Class IX - Mathematics Practice Paper - 1

Subject: Mathematics

Grade: IX

Time: 3 hours

MM: 80

General Instructions:

1. This question paper contains two Parts ‘A’ and ‘B’. Each part is compulsory. Part ‘A’ carries 48 marks and Part ‘B’ carries 32 marks.

2. Both Part ‘A’ and Part ‘B’ have internal choices

Part ‘A’:

1. It consists of three Sections – I, II and III

2. Section – I has 16 Objective Type questions of 1 mark each. Internal choice is provided in 5 questions.

3. Section – II has 4 Questions (Q. No 17 to 20) on case study. Each case study has 5 case-based sub-parts. An examinee is to attempt any 4 out of 5 sub-parts.

4. Section – III comprises of 16 Multiple Choice Questions of 1 mark each. Internal choice is provided in 5 questions.

Part ‘B’

1. It consists three Sections – IV, V and VI

2. Section – IV (Q. No 37 to 40) are Short Answer questions of 2 marks each.

3. Section – V (Q. No 41 to 43) are Short Answer questions of 3 marks each.

4. Section – VI (Q. No 44 to 46) are Long Answer questions of 5 marks each.

5. Internal choice is provided in 2 questions of 2 marks, 1 question of 3 marks and 1 question of 5 marks.

Part ‘A’

Section – I

Question numbers 1 to 16 are of one mark each.

1. Find the value of √5 – 3√12 + 2√75.

2. Is x² + 4x^3/2/√x a polynomial? Justify your answer.

Find the remainder when x³ – 2x² + x + 1 is divided by (x – 1)

3. How many solutions will y – 13x + 8 have?

4. In which quadrant or on which axis does the point (3, -1) lie?

What is the name of each part of the coordinate plane formed by two axes?

5. P, Q, R and S are points on line l. If PQ = RS, show that PR = QS and state the axiom used.

6. Express the linear equation x – y/2 – 5 = 0 in the form of ax + by + c = 1 and indicate the values of a, b and c.

7. An angle is 14° more than its complementary angle. What is its measure?

8. In quadrilateral ABCD, if ∠A : ∠C = 3 : 8 and ∠B : ∠D = 4 : 9 and ∠A + ∠C = 165°, then find all the angles of the quadrilateral.

9. The radius of a circle is 5 cm and length of one of its chords is 8 cm. Find the distance of the chord from the centre.

10. The sides of a triangle are 16 cm, 30 cm, 34 cm. Find its area.

11. Find the volume of right circular cone of radius 6 cm and height 7 cm.

Find the total surface area of a solid hemisphere of radius 5.6 cm

12. Consider the following frequency distribution:

Class Interval	5 – 10	10-15	15-25	25-45	45-75
Frequency	6	12	10	8	15

Find the adjusted frequency to draw the histogram.

In a frequency distribution, the mid-value of a class is 15 and width of the class intervals is 4. Find the lower limit of the class.

13. ADC is a line such that ∠ADB = 5x and ∠BDC = 4x. Find the value of x.

14. In figure, D and E are points on side BC of a ΔABC, such that BD = CE and AD = AE. Show that ΔABD ≅ ΔACE

15. the area of an equilateral triangle is 16√3 cm². Find the length of its side.

16. In figure, p || q and M is the midpoint of AB. Prove that M is the mid-point of CD

In the figure, if ∠A = 40° and ∠B = 70°, then find ∠BCE

Section II

Case Study based questions (Q. No 17 to 20) are compulsory. Attempt any 4 sub-parts from each question. Each sub-part carries 1 mark.

17. Case Study – 1

Every year there are floods and droughts in the country. Victims suffer a lot. The government does a lot to help the people, but it is not enough. Society also must do its bit.

A group of students of Class XII collect clothes, money and medicines from their neighbourhood to help such victims. The amount collected by them is given by the polynomial p(x) = x2 – 350x + 25000 which is the product of the donation by number of people and they collected equal amount from different people.

Answer the following questions based on above information:

(i) Coefficient of x in the given polynomial is

(a) 1

(b) 350

(d) -350

(ii) Total amount collected if x = 500 is

(a) ₹ 1,00,000

(b) ₹75,000

(d) ₹ 1,20,000

(iii) The amount donated by each person and number of people are

(a) (x + 250), (x + 100)

(b) (x + 250), (x – 100)

(d) (x – 250), (x + 100)

(iv) Name the polynomial of amounts invested by each partner.

(a) Linear

(b) Quadratic

(d) None of these

(v) Find the value of x, if the total amount collected is equal to 0.

(a) 100

(b) 250

(d) None of these

18. Case Study – 2

Burkhamal, a zamindar sells his produce of crops at auction. He forms 12 heaps of rice and wheat of diameter 24 m height 3.5 m

Answer the following questions based on above information:

(i) What is the base area covered by one heap

(a) 415.50 m²

(b) 315.25 m²

(d) 550 m²

(ii) What is the volume of each heap of crop?

(a) 1056 m³

(b) 2112 m³

(d) 12672 m³

(iii) He covers each heap of crop with plastic sheet. How much sheet does he require to cover each heap?

(a) 425.53 m²

(b) 471.43 m²

(d) 880 m²

(iv) He purchases 1.5 m wide plastic sheet to cover 12 heaps of crop. How much length of plastic sheet does he require to purchase?

(a) 4000 m

(b) 3572 m

(d) 3772 m

(v) What is the formula to find total surface area of cone?

(a) πr(l + r)

(b) π(h + r)

(d) Both (a) and (c)

19 – Case Study – 3

A farmer has circular garden as shown in the picture below. He has a different type of trees, plants and flower plants in his garden. In the garden, there are two mango trees A and B at a distance of 10 m. Similarly, he has two Ashoka trees at a distance of 10 m as shown at C and D.

AB subtends ∠AOB = 120° at the centre O. The perpendicular distance of AC from centre is 5 m. The radius of the circle is 13 m.

Answer the following questions based on above information:

(i) What is the measure of ∠COD?

(a) 60°

(b) 80°

(d) 120°

(ii) What is the distance between mango tree A and Ashoka tree C?

(a) 12 m

(b) 13 m

(d) 15 m

(iii) What is the measure of ∠OCD?

(a) 30°

(b) 45°

(d) 90°

(iv) In ∠AOC = 80°, then what is the measure of ∠BOD?

(a) 140°

(b) 80°

(d) 40°

(v) How much distance is covered by the farmer if he goes to water all the trees starting from Tree B to Tree A then to Tree C and then to Tree D.

(a) 20 m

(b) 24 m

(d) 44 m

20. Case Study – 4

Ajay decorated one of his bed room wall as shown in the picture. He was having rectangular wooden pieces of different colours to fix on the wall.

Answer the following questions based on above information:

(i) Name the type of graphical representation.

(a) Pictograph

(b) Bar graph

(d) Frequency polygon

(ii) Which two bars are equal in length?

(a) Yellow and Orange

(b) Orange and Pink

(d) White and Green

(iii) How much width is used in fixing one set of all five colours?

(a) 60 cm

(b) 50 cm

(d) 40 cm

(iv) What is class mark of class interval 30-40?

(a) 25

(b) 30

(d) 40

(v) What is the class size of class interval 10-20?

(a) 10

(b) 20

(d) 30

Section – III

Q. No 21 to 36 are multiple choice questions. Select the most appropriate answer from the given options.

21. The sides of a triangle are 50 cm, 78 cm and 112 cm. The smallest altitude is

(a) 20 cm

(b) 30 cm

(d) 50 cm

If the perimeter of an equilateral triangle is 36 cm, its area is given by

(a) 98√3 cm²

(b) 8√3 cm²

(d) 36√3 cm²

22. If p(x) = x2 – 2√2x + 1, then p(2√2) = ?

(a) 0

(b) 1

(d) -1

23. x = 2, y = -1 is a solution of the linear equation

(a) x + 2y = 0

(b) x + 2y = 4

(d) 2x + y = 5

The point of the form (a, a), a≠ 0 lies on

(a) x-axis

(b) y-axis

(d) the line y + x = 0

24. A point lies on the negative side of x-axis. Its distance from origin is 10 units. The coordinates of points are

(a) (10, 0)

(b) (-10, 0)

(d) (0, -10)

25. One of the factors of (9x² – 1) – (1 + 3x)² is

(a) 3 + x

(b) 3 – x

(d) 3x + 1

26. In figure, ∠DBC equals

(a) 40°

(b) 60°

(d) 100°

In figure, x is greater than y by one third of a right angle. The values of x and y are

(a) 105°, 75°

(b) 100°, 80°

(d) 110°, 70°

27. In triangles ABC and DEF, ∠A = ∠F, ∠B = ∠D and AB = DF, then which of the following congruence criterion applies:

(a) SAS

(b) ASA

(d) RHS

28. In figure, ABCD is a rhombus in which ∠BCD = 110. Then (x + y) equals

(a) 40°

(b) 50°

(d) 70°

The sides of a quadrilateral are extended in order to form exterior angle. The sum of these exterior angles is:

(a) 180°

(b) 270°

(d) 360°

29. In the given figure, if ∠AOB = 80 and ∠ABC = 30, then m∠CAO = ?

(a) 20°

(b) 60°

(d) 30°

30. The area of an isosceles triangle with base 2 cm and length of one of the equal sides 4 cm, is

(a) √15 cm²

(b) √15/2 cm²

(d) 4√15 cm²

31. If the ration of volumes of two spheres is 1 : 8, then the ration of their surface area is

(a) 1 : 2

(b) 1 : 4

(d) 1 : 16

A cone and a hemisphere have equal bases and equal volumes, then the ratio of their heights is

(a) 1 : 2

(b) 2 : 1

(d) √2 : 1

32. Figure shows the bar graph of number of boys and number of girls in a school from 2018 and 2021:

Find the ratio between the number of students in the year 2018 and 2020.

(a) 107 : 145

(b) 29 : 23

(d) 107 : 127

33. The graph of the linear equation 3x + 5y = 6 cuts the x-axis at the point

(a) (2, 0)

(b) (0, 2)

(d) (6/5, 0)

34. In figure, if l1 || l2, what is the value of x

(a) 36°

(b) 76°

(d) 72°

35. In a histogram the class interval or the groups are taken along

(a) x-axis

(b) y-axis

(d) both x-axis and y-axis

36. In figure, if AC is the bisector of ∠BAD such that AB = 3 cm and AC = 5 cm, then CD = ______

(a) 2 cm

(b) 3 cm

(d) 5 cm

Part ‘B’

Section – IV

Questions numbers 37 to 40 carry 2 marks each.

37.

If 3^4x = (81)^-1 and (10)^1/y = 0.0001, find the value of 2 ^{–x + 4y}.

38. A cab driver charges ₹ 10 for first kilometer and ₹ 7 for every subsequent kilometer. For a distance of x km, an amount of ₹ y is paid. Write the linear equation representing the above information.

39. In which quadrant do these points lie if

(i) ordinate is 6 and abscissa is -2?

(ii) abscissa is -3 and ordinate is -5?

40. Prove that the equal chords of a circle subtend equal angles at the centre.

In the given figure, find the value of x.

Section – V

Question numbers 41 to 43 carry 3 marks each.

41. In the figure, if AB and CD are parallel, find the value of x.

If two lines intersect, prove that vertically opposite angles are equal.

42. Find the area of a triangle whose perimeter is 180 cm and two of its sides are 80 cm and 18 cm. Calculate the altitude of triangle corresponding to its shortest side.

43. Draw histogram and frequency polygon for the following data:

Marks	10 – 20	20 – 30	30 – 40	40 – 50	50 – 60	60 – 70	70 – 80
No of candidates	2	5	6	4	8	10	5

Section – VI

Question numbers 44 to 46 carry 5 marks each.

44. State Factor theorem. Using Factor theorem, factorise x³ – 3x² – x + 3

45. If x = (√2 + 1)/(√2 – 1) and y = (√2 – 1)/( √2 + 1), find the value of x² + y² + xy

Prove that: (x^a/x^b)^{a2 + ab + b2}. (x^b/x^c)^{b2 + bc + c2}. (x^c/x^a)^{c2 + ca + a2}

46. PQ and RS are two parallel chords of a circle whose centre is O and radius is 10 cm. If PQ = 16 cm and RS = 12 cm, find the distance between PQ and RS when they lie,

(i) on the same side of centre O

(ii) on the opposite sides of centre O

Class IX – Mathematics Practice Paper – 1