Class 9 Maths Test Paper Chapter 1, 2 AND 3 | CBSE Board | Detailed Solutions & Marking Scheme

 

VEDANT SKILL ASSESSMENT SERIES - ACADEMIC YEAR 2026-27

CLASS: IX SUBJECT: MATHEMATICS MAX. MARKS: 30 | TIME: 1 HOUR Syllabus: Number System, Polynomials, Coordinate Geometry, Real-Life Linear Polynomials

GENERAL INSTRUCTIONS

1. Read all questions carefully before answering. - Misreading the question and then blaming the paper will not increase your marks.

2. Show proper steps wherever required. - “Sir, answer toh yahi aana tha” is not an accepted mathematical method.

3. Write neatly and clearly. - If your handwriting requires a decoder machine, checking may become an adventure.

4. Manage your time wisely. - Spending 45 minutes on one question and calling the rest “optional” is not a strategy.

5. If you do not know the answer, you may cry silently. - Loud crying, emotional speeches, and negotiations for hints are strictly prohibited.

SECTION A: Multiple Choice & Assertion-Reasoning Questions (7 Marks)

Q1. Which of the following is an irrational number between 2 and 2.5? [1 Mark] A. √11

B. √5

C. 2.101001000...

D. Both B and C

Q2. If p(x) = x² - 2√2x + 1, then the value of p(2√2) is equal to: [1 Mark] A. 0

B. 1

C. 4√2

D. -1

Q3. The point whose ordinate is 4 and which lies on the y-axis is given by the coordinates: [1 Mark] A. (4, 0)

B. (0, 4)

C. (1, 4)

D. (4, 4)

Q4. A cab driver charges a fixed insurance premium fee of Rs 20 and an additional Rs 12 per kilometer driven. If the total cost is represented by a linear polynomial C(x) where x is the distance in kilometers, what is the correct formulation? [1 Mark] A. C(x) = 20x + 12

B. C(x) = 12x + 20

C. C(x) = 32x

D. C(x) = 12x - 20

Q5. If the coordinates of a point are P(-3, 5), then its perpendicular distance from the x-axis is: [1 Mark] A. 3 units

B. -3 units

C. 5 units

D. √34 units

Directions for Q6-Q7: Select the correct option from the following regular options:

A. Both A and R are true, R is correct explanation of A B. Both A and R are true, R is not correct explanation of A C. A is true, R is false D. A is false, R is true Q6. Assertion (A): The real number x = 0.333... can be expressed in the rational form p/q as 1/3.

Reason (R): Every terminating or non-terminating repeating decimal expansion represents a rational number. [1 Mark] Q7. Assertion (A): The point M(-2, -5) lies entirely in the Quadrant III of the Cartesian plane.

Reason (R): In Quadrant III, both the x-coordinate (abscissa) and the y-coordinate (ordinate) have negative mathematical signs. [1 Mark] ---

SECTION B: Very Short Answer Questions (8 Marks)

Q8. Simplify the expression by rationalizing the denominator: 4 / (√7 + √3). Show every calculation step clearly. [2 Marks]

Q9. Without executing the long direct expansion or multiplication process, compute the exact numerical value of (103)³ using a suitable algebraic identity.

OR Represent Root2 and Root3 on number line  [2 Marks] 

Q10. An online delivery portal computes its final billing amount through a linear setup: P(x) = 40x + 25, where x represents the total item weight in kilograms, and P(x) is the price in Rupees. Find the delivery bill if a customer orders packages weighing exactly 4.5 kilograms. [2 Marks] 

Q11. Three vertices of a rectangle ABCD are plotted on a graph sheet at A(-1, 2), B(5, 2), and C(5, -3). Write down the precise coordinates of the missing fourth vertex D. [2 Marks] ---

SECTION C: Short Answer Questions (6 Marks)

Q12. If both (x - 2) and (x - 1/2) are confirmed factors of the polynomial expression given by f(x) = px² + 5x + r, show by mathematical steps that p = r. [3 Marks] Q13. Locate and write the coordinates of the points after performing the following operations: [3 Marks] (i) A point K(4, -3) is reflected across the x-axis.

(ii) A point L(-2, -1) is reflected across the y-axis.

(iii) Find the distance between the point B(0, -5) and the Origin O(0, 0).

SECTION D: Long Answer Question (5 Marks)

Q14. Express the non-terminating repeating decimal value 0.2353535... (where the block '35' repeats indefinitely) in its structural fractional form p / q, where p and q are coprime integers and q ≠ 0.

Hence, if this value represents the scaling coefficient of a linear polynomial model L(y) = (q - 44p)y + 7, evaluate the value of L(2). [5 Marks] ---

SECTION E: Case-Based / Competency Assessment (4 Marks)

Q15. Real-Life Logistics Management Setup: A modern solar energy layout company is laying down primary power supply nodes on a simulated grid layout map. The central supervisor node is established at the reference point Origin O(0, 0). Two crucial power sub-stations, Sub-station Alpha and Sub-station Beta, are assigned structured coordinate tracks on this grid layout to manage the voltage dispatching mechanism efficiently.

Sub-station Alpha is placed at coordinates A(3, 4), and Sub-station Beta is stationed at coordinates B(-4, 3) on the identical corporate planning sheet layout.

Sub-questions: (i) Determine the perpendicular distance of Sub-station Alpha from the y-axis grid line. [1 Mark] (ii) State the specific Quadrant locations where Sub-station Alpha and Sub-station Beta reside. [1 Mark] (iii) Calculate the values of (Abscissa of A) - (Ordinate of B), and then calculate the absolute straight-line structural path distance of Sub-station Alpha from the Origin O(0,0). [2 Marks]

This is a long, reusable document that you’ll likely print/share with students or staff, so I’m putting it in an easy-to-copy format with proper symbols, powers, and marking scheme.

VEDANT SKILL ASSESSMENT SERIES

ACADEMIC YEAR 2026–27

CLASS: IX | SUBJECT: MATHEMATICS

ANSWER KEY WITH MARKING SCHEME

Max. Marks: 30 | Time: 1 Hour
Syllabus: Number System, Polynomials, Coordinate Geometry, Real-Life Linear Polynomials

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SECTION A: Multiple Choice & Assertion-Reasoning Questions (7 Marks)

Q1. Which of the following is an irrational number between 2 and 2.5? [1 Mark]

Correct Answer: D. Both B and C

Explanation:

• √5 = 2.236... → irrational and lies between 2 and 2.5

• 2.101001000... → non-terminating non-repeating decimal, hence irrational and lies between 2 and 2.5

Marking Scheme:
✔ Correct option = 1 mark

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Q2. If p(x) = x² − 2√2x + 1, find p(2√2). [1 Mark]

Given:

p(x) = x² − 2√2x + 1

Substitute x = 2√2

p(2√2) = (2√2)² − 2√2(2√2) + 1

= 8 − 8 + 1

= 1

Correct Answer: B. 1

Marking Scheme:
✔ Correct option = 1 mark

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Q3. Point whose ordinate is 4 and lies on y-axis. [1 Mark]

• Ordinate = y-coordinate = 4

• Point on y-axis ⇒ x = 0

Therefore coordinates = (0, 4)

Correct Answer: B. (0, 4)

Marking Scheme:
✔ Correct option = 1 mark

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Q4. Cab driver cost polynomial. [1 Mark]

Fixed charge = Rs 20
Additional charge = Rs 12 per km

Therefore:

C(x) = 12x + 20

Correct Answer: B. C(x) = 12x + 20

Marking Scheme:
✔ Correct option = 1 mark

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Q5. Distance of P(−3, 5) from x-axis. [1 Mark]

Distance from x-axis = absolute value of y-coordinate

= |5|

= 5 units

Correct Answer: C. 5 units

Marking Scheme:
✔ Correct option = 1 mark

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Q6. Assertion–Reason [1 Mark]

Assertion (A):
0.333... = 1/3 → True

Reason (R):
Every terminating or non-terminating recurring decimal is rational → True

Reason correctly explains assertion.

Correct Answer:
A. Both A and R are true, and R is the correct explanation of A

Marking Scheme:
✔ Correct option = 1 mark

---

Q7. Assertion–Reason [1 Mark]

Point M(−2, −5)

Since x < 0 and y < 0, point lies in Quadrant III

Assertion: True

In Quadrant III, both coordinates are negative.

Reason: True

Reason correctly explains assertion.

Correct Answer:
A. Both A and R are true, and R is the correct explanation of A

Marking Scheme:
✔ Correct option = 1 mark

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SECTION B: Very Short Answer Questions (8 Marks)

Q8. Rationalize:

4 / (√7 + √3) [2 Marks]

Multiply numerator and denominator by conjugate:

= 4 / (√7 + √3) × (√7 − √3)/(√7 − √3)

= 4(√7 − √3) / (7 − 3)

= 4(√7 − √3) / 4

= √7 − √3

Final Answer: √7 − √3

Marking Scheme:

• Correct conjugate used = 1 mark

• Correct simplification and answer = 1 mark

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Q9. Find (103)³ using identity. [2 Marks]

Using identity:

(a + b)³ = a³ + 3a²b + 3ab² + b³

Here,

a = 100, b = 3

(103)³

= (100 + 3)³

= 100³ + 3(100²)(3) + 3(100)(3²) + 3³

= 1000000 + 90000 + 2700 + 27

= 1092727

Final Answer: 1092727

Marking Scheme:

• Correct identity = 1 mark

• Correct calculation = 1 mark

or

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Q10. P(x) = 40x + 25, find bill for x = 4.5 kg [2 Marks]

P(4.5)

= 40(4.5) + 25

= 180 + 25

= 205

Final Answer: Rs 205

Marking Scheme:

• Correct substitution = 1 mark

• Correct answer = 1 mark

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Q11. Find fourth vertex D of rectangle. [2 Marks]

Given:

A(−1, 2)

B(5, 2)

C(5, −3)

Since rectangle sides are parallel to axes:

D = (−1, −3)

Final Answer: D(−1, −3)

Marking Scheme:

• Understanding rectangle coordinates = 1 mark

• Correct coordinate = 1 mark

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SECTION C: Short Answer Questions (6 Marks)

Q12. Show that p = r [3 Marks]

Given:

f(x) = px² + 5x + r

Factors are:

(x − 2) and (x − 1/2)

Therefore,

f(2) = 0

4p + 10 + r = 0

4p + r = −10 ...(1)

Also,

f(1/2) = 0

p(1/4) + 5(1/2) + r = 0

p/4 + 5/2 + r = 0

Multiply by 4:

p + 10 + 4r = 0

p + 4r = −10 ...(2)

Subtract equation (2) from equation (1):

(4p + r) − (p + 4r) = 0

3p − 3r = 0

p − r = 0

Therefore,

p = r

Marking Scheme:

• Applying factor theorem = 1 mark

• Forming equations = 1 mark

• Showing p = r = 1 mark

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Q13. Coordinate operations [3 Marks]

(i) Reflection of K(4, −3) across x-axis

Rule:

(x, y) → (x, −y)

K′ = (4, 3)

(ii) Reflection of L(−2, −1) across y-axis

Rule:

(x, y) → (−x, y)

L′ = (2, −1)

(iii) Distance between B(0, −5) and Origin O(0, 0)

Distance = |y-coordinate|

= |−5|

= 5 units

Final Answers:

(i) (4, 3)

(ii) (2, −1)

(iii) 5 units

Marking Scheme:

• Part (i) correct = 1 mark

• Part (ii) correct = 1 mark

• Part (iii) correct = 1 mark

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SECTION D: Long Answer Question (5 Marks)

Q14. Convert 0.2353535... into p/q and evaluate L(2)

Let

x = 0.2353535...

Separate non-repeating and repeating part:

x = 0.2 + 0.0353535...

Let

y = 0.353535...

Multiply by 100:

100y = 35.353535...

Subtract:

100y − y = 35.353535... − 0.353535...

99y = 35

y = 35/99

Now:

x = 0.2 + 0.0(35/99)

= 1/5 + 35/990

LCM = 990

= 198/990 + 35/990

= 233/990

Thus,

p = 233, q = 990

Now:

L(y) = (q − 44p)y + 7

Substitute values:

= (990 − 44 × 233)y + 7

= (990 − 10252)y + 7

= −9262y + 7

Find L(2):

L(2)

= −9262(2) + 7

= −18524 + 7

= −18517

Final Answer:

p/q = 233/990

L(2) = −18517

Marking Scheme:

• Correct recurring decimal conversion = 2 marks

• Correct values of p and q = 1 mark

• Correct polynomial substitution = 1 mark

• Correct final answer = 1 mark

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SECTION E: Case-Based / Competency Assessment (4 Marks)

Q15. Solar Energy Grid Setup

Given:

A(3, 4)

B(−4, 3)

Origin O(0, 0)

(i) Distance of A from y-axis [1 Mark]

Distance from y-axis = |x-coordinate|

= |3|

= 3 units

(ii) Quadrants [1 Mark]

A(3, 4)

x > 0, y > 0

⇒ Quadrant I

B(−4, 3)

x < 0, y > 0

⇒ Quadrant II

(iii) Required calculations [2 Marks]

(Abscissa of A) − (Ordinate of B)

= 3 − 3

= 0

Distance of A from Origin:

= √[(3 − 0)² + (4 − 0)²]

= √(9 + 16)

= √25

= 5 units

Final Answers:

(i) 3 units

(ii) A → Quadrant I, B → Quadrant II

(iii) 0 and 5 units

Marking Scheme:

• Part (i) = 1 mark

• Part (ii) = 1 mark

• First calculation in (iii) = 1 mark

• Distance calculation = 1 mark

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FINAL ANSWER KEY (MCQ ONLY)

Q1 → D
Q2 → B
Q3 → B
Q4 → B
Q5 → C
Q6 → A
Q7 → A

Total Marks = 30