Subject: Mathematics
Class IX
Time Allowed: 3 hours
Max. Marks: 80
General Instructions:
- This Question Paper has 5 Sections A – E
- Section A has 20 MCQs carrying 1 mark each
- Section B has 5 questions carrying 2 marks each
- Section C has 6 questions carrying 3 marks each
- Section D has 4 questions carrying 5 marks each
- Section E has 3 case based integrated units of assessment (4 marks each) with subparts of the values of 1, 1 and 2 marks each respectively
- All questions are compulsory. However, an internal choice in 2 Qs of 5 marks, 2 Qs of 3 marks and 2 Qs of 2 marks has been provided. An internal choice has been provided in the 2 marks questions of Section E.
- Draw neat figure wherever required. Take π = 22/7 wherever required if not stated.
SECTION A
1. The value of k if x = 3 and y = -2 is a solution of the equation 2x – 13y = k is
a) 31
b) 23
c) 32
d) 30
1. If √2 = 1.41 then 1/√2 = ______
a) 0.709
b) 7.05
c) 0.75
d) 0.075
3. The co-ordinates of a point below the x-axis lying on y-axis at a distance of 4 units are
a) (-4, 0)
b) (0, 4)
c) (0, -4)
d) (4, 0)
4. Express y in terms of x in the equation 5y – 3x – 10 = 0
5. In the given graph, the number of students who scored 60 or more marks is
a) 22
b) 20
c) 21
d) 19
6. In the adjoining figure, the value of x is:
a) 150
b) 100
c) 120
d) 180
7. If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than 1800, then the two straight lines, if produced indefinitely, meet on that side on which the angles taken together are
a) < 1800
b) = 1800
c) > 1800
d) None of these
8. In ΔABC, E is the midpoint of median AD such that BE produced meets AC at F. If AC = 10.5 cm, then AF =
a) 2.5 cm
b) 5 cm
c) 3 cm
d) 3.5 cm
9. How many lines pass through one point?
a) one
b) three
c) two
d) many
10. When p(x) = x3 + ax2 + 2x + a is divided by x + a, the remainder is
a) a
b) 0
c) 1
d) –a
11. In the adjoining figure, BC = AC. If ∠ACD = 1150, the ∠A is
a) 500
b) 650
c) 57.50
d) 700
12. E and F are the midpoints of the sides AB and AC of a ΔABC. If AB = 6 cm, BC = 5 cm and AC = 6 cm, then EF is equal to
a) 4 cm
b) 3 cm
c) 2.5 cm
d) None of these
13. If x = 3 + 2√2, then the value of x + 1/x is
a) 0
b) 3
c) 1
d) 6
14. In the given figure, ACD is a cyclic quadrilateral, ∠CBQ = 480 and a = 2b. The, b is equal to
a) 480
b) 180
c) 380
d) 280
15. The factors x3 – 1 + y3 + 3xy are
16. The cost of turfing a triangular field at the rate of Rs. 45 per 100m2 is Rs 900. If the double the base of the triangle is 5 times its height, then its height is
a) 40 m
b) 42 m
c) 32 m
d) 44 m
17. Which of the following is a binomial?
a) x + 3 + 1/x
b) x2 + 4
c) 2x2
d) x2 + x + 3
18. Assertion (A): The perimeter of a right angled triangle is 60 cm and its hypotenuse is 26 cm. The other sides of the triangles are 10 cm and 24 cm area of the triangle is 120 cm2
Reason (R): (Base)2 + (Perpendicular)2 = (Hypotenuse)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
19. A right triangle with sides 3 cm, 4 cm and 5 cm is rotated about the side of 3 cm to form a cone. The volume of the cone so formed is
a) 12π cm3
b) 20π cm3
c) 16π cm3
d) 15π cm3
20. Assertion (A): There are infinite number of lines which passes through (2, 14)
Reason (R): A linear equation in two variables has infinitely many solutions.
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
SECTION B
21. Factorize: 6ab – b2 + 12ac -2bc
22. An isosceles triangle has perimeter 30 cm and each of the equal sides is 12 cm. Find the area of the triangle.
23. What must be added to 2x2 – 5x + 6 to get x3 – 3x2 + 3x – 5
OR
Find whether polynomial g(x) is a factor of polynomial f(x) of not: f(x) = x3 – 6x2 – 19x + 84, g(x) = x – 7
24. The height of a cone is 15 cm. If its volume is 1570 cm3. Find the radius of the base.
25. Write the linear equation represented by line AB and PQ. Also find the co-ordinates of intersection of line AB and PQ.
OR
Express x in terms of y for the linear equation 2/3x + 4y = -7
SECTION C
26. Locate √3 on the number line.
27. Find the area of the shaded region in figure.
OR
The perimeter of a triangle is 480 meters and its sides are in the ratio of 1:2:3. Find the area of the triangle.
28. Find at least 3 solutions for the linear equation 2x – 3y + 7 = 0.
29. Without actually calculating the cubes, find the value of (-12)3 + (7)3 + (5)3
30. In fig. write the co-ordinates of the points and if we join the points write the name of fig. formed. Also write co-ordinates of intersection point of AC and BD.
31. ABCD is a square and DEC is an equilateral triangle. Prove that AE = BE
OR
In ΔABC, if ∠A + ∠B = 1250 and ∠A + ∠C = 1130, find ∠A, ∠B, ∠C
SECTION D
32. Visualize the representation on the number line upto 5 decimal places that is, upto 5.37777
OR
If x = 2 – √3, find the value of (x – 1/x)3
33. If two lines intersect, prove that the vertically opposite angles are equal.
OR
Fig. AB || CD and CD ||EF. Also EA is perpendicular to AB. If ∠BEF = 550, find the values of x, y and z.
34. In the adjoining figure, name:
i. Six points
ii. Five line segments
iii. Four rays
iv. four lines
v. Four collinear points
35. In a study of diabetic patients in a village, the following observations were noted:
Represent the above data by a frequency polygon.
SECTION E
36. Read the text carefully and answer the questions:
The front compound wall of a house is decorated by wooden spheres of diameter 21 cm, placed on small supports as shown in figure. 25 such spheres are used for this purpose and are to be painted silver. Each support is a cylinder and is to be painted black.
(i) What will be the total surface area of the spheres all around the wall?
(ii) Find the cost of orange paint required if this paint costs 20 paise per cm2
OR
What will be the volume of total spheres all around the wall?
(iii) How much orange paint in liters is required for painting the supports if the paint required is 3 ml per cm2
37. Read the text carefully and answer the questions:
Sanjay and his mother visited in a mall. He observes that three shops are situated at P, Q, R as shown in the figure from where they have to purchase things according to their need. Distance between shop P and Q is 8 m and between shop P and R is 6 m.
Considering O as the center of the circles.
(i) Find the measure of ∠QPR
(ii) Find the radius of the circle
(iii) Find the measure of ∠QSR
OR
Find the area of ΔPQR
38. Read the text carefully and answer the questions:
There is a Diwali celebration in the DPS school Janakpuri New Delhi. Girls are asked to prepare Rangoli in a triangular shape. They made a rangoli in the shape of triangle ABC. Dimensions of ΔABC are 26 cm, 28 cm and 25 cm.
(i) In fig R and Q are mid points of AB and AC respectively. Find the length of RQ
(ii) Find the length of Garland which is to be placed along the side of ΔQPR
OR
R, P, Q are the mid-points of corresponding sides AB, BC, CA in ΔABC, then name the figure so obtained BPQR
(iii) R, P and Q are the mid-points of AB, BC and AC respectively. Then find the relation between area of ΔPQR and area of ΔABC