section-a 4 2 marks section-c 0

of 6

MATHEMATICS Class – IX

Time allowed : 3 hours Maximum Marks : 90

General Instructions : (i) All questions are compulsory. (ii) The question paper consists of 31 questions divided into five sections A, B, C ,D and E.

Section-A comprises of 4 questions of 1 mark each, Section-B comprises of 6 questions of 2 marks each, Section-C comprises of 8 questions of 3 marks each and Section-D comprises of 10 questions of 4 marks each. Section E comprises of two questions of 3 marks each and 1 question of 4 marks from Open Text theme.

(iii) There is no overall choice. (iv) Use of calculator is not permitted.

SECTION-A

Question numbers 1 to 4 carry one mark each.

1 Express y + 3 = 0, in the form of ax + by + c = 0.

1

2 How many solution(s) the equation x=−1 has, if it is treated as an equation in two variables ?

1

3 A rectangle and a parallelogram are on same base and between same parallels. If height of parallelogram is 4 cm and length of base of rectangle is 8 cm, find the area of parallelogram.

1

4 Find the number of small cubes with edge 20 cm that can be accommodated in a cubical box of 2 meter edge.

1

SECTION-B

Question numbers 5 to 10 carry two marks each.

5 In the given figure, P is any point on the diagonal AC of the parallelogram ABCD. Show that ar (∆ADP)= ar (∆ABP).

2

6 In the figure, O is centre of the circle passing through P, Q, R and S.

If ∠SOQ=150�, find the values of x and y.

2

Downloaded from www.studiestoday.com


www.stud

iestod

ay.co

m

of 6

7 In the figure, the lengths of the diagonals AC and BD of rhombus ABCD are 16 cm and 12 cm respectively. Find the length of each side of the rhombus.

2

8 The volume of open cylindrical can is 6358 cm3 and its height is 28 cm. Find its radius and curved surface area.

2

9 A group of 75 students are selected of class X and asked for their choice of subject to be taken in class XI, which is recorded as below :

Stream PCM PCB Comm Humanities Total

Number of Students 28 17 18 12 75

If a student is chosen at random find the probability that it is : (i) A student of science stream (ii) A student of commerce stream

2

10 A store stocked some bags of wheat flour containing the following weights of flour (in kg): 4.97, 5.05, 5.08, 4.85, 5.11, 5.03, 5.00, 5.06, 5.08, 5.07, 5.04, 5.00, 4.98. Find the probability that any of these bags chosen at random contains more than 5 kg of flour.

2

SECTION-C

Question numbers 11 to 18 carry three marks each.

11 Draw the graph of 3 8 0x y+ − = in a Cartesian plane, Find whether (2, 2) is a solution or not.

3

12 PQR is a triangle as shown in the figure.

(a) Find co-ordinates of the vertices of ∆ PQR.

(b) Write equation of QR and also find its length.

(c) Determine the area of ∆ PQR.

3



www.stud

iestod

ay.co

m

of 6

13 In the given figure, BE and CD are the medians of ∆ABC. Prove that ar (∆BFC)=ar (quad DFEA).

3

14 In the given figure, points A, D, P, C and B lie on a circle with centre O. If ∠BOD=150�, find

the measures of ∠BPD, ∠BCD and ∠BAD.

3

15 Draw an angle of 120� using protractor. Bisect it and verify the result.

3

16 ABCD is a parallelogram and M is the mid-point of CD. Through D, a line segment is drawn parallel to MB to meet CB produced at O and AB at N as shown in the figure. Prove that :

(i) AD =1

2CO

(ii) DO =2 BM

3

17 Two equal chords PQ and RS of a circle with centre O, when produced meet at a point M as shown in the figure. Prove that QM=SM

3



www.stud

iestod

ay.co

m

of 6

18 The surface area of the sphere of radius 5 cm is five times the curved surface area of a cone of radius 4

cm. find the volume of the come.

3

SECTION-D

Question numbers 19 to 28 carry four marks each.

19 Find three integral solutions of 13x+y=13. Represent this equation by a graph. Does it pass through origin ?

4

20 A student wrote the equations of the lines a and b drawn in the following graph as y=1 and 2x+3y=6. Is he right ? If yes, write coordinates of point of intersection lines a and b.

Also, find the area enclosed between these lines and y-axis.

4

21 XD is a median of ∆XYZ. E is a point on XD such that XE=ED. Find ar(∆YXD) : ar(∆XEZ).

4

22 AB and CD are two parallel chords of a circle whose centre is O and radius is 20 cm. If AB=24 cm and CD=32 cm, find the distance between AB and CD, if its given that they lie. (i) On the opposite sides of centre O. (ii) On the same side of centre O.

4



www.stud

iestod

ay.co

m

of 6

23 Construct ∆ABC in which BC =7.4 cm, AC=6.2 cm and the included angle is 105�. Construct

angle bisectors of other two angles of ∆ABC. Measure the angle formed at point of intersection of the bisecting rays.

4

24 In the figure, ABCD is a square and EF��BD. M is the mid−point of EF. Prove that AM bisects

∠BAD.

4

25 In a women’s self-defense training camp, the soup is prepared in a cylindrical utensil of radius 10 cm.

If there are 9 women in camp who take soup in hemispherical bowl of radius 5 cm then, how much

soup should be made ? What would be the height of cylindrical utensil of soup? What value is

depicted here ?

4

26 A bigger cube is formed from the material obtained by melting three smaller cubes of 3 cm, 4 cm, and 5 cm side. What is the ratio of the total surface areas of the smaller cubes and the bigger cube ?

4

27 Find the number of cylindrical glasses of diameter 6 cm and height 80 mm that can be filled with juice from a cylindrical vessel of base diameter 30 cm, given that the vessel is filled with juice upto a height of 32 cm.

4

28 A survey of 200 people was conducted about their preference of visting various pavilion during trade fair.

Pavilion Good living

Delhi pavilion

Toy pavilion

Defence pavilion

Number of People

95 45 40 20

Find the probability that a randomly selected person visited : (a) both good living and Delhi pavilion (b) only the defence pavilion (c) only the toy pavilion (d) both toy pavilion and defence pavilion

4

SECTION-E

(Open Text) (* Please ensure that open text of the given theme is supplied with this question paper.)

Theme : Empower to learn

29 How Harshit gets benefitted with these sites ?

3

30 Represent the data in the Table -2 as frequency distribution table. Find the class mark of each 3



www.stud

iestod

ay.co

m

of 6

class interval and cumulative frequency.

31 In figure 1, number of students in age group 05 to 08 are .4 millions i.e, 4 Lakhs. similarly take number of students from all age groups (in total 7). Find mean and mode.

4

-o0o0o0o-



www.stud

iestod

ay.co

m

of 17

Marking Scheme Mathematics (Class – IX)

General Instructions:

1. The Marking Scheme provides general guidelines to reduce subjectivity and maintain uniformity.

The answers given in the marking scheme are the best suggested answers.

2. Marking be done as per the instructions provided in the marking scheme. (It should not be done

according to one’s own interpretation or any other consideration).

3. Alternative methods be accepted. Proportional marks be awarded.

4. If a question is attempted twice and the candidate has not crossed any answer, only first attempt be

evaluated and ‘EXTRA’ be written with the second attempt.

5. In case where no answers are given or answers are found wrong in this Marking Scheme,

correct answers may be found and used for valuation purpose.

/ SECTION-A

1 4 1

Question numbers 1 to 4 carry one mark each.

1 0xy30

1

2 Infinite

1

3 As rectangle and parallelogram lie on the same base and between same parallels.

ar(rectangle)ar(parallelogram)

[ parallelogram on same base and between same parallels are equal in area]

i.e. lbar(parallelogram)

1



www.stud

iestod

ay.co

m

of 17

ar(parallelogram)84

32 cm2

4 1000

1

/ SECTION-B

5 10 2

Question numbers 5 to 10 carry two marks each.

5 ar (AOD)ar (AOB) ___________ (1) (AO is the median)

ar (POD)ar (POB) ___________ (2) (PO is the median)

(1) & (2)

ar (ADP)ar(ABP)

2

6 x

1

2SOQ (angle at centre is

1

2 angle at the circumference)

1

215075

2



www.stud

iestod

ay.co

m

of 17

xy180 (opposite S of cyclic quadrilateral)

y18075

105

7

1 1AO AC 16 8 cm

2 2

1 1BO BD 12 6 cm

2 2

AOB90

(Diagonals bisect each other at right angles)

In rt. AOB,

2 2

2 2

AB AO BO By pathogoras theo

8 6

64 36

100

10 cm

Length of each side 10 cm

2

8 Volume 6358 cm3, h28 cm

r2h6358

22

7r2

28 6358

2



www.stud

iestod

ay.co

m

of 17

r 17

28.5 cm

C.S.A 2rh

222

78.528

1496 cm2

9 (i) A student of science stream

P (E) 45 3

75 5

(ii) A student of commerce stream

P (E) 18 6

75 25

2

10 Total bags13

Favourable outcomes8

8P(E) =

13

2

/ SECTION-C

11 18 3

Question numbers 11 to 18 carry three marks each.

11 x = 8 — 3y x 5 2

y 1 2

Yes (2, 2) is a solution Graph.

3



www.stud

iestod

ay.co

m

of 17

12 (a) P(0, 2), Q(2, 0) and R(3,0)

(b) Equation of QR is y0 and its length is 5 units

(c) as (PQR) 1

2525 sq units

3

13

Since Median of a divides it into two ‘s of equal area

ar (BCE) 1

2 ar (ABC) ___________ (1)

ar (ACD) 1

2 ar (ABC) ___________ (2)

RHS (1) RHS (2)

ar (BCE)ar (ACD) __________ (3)

Subtracting ar (FCE) from both sides.

ar (BCE) ar (FCE)ar (ACD) ar (FCE)

ar (BFC) ar (quad DFEA)

3

14 For major arc DB

reflex DOB2BPD (angle subtended by an arc at the centre is twice the angle at

remaining circumference)

3



www.stud

iestod

ay.co

m

of 17

BPD1

2 (360150)

1

2210

105

BCDBPD105 (angles in the same segment)

BADBCD180 (opp. angles of cyclic quadrilateral)

BAD105180

BAD18010575

15 Construct angle of 120Bisect the angle Verify the result

3

16 In CDO,

M is the mid-point of CD Given

and BM DO

B is the mid-point of CO ( By converse of mid-point theorem)

and BM

1

2DO

i.e BC

1

2CO

But BCAD (Opposite sides of gm )

ADBC 1

2CO

(II) AsBM 1

2DO (Proved above)

3



www.stud

iestod

ay.co

m

of 17

i.e. DO2BM

17

Construction : Join OM. Draw OL PM and ON RM.

Proof : PQRS

In OLM and ONM

OLON (equal chords are equidistant from centre)

OLMONM90

OMOM (common)

OLM ONM (RHS)

LMNM (cpct) ------ (1)

Since PQRS

1

2PQ

1

2RS

QLSN (perpendicular from centre bisects the chord) ----- (2)

Subtracting (2) from (1)

LMQLNMSN

QMSM

3



www.stud

iestod

ay.co

m

of 17

18 Surface area of sphere4r2

Curved Surface area of conerl

SA of sphere of r5 is 4(5)2

C.S.A of cone of radius 4 cm(4)l

100 5 [4l] l5

l5, r4 height of cone (h)3

so, volume of cone 2 31 1 22r h 16 3 50.3 cm

3 3 7

3

/ SECTION-D

19 28 4

Question numbers 19 to 28 carry four marks each.

19 13xy13 4



www.stud

iestod

ay.co

m

of 17

y13(1x)

X 1 0 1

Y 0 13 26

No, it does not pass through origin

20 a : y1

b : 2x3y6

student is right

intersection point3

1,2

Area1

21

3

2

3

4 sq. units

4



www.stud

iestod

ay.co

m

of 17

21

X D is the median

ar(YXD)ar(ZXD) (1) [ median divides a triangle into two triangles of equal area]

Since XEED

E is the mid-point of XD

So, EZ is the median to XDZ

ar(XEZ)ar(DEZ)1

2 ar(ZXD)

i.e, 2ar(XEZ)ar(ZXD) (2)

using (2) in (1) we get

ar(YXD)2ar(XEZ)

i.e,

ar YXD 2

ar XEZ 1

ar(YXD): ar(XEZ)2:1

4

22

AB24 cm and CD34 cm

CASE I : AB and CD lie on opposite sides of centre O.

Draw OLAB and OMCD

OM and OL will be in same line as ABCD

4



www.stud

iestod

ay.co

m

of 17

Join OA and OC

Let OLx cm

OMy cm

In OAL

OA20 cm (radius)

AL1

2AB (perpendicular from centre bisect the chord)

12 cm

By Pythagoras theorem : OA2OL2

AL2

202122

x2

x2400144

256 16 cmx

In OCM

OC20 cm (radius)

1CM

2(perpendicular from centre bisects the chord)

1 32 16 cm2

By Pythagoras theorem : OC2OM2

CM2

(20)2162

y2

400256y2

144 12 cmy

Distance between AB and CDxy

1612

28 cm



www.stud

iestod

ay.co

m

of 17

CASE II : AB and CD lie on the same side of centre O

Distances between AB and CD

yx

1612

4 cm

23 Construction 2 marks

Steps of construction 1 mark ; measure of angle142 1

2 1 marks

4

24

BD (Opposite angles are equal)

1

2B

1

2D

ie, 13 (Diagonal BD of a square bisect B and D)

C90

1345 (By Angle sum property of a )

4



www.stud

iestod

ay.co

m

of 17

EFBD (Given)

12 Corresponding angles

and 34

2445

CFCE (Sides opposite to equal angles)

But CDBC (Sides of square are equal)

CDCFBCCE

DFBE

In ABE and ADF,

ABAD (Sides of square are equal)

BD90 (Each angle of square90)

BEDF (Proved above)

ABE ADF By SAS Congruence Rule)

56 -----(1) By cpct

and AEAF

In AEM and AFM,

AEAF (Proved above)

MEMF (M is Midpoint of EF)

AMAM (Common)

AEM AFM (By SSS Congruence Rule)

78 -----(2) (By cpct)

Adding (1) and (2),

5768

ie, BAMDAM

AM bisects BAD



www.stud

iestod

ay.co

m

of 17

25 (a) r5 cm

Vol. of bowl 2

3r3

3250 cm

3

Vol. of 9 bowl 3250 9 cm

3

750 cm3

Vol. of utensilVol. of 9 bowl

r2h750 100h750 h7.5 cm (b) Sharing Caring

4

26

3

2

2 2 2

Vol of bigger cube 27 64 125

216 cm

Side 6 cm

.S.A. of bigger cube 6 6

216 Sq m

.S.A. of smaller cubes 6 3 6 4 6 5

6 9 16 25

.

6 50

300

300Ratio of T.S.A of smaller cubes and bigger cube

5025

36216

18

25 18:

4

27 Diameter of glass = 6cm 4



www.stud

iestod

ay.co

m

of 17

Radius of base of the glass = 3cm (= r)

Height of the glass = 80mm = 8cm (=h)

Radius of vessel, R = 15cm (= 30/2)

Height of water in the vessel ‘h’ 32cm

Volume of water in the cylindrical vessel, V = R2h1

2 15 32

= 7200 cm3

Volume of glass, V1 = r2h

= 2 3 3 8 72 cm

Number of glasses 1

V

V

7200

10072

28 Total number of people = 200

(a) Let E be an event of selecting a person visiting Delhi Pavillion and good living parillion.

No. of favourable outcomes = 140 = (95 + 45)

Probability of an event = 140 7

200 10

(b) Probability of visiting only defence Pavilion

= 20 1

200 10

(c) Probability of visiting a toy Pavillion

= 40 1

=200 5

4



www.stud

iestod

ay.co

m

of 17

(d) Probability of visiting both toy pavillion and Defence pavillion

=

40 20 3

200 10

/SECTION-E

( /Open Text)

(* Please ensure that open text of the given theme is supplied with this question paper.)

Theme : Empower to learn

29 Technology enhances the performance of an individual : (i) because he can interactive with academic experts across the world. (ii) because he can ask questions at any point of time and clear his doubts. (iii) because he can get assistance in any subject. (iv) because it helps him in using his time and energy at his own pace but with global

inputs. (Any three)

3

30 Cumulative Frequency

Class marks

3

31 Mean : Number of students ( in lakhs) are 4, 5, 12, 28, 15, 12, 2

Mean =

Mode : data 12 comes two times so, mode = 12 lakhs

4



www.stud

iestod

ay.co

m

section-a 4 2 marks section-c 0

Documents