MATHEMATICS
Class: IX Maximum marks: 80
Subject: Mathematics Time : 3 hours
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GENERAL INSTRUCTIONS:
1. The question paper has 5 sections A, B, C, D and E.
2. Section A has 20 multiple choice questions (MCQs) carrying 1 mark each.
3. Section B has 5 short answer – I (SA-I) type questions carrying 2 marks each.
4. Section C has 6 short answer – II (SA-II) type questions carrying 3 marks each.
5. Section D has 4 long answer – II (LA) type questions carrying 5 marks each.
6. Section E has 3 Case based integrated units of assessment (4 marks each) with
sub parts of the values of 1,1 and 2 marks each respectively.
7. All questions are compulsory. However, an internal choice in 2 questions of 2
marks, 2 questions in 3 marks and 2 questions in 5 marks has been provided.
An internal choice has been provided in the 2 marks questions of section E.
8. Draw neat figure wherever required. Take π = 22/7 wherever required if not
stated.
Section A
1 The number obtained on rationalizing the denominator of is________
√
[1]
√ √ √ √
a) b) c) d)
2 Between two rational numbers __________
a) there are only rational numbers and no irrational number
b) there are infinitely many rational numbers [1]
c) there is exactly one rational number
d) there is no rational number
3 The value of the polynomial 5x − 4x + 3 , when x = −1 is ______
[1]
a) − 6 b) 6 c) 1 d) − 1
4 The factorization of 4x + 8x + 3 is___________
a) (2x – 1) (2x – 3) b) (2x + 2) (2x + 5) [1]
c) (x + 1) (x + 3) d) (2x + 1) (2x + 3)
5 The linear equation 3x – y = x – 1 has__________
a) A unique solution b) Two solutions [1]
c) No solution d) Infinitely many solutions
6 The graph of the linear equation y = x passes through the point_______
[1]
a) , b) 0 , c) , d) (1 , 1)
7 Point (–3, 5) lies in the_________ [1]
a) second quadrant b) fourth quadrant c) third quadrant d) first quadrant
8 If the y co-ordinate of a point is zero, then this point always lies_____ [1]
a) in quadrant I b) on y - axis c) on x - axis d) in quadrant II
9 The angles of a triangle are in the ratio 5: 3: 7, the triangle is_____
a) An isosceles triangle. b) An obtuse angled triangle [1]
c) A right triangle d) An acute angled triangle
10 In the given figure (not drawn to scale), if AB || CD, then x and y respectively are ____.
[1]
a) 40°, 30° b) 30°, 45° c) 90°, 30° d) 50°, 77°
11 It is given that △ ABC ≅ △ FDE and AB = 5 cm, ∠ B = 40° and ∠ A = 80°. Then
which of the following is true? [1]
a) DE = 5 cm, ∠ E = 60° b) DF = 5 cm, ∠ E = 60°
c) DF = 5 cm, ∠ F = 60° d) DE = 5 cm, ∠ D = 40°
12 ABCD is a Rhombus such that ∠ ACB = 40∘ , then ∠ ADB is______
[1]
a) 100∘ b) 40∘ c) 60∘ d) 50∘
13 Diagonals of a Parallelogram ABCD intersect at O. If ∠ BOC = 90∘ , ∠ BDC = 50∘ then
∠ OAB is_______ [1]
a) 10∘ b) 40∘ c) 90∘ d) 50∘
14 The sides of a triangle are 56 cm, 60 cm and 52 cm long. Then the area of the triangle
is___________ [1]
a) 1311cm b) 1344cm c) 1322cm d) 1392cm
15
In the given figure, AB = 8 cm, OM = ON = 4 cm. Then CD is ________.
[1]
a) 3.5 cm b) 4.5 cm c) 8 cm d) 3 cm
16 In the given figure, ∠ ABD = 70°, ∠ ADB = 30°. Then, ∠ BCD is ________.
[1]
a) 100° b) 90° c) 120° d) 80°
17 If the TSA of a solid hemisphere is 12π sq. cm, then its CSA is______
[1]
a) 16π sq. cm b) 12π sq. cm c) 24π sq. cm d) 8π sq. cm
18 To draw a histogram to represent the following frequency distribution, the adjusted
frequency for the class 25 - 45 is________
Class 5 – 10 10 – 15 15 – 25 25 – 45 45 – 75
[1]
Frequency 6 12 10 8 15
a) 6 b) 5 c) 2 d) 3
19 Assertion (A): The point (2, −3) lies on the on the line x + y = 5.
Reason (R): A point which satisfies the linear equation lies on the line representing it.
a) Both A and R are true and R is the correct explanation of A. [1]
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.
20 Assertion (A): The measure of ∠ AOC = 60
Reason (R): Angle subtended by an arc of a circle at the centre of the circle is double
the angle subtended by arc on the circumference.
[1]
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 Name the quadrant in which the following points lie:
[2]
(i) A (2, 9) (ii) B (–3, 5) (iii) C (–4, –7) (iv) D (3, –2)
22 Write any two of the Euclid’s Axioms. [2]
23 In Figure, line segment AB is parallel to another line segment CD. O is the mid-point of
AD. Show that:
6. △ AOB ≅ △ DOC
7. O is also the mid - point of BC.
[2]
OR
If DA and CB are equal perpendiculars to a line segment AB. Show that CD bisects AB.
24 Find the capacity in litres of a conical vessel with height 12 cm, slant height 13 cm.
OR
The radii of two cones are in the ratio 2:1 and their volumes are equal. What is the ratio [2]
of their heights?
25 A conical tent is 10 m high and the radius of its base is 24 m. Find
1. Slant height of the tent. [2]
2. Cost of the canvas required to make the tent, if the cost of 1 m canvas is ₹ 70.
Section C
26 Locate √10 on the number line.
OR [3]
Express 0. 125 in the form, where p and q are integers and q ≠ 0.
27 Factorize: 3x − x − 4 [3]
28 Find 3 solutions for the linear equation 2x – 3y + 7 = 0. [3]
29 In figure, if PQ || RS,∠ MXQ = 135° and ∠ MYR = 40°, find ∠ XMY.
[3]
OR
In figure, if PQ || ST, ∠ PQR = 110 and ∠ RST = 130 ,find ∠ QRS.
30 In the given figure, the side BC of △ ABC has been produced to a point D. If the
bisectors of ∠ ABC and ∠ ACD meet at point E then prove that ∠ BEC = ∠ BAC .
[3]
31
The triangular side walls of a flyover have been used for advertisements. The sides of
the walls are 122 m, 22 m and 120 m (see figure). The advertisements yield earnings
of ₹ 5000 per m per year. A company hired one of its walls for 3 months. How much
rent did it pay?
[3]
Section D
32
√ √
Find the values of a and b if, − = a + b√5 . [5]
√ √
33
If a + b + c = 5 and ab + bc + ca = 10, then prove that, a + b + c − 3abc = −25
OR [5]
Using factor theorem, factorize the polynomial: x + 2x – x – 2
34 Two parallel lines l and m are intersected by a transversal p. Show that the quadrilateral
formed by the bisectors of interior angles is a rectangle.
OR [5]
ABC is a triangle right angled at C. A line through the mid-point M of hypotenuse AB
and parallel to BC intersects AC at D. show that
8. D is mid - point of AC
9. MD ⊥ AC
10. CM = MA = AB
35 The following table gives the distribution of students of two sections according to the
marks obtained by them:
Section A Section B
Marks Frequency Marks Frequency
0 – 10 3 0 – 10 5
10 – 20 9 10 – 20 19
20 – 30 17 20 – 30 15
30 – 40 12 30 – 40 10
40 – 50 9 40 – 50 1
Represent the marks of the students of both the sections on the same graph by frequency
polygons.
Section E
36 Read the text carefully and answer the questions:
Beti Bachao, Beti Padhao is a personal campaign of the Government of India that aims
to generate awareness and improve the efficiency of welfare services intended for girls.
In a school, a group of (x+ y) teachers, (x + y ) girls and (x + y ) boys organised a
campaign on Beti Bachao, Beti Padhao.
[1]
11. How many teachers are there in the group if there are 63 girls (given xy= 9)?
[1]
12. What is the value of (x − y ) if the number of teaches are 10 ?
[given (x − y) = 23]
13. How many girls are there in the group if there are 10 teachers and 370 boys? [2]
OR
How many boys are there in the group if there are 10 teachers and 58 girls?
37 Read the text carefully and answer the questions:
Ankit visited in a mall with his father. He sees 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.
[1]
(i) Identify the type of quadrilateral formed by joining P, S, Q and R in a
sequence. [1]
(ii) Find measure of ∠QPR. [2]
(iii) Find the area of ∆PQR.
OR
Find the radius of the circle.
38 Read the text carefully and answer the questions:
Mathematics teacher of a school took her 9th standard students to show Red fort. There
are 2 pillars which are cylindrical in shape having radius of the base as 7 m and height
10 m. There are also 2 domes at the corners which are hemispherical and 7 smaller
domes at the centre. All these domes are identical. Flag hoisting ceremony on
Independence Day takes place near these domes.
[1]
14. What is the volume of each dome? [1]
15. What is the volume of both the pillars?
16. If the outer side of each of all the domes is to be white-washed and the cost of
white-washing is ₹ 50 per m , what will be the cost of white-washing all these [2]
domes?
OR
Find the curved surface area of each of the pillars.
