

Find the square root of the following rational expression

(121("a" + "b")^8 (x + y)^8 ("b" - "c")^8)/(81("b" - "c")^4 ("a" - "b")^12 ("b" - "c")^4)

Exercise 3.7 | Q 2. (i) | Page 105

Find the square root of the following

4x2 + 20x + 25

Exercise 3.7 | Q 2. (ii) | Page 105

Find the square root of the following

9x2 – 24xy + 30xz – 40yz + 25z2 + 16y

Exercise 3.7 | Q 2. (iii) | Page 105

Find the square root of the following

1 + 1/(x^6) + 2/(x^3)

Exercise 3.7 | Q 2. (iv) | Page 105

Find the square root of the following

(4x2 – 9x + 2)(7x2 – 13x – 2)(28x2 – 3x – 1)

Exercise 3.7 | Q 2. (v) | Page 105

Find the square root of the following

(2x^2 + 17/6 x + 1) (3/2 x^2 + 4x + 2) (4/3 x^2 + 11/3 x + 2)

Exercise 3.8 [Page 106]

### Samacheer Kalvi solutions for Mathematics Class 10 SSLC Tamil Nadu State Board Chapter 3 Algebra Exercise 3.8 [Page 106]

Exercise 3.8 | Q 1. (i) | Page 106

Find the square root of the following polynomials by division method

x4 – 12x3 + 42x2 – 36x + 9

Exercise 3.8 | Q 1. (ii) | Page 106

Find the square root of the following polynomials by division method

37x2 – 28x3 + 4x4 + 42x + 9

Exercise 3.8 | Q 1. (iii) | Page 106

Find the square root of the following polynomials by division method

16x4 + 8x2 + 1

Exercise 3.8 | Q 1. (iv) | Page 106

Find the square root of the following polynomials by division method

121x4 – 198x3 – 183x2 + 216x + 144

Exercise 3.8 | Q 2. (i) | Page 106

Find the values of a and b if the following polynomial is a perfect square

4x4 – 12x3 + 37x2 + bx + a

Exercise 3.8 | Q 2. (ii) | Page 106

Find the values of a and b if the following polynomial is a perfect square

ax4 + bx3 + 361x2 + 220x + 100

Exercise 3.8 | Q 3. (i) | Page 106

Find the values of m and n if the following polynomial is a perfect square

1/(x^4) - 6/(x^3) + 13/(x^2) + "m"/x + "n"

Exercise 3.8 | Q 3. (ii) | Page 106

Find the values of m and n if the following polynomial is a perfect square

x4 – 8x3 + mx2 + nx + 16

Exercise 3.9 [Page 109]

### Samacheer Kalvi solutions for Mathematics Class 10 SSLC Tamil Nadu State Board Chapter 3 Algebra Exercise 3.9 [Page 109]

Exercise 3.9 | Q 1. (i) | Page 109

Determine the quadratic equation, whose sum and product of roots are – 9, 20

Exercise 3.9 | Q 1. (ii) | Page 109

Determine the quadratic equation, whose sum and product of roots are 5/3, 4

Exercise 3.9 | Q 1. (iii) | Page 109

Determine the quadratic equation, whose sum and product of roots are (-3)/2, – 1

Exercise 3.9 | Q 1. (iv) | Page 109

Determine the quadratic equation, whose sum and product of roots are – (2 – a)2, (a + 5)2

Exercise 3.9 | Q 2. (i) | Page 109

Find the sum and product of the roots for the following quadratic equation

x2 + 3x – 28 = 0

Exercise 3.9 | Q 2. (ii) | Page 109

Find the sum and product of the roots for the following quadratic equation

x2 + 3x = 0

Exercise 3.9 | Q 2. (iii) | Page 109

Find the sum and product of the roots for the following quadratic equation

3 + 1/"a" = 10/"a"^2

Exercise 3.9 | Q 2. (iv) | Page 109

Find the sum and product of the roots for the following quadratic equation

3y2 – y – 4 = 0

Exercise 3.10 [Page 111]

### Samacheer Kalvi solutions for Mathematics Class 10 SSLC Tamil Nadu State Board Chapter 3 Algebra Exercise 3.10 [Page 111]

Exercise 3.10 | Q 1. (i) | Page 111

Solve the following quadratic equation by factorization method

4x2 – 7x – 2 = 0

Exercise 3.10 | Q 1. (ii) | Page 111

Solve the following quadratic equation by factorization method

3(p2 – 6) = p(p + 5)

Exercise 3.10 | Q 1. (iii) | Page 111

Solve the following quadratic equation by factorization method

sqrt("a"("a" - 7)) = 3sqrt(2)

Exercise 3.10 | Q 1. (iv) | Page 111

Solve the following quadratic equation by factorization method

sqrt(2)x^2 + 7x + 5sqrt(2) = 0

Exercise 3.10 | Q 1. (v) | Page 111

Solve the following quadratic equation by factorization method

2x^2 - x + 1/8 = 0

Exercise 3.10 | Q 2 | Page 111

The number of volleyball games that must be scheduled in a league with n teams is given by G(n) = ("n"^2 - "n")/2 where each team plays with every other team exactly once. A league schedules 15 games. How many teams are in the league?

Exercise 3.11 [Page 114]

### Samacheer Kalvi solutions for Mathematics Class 10 SSLC Tamil Nadu State Board Chapter 3 Algebra Exercise 3.11 [Page 114]

Exercise 3.11 | Q 1. (i) | Page 114

Solve the following quadratic equation by completing the square method

9x2 – 12x + 4 = 0

Exercise 3.11 | Q 1. (ii) | Page 114

Solve the following quadratic equation by completing the square method

(5x + 7)/(x - 1) = 3x + 2

Exercise 3.11 | Q 2. (i) | Page 114

Solve the following quadratic equation by formula method

2x2 – 5x + 2 = 0

Exercise 3.11 | Q 2. (ii) | Page 114

Solve the following quadratic equation by formula method

sqrt(2)"f"^2 - 6"f" + 3sqrt(2) = 0

Exercise 3.11 | Q 2. (iii) | Page 114

Solve the following quadratic equation by formula method

3y2 – 20y – 23 = 0

Exercise 3.11 | Q 2. (iv) | Page 114

Solve the following quadratic equation by formula method

36y2 – 12ay + (a2 – b2) = 0

Exercise 3.11 | Q 3 | Page 114

A ball rolls down a slope and travels a distance d = t2 – 0.75t feet in t seconds. Find the time when the distance travelled by the ball is 11.25 feet

Exercise 3.12 [Pages 116 - 117]

### Samacheer Kalvi solutions for Mathematics Class 10 SSLC Tamil Nadu State Board Chapter 3 Algebra Exercise 3.12 [Pages 116 - 117]

Exercise 3.12 | Q 1 | Page 116

If the difference between a number and its reciprocal is 24/5, find the number

Exercise 3.12 | Q 2 | Page 116

A garden measuring 12m by 16m is to have a pedestrian pathway that is ‘ω’ meters wide installed all the way around so that it increases the total area to 285 m2. What is the width of the pathway?

Exercise 3.12 | Q 3 | Page 116

A bus covers a distance of 90 km at a uniform speed. Had the speed been (15"km")/"hour" more it would have taken 30 minutes less for the journey. Find the original speed of the bus

Exercise 3.12 | Q 4 | Page 116

A girl is twice as old as her sister. Five years hence, the product of their ages (in years) will be 375. Find their present ages.

Exercise 3.12 | Q 5 | Page 116

A pole has to be erected at a point on the boundary of a circular ground of diameter 20 m in such a way that the difference of its distances from two diametrically opposite fixed gates P and Q on the boundary is 4 m. Is it possible to do so? If answer is yes at what distance from the two gates should the pole be erected?

Exercise 3.12 | Q 6 | Page 116

From a group of 2x2 black bees, square root of half of the group went to a tree. Again eight-ninth of the bees went to the same tree. The remaining two got caught up in a fragrant lotus. How many bees were there in total?

Exercise 3.12 | Q 7 | Page 117

Music is been played in two opposite galleries with certain group of people. In the first gallery a group of 4 singers were singing and in the second gallery 9 singers were singing. The two galleries are separated by the distance of 70 m. Where should a person stand for hearing the same intensity of the singers voice?

(Hint: The ratio of the sound intensity is equal to the square of the ratio of their corresponding distances)

Exercise 3.12 | Q 8 | Page 117

There is a square field whose side is 10 m. A square flower bed is prepared in its centre leaving a gravel path all round the flower bed. The total cost of laying the flower bed and gravelling the path at ₹ 3 and ₹ 4 per square metre respectively is ₹ 364. Find the width of the gravel path

Exercise 3.12 | Q 9 | Page 117

The hypotenuse of a right angled triangle is 25 cm and its perimeter 56 cm. Find the length of the smallest side

Exercise 3.13 [Page 119]

### Samacheer Kalvi solutions for Mathematics Class 10 SSLC Tamil Nadu State Board Chapter 3 Algebra Exercise 3.13 [Page 119]

Exercise 3.13 | Q 1. (i) | Page 119

Determine the nature of the roots for the following quadratic equation

15x2 + 11x + 2 = 0

Exercise 3.13 | Q 1. (ii) | Page 119

Determine the nature of the roots for the following quadratic equation

x2 – x – 1 = 0

Exercise 3.13 | Q 1. (iii) | Page 119

Determine the nature of the roots for the following quadratic equation

sqrt(2)"t"^2 - 3"t" + 3sqrt(2) = 0

Exercise 3.13 | Q 1. (iv) | Page 119

Determine the nature of the roots for the following quadratic equation

9y^2 - 6sqrt(2)y + 2 = 0

Exercise 3.13 | Q 1. (v) | Page 119

Determine the nature of the roots for the following quadratic equation

9a2b2x2 – 24abcdx + 16c2d2 = 0, a ≠ 0, b ≠ 0

Exercise 3.13 | Q 2. (i) | Page 119

Find the value of ‘k’ to identify the roots of the following equation is real and equal

(5k – 6)x2 + 2kx + 1 = 0

Exercise 3.13 | Q 2. (ii) | Page 119

Find the value of ‘k’ to identify the roots of the following equation is real and equal

kx2 + (6k + 2)x + 16 = 0

Exercise 3.13 | Q 3 | Page 119

If the roots of (a – b)x2 + (b – c)x + (c – a) = 0 are real and equal, then prove that b, a, c are in arithmetic progression

Exercise 3.13 | Q 4 | Page 119

If a, b are real then show that the roots of the equation (a – b)x2 – 6(a + b)x – 9(a – b) = 0 are real and unequal

Exercise 3.13 | Q 5 | Page 119

If the roots of the equation (c2 – ab)x2 – 2(a2 – bc)x + b2 – ac = 0 are real and equal prove that either a = 0 (or) a3 + b3 + c3 = 3abc

Exercise 3.14 [Page 122]

### Samacheer Kalvi solutions for Mathematics Class 10 SSLC Tamil Nadu State Board Chapter 3 Algebra Exercise 3.14 [Page 122]

Exercise 3.14 | Q 1. (i) | Page 122

Write the following expression in terms of α + β and αβ

alpha/(3beta) + beta/(3alpha)

Exercise 3.14 | Q 1. (ii) | Page 122

Write the following expression in terms of α + β and αβ

1/(alpha^2beta) + 1/(beta^2alpha)

Exercise 3.14 | Q 1. (iii) | Page 122

Write the following expression in terms of α + β and αβ

(3α – 1) (3β – 1)

Exercise 3.14 | Q 1. (iv) | Page 122

Write the following expression in terms of α + β and αβ

(alpha + 3)/beta + (beta + 3)/alpha

Exercise 3.14 | Q 2. (i) | Page 122

The roots of the equation 2x2 – 7x + 5 = 0 are α and β. Without solving for the roots, find 1/alpha + 1/beta

Exercise 3.14 | Q 2. (ii) | Page 122

The roots of the equation 2x2 – 7x + 5 = 0 are α and β. Without solving for the roots, find alpha/beta + beta/alpha

Exercise 3.14 | Q 2. (iii) | Page 122

The roots of the equation 2x2 – 7x + 5 = 0 are α and β. Without solving for the roots, find (alpha + 2)/(beta + 2) + (beta + 2)/(alpha + 2)

Exercise 3.14 | Q 3. (i) | Page 122

The roots of the equation x2 + 6x – 4 = 0 are α, β. Find the quadratic equation whose roots are α2 and β2

Exercise 3.14 | Q 3. (ii) | Page 122

The roots of the equation x2 + 6x – 4 = 0 are α, β. Find the quadratic equation whose roots are 2/alpha and 2/beta

Exercise 3.14 | Q 3. (iii) | Page 122

The roots of the equation x2 + 6x – 4 = 0 are α, β. Find the quadratic equation whose roots are α2β and β2α

Exercise 3.14 | Q 4 | Page 122

If α, β are the roots of 7x2 + ax + 2 = 0 and if β – α = (-13)/7 Find the values of a

Exercise 3.14 | Q 5 | Page 122

If one root of the equation 2y2 – ay + 64 = 0 is twice the other then find the values of a

Exercise 3.14 | Q 6 | Page 122

If one root of the equation 3x2 + kx + 81 = 0 (having real roots) is the square of the other then find k

Exercise 3.16 [Page 137]

### Samacheer Kalvi solutions for Mathematics Class 10 SSLC Tamil Nadu State Board Chapter 3 Algebra Exercise 3.16 [Page 137]

Exercise 3.16 | Q 1. (i) | Page 137

Graph the following quadratic equation and state their nature of solution

x2 – 9x + 20 = 0

Exercise 3.16 | Q 1. (ii) | Page 137

Graph the following quadratic equation and state their nature of solution

x2 – 4x + 4 = 0

Exercise 3.16 | Q 1. (iii) | Page 137

Graph the following quadratic equation and state their nature of solution

x2 + x + 7 = 0

Exercise 3.16 | Q 1. (iv) | Page 137

Graph the following quadratic equation and state their nature of solution

x2 – 9 = 0

Exercise 3.16 | Q 1. (v) | Page 137

Graph the following quadratic equation and state their nature of solution

x2 – 6x + 9 = 0

Exercise 3.16 | Q 1. (vi) | Page 137

Graph the following quadratic equation and state their nature of solution

(2x – 3) (x + 2) = 0

Exercise 3.16 | Q 2 | Page 137

Draw the graph of y = x2 – 4 and hence solve x2 – x – 12 = 0

Exercise 3.16 | Q 3 | Page 137

Draw the graph of y = x2 + x and hence solve x2 + 1 = 0

Exercise 3.16 | Q 4 | Page 137

Draw the graph of y = x2 + 3x + 2 and use it to solve x2 + 2x + 1 = 0

Exercise 3.16 | Q 5 | Page 137

Draw the graph of y = x2 + 3x – 4 and hence use it to solve x2 + 3x – 4 = 0

Exercise 3.16 | Q 6 | Page 137

Draw the graph of y = x2 – 5x – 6 and hence solve x2 – 5x – 14 = 0

Exercise 3.16 | Q 7 | Page 137

Draw the graph of y = 2x2 – 3x – 5 and hence solve 2x2 – 4x – 6 = 0

Exercise 3.16 | Q 8 | Page 137

Draw the graph of y = (x – 1) (x + 3) and hence solve x2 – x – 6 = 0

Exercise 3.17 [Page 144]

### Samacheer Kalvi solutions for Mathematics Class 10 SSLC Tamil Nadu State Board Chapter 3 Algebra Exercise 3.17 [Page 144]

Exercise 3.17 | Q 1. (i) | Page 144

In the matrix A = [(8, 9, 4, 3),(- 1, sqrt(7), sqrt(3)/2, 5),(1, 4, 3, 0),(6, 8, -11, 1)], write The number of elements

Exercise 3.17 | Q 1. (ii) | Page 144

In the matrix A = [(8, 9, 4, 3),(- 1, sqrt(7), sqrt(3)/2, 5),(1, 4, 3, 0),(6, 8, -11, 1)], write The order of the matrix

Exercise 3.17 | Q 1. (iii) | Page 144

In the matrix A = [(8, 9, 4, 3),(- 1, sqrt(7), sqrt(3)/2, 5),(1, 4, 3, 0),(6, 8, -11, 1)], Write the elements a22, a23, a24, a34, a43, a44

Exercise 3.17 | Q 2 | Page 144

If a matrix has 18 elements, what are the possible orders it can have? What if it has 6 elements?

Exercise 3.17 | Q 3. (i) | Page 144

Construct a 3 × 3 matrix whose elements are given by aij = |i – 2j|

Exercise 3.17 | Q 3. (ii) | Page 144

Construct a 3 × 3 matrix whose elements are given by aij = ("i" + "j")^3/3

Exercise 3.17 | Q 4 | Page 144

If A = [(5, 4, 3),(1, -7, 9),(3, 8, 2)] then find the transpose of A

Exercise 3.17 | Q 5 | Page 144

If A = [(sqrt(7), - 3),(- sqrt(5), 2),(sqrt(3), -5)] then find the transpose of – A

Exercise 3.17 | Q 6 | Page 144

If A = [(5, 2, 2),(-sqrt(17), 0.7, 5/2),(8, 3, 1)] then verify (AT)T = A

Exercise 3.17 | Q 7. (i) | Page 144

Find the values of x, y and z from the following equation

[(12, 3),(x, 3/2)] = [(y, z),(3, 5)]

Exercise 3.17 | Q 7. (ii) | Page 144

Find the values of x, y and z from the following equation

[(x + y, 2),(5 + z, xy)] = [(6, 2),(5, 8)]

Exercise 3.17 | Q 7. (iii) | Page 144

Find the values of x, y and z from the following equation

[(x + y + z),(x + z),(y + z)] = [(9),(5),(7)]

Exercise 3.18 [Pages 148 - 149]

### Samacheer Kalvi solutions for Mathematics Class 10 SSLC Tamil Nadu State Board Chapter 3 Algebra Exercise 3.18 [Pages 148 - 149]

Exercise 3.18 | Q 1. (i) | Page 148

If A = [(1, 9),(3, 4),(8, -3)], B = [(5, 7),(3, 3),(1, 0)] then verify that A + B = B + A

Exercise 3.18 | Q 1. (ii) | Page 148

If A = [(1, 9),(3, 4),(8, -3)], B = [(5, 7),(3, 3),(1, 0)] then verify that A + (– A) = (– A) + A = 0

Exercise 3.18 | Q 2 | Page 148

If A = [(4, 3, 1),(2, 3, -8),(1, 0, -4)], B = [(2, 3, 4),(1, 9, 2),(-7, 1, -1)] and C = [(8, 3, 4),(1, -2, 3),(2, 4, -1)] then verify that A + (B + C) = (A + B) + C

Exercise 3.18 | Q 3 | Page 149

Find X and Y if X + Y = [(7, 0),(3, 5)] and X – Y = [(3, 0),(0, 4)]

Exercise 3.18 | Q 4. (i) | Page 149

If A = [(0, 4, 9),(8, 3, 7)], B = [(7, 3, 8),(1, 4, 9)] find the value of B – 5A

Exercise 3.18 | Q 4. (ii) | Page 149

If A = [(0, 4, 9),(8, 3, 7)], B = [(7, 3, 8),(1, 4, 9)] find the value of 3A – 9B

Exercise 3.18 | Q 5. (i) | Page 149

Find the values of x, y, z if [(x - 3, 3x - z),(x + y + 7, x + y + z)] = [(1, 0),(1, 6)]

Exercise 3.18 | Q 5. (ii) | Page 149

Find the values of x, y, z if [(x), (y  – z), (z + 3)] + [(y), (4), (3)] = [(4), (8), (16)]

Exercise 3.18 | Q 6 | Page 149

Find x and y if x[(4),(-3)] + y[(-2),(3)] = [(4),(6)]

Exercise 3.18 | Q 7 | Page 149

Find the non-zero values of x satisfying the matrix equation

x[(2x, 2),(3, x)] + 2[(8, 5x),(4, 4x)] = 2[(x^2 + 8, 24),(10, 6x)]

Exercise 3.18 | Q 8 | Page 149

Solve for x, y : [(x^2),(y^2)] + 2[(-2x),(-y)] = [(5),(8)]

Exercise 3.19 [Pages 153 - 154]

### Samacheer Kalvi solutions for Mathematics Class 10 SSLC Tamil Nadu State Board Chapter 3 Algebra Exercise 3.19 [Pages 153 - 154]

Exercise 3.19 | Q 1 | Page 153

Find the order of the product matrix AB if

 (i) (ii) (iii) (iv) (v) Order of A 3 × 3 4 × 3 4 × 2 4 × 5 1 × 1 Order of B 3 × 3 3 × 2 2 × 2 5 × 1 1 × 3
Exercise 3.19 | Q 2 | Page 153

If A is of order p × q and B is of order q × r what is the order of AB and BA?

Exercise 3.19 | Q 3 | Page 153

A has ‘a’ rows and ‘a + 3’ columns. B has ‘b’ rows and ‘17 − b’ columns, and if both products AB and BA exist, find a, b?

Exercise 3.19 | Q 4 | Page 153

If A = [(2, 5),(4, 3)], B = [(1, -3),(2, 5)] find AB, BA and verify AB = BA?

Exercise 3.19 | Q 5 | Page 153

Given that A = [(1, 3),(5, -1)], B = [(1, -1, 2),(3, 5, 2)], C = [(1, 3, 2),(-4, 1, 3)] verify that A(B + C) = AB + AC

Exercise 3.19 | Q 6 | Page 154

Show that the matrices A = [(1, 2),(3, 1)], B = [(1, -2),(-3, 1)] satisfy commutative property AB = BA

Exercise 3.19 | Q 7. (i) | Page 154

Let A = [(1, 2),(1, 3)], B = [(4, 0),(1, 5)], C = [(2, 0),(1, 2)] Show that A(BC) = (AB)C

Exercise 3.19 | Q 7. (ii) | Page 154

Let A = [(1, 2),(1, 3)], B = [(4, 0),(1, 5)], C = [(2, 0),(1, 2)] Show that (A – B)C = AC – BC

Exercise 3.19 | Q 7. (iii) | Page 154

Let A = [(1, 2),(1, 3)], B = [(4, 0),(1, 5)], C = [(2, 0),(1, 2)] Show that (A – B)T = AT – BT

Exercise 3.19 | Q 8 | Page 154

If A = [(costheta, theta),(0, costheta)], B = [(sintheta, 0),(0, sintheta)] then show that A2 + B2 = I

Exercise 3.19 | Q 9 | Page 154

If A = [(costheta, sintheta),(-sintheta, costheta)] prove that AAT = I

Exercise 3.19 | Q 10 | Page 154

Verify that A2 = I when A = [(5, -4),(6, -5)]

Exercise 3.19 | Q 11 | Page 154

If A = [("a", "b"),("c", "d")] and I = [(1, 0),(0, 1)] show that A2 – (a + d)A = (bc – ad)I2

Exercise 3.19 | Q 12 | Page 154

If A = [(5, 2, 9),(1, 2, 8)], B = [(1, 7),(1, 2),(5, -1)] verify that (AB)T = BT AT

Exercise 3.19 | Q 13 | Page 154

If A = [(3, 1),(-1, 2)] show that A2 – 5A + 7I2 = 0

Exercise 3.20 [Pages 154 - 156]

### Samacheer Kalvi solutions for Mathematics Class 10 SSLC Tamil Nadu State Board Chapter 3 Algebra Exercise 3.20 [Pages 154 - 156]

#### Multiple choice questions

Exercise 3.20 | Q 1 | Page 154

A system of three linear equation in three variables is inconsistent if their planes

• intersect only at a point

• intersect in a line

• coincides with each other

• do not intersect

Exercise 3.20 | Q 2 | Page 154

The solution of the system x + y – 3z = – 6, – 7y + 7z = 7, 3z = 9 is

•  x = 1, y = 2, z = 3

• x = – 1, y = 2, z = 3

• x = – 1, y = – 2, z = 3

• x = 1, y = − 2, z = 3

Exercise 3.20 | Q 3 | Page 154

If (x – 6) is the HCF of x2 – 2x – 24 and x2 – kx – 6 then the value of k is

• 3

• 5

• 6

• 8

Exercise 3.20 | Q 4 | Page 154

(3y - 3)/y ÷ (7y - 7)/(3y^2) is

• (9y)/7

• (9y^3)/((21y - 21))

• (21y^2 - 42y + 21)/(3y^3)

• (7(y^2 - 2y + 1))/(y^2)

Exercise 3.20 | Q 5 | Page 155

y^2 + 1/y^2 is not equal to

• (y^4 + 1)/(y^2)

• (y + 1/y)^2

• (y - 1/y)^2 + 2

• (y + 1/2)^2 - 2

Exercise 3.20 | Q 6 | Page 155

x/(x^2 - 25) - 8/(x^2 + 6x + 5) gives

• (x^2 - 7x + 40)/((x - 5)(x + 5))

• (x^2 + 7x + 40)/((x - 5)(x + 5)(x + 1))

• (x^2 - 7x + 40)/((x^2 - 25)(x + 1))

• (x^2 + 10)/((x^2 - 25)(x + 1))

Exercise 3.20 | Q 7 | Page 155

The square root of (256x^8y^4z^10)/(25x^6y^6z^6) is equal to

• 16/5|(x^2z^4)/y^2|

• 16|y^2/(x^2z^4)|

• 16/5|y/(xz^2)|

• 16/5|(xz^2)/y|

Exercise 3.20 | Q 8 | Page 155

Which of the following should be added to make x4 + 64 a perfect square

• 4x2

• 16x2

• 8x2

• – 8x2

Exercise 3.20 | Q 9 | Page 155

The solution of (2x – 1)2 = 9 is equal to

• – 1

• 2

• – 1, 2

• None of these

Exercise 3.20 | Q 10 | Page 155

The values of a and b if 4x4 – 24x3 + 76x2 + ax + b is a perfect square are

• 100, 120

• 10, 12

• – 120, 100

• 12, 10

Exercise 3.20 | Q 11 | Page 155

If the roots of the equation q2x2 + p2x + r2 = 0 are the squares of the roots of the equation qx2 + px + r = 0, then q, p, r are in __________

• A.P.

•  G.P.

• Both A.P. and G.P.

• none of these

Exercise 3.20 | Q 12 | Page 155

Graph of a linear equation is a _______

• straight line

• circle

• parabola

• hyperbola

Exercise 3.20 | Q 13 | Page 155

The number of points of intersection of the quadratic polynomial x2 + 4x + 4 with the X axis is

• 0

• 1

• 0 or 1

• 2

Exercise 3.20 | Q 14 | Page 155

For the given matrix A = [(1, 3, 5, 7),(2, 4, 6, 8),(9, 11, 13, 15)] the order of the matrix AT is

• 2 × 3

• 3 × 2

• 3 × 4

• 4 × 3

Exercise 3.20 | Q 15 | Page 155

If A is a 2 × 3 matrix and B is a 3 × 4 matrix, how many columns does AB have

• 3

• 4

• 2

• 5

Exercise 3.20 | Q 16 | Page 156

If the number of columns and rows are not equal in a matrix then it is said to be a

• diagonal matrix

• rectangular matrix

• square matrix

• identity matrix

Exercise 3.20 | Q 17 | Page 156

Transpose of a column matrix is

• unit matrix

• diagonal matrix

• column matrix

• row matrix

Exercise 3.20 | Q 18 | Page 156

Find the matrix X if 2X + [(1, 3),(5, 7)] = [(5, 7),(9, 5)]

• [(-2, -2),(2, -1)]

• [(2, 2),(2, -1)]

• [(1, 2),(2, 2)]

• [(2, 1),(2, 2)]

Exercise 3.20 | Q 19 | Page 156

Which of the following can be calculated from the given matrices A = [(1, 2),(3, 4),(5, 6)], B = [(1, 2, 3),(4, 5, 6),(7, 8, 9)],
(i) A2
(ii) B2
(iii) AB
(iv) BA

• (i) and (ii) only

• (ii) and (iii) only

• (ii) and (iv) only

• all of these

Exercise 3.20 | Q 20 | Page 156

If A = [(1, 2, 3),(3, 2, 1)], B = [(1, 0),(2, -1),(0, 2)] and C = [(0, 1),(-2, 5)] Which of the following statements are correct?

(i) AB + C = [(5, 5),(5, 5)]

(ii) BC = [(0, 1),(2, -3),(-4, 10)]

(iii) BA + C = [(2, 5),(3, 0)]

(iv) (AB)C = [(-8, 20),(-8, 13)]

• (i) and (ii) only

• (ii) and (iii) only

• (iii) and (iv) only

• all of these

Unit Exercise – 3 [Pages 156 - 157]

### Samacheer Kalvi solutions for Mathematics Class 10 SSLC Tamil Nadu State Board Chapter 3 Algebra Unit Exercise – 3 [Pages 156 - 157]

Unit Exercise – 3 | Q 1 | Page 156

Solve 1/3 (x + y – 5) = y – z = 2x – 11 = 9 – (x + 2z)

Unit Exercise – 3 | Q 2 | Page 156

One hundred and fifty students are admitted to a school. They are distributed over three sections A, B and C. If 6 students are shifted from section A to section C, the sections will have equal number of students. If 4 times of students of section C exceeds the number of students of section A by the number of students in section B, find the number of students in the three sections

Unit Exercise – 3 | Q 3 | Page 157

In a three-digit number, when the tens and the hundreds digit are interchanged the new number is 54 more than three times the original number. If 198 is added to the number, the digits are reversed. The tens digit exceeds the hundreds digit by twice as that of the tens digit exceeds the unit digit. Find the original number

Unit Exercise – 3 | Q 4 | Page 157

Find the least common multiple of xy(k2 + 1) + k(x2 + y2) and xy(k2 – 1) + k(x2 – y2)

Unit Exercise – 3 | Q 5 | Page 157

Find the GCD of the following by division algorithm

2x4 + 13x3 + 27x2 + 23x + 7, x3 + 3x2 + 3x + 1, x2 + 2x + 1

Unit Exercise – 3 | Q 6. (i) | Page 157

Reduce the given Rational expression to its lowest form

(x^(3"a") - 8)/(x^(2"a") + 2x^"a" + 4)

Unit Exercise – 3 | Q 6. (ii) | Page 157

Reduce the given Rational expression to its lowest form

(10x^3 - 25x^2 + 4x - 10)/(-4 - 10x^2)

Unit Exercise – 3 | Q 7 | Page 157

Simplify (1/("p") + 1/("q" + "r"))/(1/"p" - 1/("q" + "r")) xx [1 + ("q"^2 + "r"^2 - "p"^2)/(2"qr")]

Unit Exercise – 3 | Q 8 | Page 157

Arul, Madan and Ram working together can clean a store in 6 hours. Working alone, Madan takes twice as long to clean the store as Arul does. Ram needs three times as long as Arul does. How long would it take each if they are working alone?

Unit Exercise – 3 | Q 9 | Page 157

Find the square root of 289x4 – 612x3 + 970x2 – 684x + 361

Unit Exercise – 3 | Q 10 | Page 157

Solve sqrt(y + 1) + sqrt(2y - 5) = 3

Unit Exercise – 3 | Q 11 | Page 157

A boat takes 1.6 hours longer to go 36 kms up a river than down the river. If the speed of the water current is 4 km per hr, what is the speed of the boat in still water?

Unit Exercise – 3 | Q 12 | Page 157

Is it possible to design a rectangular park of perimeter 320 m and area 4800 m2? If so find its length and breadth.

Unit Exercise – 3 | Q 13 | Page 157

At t minutes past 2 pm, the time needed to 3 pm is 3 minutes less than ("t"^2)/4. Find t.

Unit Exercise – 3 | Q 14 | Page 157

The number of seats in a row is equal to the total number of rows in a hall. The total number of seats in the hall will increase by 375 if the number of rows is doubled and the number of seats in each row is reduced by 5. Find the number of rows in the hall at the beginning.

Unit Exercise – 3 | Q 15. (i) | Page 157

If α and β are the roots of the polynomial f(x) = x2 – 2x + 3, find the polynomial whose roots are α + 2, β + 2

Unit Exercise – 3 | Q 15. (ii) | Page 157

If α and β are the roots of the polynomial f(x) = x2 – 2x + 3, find the polynomial whose roots are (alpha - 1)/(alpha + 1), (beta - 1)/(beta + 1)

Unit Exercise – 3 | Q 16 | Page 157

If – 4 is a root of the equation x2 + px – 4 = 0 and if the equation x2 + px + q = 0 has equal roots, find the values of p and q.

Unit Exercise – 3 | Q 17. (i) | Page 157

Two farmers Thilagan and Kausigan cultivates three varieties of grains namely rice, wheat and ragi. If the sale (in ₹) of three varieties of grains by both the farmers in the month of April is given by the matrix.

{:("April sale in"  ₹)/("rice"   "wheat"   "ragi"):}

A = [(500, 1000, 1500),(2500, 1500, 500)]"Thilagan"/"Kausigan"

and the May month sale (in ₹) is exactly twice as that of the April month sale for each variety.

What is the average sales for the months of April and May

Unit Exercise – 3 | Q 17. (ii) | Page 157

Two farmers Thilagan and Kausigan cultivates three varieties of grains namely rice, wheat and ragi. If the sale (in ₹) of three varieties of grains by both the farmers in the month of April is given by the matrix.

{:"April sale in"  ₹)/("rice"   "wheat"   "ragi":}

A = [(500, 1000, 1500),(2500, 1500, 500)]"Thilagan"/"Kausigan"

and the May month sale (in ₹) is exactly twice as that of the April month sale for each variety.

If the sales continue to increase in the same way in the successive months, what will be sales in the month of August?

Unit Exercise – 3 | Q 18 | Page 157

If cos theta [(cos theta, sin theta),(-sin theta, cos theta)] + sin theta[(x, -cos theta),(cos theta, x)] = I2, find x.

Unit Exercise – 3 | Q 19 | Page 157

Given A = [("p", 0),(0, 2)], B = [(0, -"q"),(1, 0)], C = [(2, -2),(2, 2)] and if BA = C2, find p and q.

Unit Exercise – 3 | Q 20 | Page 157

A = [(3, 0),(4, 5)], B = [(6, 3),(8, 5)], C = [(3, 6),(1, 1)] find the matrix D, such that CD – AB = 0

## Chapter 3: Algebra

Exercise 3.1Exercise 3.2Exercise 3.3Exercise 3.4Exercise 3.5Exercise 3.6Exercise 3.7Exercise 3.8Exercise 3.9Exercise 3.10Exercise 3.11Exercise 3.12Exercise 3.13Exercise 3.14Exercise 3.16Exercise 3.17Exercise 3.18Exercise 3.19Exercise 3.20Unit Exercise – 3

## Samacheer Kalvi solutions for Mathematics Class 10 SSLC Tamil Nadu State Board chapter 3 - Algebra

Samacheer Kalvi solutions for Mathematics Class 10 SSLC Tamil Nadu State Board chapter 3 (Algebra) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the Tamil Nadu Board of Secondary Education Mathematics Class 10 SSLC Tamil Nadu State Board solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Mathematics Class 10 SSLC Tamil Nadu State Board chapter 3 Algebra are Introduction to Algebra, Simultaneous Linear Equations in Three Variables, GCD and LCM of Polynomials, Rational Expressions, Square Root of Polynomials, Quadratic Equations, Graph of Variations, Quadratic Graphs, Matrices.

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