Tamil Nadu Board of Secondary EducationHSC Science Class 11th

# Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide chapter 2 - Basic Algebra [Latest edition]

## Chapter 2: Basic Algebra

Exercise 2.1Exercise 2.2Exercise 2.3Exercise 2.4Exercise 2.5Exercise 2.6Exercise 2.7Exercise 2.8Exercise 2.9Exercise 2.10Exercise 2.11Exercise 2.12Exercise 2.13
Exercise 2.1 [Page 55]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 2 Basic Algebra Exercise 2.1 [Page 55]

Exercise 2.1 | Q 1 | Page 55

Classify each element of {sqrt(7), (-1)/4, 0, 3, 1, 4, 4, 22/7} as a member of N, Q, R − Q or Z

Exercise 2.1 | Q 2 | Page 55

Prove that sqrt(3) is an irrational number.
(Hint: Follow the method that we have used to prove sqrt(2) ∉ Q)

Exercise 2.1 | Q 3 | Page 55

Are there two distinct irrational numbers such that their difference is a rational number? Justify

Exercise 2.1 | Q 4 | Page 55

Find two irrational numbers such that their sum is a rational number. Can you find two irrational numbers whose product is a rational number

Exercise 2.1 | Q 5 | Page 55

Find a positive number smaller than 2^(1/1000). Justify

Exercise 2.2 [Page 57]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 2 Basic Algebra Exercise 2.2 [Page 57]

Exercise 2.2 | Q 1. (i) | Page 57

Solve for x: |3 − x| < 7

Exercise 2.2 | Q 1. (ii) | Page 57

Solve for x: |4x − 5| ≥ −2

Exercise 2.2 | Q 1. (iii) | Page 57

Solve for x: |3 - 3/4x| ≤ 1/4

Exercise 2.2 | Q 1. (iv) | Page 57

Solve for x: |x| − 10 < −3

Exercise 2.2 | Q 2 | Page 57

Solve 1/(|2x - 1|) < 6 and express the solution using the interval notation

Exercise 2.2 | Q 3 | Page 57

Solve −3|x| + 5 ≤ −2 and graph the solution set in a number line

Exercise 2.2 | Q 4 | Page 57

Solve 2|x + 1| − 6 ≤ 7 and graph the solution set in a number line

Exercise 2.2 | Q 5 | Page 57

Solve 1/5|10x - 2| < 1

Exercise 2.2 | Q 6 | Page 57

Solve |5x − 12| < −2

Exercise 2.3 [Page 59]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 2 Basic Algebra Exercise 2.3 [Page 59]

Exercise 2.3 | Q 1. (i) | Page 59

Represent the following inequalities in the interval notation:

x ≥ −1 and x < 4

Exercise 2.3 | Q 1. (ii) | Page 59

Represent the following inequalities in the interval notation:

x ≤ 5 and x ≥ − 3

Exercise 2.3 | Q 1. (iii) | Page 59

Represent the following inequalities in the interval notation:

x < −1 or x < 3

Exercise 2.3 | Q 1. (iv) | Page 59

Represent the following inequalities in the interval notation:

−2x > 0 or 3x − 4 < 11

Exercise 2.3 | Q 2. (i) | Page 59

Solve 23x < 100 when x is a natural number

Exercise 2.3 | Q 2. (ii) | Page 59

Solve 23x < 100 when x is an integer

Exercise 2.3 | Q 3. (i) | Page 59

Solve −2x ≥ 9 when x is a real number

Exercise 2.3 | Q 3. (ii) | Page 59

Solve −2x ≥ 9 when x is an integer

Exercise 2.3 | Q 3. (iii) | Page 59

Solve −2x ≥ 9 when x is a natural number

Exercise 2.3 | Q 4. (i) | Page 59

Solve: (3(x - 2))/5 ≤ (5(2 - x))/3

Exercise 2.3 | Q 4. (ii) | Page 59

Solve: (5 - x)/3 < x/2 - 4

Exercise 2.3 | Q 5 | Page 59

To secure A grade one must obtain an average of 90 marks or more in 5 subjects each of maximum 100 marks. If one scored 84, 87, 95, 91 in first four subjects, what is the minimum mark one scored in the fifth subject to get A grade in the course?

Exercise 2.3 | Q 6 | Page 59

A manufacturer has 600 litres of a 12 percent solution of acid. How many litres of a 30 percent acid solution must be added to it so that the acid content in the resulting mixture will be more than 15 percent but less than 18 percent?

Exercise 2.3 | Q 7 | Page 59

Find all pairs of consecutive odd natural numbers both of which are larger than 10 and their sum is less than 40

Exercise 2.3 | Q 8 | Page 59

A model rocket is launched from the ground. The height h reached by the rocket after t seconds from lift off is given by h(t) = −5t2 +100t, 0 ≤ t ≤ 20. At what time the rocket is 495 feet above the ground?

Exercise 2.3 | Q 9 | Page 59

A plumber can be paid according to the following schemes: In the first scheme he will be paid rupees 500 plus rupees 70 per hour, and in the second scheme he will be paid rupees 120 per hour. If he works x hours, then for what value of x does the first scheme give better wages?

Exercise 2.3 | Q 10 | Page 59

A and B are working on similar jobs but their monthly salaries differ by more than Rs 6000. If B earns rupees 27000 per month, then what are the possibilities of A’s salary per month?

Exercise 2.4 [Page 62]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 2 Basic Algebra Exercise 2.4 [Page 62]

Exercise 2.4 | Q 1 | Page 62

Construct a quadratic equation with roots 7 and −3

Exercise 2.4 | Q 2 | Page 62

A quadratic polynomial has one of its zeros 1 + sqrt(5) and it satisfies p(1) = 2. Find the quadratic polynomial

Exercise 2.4 | Q 3 | Page 62

If α and β are the roots of the quadratic equation x^2 + sqrt(2)x + 3 = 0, form a quadratic polynomial with zeroes 1/α, 1/β

Exercise 2.4 | Q 4 | Page 62

If one root of k(x − 1)2 = 5x − 7 is double the other root, show that k = 2 or −25

Exercise 2.4 | Q 5 | Page 62

If the difference of the roots of the equation 2x2 − (a + 1)x + a − 1 = 0 is equal to their product, then prove that a = 2

Exercise 2.4 | Q 6. (i) | Page 62

Find the condition that one of the roots of ax2 + bx + c may be negative of the other

Exercise 2.4 | Q 6. (ii) | Page 62

Find the condition that one of the roots of ax2 + bx + c may be thrice the other

Exercise 2.4 | Q 6. (iii) | Page 62

Find the condition that one of the roots of ax2 + bx + c may be reciprocal of the other

Exercise 2.4 | Q 7 | Page 62

If the equations x2 − ax + b = 0 and x2 − ex + f = 0 have one root in common and if the second equation has equal roots, then prove that ae = 2(b + f)

Exercise 2.4 | Q 8. (I) | Page 62

Discuss the nature of roots of − x2 + 3x + 1 = 0

Exercise 2.4 | Q 8. (ii) | Page 62

Discuss the nature of roots of 4x2 − x − 2 = 0

Exercise 2.4 | Q 8. (iii) | Page 62

Discuss the nature of roots of 9x2 + 5x = 0

Exercise 2.4 | Q 9. (i) | Page 62

Without sketching the graph, find whether the graph of the following function will intersect the x-axis and if so in how many points
y = x2 + x + 2

Exercise 2.4 | Q 9. (ii) | Page 62

Without sketching the graph, find whether the graph of the following function will intersect the x-axis and if so in how many points
y = x2 − 3x − 7

Exercise 2.4 | Q 9. (iii) | Page 62

Without sketching the graph, find whether the graph of the following function will intersect the x-axis and if so in how many points
y = x2 + 6x + 9

Exercise 2.4 | Q 10 | Page 62

Write f(x) = x2 + 5x + 4 in completed square form

Exercise 2.5 [Page 63]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 2 Basic Algebra Exercise 2.5 [Page 63]

Exercise 2.5 | Q 1 | Page 63

Solve 2x2 + x – 15 ≤ 0

Exercise 2.5 | Q 2 | Page 63

Solve – x2 + 3x – 2 ≥ 0

Exercise 2.6 [Page 66]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 2 Basic Algebra Exercise 2.6 [Page 66]

Exercise 2.6 | Q 1 | Page 66

Find the zeros of the polynomial function f(x) = 4x2 − 25

Exercise 2.6 | Q 2 | Page 66

If x = −2 is one root of x3 − x2 − 17x = 22, then find the other roots of equation

Exercise 2.6 | Q 3 | Page 66

Find the real roots of x4 = 16

Exercise 2.6 | Q 4 | Page 66

Solve (2x + 1)2 − (3x + 2)2 = 0

Exercise 2.7 [Page 68]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 2 Basic Algebra Exercise 2.7 [Page 68]

Exercise 2.7 | Q 1 | Page 68

Factorize: x4 + 1. (Hint: Try completing the square)

Exercise 2.7 | Q 2 | Page 68

If x2 + x + 1 is a factor of the polynomial 3x3 + 8x2 + 8x + a, then find the value of a

Exercise 2.8 [Page 69]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 2 Basic Algebra Exercise 2.8 [Page 69]

Exercise 2.8 | Q 1 | Page 69

Find all values of x for which (x^3(x - 1))/((x - 2)) > 0

Exercise 2.8 | Q 2 | Page 69

Find all values of x that satisfies the inequality (2x - 3)/((x - 2)(x - 4)) < 0

Exercise 2.8 | Q 3 | Page 69

Solve (x^2 - 4)/(x^2 - 2x - 15) ≤ 0

Exercise 2.9 [Page 71]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 2 Basic Algebra Exercise 2.9 [Page 71]

Exercise 2.9 | Q 1 | Page 71

Resolve the following rational expressions into partial fractions

1/(x^2 - "a"^2)

Exercise 2.9 | Q 2 | Page 71

Resolve the following rational expressions into partial fractions

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

Exercise 2.9 | Q 3 | Page 71

Resolve the following rational expressions into partial fractions

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

Exercise 2.9 | Q 4 | Page 71

Resolve the following rational expressions into partial fractions

x/((x - 1)^3

Exercise 2.9 | Q 5 | Page 71

Resolve the following rational expressions into partial fractions

1/(x^4 - 1)

Exercise 2.9 | Q 6 | Page 71

Resolve the following rational expressions into partial fractions

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

Exercise 2.9 | Q 7 | Page 71

Resolve the following rational expressions into partial fractions

(x^2 + x + 1)/(x^2 - 5x + 6)

Exercise 2.9 | Q 8 | Page 71

Resolve the following rational expressions into partial fractions

(x^3 + 2x + 1)/(x^2 + 5x + 6)

Exercise 2.9 | Q 9 | Page 71

Resolve the following rational expressions into partial fractions

(x + 12)/((x + 1)^2 (x - 2))

Exercise 2.9 | Q 10 | Page 71

Resolve the following rational expressions into partial fractions

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

Exercise 2.9 | Q 11 | Page 71

Resolve the following rational expressions into partial fractions

(2x^2 + 5x - 11)/(x^2 + 2x - 3)

Exercise 2.9 | Q 12 | Page 71

Resolve the following rational expressions into partial fractions

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

Exercise 2.10 [Page 73]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 2 Basic Algebra Exercise 2.10 [Page 73]

Exercise 2.10 | Q 1 | Page 73

Determine the region in the plane determined by the inequalities:

x ≤ 3y, x ≥ y

Exercise 2.10 | Q 2 | Page 73

Determine the region in the plane determined by the inequalities:

y ≥ 2x, −2x + 3y ≤ 6

Exercise 2.10 | Q 3 | Page 73

Determine the region in the plane determined by the inequalities:

3x + 5y ≥ 45, x ≥ 0, y ≥ 0

Exercise 2.10 | Q 4 | Page 73

Determine the region in the plane determined by the inequalities:

2x + 3y ≤ 35, y ≥ 2, x ≥ 5.

Exercise 2.10 | Q 5 | Page 73

Determine the region in the plane determined by the inequalities:

2x + 3y ≤ 6, x + 4y ≤ 4, x ≥ 0, y ≥ 0

Exercise 2.10 | Q 6 | Page 73

Determine the region in the plane determined by the inequalities:

x − 2y ≥ 0, 2x − y ≤ −2, x ≥ 0, y ≥ 0

Exercise 2.10 | Q 7 | Page 73

Determine the region in the plane determined by the inequalities:

2x + y ≥ 8, x + 2y ≥ 8, x + y ≤ 6

Exercise 2.11 [Page 77]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 2 Basic Algebra Exercise 2.11 [Page 77]

Exercise 2.11 | Q 1. (i) | Page 77

Simplify: (125)^(2/3)

Exercise 2.11 | Q 1. (ii) | Page 77

Simplify: 16^((-3)/4)

Exercise 2.11 | Q 1. (iii) | Page 77

Simplify: (- 1000)^((-2)/3)

Exercise 2.11 | Q 1. (iv) | Page 77

Simplify: (3^-6)^(1/3)

Exercise 2.11 | Q 1. (v) | Page 77

Simplify: (27^((-2)/3))/(27^((-1)/3))

Exercise 2.11 | Q 2 | Page 77

Evaluate [((256)^(-1/2))^((-1)/4)]^3

Exercise 2.11 | Q 3 | Page 77

If (x^(1/2) + x^(- 1/2))^2 = 9/2 then find the value of (x^(1/2) - x^(-1/2)) for x >1

Exercise 2.11 | Q 4 | Page 77

Simplify and hence find the value of n:

(3^(2"n")*9^2*3^(-"n"))/(3^(3"n")) = 27

Exercise 2.11 | Q 5 | Page 77

Find the radius of the spherical tank whose volume is (32pi)/3 units

Exercise 2.11 | Q 6 | Page 77

. Simplify by rationalising the denominator (7 + sqrt(6))/(3 - sqrt(2))

Exercise 2.11 | Q 7 | Page 77

Simplify 1/(3 - sqrt(8)) - 1/(sqrt(8) - sqrt(7)) + 1/(sqrt(7) - sqrt(6)) - 1/(sqrt(6) - sqrt(5)) + 1/(sqrt(5) - 2)

Exercise 2.11 | Q 8 | Page 77

If x = sqrt(2) + sqrt(3) find (x^2 + 1)/(x^2 - 2)

Exercise 2.12 [Pages 80 - 81]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 2 Basic Algebra Exercise 2.12 [Pages 80 - 81]

Exercise 2.12 | Q 1 | Page 80

Let b > 0 and b ≠ 1. Express y = bx in logarithmic form. Also state the domain and range of the logarithmic function

Exercise 2.12 | Q 2 | Page 80

Compute log9 27 – log27 9

Exercise 2.12 | Q 3 | Page 80

Solve log8x + log4x + log2x = 11

Exercise 2.12 | Q 4 | Page 80

Solve log28x = 2log28

Exercise 2.12 | Q 5 | Page 80

If a2 + b2 = 7ab, show that log  ("a" + "b")/3 = 1/2(log"a" + log "b")

Exercise 2.12 | Q 6 | Page 80

Prove log  "a"^2/"bc" + log  "b"^2/"ca" + log  "c"^2/"ab" = 0

Exercise 2.12 | Q 7 | Page 80

Prove that log 2 + 16log  16/15 + 12log  25/24 + 7log  81/80 = 1

Exercise 2.12 | Q 8 | Page 80

Prove that loga2 a + logb2 b + logc2 c = 1/8

Exercise 2.12 | Q 9 | Page 80

Prove log a + log a2 + log a3 + · · · + log an = ("n"("n" + 1))/2 log "a"

Exercise 2.12 | Q 10 | Page 81

If log x/(y - z) = logy/(z - x) = logz/(x - y), then prove that xyz = 1

Exercise 2.12 | Q 11 | Page 81

Solve log_2 x − 3 log_(1/2) x = 6

Exercise 2.12 | Q 12 | Page 81

Solve log5 – x (x2 – 6x + 65) = 2

Exercise 2.13 [Pages 81 - 83]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 2 Basic Algebra Exercise 2.13 [Pages 81 - 83]

#### MCQ

Exercise 2.13 | Q 1 | Page 81

Choose the correct alternative:
If |x + 2| ≤ 9, then x belongs to

• (−∞, −7)

• [−11, 7]

• (−∞, −7) ∪ [11, ∞)

• (−11, 7)

Exercise 2.13 | Q 2 | Page 81

Choose the correct alternative:
Given that x, y and b are real numbers x < y, b > 0, then

• xb < yb

• xb > yb

• xb ≤ yb

• x/"b" ≥ y/"b"

Exercise 2.13 | Q 3 | Page 81

Choose the correct alternative:
If  |x - 2|/(x - 2) ≥ 0, then x belongs to

• [2, ∞)

• (2, ∞)

• (−∞, 2)

• (−2, ∞)

Exercise 2.13 | Q 4 | Page 81

Choose the correct alternative:
The solution of 5x − 1 < 24 and 5x + 1 > −24 is

• (4, 5)

• (−5, −4)

• (−5, 5)

• (−5, 4)

Exercise 2.13 | Q 5 | Page 81

Choose the correct alternative:
The solution set of the following inequality |x − 1| ≥ |x − 3| is

• [0, 2]

• [2, ∞)

• (0, 2)

• (−∞, 2)

Exercise 2.13 | Q 6 | Page 82

Choose the correct alternative:
The value of log_(sqrt(5)) 512 is

• 16

• 18

• 9

• 12

Exercise 2.13 | Q 7 | Page 82

Choose the correct alternative:
The value of log_3  1/81 is

• −2

• −8

• −4

• −9

Exercise 2.13 | Q 8 | Page 82

Choose the correct alternative:
If log_(sqrt(x) 0.25 = 4, then the value of x is

• 0.5

• 2.5

• 1.5

• 1.25

Exercise 2.13 | Q 9 | Page 82

Choose the correct alternative:
The value of logab  logbc  logca is

• 2

• 1

• 3

• 4

Exercise 2.13 | Q 10 | Page 82

Choose the correct alternative:
If 3 is the logarithm of 343, then the base is

• 5

• 7

• 6

• 9

Exercise 2.13 | Q 11 | Page 82

Choose the correct alternative:
Find a so that the sum and product of the roots of the equation 2x2 + (a − 3)x + 3a − 5 = 0 are equal is

• 1

• 2

• 0

• 4

Exercise 2.13 | Q 12 | Page 82

Choose the correct alternative:
If a and b are the roots of the equation x2 − kx + 16 = 0 and satisfy a2 + b2 = 32, then the value of k is

• 10

• −8

• −8, 8

• 6

Exercise 2.13 | Q 13 | Page 82

Choose the correct alternative:
The number of solutions of x2 + |x − 1| = 1 is

• 1

• 0

• 2

• 3

Exercise 2.13 | Q 14 | Page 82

Choose the correct alternative:
The equation whose roots are numerically equal but opposite in sign to the roots of 3x2 − 5x − 7 = 0 is

• 3x2 − 5x − 7 = 0

• 3x2 + 5x − 7 = 0

• 3x2 − 5x + 7 = 0

• 3x2 + x − 7 = 0

Exercise 2.13 | Q 15 | Page 82

Choose the correct alternative:
If 8 and 2 are the roots of x2 + ax + c = 0 and 3, 3 are the roots of x2 + dx + b = 0, then the roots of the equation x2 + ax + b = 0 are

• 1, 2

• −1, 1

• 9, 1

• −1, 2

Exercise 2.13 | Q 16 | Page 82

Choose the correct alternative:
If a and b are the real roots of the equation x2 − kx + c = 0, then the distance between the points (a, 0) and (b, 0) is

• sqrt("k"^2 - 4"c")

• sqrt(4"k"^2 - "c")

• sqrt(4"c" - "k"^2)

• sqrt("k" - 8"c")

Exercise 2.13 | Q 17 | Page 83

Choose the correct alternative:
If ("k"x)/((x + 2)(x - 1)) = 2/(x + 2) + 1/(x - 1), then the value of k is

• 1

• 2

• 3

• 4

Exercise 2.13 | Q 18 | Page 83

Choose the correct alternative:
If (1 - 2x)/(3 + 2x - x^2) = "A"/(3 - x) + "B"/(x + 1), then the value of A + B is

• (-1)/2

• (-2)/3

• 1/2

• 2/3

Exercise 2.13 | Q 19 | Page 83

Choose the correct alternative:
The number of roots of (x + 3)4 + (x + 5)4 = 16 is

• 4

• 2

• 3

• 0

Exercise 2.13 | Q 20 | Page 83

Choose the correct alternative:
The value of log3 11 . log11 13 . log13 15 . log15 27 . log27 81 is

• 1

• 2

• 3

• 4

## Chapter 2: Basic Algebra

Exercise 2.1Exercise 2.2Exercise 2.3Exercise 2.4Exercise 2.5Exercise 2.6Exercise 2.7Exercise 2.8Exercise 2.9Exercise 2.10Exercise 2.11Exercise 2.12Exercise 2.13

## Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide chapter 2 - Basic Algebra

Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide chapter 2 (Basic 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 Class 11th Mathematics Volume 1 and 2 Answers Guide solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 11th Mathematics Volume 1 and 2 Answers Guide chapter 2 Basic Algebra are Introduction to Basic Algebra, Real Number System, Absolute Value, Linear Inequalities, Quadratic Functions, Polynomial Functions, Rational Functions, Exponents and Radicals, Logarithms, Application of Algebra in Real Life.

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