Half Yearly Class 9 Mathematics Set 3 (Solutions) | Mathematics (Maths) Class 9 PDF Download

Time: 3 Hours
Maximum Marks: 80
General Instructions:
(i) The question paper comprises four sections: A, B, C, and D. 
(ii) All questions are compulsory. However, internal choices are provided in some questions. 
(iii) Section A has 10 Questions carrying 1 mark each.
(iv) Section B has 5 Questions carrying 2 marks each.
(v) Section C has 10 Questions carrying 3 marks each.
(vi) Section D has 5 Questions carrying 6 marks each.
(vii) Use of calculators is not permitted.

Section A

Q1. Is √8 rational or irrational?    (1 Mark) 

Ans: √8 = 2√2, irrational as √2 is irrational.

Q2. What is the degree of a zero polynomial?    (1 Mark) 

Ans: Zero polynomial: degree undefined or −∞.

Q3. What is the y-coordinate of a point on the x-axis?    (1 Mark) 

Ans: On x-axis, y-coordinate is 0.

Q4. If x − 2y = 4, what is y when x = 0?    (1 Mark) 

Ans: x − 2y = 4, x = 0: −2y = 4 ⇒  y = −2

Q5. State Euclid’s fourth postulate.    (1 Mark) 

Ans: Euclid’s fourth postulate: All right angles are equal.

Q6. If ∠A = 60° in △ABC and AB = AC , what is ∠B?    (1 Mark) 

Ans: AB = AC implies ∠B = ∠C.
Then, 60°  + 2∠B = 180°
⇒∠B = 60°

Q7. Simplify: √75.    (1 Mark) 

Ans: √75 =

Q8. If p(−1) = 0 for p(x) = x² + kx − 3, find k.    (1 Mark) 

Ans: p(−1) = (−1)² + k(−1) − 3 = 0  
⇒ 1 − k − 3 = 0
⇒ k = −2.

Q9. Does the point (0, 5) lie on the x-axis or y-axis?    (1 Mark) 

Ans: (0, 5) lies on the y-axis.

Q10. If a transversal intersects two parallel lines, what is the sum of interior angles on the same side?    (1 Mark) 

Ans: Sum of interior angles on the same side is 180°.

Section B

Q1. Rationalize:    (2 Marks)

Ans: 

Q2. Find the product of the zeroes of x² − 6x + 8.   (2 Marks)

Ans: For x² − 6x + 8, product of zeroes = c/a = 8/1 = 8.

Q3. Find the distance between A(0, 2) and B(3, −2).   (2 Marks)

Ans:  Distance:

Q4. Solve 4x + y = 10 for x when y = 2.   (2 Marks)

Ans: 4x + y = 10, y = 2: 4x + 2 = 10
⇒ 4x = 8
⇒ x = 2

Q5. If ∠X Y Z = 110° and X Y ∥ Z W, find the corresponding angle to ∠X Y Z.   (2 Marks)

Ans: Corresponding angle to ∠X Y Z = 110° is 110°.

Section C

Q1. Prove that √2 is irrational.   (3 Marks) 

Ans: Assume √2 = a/b, a, b coprime. Then, 2 = a²/b² ⇒ a² = 2b².
So, 2 divides a. 
Let a = 2k. Then, 4k² = 2b²
⇒ b² = 2k², so 2 divides b.  

Q2. Find the zeroes of x² − 3x − 10.   (3 Marks) 

Ans: x² − 3x − 10 = (x − 5)(x + 2) = 0. Zeroes: x = 5, −2.

Q3. Find the area of the triangle formed by points A(1, 2), B(4, 2), C (2, 5).   (3 Marks) 

Ans: Area: 1/2| 1(2 - 5) + 4 (5 - 2) + 2 (2 - 2) = 1/2 |- 3 + 12 + 0| = 9/2sq.

Q4. Solve: x + 2y = 7 and 3x − y = 4 graphically.   (3 Marks) 

Ans: For x + 2y = 7: (7, 0), (1, 3).
For 3x − y = 4: (2, 2), (1, −1).
Intersection: x = 3, y = 2.

Q5. Using Euclid’s postulates, prove that every line segment has a unique midpoint.   (3 Marks) 

Ans: Take points A and B. By Euclid’s postulates, draw line AB. Construct perpendicular bisector using equal circles (postulate 3). It intersects AB at one point (unique midpoint).

Q6.In △DEF, if DE ∥ FG and ∠D = 45°, find ∠EFG   (3 Marks) 

Ans: Since DE ∥ FG, ∠EFG = ∠D = 45°(corresponding angles).

 Q7. Find k such that 2x²  + kx + 5 has equal roots.   (3 Marks) 

Ans: Discriminant: k² − 4 · 2 · 5 = 0
⇒ k² = 40
⇒ k = ±2√ 10.   

Q8. Express as a fraction.   (3 Marks) 

Ans:
Subtract: 90x = 15
⇒ x = 15/90 = 1/6

Q9. Solve: 5x − 2y = 8 and x + y = 4 using graphical method.   (3 Marks) 

Ans: For 5x − 2y = 8: (2, 1), (0, −4).
For x + y = 4: (4, 0), (0, 4).  
Intersection: x = 2, y = 2.

Q10. Find the area of the triangle whose sides are 42 cm, 34 cm and 20 cm in length. Hence, find the height corresponding to the longest side.   (3 Marks) 

Ans: Let:

By Heron's formula, we have:

Area of triangle = 336 cm²
We know that the longest side is 42 cm.
Thus, we can find out the height of the triangle corresponding to 42 cm.
We have:
Area of triangle = 336 cm²

Section D

Q1. Find the quotient and remainder when x³  − 6x²  + 11x − 6 is divided by x − 1.   (6 Marks) 

Ans: Synthetic division: 1, −6, 11, −6 with x = 1 : 1, 1, 1 − 5, −5 + 6, 1 − 6
⇒ x² − 5x + 6, remainder 0. 

Q2. For points A(−1, 0),  B(3, 0),  C (0, 4),  find the area of △ABC  and check if it’s isosceles.   (6 Marks) 

Ans: Area: 1/2 |− 1(0 − 4) + 3(4 − 0) + 0(0 − 0)| = 1/2 |4 + 12| = 8 sq units. 
Sides: AB = 4, AC = √17, BC = √17 Isosceles

Q3. Prove that the sum of angles in a triangle is 180° using parallel lines and a transversal.   (6 Marks) 

Ans: Extend BC to D, draw C E || AB. ∠BAC = ∠ACE, ∠ABC = ∠BCE (correspond-ing).
∠ACE + ∠ACB + ∠BCE = 180°, so sum is 180°.

Q4. Solve: x − y = 3 and 2x + y = 9 graphically. Find the area of the triangle formed by these lines and the y-axis.   (6 Marks) 

Ans: For x − y = 3:  (3, 0), (0, −3).  For 2x + y = 9: (4, 1), (3, 3).
Intersection:  x = 4, y = 1.
Triangle vertices:  (0, −3), (0, 9), (4, 1).
Area: 1/2 · 4 · 12 = 24 sq. units.

Q5. Solve:  4x + 3y = 14 and 2x − y = 2 using elimination method.  Verify the solution.   (6 Marks) 

Ans: Multiply second by 3: 6x − 3y = 6.
Add to first: 4x + 3y + 6x − 3y = 14 + 6
⇒ 10x = 20
⇒ x = 2, y = 2.
Verify: 4 · 2 + 3 · 2 = 14, 
2 · 2 − 2 = 2.