Question Bank — Introduction to Linear Polynomials

NCERT Class 9 Mathematics · Chapter 2 · CBSE Board Pattern

Class 9 Mathematics Chapter 2 Question Bank CBSE Board Pattern

Section A — MCQs (10 questions, 1 mark each)

1. Which of the following is a linear polynomial?
(a) \(x^2 + 5x + 1\)
(b) \(3z + 7\)
(c) \(5y^3 + y^2 + 2y - 1\)
(d) \(8\)

2. In the expression \(4x + 5y + 3\), the coefficient of \(x\) is:
(a) 4
(b) 5
(c) 3
(d) \(x\)

3. The degree of the polynomial \(-9\) is:
(a) 3
(b) 1
(c) 0
(d) undefined

4. The constant term in the polynomial \(9x^3 + 5x^2 - 8x - 10\) is:
(a) 9
(b) 5
(c) \(-8\)
(d) \(-10\)

5. Which expression represents a linear polynomial?
(a) \(2x^2 - 5x + 3\)
(b) \(y^3 + 2y - 1\)
(c) \(4z - 3\)
(d) \(x^4 - 3x^3 + 6x^2 - 2x + 7\)

6. Assertion-Reason

Assertion (A): The expression \(200l + 160w + 50lw\) is a polynomial of degree 2.
Reason (R): The highest power of any variable in the expression is 2.
(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.

7. Assertion-Reason

Assertion (A): \(2n - 1\) is a linear polynomial that represents a linear pattern.
Reason (R): In a linear pattern, the difference between consecutive terms is constant.
(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.

8. In the linear relationship \(y = ax + b\), \(b\) represents:
(a) the slope
(b) the y-intercept
(c) the degree
(d) the constant term only when \(a = 0\)

9. The expression for the perimeter of a square of side \(x\) is:
(a) \(2x\)
(b) \(4x\)
(c) \(x^2\)
(d) \(x + 4\)

10. Linear growth is represented graphically by a straight line with:
(a) negative slope
(b) zero slope
(c) positive slope
(d) undefined slope

Section B — Very Short Answer (6 questions, 2 marks each)

1. Identify the terms, variables and coefficients in the expression \(4x + 5y + 3\).

2. Find the degree of each polynomial:
(i) \(2x^2 - 5x + 3\)
(ii) \(4z - 3\)

3. Write the coefficient of \(z^3\) and the constant term in \(4z^3 + 5z^2 - 11\).

4. Evaluate the linear polynomial \(5x - 3\) when \(x = 0\), \(x = -1\) and \(x = 2\).

5. State whether each of the following is a linear polynomial. Give reason.
(i) \(x^2 + 5x + 1\)
(ii) \(3z + 7\)

6. What is the y-intercept of the line \(y = 2x - 1\)?

Section C — Short Answer (5 questions, 3 marks each)

1. A rectangular garden has length \(l\) metres and width \(w\) metres. The total cost of fencing and sowing seeds is given by \(200l + 160w + 50lw\). Identify the terms, variables, coefficients and constant term(s) in this expression.

2. The amount left with Bela after spending ₹5 every day from her initial pocket money of ₹100 is given by \(100 - 5n\). Find the amount left on the 12th day and explain why this represents linear decay.

3. A wire of length 20 cm is bent to form a rectangle with length \(x\) cm. Write the expression for its area and find the area when \(x = 6\) cm.

4. Find the values of \(a\) and \(b\) if the linear relationship \(y = ax + b\) satisfies the conditions: when \(x = 10\), \(y = 350\) and when \(x = 20\), \(y = 550\).

5. A plant has height 1.75 feet and grows by 0.5 feet each month. Write a linear expression for its height \(h\) after \(t\) months and find its height after 7 months.

Section D — Long Answer (3 questions, 5 marks each)

1. (Multi-step) A student observes that a telecom company charges a fixed monthly fee plus an additional cost per GB of data. When 10 GB is used the bill is ₹350 and when 20 GB is used the bill is ₹550.
(i) Form the linear relationship \(y = ax + b\).
(ii) Find the values of \(a\) and \(b\).
(iii) Write the equation and state what \(a\) and \(b\) represent.
(iv) Find the bill when 15 GB of data is used.

2. Draw the graphs of \(y = 2x - 1\), \(y = 2x + 1\) and \(y = 2x + 5\) on the same coordinate axes. Identify the slope and y-intercept of each line. Explain the effect of changing the value of \(b\) while keeping \(a\) fixed.

3. A farmer cuts a 300-foot fence into two pieces where the longer piece is four times the shorter piece.
(i) Form a linear equation to represent the situation.
(ii) Solve the equation to find the lengths of both pieces.
(iii) Verify that the sum of the lengths equals 300 feet.

Section E — Case/Source-Based (2 questions, 4 marks each)

Case 1: Pocket Money Spending

Bela starts with ₹100 and spends ₹5 every day. The amount left on the \(n\)th day is given by \(100 - 5n\).

(i) Write the linear polynomial representing the amount left.
(ii) Find the amount left on the 12th day.
(iii) After how many days will the amount left be ₹40?
(iv) Does this situation represent linear growth or linear decay? Justify.

Case 2: Auto-rickshaw Fare

An auto-rickshaw fare starts at ₹25 for the first 2 km and then increases by ₹15 per additional km. The fare for \(n\) km (\(n \geq 2\)) is given by \(15n - 5\).

(i) Verify that the expression gives ₹25 when \(n = 2\).
(ii) Find the fare for 10 km.
(iii) For how many km will the fare be ₹130?
(iv) Explain why the fare function represents a linear relationship.

▶ Answer Key Attempt all questions first,
then tap to reveal

Section A

  1. (b) — degree 1 (highest power of variable is 1).
  2. (a) — coefficient of \(x\) is the number multiplying \(x\).
  3. (c) — constant can be written as \(-9x^0\), degree 0.
  4. (d) — term without variable.
  5. (c) — highest power of variable is 1.
  6. (d) — A is false (degree 2 due to \(lw\)), R is true.
  7. (a) — both true; constant difference makes it linear.
  8. (b) — y-intercept is the point where line cuts y-axis.
  9. (b) — perimeter = 4 × side.
  10. (c) — positive slope shows increase by constant amount.

Section B

  1. Terms: \(4x, 5y, 3\); variables: \(x, y\); coefficients: 4 (of \(x\)), 5 (of \(y\)).
  2. (i) 2 (ii) 1.
  3. Coefficient of \(z^3\): 4; constant term: \(-11\).
  4. When \(x=0\): \(-3\); \(x=-1\): \(-8\); \(x=2\): 7.
  5. (i) No (degree 2) (ii) Yes (degree 1).
  6. \(-1\) (value of \(y\) when \(x=0\)).

Section C

  1. Terms: \(200l, 160w, 50lw\); variables: \(l, w\); coefficients: 200, 160, 50; no constant term.
  2. On 12th day: ₹40. Linear decay because amount decreases by fixed ₹5 each day.
  3. Area = \(x(10-x)\). When \(x=6\): 24 cm².
  4. \(a=20\), \(b=150\) (solving simultaneous equations).
  5. \(h=1.75+0.5t\); after 7 months: 5.25 feet.

Section D

  1. (i) \(y=ax+b\) (ii) \(a=20\), \(b=150\) (iii) \(a\) = cost per GB, \(b\) = fixed fee (iv) ₹450.
  2. All lines have slope 2; y-intercepts −1, 1, 5 respectively. Changing \(b\) shifts lines parallel to each other.
  3. Let shorter piece = \(x\), longer = \(4x\); \(5x=300\); lengths 60 ft and 240 ft.

Section E

Case 1: (i) \(100-5n\) (ii) ₹40 (iii) 12 days (iv) Linear decay (constant decrease).
Case 2: (i) When \(n=2\), fare = ₹25 (ii) ₹145 (iii) 8 km (iv) Fare increases by fixed amount per km.

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All questions are answerable from the NCERT chapter text. Reviewed by GFIS faculty.