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Mathematics (Maths) Class 9

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 Page 1


  
 
CBSE IX | Mathematics 
Sample Paper 3 
 
     
CBSE Board 
Class IX Mathematics 
Sample Paper 3 
Time: 3 hrs  Total Marks: 80 
      
General Instructions:  
1. All questions are compulsory. 
2. The question paper consists of 30 questions divided into four sections A, B, C, and D. 
Section A comprises of 6 questions of 1 mark each, Section B comprises of 6 questions 
of 2 marks each, Section C comprises of 10 questions of 3 marks each and Section D 
comprises of 8 questions of 4 marks each. 
3. Use of calculator is not permitted. 
 
Section A 
(Questions 1 to 6 carry 1 mark each) 
 
1. If 
? ?
?
2
56 = a + b 30 then find the values of a and b. 
 
2. p(x) = cx + d is a zero polynomial. What is the value of x? 
 
3. In the given figure, PQRS is a parallelogram having base PQ = 6 cm and perpendicular 
height is also 6 cm, Find the area of ?PTQ?  
 
 
OR 
ABCD is a parallelogram having an area of 60 cm
2
. P is a point on CD. Calculate the 
area of ? APB. 
 
4. Check whether (
1
2
, 0) is the solution of the equation 2x + y = 1? 
OR 
If (4, 19) is a solution of the equation y = ax + 3 then find the value of a. 
 
5. Define Median of a triangle. 
 
 
 
Page 2


  
 
CBSE IX | Mathematics 
Sample Paper 3 
 
     
CBSE Board 
Class IX Mathematics 
Sample Paper 3 
Time: 3 hrs  Total Marks: 80 
      
General Instructions:  
1. All questions are compulsory. 
2. The question paper consists of 30 questions divided into four sections A, B, C, and D. 
Section A comprises of 6 questions of 1 mark each, Section B comprises of 6 questions 
of 2 marks each, Section C comprises of 10 questions of 3 marks each and Section D 
comprises of 8 questions of 4 marks each. 
3. Use of calculator is not permitted. 
 
Section A 
(Questions 1 to 6 carry 1 mark each) 
 
1. If 
? ?
?
2
56 = a + b 30 then find the values of a and b. 
 
2. p(x) = cx + d is a zero polynomial. What is the value of x? 
 
3. In the given figure, PQRS is a parallelogram having base PQ = 6 cm and perpendicular 
height is also 6 cm, Find the area of ?PTQ?  
 
 
OR 
ABCD is a parallelogram having an area of 60 cm
2
. P is a point on CD. Calculate the 
area of ? APB. 
 
4. Check whether (
1
2
, 0) is the solution of the equation 2x + y = 1? 
OR 
If (4, 19) is a solution of the equation y = ax + 3 then find the value of a. 
 
5. Define Median of a triangle. 
 
 
 
  
 
CBSE IX | Mathematics 
Sample Paper 3 
 
     
6. ABCD is a parallelogram. If OA and OB are the angle bisectors of the consecutive 
angles, then m ?AOB =? 
 
Section B 
(Questions 7 to 12 carry 2 marks each) 
 
7. Express 0.975in the form
p
q
, where p and q are integers and q ? 0. 
 
8. Factorise:
  
2
2
12
x 2 2x
x x
? ? ? ? 
 
9. In the figure below, BC = AC = AD and ? DAE = 75°. Find the value of y. 
 
 
 
10. In the figure, AD is the bisector of ?A; prove that AB > BD.            
 
OR 
 
In ?PQR, PR > PQ and T is a point on PR such that PT = PQ. Prove that QR > TR. 
Page 3


  
 
CBSE IX | Mathematics 
Sample Paper 3 
 
     
CBSE Board 
Class IX Mathematics 
Sample Paper 3 
Time: 3 hrs  Total Marks: 80 
      
General Instructions:  
1. All questions are compulsory. 
2. The question paper consists of 30 questions divided into four sections A, B, C, and D. 
Section A comprises of 6 questions of 1 mark each, Section B comprises of 6 questions 
of 2 marks each, Section C comprises of 10 questions of 3 marks each and Section D 
comprises of 8 questions of 4 marks each. 
3. Use of calculator is not permitted. 
 
Section A 
(Questions 1 to 6 carry 1 mark each) 
 
1. If 
? ?
?
2
56 = a + b 30 then find the values of a and b. 
 
2. p(x) = cx + d is a zero polynomial. What is the value of x? 
 
3. In the given figure, PQRS is a parallelogram having base PQ = 6 cm and perpendicular 
height is also 6 cm, Find the area of ?PTQ?  
 
 
OR 
ABCD is a parallelogram having an area of 60 cm
2
. P is a point on CD. Calculate the 
area of ? APB. 
 
4. Check whether (
1
2
, 0) is the solution of the equation 2x + y = 1? 
OR 
If (4, 19) is a solution of the equation y = ax + 3 then find the value of a. 
 
5. Define Median of a triangle. 
 
 
 
  
 
CBSE IX | Mathematics 
Sample Paper 3 
 
     
6. ABCD is a parallelogram. If OA and OB are the angle bisectors of the consecutive 
angles, then m ?AOB =? 
 
Section B 
(Questions 7 to 12 carry 2 marks each) 
 
7. Express 0.975in the form
p
q
, where p and q are integers and q ? 0. 
 
8. Factorise:
  
2
2
12
x 2 2x
x x
? ? ? ? 
 
9. In the figure below, BC = AC = AD and ? DAE = 75°. Find the value of y. 
 
 
 
10. In the figure, AD is the bisector of ?A; prove that AB > BD.            
 
OR 
 
In ?PQR, PR > PQ and T is a point on PR such that PT = PQ. Prove that QR > TR. 
  
 
CBSE IX | Mathematics 
Sample Paper 3 
 
     
 
11. The total surface area of a cube is 294 cm
2
. Find its volume. 
OR 
Find the volume of a cube whose diagonal is 48 cm. 
12. Check which of the following are solutions of the equation 7x – 5y = –3. 
i.     (–1, –2) 
ii.     (–4, –5) 
 
Section C 
(Questions 13 to 22 carry 3 marks each) 
 
13. Evaluate: 
2
3
(343)
?
 
OR 
Evaluate 
12
23
1
0.01 27
4
 
14. What is the zero of the polynomial p(x) = (a
2
 + b
2
) x + (a – b)
2
 + (a + b)
2
? 
 
15. Use a suitable identity to factorise 27p
3
 + 8q
3
 + 54p
2
q + 36p q
2
. 
  
16. In the figure, sides QP and RQ of  ? PQR are produced to points S and T respectively. 
If ? SPR = 135° and ? PQT = 110°, then find ? PRQ. 
 
 
17. Prove that in an isosceles triangle the angles opposite to the equal sides are equal. 
OR 
Prove that the medians corresponding to equal sides of an isosceles triangle are equal. 
 
Page 4


  
 
CBSE IX | Mathematics 
Sample Paper 3 
 
     
CBSE Board 
Class IX Mathematics 
Sample Paper 3 
Time: 3 hrs  Total Marks: 80 
      
General Instructions:  
1. All questions are compulsory. 
2. The question paper consists of 30 questions divided into four sections A, B, C, and D. 
Section A comprises of 6 questions of 1 mark each, Section B comprises of 6 questions 
of 2 marks each, Section C comprises of 10 questions of 3 marks each and Section D 
comprises of 8 questions of 4 marks each. 
3. Use of calculator is not permitted. 
 
Section A 
(Questions 1 to 6 carry 1 mark each) 
 
1. If 
? ?
?
2
56 = a + b 30 then find the values of a and b. 
 
2. p(x) = cx + d is a zero polynomial. What is the value of x? 
 
3. In the given figure, PQRS is a parallelogram having base PQ = 6 cm and perpendicular 
height is also 6 cm, Find the area of ?PTQ?  
 
 
OR 
ABCD is a parallelogram having an area of 60 cm
2
. P is a point on CD. Calculate the 
area of ? APB. 
 
4. Check whether (
1
2
, 0) is the solution of the equation 2x + y = 1? 
OR 
If (4, 19) is a solution of the equation y = ax + 3 then find the value of a. 
 
5. Define Median of a triangle. 
 
 
 
  
 
CBSE IX | Mathematics 
Sample Paper 3 
 
     
6. ABCD is a parallelogram. If OA and OB are the angle bisectors of the consecutive 
angles, then m ?AOB =? 
 
Section B 
(Questions 7 to 12 carry 2 marks each) 
 
7. Express 0.975in the form
p
q
, where p and q are integers and q ? 0. 
 
8. Factorise:
  
2
2
12
x 2 2x
x x
? ? ? ? 
 
9. In the figure below, BC = AC = AD and ? DAE = 75°. Find the value of y. 
 
 
 
10. In the figure, AD is the bisector of ?A; prove that AB > BD.            
 
OR 
 
In ?PQR, PR > PQ and T is a point on PR such that PT = PQ. Prove that QR > TR. 
  
 
CBSE IX | Mathematics 
Sample Paper 3 
 
     
 
11. The total surface area of a cube is 294 cm
2
. Find its volume. 
OR 
Find the volume of a cube whose diagonal is 48 cm. 
12. Check which of the following are solutions of the equation 7x – 5y = –3. 
i.     (–1, –2) 
ii.     (–4, –5) 
 
Section C 
(Questions 13 to 22 carry 3 marks each) 
 
13. Evaluate: 
2
3
(343)
?
 
OR 
Evaluate 
12
23
1
0.01 27
4
 
14. What is the zero of the polynomial p(x) = (a
2
 + b
2
) x + (a – b)
2
 + (a + b)
2
? 
 
15. Use a suitable identity to factorise 27p
3
 + 8q
3
 + 54p
2
q + 36p q
2
. 
  
16. In the figure, sides QP and RQ of  ? PQR are produced to points S and T respectively. 
If ? SPR = 135° and ? PQT = 110°, then find ? PRQ. 
 
 
17. Prove that in an isosceles triangle the angles opposite to the equal sides are equal. 
OR 
Prove that the medians corresponding to equal sides of an isosceles triangle are equal. 
 
  
 
CBSE IX | Mathematics 
Sample Paper 3 
 
     
18. Fifty seeds each were selected at random from 5 bags of seeds, and were kept under 
standardized conditions favorable to germination. After 20 days, the number of seeds 
which had germinated in each collection were counted and recorded as follows: 
Bags 1 2 3 4 5 
Number of germinated seeds 40 48 42 39 41 
 
What is the probability of  
i.    More than 40 seeds germinating in a bag? 
ii.    49 seeds germinating in a bag? 
iii.    More than 35 seeds germinating in a bag? 
OR 
A survey was undertaken in 30 classes at a school to find the total number of fail 
students in each class. The table below shows the results:   
No. of fail students 0 1 2 3 4 5 
Frequency (no. of 
classes) 
1 2 5 12 8 2 
 A class was selected at random.  
(a) Find the probability that the class has 2 fail students.  
(b) What is the probability that the class has at least 3 fail students?  
(c) Given that the total number of students in the 30 classes is 960, find the 
probability that a student randomly chosen from these 30 classes is fail. 
 
 
19. In the figure, O is the centre of the circle, OM ? BC, OL ? AB, ON ? AC and OM = ON = 
OL. 
 
Is ?ABC equilateral? Give reasons.  
 
20. Show that if the diagonals of a quadrilateral are equal and bisect each other at right 
angles, then it is a square. 
 
21. The relative humidity (in %) of a certain city for a month of 30 days was as follows: 
98.1 98.6 99.2 90.3 86.5 95.3 92.9 96.3 94.2 95.1 
97.3 89.2 92.3 97.1 93.5 92.7 95.1 97.2 93.3 95.2 
89 96.2 92.1 84.9 90.2 95.7 98.3 97.3 96.1 92.1 
Page 5


  
 
CBSE IX | Mathematics 
Sample Paper 3 
 
     
CBSE Board 
Class IX Mathematics 
Sample Paper 3 
Time: 3 hrs  Total Marks: 80 
      
General Instructions:  
1. All questions are compulsory. 
2. The question paper consists of 30 questions divided into four sections A, B, C, and D. 
Section A comprises of 6 questions of 1 mark each, Section B comprises of 6 questions 
of 2 marks each, Section C comprises of 10 questions of 3 marks each and Section D 
comprises of 8 questions of 4 marks each. 
3. Use of calculator is not permitted. 
 
Section A 
(Questions 1 to 6 carry 1 mark each) 
 
1. If 
? ?
?
2
56 = a + b 30 then find the values of a and b. 
 
2. p(x) = cx + d is a zero polynomial. What is the value of x? 
 
3. In the given figure, PQRS is a parallelogram having base PQ = 6 cm and perpendicular 
height is also 6 cm, Find the area of ?PTQ?  
 
 
OR 
ABCD is a parallelogram having an area of 60 cm
2
. P is a point on CD. Calculate the 
area of ? APB. 
 
4. Check whether (
1
2
, 0) is the solution of the equation 2x + y = 1? 
OR 
If (4, 19) is a solution of the equation y = ax + 3 then find the value of a. 
 
5. Define Median of a triangle. 
 
 
 
  
 
CBSE IX | Mathematics 
Sample Paper 3 
 
     
6. ABCD is a parallelogram. If OA and OB are the angle bisectors of the consecutive 
angles, then m ?AOB =? 
 
Section B 
(Questions 7 to 12 carry 2 marks each) 
 
7. Express 0.975in the form
p
q
, where p and q are integers and q ? 0. 
 
8. Factorise:
  
2
2
12
x 2 2x
x x
? ? ? ? 
 
9. In the figure below, BC = AC = AD and ? DAE = 75°. Find the value of y. 
 
 
 
10. In the figure, AD is the bisector of ?A; prove that AB > BD.            
 
OR 
 
In ?PQR, PR > PQ and T is a point on PR such that PT = PQ. Prove that QR > TR. 
  
 
CBSE IX | Mathematics 
Sample Paper 3 
 
     
 
11. The total surface area of a cube is 294 cm
2
. Find its volume. 
OR 
Find the volume of a cube whose diagonal is 48 cm. 
12. Check which of the following are solutions of the equation 7x – 5y = –3. 
i.     (–1, –2) 
ii.     (–4, –5) 
 
Section C 
(Questions 13 to 22 carry 3 marks each) 
 
13. Evaluate: 
2
3
(343)
?
 
OR 
Evaluate 
12
23
1
0.01 27
4
 
14. What is the zero of the polynomial p(x) = (a
2
 + b
2
) x + (a – b)
2
 + (a + b)
2
? 
 
15. Use a suitable identity to factorise 27p
3
 + 8q
3
 + 54p
2
q + 36p q
2
. 
  
16. In the figure, sides QP and RQ of  ? PQR are produced to points S and T respectively. 
If ? SPR = 135° and ? PQT = 110°, then find ? PRQ. 
 
 
17. Prove that in an isosceles triangle the angles opposite to the equal sides are equal. 
OR 
Prove that the medians corresponding to equal sides of an isosceles triangle are equal. 
 
  
 
CBSE IX | Mathematics 
Sample Paper 3 
 
     
18. Fifty seeds each were selected at random from 5 bags of seeds, and were kept under 
standardized conditions favorable to germination. After 20 days, the number of seeds 
which had germinated in each collection were counted and recorded as follows: 
Bags 1 2 3 4 5 
Number of germinated seeds 40 48 42 39 41 
 
What is the probability of  
i.    More than 40 seeds germinating in a bag? 
ii.    49 seeds germinating in a bag? 
iii.    More than 35 seeds germinating in a bag? 
OR 
A survey was undertaken in 30 classes at a school to find the total number of fail 
students in each class. The table below shows the results:   
No. of fail students 0 1 2 3 4 5 
Frequency (no. of 
classes) 
1 2 5 12 8 2 
 A class was selected at random.  
(a) Find the probability that the class has 2 fail students.  
(b) What is the probability that the class has at least 3 fail students?  
(c) Given that the total number of students in the 30 classes is 960, find the 
probability that a student randomly chosen from these 30 classes is fail. 
 
 
19. In the figure, O is the centre of the circle, OM ? BC, OL ? AB, ON ? AC and OM = ON = 
OL. 
 
Is ?ABC equilateral? Give reasons.  
 
20. Show that if the diagonals of a quadrilateral are equal and bisect each other at right 
angles, then it is a square. 
 
21. The relative humidity (in %) of a certain city for a month of 30 days was as follows: 
98.1 98.6 99.2 90.3 86.5 95.3 92.9 96.3 94.2 95.1 
97.3 89.2 92.3 97.1 93.5 92.7 95.1 97.2 93.3 95.2 
89 96.2 92.1 84.9 90.2 95.7 98.3 97.3 96.1 92.1 
  
 
CBSE IX | Mathematics 
Sample Paper 3 
 
     
 
i. Construct a grouped frequency distribution table with classes 84 - 86,     86 - 88 
ii. Which month or season do you think this data is about? 
iii. What is the range of this data? 
 
22. A hemispherical bowl, made of steel, is 0.25 cm thick. The inner radius of the bowl is 5 
cm. Find the outer curved surface area of the bowl. 
OR 
50 cylindrical pillars of a hall are to be painted. The diameter of each pillar is 5 m and the 
height is 21 m, what will be the cost of painting them at the rate of Rs 4.50 per m
2
? 
 
Section D 
(Questions 23 to 30 carry 4 marks each) 
 
23. Find the value of ? ? ? ?
? ? ? ? ?
1 1 1 1 1
3 8 8 7 7 6 6 5 5 2
 
 
24. How does Euclid's fifth postulate imply the existence of parallel lines? Give a 
mathematical proof.  
 
25. Find x
3
 + y
3
 when x = 
?
1
3 2 2
 and y = 
?
1
3 2 2
.  
OR 
If p + q = 8 and p – q = 4, find:  
         (i) pq, (ii) p
2
 + q
2
 
26. In the given figure, AC = AE, AB = AD and ? BAD = ?EAC. Prove that BC = DE. 
 
 
OR 
 
In ? ABC, AB = AC and the bisectors of angles B and C intersect at point O. Prove that 
BO = CO and the ray AO is the bisector of angle BAC. 
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