Transcript
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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
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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
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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
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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
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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
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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-
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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(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
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