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📚 Class IX Maths 🧩 Worksheet Chapter 2: Introduction to Linear Polynomials

Polynomials Worksheet Class 9 – Ganita Manjari Ch 2, 75 Qs

Class 9 Maths Linear Polynomials worksheet with answers — 75 questions with step-by-step solutions. Ganita Manjari Ch 2. Free PDF & online practice.

This free Worksheet for CBSE Class IX Maths, Chapter 2: Introduction to Linear Polynomials, contains a structured worksheet with MCQs, short answer, case-based and HOTS questions in one place. It has been prepared by Sumeet Sahu at Unique Study Point, Indore, strictly following the latest NCERT syllabus for Session 2026-27.

📌 How to use this Worksheet

Introduction to Linear Polynomials — Class 9
UNIQUE STUDY POINT BY SUMEET SAHU

Introduction to Linear Polynomials

Class 9 · Maths (Ganita Manjari) · Practice Worksheet with Solutions

75 Questions
Tap any question's "Show Answer" button to reveal the full step-by-step solution.

Section A · Objective Type Questions (Q1–Q30)

1 Mark each
1
Which of the following expressions is not a polynomial?MCQ
  • a) 5x² − √2 x + ⅔
  • b) 5x³ − 3x² − x + ⅗
  • c) √2 x³ − √5 x + 7
  • d) 5x³ − 3x² − 1√x + 2
✓ Correct Answer: (d) 5x³ − 3x² − 1√x + 2
  1. In a polynomial, every term must be a constant times the variable raised to a whole-number power (0, 1, 2, 3, …).
  2. The term 1√x can be written as x−12, and the power −½ is neither a positive integer nor zero.
  3. So this expression fails the polynomial condition, while options (a), (b), (c) all have valid integer powers.
2
What is a linear pattern in the context of sequences?MCQ
  • a) A sequence where the ratio between consecutive terms is constant.
  • b) A sequence where terms are randomly generated.
  • c) A sequence where terms increase by multiplication.
  • d) A sequence of numbers where the difference between two consecutive terms is constant.
✓ Correct Answer: (d) A sequence where the difference between consecutive terms is constant
  1. This constant difference is exactly what makes a sequence "linear" — each term is obtained by adding the same fixed amount to the one before it.
3
A rectangular plot has a fixed breadth of 12 m. If its length is 'l' meters, what is the linear pattern representing the perimeter of the plot?MCQ
  • a) 2l + 12
  • b) 12l
  • c) 2(l + 12)
  • d) 24 + l
✓ Correct Answer: (c) 2(l + 12)
Perimeter = 2 × (Length + Breadth)
  1. Given breadth = 12 m, length = l meters.
  2. Substituting: Perimeter = 2 × (l + 12) = 2(l + 12).
4
A chef buys 'n' kilograms of flour. The total cost is represented by the linear polynomial 3.5n + 25, where ₹25 is a fixed delivery charge. If the variable 'n' is increased by 1.5 kg, what is the corresponding increase in the total cost?MCQ
  • a) ₹3.5
  • b) ₹28.5
  • c) ₹25
  • d) ₹5.25
✓ Correct Answer: (d) ₹5.25
  1. Original cost = 3.5n + 25.
  2. New amount of flour = n + 1.5, so new cost = 3.5(n + 1.5) + 25 = 3.5n + 5.25 + 25 = 3.5n + 30.25.
  3. Increase = (3.5n + 30.25) − (3.5n + 25) = ₹5.25.
5
√2 is a polynomial of degreeMCQ
  • a) 2
  • b) 1
  • c) ½
  • d) 0
✓ Correct Answer: (d) 0
  1. √2 is just a constant number (no variable attached), so it is a constant term.
  2. The degree of any nonzero constant is always 0.
6
A store offers a discount on bulk purchases. The total price 'y' for 'x' items can be represented by a linear relationship y = ax + b. If buying 10 items costs ₹320 and buying 25 items costs ₹770, what is the price per item ('a') and what is the fixed discount or charge ('b')?MCQ
  • a) a = 30, b = 20
  • b) a = 25, b = 70
  • c) a = 20, b = 30
  • d) a = 35, b = −30
✓ Correct Answer: (a) a = 30, b = 20
  1. From the given data: 320 = 10a + b …(1), and 770 = 25a + b …(2).
  2. Subtracting (1) from (2): 450 = 15a  ⇒  a = 30.
  3. Substituting a = 30 into (1): 320 = 300 + b  ⇒  b = 20.
✓ a = 30, b = 20
7
What is the value of P(x) = 2x + 1 when x = 3?MCQ
  • a) 7
  • b) 5
  • c) 6
  • d) 8
✓ Correct Answer: (a) 7
  1. Substitute x = 3: P(3) = 2(3) + 1 = 6 + 1 = 7.
8
The coefficient of x³ in 3x² + x² − 5x³ + x⁴ isMCQ
  • a) 1
  • b) 2
  • c) 3
  • d) −5
✓ Correct Answer: (d) −5
  1. The only term containing x³ is −5x³, so its coefficient is −5.
9
The relationship between two units of measurement, 'A' and 'B', can be expressed as a linear equation A = aB + b. If 'A' is 50 when 'B' is 10, and 'A' is 90 when 'B' is 20, find the values of 'a' and 'b'.MCQ
  • a) a = 2, b = 30
  • b) a = 3, b = 20
  • c) a = 5, b = 0
  • d) a = 4, b = 10
✓ Correct Answer: (d) a = 4, b = 10
  1. From the data: 50 = 10a + b …(1), and 90 = 20a + b …(2).
  2. Subtracting (1) from (2): 40 = 10a  ⇒  a = 4.
  3. Substituting a = 4 into (1): 50 = 40 + b  ⇒  b = 10.
✓ a = 4, b = 10
10
Two delivery services charge according to the following rules:
Service A: ₹40 fixed charge + ₹12 per package
Service B: ₹70 fixed charge + ₹6 per package
For how many packages will both services charge the same amount?MCQ
  • a) 6 packages
  • b) 4 packages
  • c) 3 packages
  • d) 5 packages
✓ Correct Answer: (d) 5 packages
  1. Let p be the number of packages. Cost of A = 40 + 12p, Cost of B = 70 + 6p.
  2. Equating: 40 + 12p = 70 + 6p  ⇒  6p = 30  ⇒  p = 5.
✓ Both cost the same for 5 packages
11
A production line has a fixed setup cost and a variable cost per item produced. If 100 items cost ₹800 to produce and 250 items cost ₹1700 to produce, what is the fixed setup cost?MCQ
  • a) ₹150
  • b) ₹300
  • c) ₹200
  • d) ₹250
✓ Correct Answer: (c) ₹200
  1. Let y = ax + b, where b is the fixed setup cost.
  2. From the data: 800 = 100a + b …(1), and 1700 = 250a + b …(2).
  3. Subtracting (1) from (2): 900 = 150a  ⇒  a = 6.
  4. Substituting a = 6 into (1): 800 = 600 + b  ⇒  b = 200.
✓ Fixed setup cost = ₹200
12
Which of the following equations represents a linear relationship where the y-coordinate is three times the x-coordinate?MCQ
  • a) x = 3y
  • b) y = x + 3
  • c) y = 3x
  • d) y = x3
✓ Correct Answer: (c) y = 3x
  1. "y-coordinate is three times the x-coordinate" directly translates to y = 3x.
13
Which of the following is a quadratic polynomial?MCQ
  • a) x² + 5x + 4
  • b) x³ + x
  • c) x³ + 2x + 6
  • d) x + 4
✓ Correct Answer: (a) x² + 5x + 4
  1. A quadratic polynomial has degree exactly 2.
  2. x² + 5x + 4 has highest power 2, so it is quadratic. The other options have degree 3 or 1.
14
A fruit vendor charges a base fee of ₹150 for delivery plus ₹25 for every kilogram of fruit ordered. If 'k' represents the number of kilograms of fruit, which of the following expressions represents the total cost?MCQ
  • a) 150k + 25
  • b) 150 − 25k
  • c) 175k
  • d) 25k + 150
✓ Correct Answer: (d) 25k + 150
  1. The base fee (₹150) is fixed, and ₹25 is charged per kilogram (k), giving cost dependent on kg = 25k.
  2. Total cost = 25k + 150.
15
If a relationship between two quantities x and y is given by y = 3x − 5, what is the value of y when x = 4?MCQ
  • a) 17
  • b) 7
  • c) −1
  • d) 10
✓ Correct Answer: (b) 7
  1. Substitute x = 4: y = 3(4) − 5 = 12 − 5 = 7.
16
A car rental company charges a daily base fee of ₹40 plus ₹0.15 per kilometer driven. Construct a linear expression representing the total cost for driving 'k' kilometers in one day.MCQ
  • a) 40k + 0.15
  • b) 0.15 − 40k
  • c) 40 − 0.15k
  • d) 0.15k + 40
✓ Correct Answer: (d) 0.15k + 40
  1. Base fee ₹40 is fixed, and ₹0.15 is charged per km, giving 0.15k.
  2. Total cost = 0.15k + 40.
17
Consider the polynomials P(x) = 5x + 3 and Q(x) = 2x − 1. If P(x) equals Q(x), what is the value of x?MCQ
  • a) x = −43
  • b) x = −2
  • c) x = 43
  • d) x = 2
✓ Correct Answer: (a) x = −43
  1. Set P(x) = Q(x): 5x + 3 = 2x − 1.
  2. Subtract 2x from both sides: 3x + 3 = −1.
  3. Subtract 3: 3x = −4  ⇒  x = −43.
18
The degree of a constant polynomial isMCQ
  • a) 1
  • b) 3
  • c) 0
  • d) 2
✓ Correct Answer: (c) 0
  1. Any nonzero constant c can be written as c·x⁰, so its degree is 0.
19
Which of the following is a polynomial.MCQ
  • a) 1x + 5
  • b) −4
  • c) x − √x + 2
  • d) 1x + 3
✓ Correct Answer: (b) −4
  1. −4 is a constant polynomial of degree zero — it satisfies the polynomial definition.
  2. The other options contain 1x (negative power) or √x (fractional power), which are not allowed in polynomials.
20
In the input-output process described for a polynomial, what does 'input' typically represent?MCQ
  • a) The result of the polynomial calculation
  • b) The coefficient of the variable
  • c) The constant term of the polynomial
  • d) The value of the variable
✓ Correct Answer: (d) The value of the variable
  1. The "input" is the value substituted for the variable (like x); the polynomial then produces an "output" value.
21
A polynomial of degree ____ is called a cubic polynomial.MCQ
  • a) 0
  • b) 1
  • c) 3
  • d) 2
✓ Correct Answer: (c) 3
  1. A polynomial of degree 3 is called cubic, with general form ax³ + bx² + cx + d.
22
A polynomial of degree ____ is called a quadratic polynomial.MCQ
  • a) constant
  • b) zero
  • c) degree
  • d) 2
✓ Correct Answer: (d) 2
  1. A polynomial of degree 2 is called quadratic, with general form ax² + bx + c.
23
The largest power of x in p(x) is the ________ of the polynomial.MCQ
  • a) root
  • b) coefficient
  • c) degree
  • d) constant
✓ Correct Answer: (c) degree
  1. The degree of a polynomial is defined as its greatest exponent (largest power of the variable).
24
Which of the following statements is false?
A. The degree of a zero polynomial is defined.
B. The degree of a zero polynomial is zero.
C. The zero of a zero polynomial is not defined.
D. The degree of a constant polynomial is not defined.MCQ
  • a) D
  • b) C
  • c) B
  • d) A
✓ Correct Answer: (d) A
  1. The degree of a zero polynomial is actually not defined — so statement A ("is defined") is false.
  2. Statements B, C, D are all true facts about zero or constant polynomials.
25
Choose the correct option for a polynomial:
i. 3x + 2
ii. 7x + 1 = 0
iii. 5x⁴ + 3x² + 1 = 0
iv. x³ + 3x² + 1MCQ
  • a) (i) & (iii)
  • b) (ii) & (iv)
  • c) (i) & (ii)
  • d) (i) & (iv)
✓ Correct Answer: (d) (i) & (iv)
  1. A polynomial is simply an expression, not an equation — it should not contain an "=" sign.
  2. (i) 3x + 2 and (iv) x³ + 3x² + 1 are pure expressions, so they are polynomials.
  3. (ii) and (iii) both involve equalities (they equal 0), so they are equations, not polynomials themselves.
26
Assertion (A): The number of elements in a sequence defined by a linear pattern where the first term is 7 and each subsequent term increases by 4 can be represented by the expression 4n + 3, where n is the term number.
Reason (R): A linear pattern is characterized by a constant difference between consecutive terms, and its general n-th term can be expressed in the form an + b.Assertion-Reason
  • a) Both A and R are true, and R is the correct explanation of A
  • b) Both A and R are true, but R is not the correct explanation of A
  • c) A is true, but R is false
  • d) A is false, but R is true
✓ Correct Answer: (a) Both A and R are true, and R is the correct explanation of A
  1. Assertion is true: substituting a₁ = 7 and d = 4 into the arithmetic formula yields 4n + 3.
  2. Reason is true: linear patterns have constant differences and follow the general form an + b.
  3. The Reason correctly explains how the algebraic form validates the Assertion's expression.
27
Assertion (A): The constant term in the polynomial 8x − 2x³ + 11 is 11.
Reason (R): A constant term in a polynomial is a term with a variable raised to the power of zero.Assertion-Reason
  • a) Both A and R are true, and R is the correct explanation of A
  • b) Both A and R are true, but R is not the correct explanation of A
  • c) A is true, but R is false
  • d) A is false, but R is true
✓ Correct Answer: (a) Both A and R are true, and R is the correct explanation of A
  1. Assertion correctly identifies the constant term as 11.
  2. Reason explains that a constant term is effectively a coefficient of x⁰ (since 11 = 11x⁰), which correctly explains why 11 is the constant term.
28
Assertion (A): The sequence generated by starting with 2 and repeatedly multiplying by 3 is a linear pattern.
Reason (R): In a linear pattern, each term is obtained by adding a constant value to the previous term.Assertion-Reason
  • a) Both A and R are true, and R is the correct explanation of A
  • b) Both A and R are true, but R is not the correct explanation of A
  • c) A is true, but R is false
  • d) A is false, but R is true
✓ Correct Answer: (d) A is false, but R is true
  1. Assertion is false: repeatedly multiplying by 3 gives 2, 6, 18, 54, … — this is a geometric sequence, not linear.
  2. Reason is true: linear patterns strictly require adding a constant value each time, which correctly refutes the false operation in the Assertion.
29
Assertion (A): A quantity that increases by a constant amount over equal intervals of an independent variable exhibits linear growth.
Reason (R): In a linear growth model, the rate of change of the dependent variable with respect to the independent variable is constant.Assertion-Reason
  • a) Both A and R are true, and R is the correct explanation of A
  • b) Both A and R are true, but R is not the correct explanation of A
  • c) A is true, but R is false
  • d) A is false, but R is true
✓ Correct Answer: (a) Both A and R are true, and R is the correct explanation of A
  1. Assertion correctly defines linear growth, characterized by a constant increase over equal intervals.
  2. Reason states the rate of change is constant, which is precisely why the growth is linear — this correctly explains the Assertion.
30
Assertion: P(x) = 14x³ − 2x² + 8x⁴ + 7x − 8 is a polynomial of degree 3.
Reason: The highest power of x in the polynomial p(x) is the degree of the polynomial.Assertion-Reason
  • a) Assertion and Reason both are correct statements and Reason is correct explanation for Assertion.
  • b) Assertion and Reason both are correct statements but Reason is not correct explanation for Assertion.
  • c) Assertion is correct statement but Reason is wrong statement.
  • d) Assertion is wrong statement but Reason is correct statement.
✓ Correct Answer: (d) Assertion is wrong statement but Reason is correct statement.
  1. The highest power of x in P(x) = 14x³ − 2x² + 8x⁴ + 7x − 8 is actually 4 (from the term 8x⁴), not 3.
  2. So the Assertion (claiming degree 3) is incorrect, but the Reason (defining degree as the highest power) is a correct general statement.

Section B · Short Answer Questions (Q31–Q50)

2 Marks each
31
Match the following:
(a) y − x = 1?
(b) 2x + 3 = 7?
(c) p + 2q = 8r − 2s?
(d) ax + by + cz = 0?
Options: (i) 1 variable   (ii) 2 variables   (iii) 3 variables   (iv) 4 variables
  1. (a) → (ii) y − x = 1 involves x and y — 2 variables.
  2. (b) → (i) 2x + 3 = 7 involves only x — 1 variable.
  3. (c) → (iv) p + 2q = 8r − 2s involves p, q, r, s — 4 variables.
  4. (d) → (iii) ax + by + cz = 0 involves x, y, z — 3 variables.
✓ (a)-(ii), (b)-(i), (c)-(iv), (d)-(iii)
32
Match the following table:
(a) 7x⁶ + 5x⁵ + 3x⁴ − 4x + 2?
(b) 2z² + 4z − 6?
(c) 4y³ − 2y + 5?
(d) x¹⁰ − x⁵ + 4?
Options: (i) 2   (ii) 10   (iii) 6   (iv) 3
  1. (a) → (iii) Highest power is 6, so degree = 6.
  2. (b) → (i) Highest power is 2, so degree = 2.
  3. (c) → (iv) Highest power is 3, so degree = 3.
  4. (d) → (ii) Highest power is 10, so degree = 10.
✓ (a)-(iii), (b)-(i), (c)-(iv), (d)-(ii)
33
Match the following:
(a) x³ − 3x² + 5x − 3?
(b) x² + 2x + 9?
(c) 7x + 12?
(d) 5x⁴ − 3x³ − 2x² + 3x + 5?
Options: (i) 2   (ii) 1   (iii) 4   (iv) 3
  1. (a) → (iv) Highest power is 3, so degree = 3.
  2. (b) → (i) Highest power is 2, so degree = 2.
  3. (c) → (ii) Highest power is 1, so degree = 1.
  4. (d) → (iii) Highest power is 4, so degree = 4.
✓ (a)-(iv), (b)-(i), (c)-(ii), (d)-(iii)
34
Match the following:
(a) Degree 1?
(b) Degree 2?
(c) Degree 3?
(d) Degree 4?
Options: (i) 1 − 3x³   (ii) x − 1   (iii) x⁴ − 3x² + 2 + 3x³   (iv) x² − 2x − 1
  1. (a) → (ii) x − 1 has highest power 1.
  2. (b) → (iv) x² − 2x − 1 has highest power 2.
  3. (c) → (i) 1 − 3x³ has highest power 3.
  4. (d) → (iii) x⁴ − 3x² + 2 + 3x³ has highest power 4.
✓ (a)-(ii), (b)-(iv), (c)-(i), (d)-(iii)
35
Match the following:
(a) Cubic polynomial?
(b) Quadratic polynomial?
(c) Linear polynomial?
(d) Zeros of x² − 5x + 6?
Options: (i) Degree 2   (ii) 2, 3   (iii) Degree 3   (iv) Degree 1
  1. (a) → (iii) A cubic polynomial has degree 3.
  2. (b) → (i) A quadratic polynomial has degree 2.
  3. (c) → (iv) A linear polynomial has degree 1.
  4. (d) → (ii) Factoring x² − 5x + 6 = (x−2)(x−3), so its zeros are 2 and 3.
✓ (a)-(iii), (b)-(i), (c)-(iv), (d)-(ii)
36
Find the degrees of the following polynomials:
  • 2x² − 5x + 3
  • y³ + 2y − 1
  • −9
  • 4z − 3
  1. (i) Highest power of x is 2  ⇒  Degree = 2
  2. (ii) Highest power of y is 3  ⇒  Degree = 3
  3. (iii) −9 is a constant polynomial (no variable)  ⇒  Degree = 0
  4. (iv) Highest power of z is 1  ⇒  Degree = 1
37
A chess club charges an entry structure of 200 + 50m, where m is the number of matches played. If a member pays a total of ₹750, calculate how many matches they played.
  1. Form the equation: 200 + 50m = 750.
  2. Subtract 200 from both sides: 50m = 550.
  3. Divide both sides by 50: m = 55050 = 11.
✓ The member played 11 matches
38
Identify whether the following algebraic expressions are univariate polynomials or not:
  • x² + 5x + 1
  • 4x + 5y + 3
  1. (i) x² + 5x + 1 is a univariate polynomial because it involves only one variable (x) raised to non-negative integer powers.
  2. (ii) 4x + 5y + 3 is not univariate because it contains two distinct variables (x and y).
39
A positive number is 5 times another number. If 21 is added to both the numbers, then one of the new numbers becomes twice the other new number. What are the numbers?
  1. Let the smaller number be x, so the larger number = 5x.
  2. After adding 21 to both: new numbers are x + 21 and 5x + 21.
  3. Given: 5x + 21 = 2(x + 21)  ⇒  5x + 21 = 2x + 42.
  4. 5x − 2x = 42 − 21  ⇒  3x = 21  ⇒  x = 7.
  5. Smaller number = 7, larger number = 5 × 7 = 35.
✓ The two numbers are 7 and 35
40
A prepaid mobile recharge plan costs ₹600, and the balance reduces by ₹15 each day. Formulate its balance model b(x) for x days, and find after how many days the balance will run out.
  1. The remaining balance equation is: b(x) = 600 − 15x.
  2. The balance runs out when b(x) = 0: 0 = 600 − 15x.
  3. 15x = 600  ⇒  x = 60015 = 40 days.
✓ b(x) = 600 − 15x; balance runs out after 40 days
41
What must be subtracted from x⁴ + 3x³ + 4x² − 3x − 6 to get 3x³ + 4x² − x + 3?
  1. Let p(x) be the required polynomial to be subtracted.
  2. (x⁴ + 3x³ + 4x² − 3x − 6) − p(x) = 3x³ + 4x² − x + 3
  3. p(x) = x⁴ + 3x³ + 4x² − 3x − 6 − 3x³ − 4x² + x − 3
  4. p(x) = x⁴ + (3−3)x³ + (4−4)x² + (−3+1)x + (−6−3)
✓ p(x) = x⁴ − 2x − 9
42
Evaluate the linear polynomial P(x) = 5x − 3 at x = −1.
  1. Substitute x = −1: P(−1) = 5(−1) − 3 = −5 − 3 = −8.
✓ P(−1) = −8
43
For the growing pattern of square tiles whose stages follow the sequence 1, 3, 5, 7, …, the general rule is 2n − 1. Find the number of tiles in the 26th stage.
  1. Number of tiles = 2n − 1. Substitute n = 26: Tiles = 2(26) − 1 = 52 − 1 = 51.
✓ 51 tiles
44
The graph of a linear polynomial p(x) = ax + b passes through the points (1, 5) and (3, 11). Solve for a and b to find the polynomial.
  1. Substitute the points into p(x) = ax + b: 5 = a + b …(1), and 11 = 3a + b …(2).
  2. From (1): b = 5 − a. Substituting into (2): 11 = 3a + (5 − a)  ⇒  11 = 2a + 5.
  3. 2a = 6  ⇒  a = 3. Then b = 5 − 3 = 2.
✓ p(x) = 3x + 2
45
A farmer cuts a 300 feet fence into two pieces of different sizes. The longer piece is four times as long as the shorter piece. How long are the two pieces?
  1. Let shorter piece = x feet, so longer piece = 4x feet.
  2. Total: x + 4x = 300  ⇒  5x = 300  ⇒  x = 60.
✓ Shorter piece = 60 feet, Longer piece = 240 feet
46
If we multiply a number by 52 and add 23 to the product, we get −712. Find the number.
  1. Let the number be x: 52x + 23 = −712.
  2. Subtract 23 from both sides: 52x = −712 − 812 = −1512 = −54.
  3. Multiply both sides by 25: x = −54 × 25 = −12.
✓ The number is −12
47
In the cost expression given by 200l + 160w + 50lw, identify all the terms, the coefficients of each term, and determine if it represents a single-variable expression.
  1. Terms: 200l, 160w, and 50lw.
  2. Coefficients: coefficient of l is 200, coefficient of w is 160, and coefficient of the product lw is 50.
  3. This expression involves two variables (l and w), so it is not a single-variable (univariate) expression.
48
Identify the type of polynomials given below (on the basis of degree):
  • 3x² + 4x + c
  • 3y³ − 4y² + 2y
  • 6y + 5
  • 3√2 x³ + x² − 7x
  1. (i) Highest power of x is 2, so it is a quadratic polynomial.
  2. (ii) Highest power of y is 3, so it is a cubic polynomial.
  3. (iii) Highest power of y is 1, so it is a linear polynomial.
  4. (iv) Highest power of x is 3, so it is a cubic polynomial.
49
What is the coefficient of z in the polynomial 4z³ + 5z² − 11?
  1. There is no term containing z¹ (i.e. just z) in the polynomial 4z³ + 5z² − 11.
✓ Coefficient of z is 0
50
Write polynomials of degrees 1, 2 and 3.
  1. Degree 1 (linear polynomial): Example — 2x + 5
  2. Degree 2 (quadratic polynomial): Example — x² + 3x + 1
  3. Degree 3 (cubic polynomial): Example — x³ − 2x² + x + 4

Section C · Short Answer Questions – II (Q51–Q65)

3 Marks each
51
A stationery shop sells notebooks at ₹12 each. A student also pays a fixed packing charge of ₹8. If x notebooks are purchased, write the algebraic expression for the total cost and find the cost for 7 notebooks.
  1. Total cost = 12x + 8.
  2. For x = 7: Total cost = 12(7) + 8 = 84 + 8 = 92.
✓ Expression: 12x + 8; Cost for 7 notebooks = ₹92
52
Using the expression 2n − 1, can you find out how many tiles will be there in the 15th stage and the 26th stage of the pattern? Also, which stage will contain 21 tiles and 47 tiles?
Number of tiles
  1. For 15th stage: 2(15) − 1 = 30 − 1 = 29 tiles.
  2. For 26th stage: 2(26) − 1 = 52 − 1 = 51 tiles.
Finding stage number
  1. For 21 tiles: 2n − 1 = 21  ⇒  2n = 22  ⇒  n = 11.
  2. For 47 tiles: 2n − 1 = 47  ⇒  2n = 48  ⇒  n = 24.
✓ 15th stage = 29 tiles, 26th stage = 51 tiles; 21 tiles = stage 11, 47 tiles = stage 24
53
A sequence is 6, 10, 14, 18, …
  • Find the common difference.
  • Write the nth term.
  • Find the 20th term.
  1. (i) Common difference = 10 − 6 = 4.
  2. (ii) nth term = First term + (n−1) × Common difference = 6 + (n−1)4 = 6 + 4n − 4 = 4n + 2.
  3. (iii) Substitute n = 20: 4(20) + 2 = 80 + 2 = 82.
✓ d = 4; nth term = 4n + 2; 20th term = 82
54
Suppose the length of a rectangle is 13 cm. Find the area if the breadth is (i) 12 cm, (ii) 10 cm, (iii) 8 cm. Find the linear pattern representing the area of the rectangle.
Area = Length × Breadth
  1. (i) Breadth = 12 cm: Area = 13 × 12 = 156 cm².
  2. (ii) Breadth = 10 cm: Area = 13 × 10 = 130 cm².
  3. (iii) Breadth = 8 cm: Area = 13 × 8 = 104 cm².
  4. Let breadth = x cm, then the linear pattern is Area = 13x.
✓ Areas: 156, 130, 104 cm²; Linear pattern: A = 13x
55
A club charges ₹150 as registration fee and ₹40 per game. Write a linear polynomial for the total cost and find the cost for 10 games.
  1. Let x = number of games played. Total cost = 150 + 40x.
  2. For x = 10: Total cost = 150 + 40(10) = 150 + 400 = 550.
✓ Polynomial: 150 + 40x; Cost for 10 games = ₹550
56
Find the value of the linear polynomial 8x − 5 for:
  • x = 2
  • x = 0
  • x = −1
  1. (i) x = 2: 8(2) − 5 = 16 − 5 = 11.
  2. (ii) x = 0: 8(0) − 5 = 0 − 5 = −5.
  3. (iii) x = −1: 8(−1) − 5 = −8 − 5 = −13.
✓ Values are 11, −5, and −13 respectively
57
The relation between x and y is y = 4x − 3.
  • Find y when x = 5.
  • Find y when x = 8.
  • Verify whether (3, 9) lies on the line.
  1. (i) x = 5: y = 4(5) − 3 = 20 − 3 = 17.
  2. (ii) x = 8: y = 4(8) − 3 = 32 − 3 = 29.
  3. (iii) For (3, 9): RHS = 4(3) − 3 = 12 − 3 = 9. Since LHS = 9 = RHS, the point does lie on the line.
✓ y(5) = 17, y(8) = 29; (3, 9) lies on the line
58
Find the perimeter of squares with sides 1 cm, 1.5 cm, 2 cm, 2.5 cm and 3 cm. What will happen to the perimeters if the sides increase by 0.5 cm?
Perimeter of a square = 4 × side
  1. Side = 1 cm ⇒ Perimeter = 4 × 1 = 4 cm
  2. Side = 1.5 cm ⇒ Perimeter = 4 × 1.5 = 6 cm
  3. Side = 2 cm ⇒ Perimeter = 4 × 2 = 8 cm
  4. Side = 2.5 cm ⇒ Perimeter = 4 × 2.5 = 10 cm
  5. Side = 3 cm ⇒ Perimeter = 4 × 3 = 12 cm
✓ Perimeters: 4, 6, 8, 10, 12 cm. Each time the side increases by 0.5 cm, the perimeter increases by a constant 2 cm (a linear pattern).
59
If you have ₹800 and you save ₹250 every month, find the amount you have after (i) 6 months (ii) 2 years. Express this as a linear pattern.
Amount after n months = 800 + 250n
  1. (i) After 6 months: 800 + 250(6) = 800 + 1500 = ₹2300.
  2. (ii) 2 years = 24 months: 800 + 250(24) = 800 + 6000 = ₹6800.
✓ After 6 months = ₹2300; After 2 years = ₹6800; Linear pattern: A = 800 + 250n
60
Identify the terms, variable, coefficients and constant term in the polynomial 9x + 14.
  1. Terms: 9x and 14
  2. Variable: x
  3. Coefficient of x: 9
  4. Constant term: 14
61
The sum of two numbers is 58. One number is 8 more than the other. Find the numbers.
  1. Let smaller number = x, larger number = x + 8.
  2. x + (x + 8) = 58  ⇒  2x + 8 = 58  ⇒  2x = 50  ⇒  x = 25.
  3. Larger number = 25 + 8 = 33.
✓ The numbers are 25 and 33
62
A student has ₹300. She spends ₹12 every day.
  • Write the amount left after n days.
  • Find the amount left after 15 days.
  • After how many days will ₹120 remain?
  1. (i) Amount left after n days = 300 − 12n.
  2. (ii) For n = 15: 300 − 12(15) = 300 − 180 = ₹120.
  3. (iii) Set 300 − 12n = 120  ⇒  12n = 180  ⇒  n = 15 days.
✓ Expression: 300 − 12n; After 15 days: ₹120; ₹120 remains after 15 days
63
A rectangle has length (x + 4) cm and width 5 cm. Write an expression for its perimeter and find the perimeter when x = 6.
Perimeter = 2(Length + Width)
  1. Perimeter = 2[(x + 4) + 5] = 2(x + 9) = 2x + 18.
  2. For x = 6: Perimeter = 2(6) + 18 = 12 + 18 = 30 cm.
✓ Expression: 2x + 18; Perimeter at x = 6 is 30 cm
64
Find the value of the linear polynomial 5x − 3 if:
  • x = 0
  • x = −1
  • x = 2
  1. (i) x = 0: 5(0) − 3 = −3.
  2. (ii) x = −1: 5(−1) − 3 = −5 − 3 = −8.
  3. (iii) x = 2: 5(2) − 3 = 10 − 3 = 7.
✓ Values are −3, −8, and 7 respectively
65
For a linear relation y = ax + b, the points (5, 220) and (10, 320) lie on the line. Find a and b.
  1. Substitute (5, 220): 220 = 5a + b …(1)
  2. Substitute (10, 320): 320 = 10a + b …(2)
  3. Subtract (1) from (2): 100 = 5a  ⇒  a = 20.
  4. Substitute a = 20 into (1): 220 = 100 + b  ⇒  b = 120.
✓ a = 20, b = 120

Section D · Case Study Based Questions (Q66–Q70)

4 Marks each
66
Case Study: Environmental scientists are studying a protected forest area containing 48,000 trees. Due to a planned afforestation programme, 1,200 new trees are planted every year. Researchers use mathematical models to predict future tree population and evaluate the effectiveness of conservation policies. Since the number of trees increases by a constant amount every year, the relationship can be represented by a linear polynomial.
Environmental scientists studying forest tree population
Researchers analysing the forest's tree population data
  • Write the linear equation representing the tree population after t years. (1)
  • Find the population after 10 years. (1)
  • Determine after how many years the population will become 60,000. (2)
    OR — State whether the situation represents linear growth or linear decay. (2)
  1. (a) Population after t years: P = 1200t + 48000.
  2. (b) Substitute t = 10: P = 1200(10) + 48000 = 12000 + 48000 = 60,000 trees.
  3. (c) Substitute P = 60000: 60000 = 1200t + 48000  ⇒  12000 = 1200t  ⇒  t = 10 years.
  4. OR: The situation represents linear growth because the population increases by a fixed, uniform amount (1,200 trees) every year — the slope is positive.
✓ (a) P = 1200t + 48000   (b) 60,000   (c) 10 years (Linear growth)
67
Case Study: As part of a school reading competition, Meera selected a book containing 720 pages. She decided to read 35 pages every day and maintain a consistent reading schedule. The activity helped students understand linear decay and the practical use of linear polynomials in daily life.
  • Write the linear equation representing the number of pages remaining after d days. (1)
  • Calculate the number of pages remaining after 12 days. (1)
  • Determine after how many days the book will be completely finished. (2)
    OR — State whether the situation represents linear growth or linear decay. (2)
  1. (a) Since Meera reads 35 pages every day, remaining pages decrease from the initial 720: P = 720 − 35d.
  2. (b) Substitute d = 12: P = 720 − 35(12) = 720 − 420 = 300 pages.
  3. (c) When finished, P = 0: 0 = 720 − 35d  ⇒  35d = 720  ⇒  d = 72035 ≈ 20.57, so she finishes on the 21st day.
  4. OR: The situation represents linear decay, since the remaining pages decrease at a uniform, constant rate (35 pages per day).
✓ (a) P = 720 − 35d   (b) 300 pages   (c) 21st day (Linear decay)
68
Case Study: To encourage water conservation, a school installed a rainwater harvesting system. The storage tank already contains 15,000 litres of water before the rainy season begins. During the rainy season, the tank collects 2,500 litres of rainwater every week.
Students observing rainwater harvesting system
Students observing and modelling the rainwater harvesting system
  • Write the linear equation representing the amount of water in the tank after w weeks. (1)
  • Calculate the amount of water in the tank after 8 weeks. (1)
  • Determine after how many weeks the tank will contain 35,000 litres of water. (2)
    OR — Identify the slope of the equation and explain its practical significance. Also state whether the graph represents linear growth or linear decay. (2)
  1. (a) Volume after w weeks: V = 2500w + 15000.
  2. (b) Substitute w = 8: V = 2500(8) + 15000 = 20000 + 15000 = 35,000 litres.
  3. (c) Substitute V = 35000: 35000 = 2500w + 15000  ⇒  20000 = 2500w  ⇒  w = 8 weeks.
  4. OR: The slope is 2500, meaning the water volume increases by 2,500 litres every week. Since the volume increases uniformly, the graph shows an upward-sloping straight line — representing linear growth.
✓ (a) V = 2500w + 15000   (b) 35,000 L   (c) 8 weeks (Slope = 2500, linear growth)
69
Case Study: Three linear relationships, y = 3x, y = 3x + 5 and y = 3x − 5, are plotted on the same coordinate grid below.
Graph of three parallel lines y=3x, y=3x+5, y=3x-5
The three parallel lines y = 3x, y = 3x + 5, and y = 3x − 5
  • Identify the y-intercept of each of the three lines. (1)
  • What do you notice about the slope of all three lines? What does this tell you about how they're positioned relative to one another? (1)
  • Which line crosses the y-axis at the highest point, and why? (2)
    OR — If a fourth line, y = −3x + 5, were added to this graph, would it be parallel to the others? Justify. (2)
  1. (i) Substituting x = 0 in each: y = 3x gives 0; y = 3x + 5 gives 5; y = 3x − 5 gives −5. So the y-intercepts are 0, 5, and −5 respectively.
  2. (ii) All three lines have the same slope (m = 3). Equal slopes mean the lines are parallel — equally steep, just shifted up or down by different amounts.
  3. (iii) y = 3x + 5 crosses highest, at (0, 5), because it has the largest value of b (the y-intercept) among the three.
  4. OR: No. Its slope is −3, which differs in sign from the slope 3 shared by the other three lines. Lines are parallel only when their slopes are exactly equal — same magnitude with opposite sign isn't enough.
✓ (i) 0, 5, −5   (ii) All slope = 3, so parallel   (iii) y = 3x+5 (or: not parallel, slope differs)
70
Case Study: To promote sustainable transportation, a city introduced electric buses. The operating company charges a fixed maintenance cost of ₹50,000 per month and an additional ₹2,500 for every route operated daily.
City officials planning electric bus routes
City officials planning electric bus routes using linear cost models
  • Write the linear equation representing the monthly operating cost. (1)
  • Calculate the operating cost for 20 routes. (1)
  • Find the number of routes corresponding to a monthly expenditure of ₹1,25,000. (2)
    OR — State the slope of the equation and explain its practical significance. (2)
  1. (a) Total cost = fixed maintenance + variable cost for x routes: y = 2500x + 50000.
  2. (b) Substitute x = 20: y = 2500(20) + 50000 = 50000 + 50000 = ₹1,00,000.
  3. (c) Substitute y = 125000: 125000 = 2500x + 50000  ⇒  75000 = 2500x  ⇒  x = 30 routes.
  4. OR: Comparing with y = mx + c, the slope (m) is 2500. This represents the constant rate at which the monthly cost increases per route — for every additional route, the operating cost rises by ₹2,500.
✓ (a) y = 2500x + 50000   (b) ₹1,00,000   (c) 30 routes (Slope = 2500)

Section E · Long Answer Questions (Q71–Q75)

5 Marks each
71
A mobile phone is bought for ₹10,000. Its value decreases by ₹800 every year.
  • Find the value of the phone after 3 years.
  • Make a table of values for t varying from 0 to 8 years and show how the value of the phone, v, depreciates with time.
  • Find an expression that relates v and t, and explain why it represents linear decay.
  1. (i) Decrease in 3 years = 800 × 3 = 2400. Value after 3 years = 10000 − 2400 = ₹7,600.
Table of values
t (years)012345678
v (₹)1000092008400760068006000520044003600
  1. (iii) Let value after t years = v. The expression is v = 10000 − 800t.
  2. This represents linear decay because the value decreases by a constant amount (₹800) every year.
✓ (i) ₹7,600   (iii) v = 10000 − 800t (linear decay)
72
(a) If we multiply a rational number by 32 and subtract 14 from the product, we obtain −58. Formulate a linear equation and find the number.
(b) A positive number is 4 times another number. If 15 is added to both numbers, then one of the new numbers becomes twice the other new number. Find both original numbers.
Part (a)
  1. Let the required rational number be x. According to the statement: 32x − 14 = −58.
  2. Transpose −14 to RHS: 32x = −58 + 14. Using LCM 8: 32x = −5+28 = −38.
  3. Multiply both sides by 23: x = −38 × 23 = −28 = −14.
✓ The required number is −14
Part (b)
  1. Let the smaller positive number be x, so the other number = 4x.
  2. When 15 is added to both: new smaller number = x + 15, new larger number = 4x + 15.
  3. Given: 4x + 15 = 2(x + 15)  ⇒  4x + 15 = 2x + 30.
  4. 4x − 2x = 30 − 15  ⇒  2x = 15  ⇒  x = 7.5.
  5. Larger number = 4(7.5) = 30.
✓ The two original numbers are 7.5 and 30
73
Suppose a plant has height 1.75 feet and it grows by 0.5 feet each month.
  • Find the height after 7 months.
  • Make a table of values for t varying from 0 to 10 months and show how the height, h, increases every month.
  • Find an expression that relates h and t, and explain why it represents linear growth.
  1. (i) Height after 7 months = 1.75 + (0.5 × 7) = 1.75 + 3.5 = 5.25 feet.
Table of values
t (months)012345
h (feet)1.752.252.753.253.754.25
t (months)678910
h (feet)4.755.255.756.256.75
  1. (iii) Let height after t months = h. The expression is h = 1.75 + 0.5t.
  2. This represents linear growth because the height increases by a constant amount (0.5 feet) every month.
✓ (i) 5.25 feet   (iii) h = 1.75 + 0.5t (linear growth)
74
The work done by a body on the application of a constant force is the product of the constant force and the distance travelled by the body in the direction of the force. Express this in the form of a linear equation in two variables (work w and distance d), and draw its graph by taking the constant force as 3 units. What is the work done when the distance travelled is 2 units? Verify it by plotting it on the graph.
  1. We know: Work done = Force × Distance. According to the question, work done = w, distance travelled = d, constant force = 3 units.
  2. Therefore, w = 3d — this is the required linear equation in two variables.
  3. If d = 2 units, w = 3 × 2 = 6 units. Taking w on the y-axis and d on the x-axis, we can plot the graph.
Graph of w=3d verifying point (2,6)
Graph of w = 3d, verifying the point (2, 6)
  1. The point (2, 6) lies on the straight line, so it is verified by the graph.
✓ w = 3d; Work done at d = 2 is w = 6 units (verified on graph)
75
Explain how the degree of a polynomial is determined and provide an example of a polynomial of degree 4, clearly identifying its terms and coefficients of each variable term.
  1. The degree of a polynomial is defined as the highest power of the variable present in the polynomial with a non-zero coefficient.
  2. For a general polynomial P(x) = axn + bxn−1 + … + kx + l, if a ≠ 0 then n is the degree of the polynomial.
Example: P(x) = 7x⁴ − 2x³ + 5x² + 9x − 11
  1. Terms: 7x⁴, −2x³, 5x², 9x, and −11 (constant term).
  2. Looking at the powers of x in each term: 7x⁴ has power 4, −2x³ has power 3, 5x² has power 2, 9x has power 1, and −11 has power 0 (since −11 = −11x⁰).
  3. The highest power among these is 4, so the degree of the polynomial is 4.
Coefficients of variable terms
  1. Coefficient of x⁴ is 7
  2. Coefficient of x³ is −2
  3. Coefficient of x² is 5
  4. Coefficient of x is 9
Prepared by Sumeet Sahu · Mob: 8103405051 · Unique Study Point
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📋 Details

ClassClass IX (CBSE / NCERT)
SubjectMaths
ChapterChapter 2: Introduction to Linear Polynomials
Resource TypeWorksheet
Last Updated05 September 2026
Session2026-27 (Latest NCERT Syllabus)
Downloads0+
Prepared bySumeet Sahu, Unique Study Point, Indore
CostFree
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