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/ MATHEMATICS IX / Class – IX
: 3 hours 90
Time Allowed : 3 hours Maximum Marks: 90
1.
2. 31 4
1 6 2 10
3 11 4
3.
4.
General Instructions:
1. All questions are compulsory. 2. The question paper consists of 31 questions divided into four sections A, B, C and D.
Section-A comprises of 4 questions of 1 mark each; Section-B comprises of 6 questions of 2 marks each; Section-C comprises of 10 questions of 3 marks each and Section-D comprises of 11 questions of 4 marks each.
3. There is no overall choice in this question paper. 4. Use of calculator is not permitted.
/ SECTION-A
1 4 1
Question numbers 1 to 4 carry one mark each.
1
1.111001110011001110011100…, 0.909090….,
1.11100111001110011100…., 0.191019101910….
Identify an irrational number among the following decimal expansions :
1
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1.111001110011001110011100…, 0.909090….,
1.11100111001110011100…, 0.191019101910….
2 (4)3(9)3
(5)3
Without actually calculating the cubes, find the value of (4)3(9)3
(5)3.
1
3 A40 B70 DCE
In the figure, if A40 and B70, then find DCE.
1
4 (2, 3)
In which quadrant does the point (2, 3) lie ?
1
/ SECTION-B
5 10 2
Question numbers 5 to 10 carry two marks each.
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5 7
3 3 2 2
Rationalize 7
3 3 2 2
2
6 2x2 5x 3
Factorise : 2x2 5x 3
2
7 POQ OR OS ROPSOQ80 ROS
ROS
In the figure, POQ is a line. OR and OS are two rays such that ROPSOQ80. Find
ROS and reflex ROS.
2
8 P Q R
PQQRPR
P and Q are centres of the two intersecting circles which intersect at R
(see figure). Prove that PQQRPR.
2
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9 x - (1, 2)
y - (1,2)
Write coordinates of a point which is the reflection of (1, 2) in x - axis. Also, write
coordinates of a point which is the reflection of (1,2) in y - axis.
2
10 8 cm
One side of an equilateral triangle is 8 cm. Find its area using Heron’s formula. Also, find its
altitude.
2
/ SECTION-C
11 20 3
Question numbers 11 to 20 carry three marks each.
11 30 29 28
31 30 29
2 2 2 7
102 2 2
Prove that 30 29 28
31 30 29
2 2 2 7 .
102 2 2
3
12
1
41 1
3 33
5 8 27
Simplify :
1
41 1
3 33
5 8 27
3
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13 (xy2z) (x2y2
4z2xy2yz2zx)
Find the product (xy2z) (x2y2
4z2xy2yz2zx).
3
14 x1 x313x2
32x20
Show that x1 is a factor of the polynomial x313x2
32x20. Hence Factorise the
polynomial.
3
15 ABC X Y AC BC AXCY
ACCB
In a triangle ABC, X and Y are the middle points of AC and BC respectively, such that AXCY, show that ACCB.
3
16 WXYZ O OWOXOZ
OWX50 OZW
WXYZ is a quadrilateral whose diagonals intersect each other at the point O such that OWOXOZ.
If OWX50, then find the measure of OZW.
3
17 AB, CD EF, O x, y, z u
Three lines AB, CD and EF meet at a point O, forming angles as shown in the figure. Find the values of
x, y, z and u.
3
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18 , l AB CD E F AEF EG
EFD FH ab EGFH ABCD
In the figure, a transversal l cuts two lines AB and CD at E and F respectively. EG is the
bisector of AEF and FH is the bisector of EFD such that ab. Show that EGFH and
ABCD.
3
19 (1, 1), (1, 5), (7, 9) (7, 9)
Plot the points (1, 1), (1, 5), (7, 9) and (7, 9). Join them in order. Measure the sides and
diagonals of the figure so obtained.
3
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20 ABCD AB 50 m, BC 60 m, CD 30 m, DA 90 m
BD 70 m
Find the area of the quadrilateral ABCD in which AB 50 m, BC60 m, CD 30 m, DA
90 m and BD 70 m.
3
/ SECTION-D
21 31 4
Question numbers 21 to 31 carry four marks each.
21
(i)
(ii)
(iii)
(iv)
Give an example of two irrational numbers whose :
(i) difference is an irrational number
(ii) sum is an irrational number
(iii) product is an irrational number
(iv) division is an irational number
Justify also.
4
22 1
7 6 13
Rationalise the denominator of 1
7 6 13.
4
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23 (mn), (np) (pm),
m(n2p2)n(p2
m2)p(m2n2)
Using factor theorem, show that (mn), (np) and (pm) are factors of
m(n2p2)n(p2
m2)p(m2n2).
4
24 x y 8x327y3
730 2x2y3xy215 ,
2x3y
If x and y are two positive real numbers such that 8x327y3
730 and 2x2y3xy215 , then
evaluate 2x3y.
4
25 p(x)4x411x3
2x211x6 x2
2x2
Find the quotient and remainder obtained on dividing p(x)4x411x3
2x211x6 by
x22x2 and verify remainder by using remainder theorem.
4
26 abc0
2 2 2(b c) (c a) (a b)1
3bc 3ac 3ab
If abc0, then prove that 2 2 2(b c) (c a) (a b)
13bc 3ac 3ab
4
27
Amit pledges to donate triangular land, area of which equals the area of a rectangular land required
for village school. Also the area of the rectangular land equals that of a square land used for farming
by him, then State the Euclid's axiom that best illustrates the relation of the three shapes of land ?
Which values are exhibited here ?
4
28 5
Sunil and Shyam have the same weight. If they each gain weight by
5 kg, how will their new weights be compared using the axioms ? Write the Euclid's axiom that best
4
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supports your answer. Also give two more axioms other than the axiom used in the above situation.
29 AB CD, CD EF EA AB BEF45 x, y z
In given figure AB CD, CD EF and EA AB. If BEF45 find the values of x, y and z.
4
30
Prove that the sum of three sides of triangle is greater than the sum of three medians.
4
31 (x40), (x20) 10
2
x
x
4
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The angles of a triangle are (x40), (x20) and 102
x
. Find the value of x and then the
angles of the triangle.
-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 Irrational number is 1.111001110011001110011100 --------- as it is non-terminating and non-repeating.
1
2 4950
(4)3(9)3
(5)30
1
3 ACB180(7040)70
ECDACB70
1
4 I quadrant
1
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/ SECTION-B
5 10 2
Question numbers 5 to 10 carry two marks each.
5
2
6 2x2 5x 3
2x2 6xx 3
2x(x3) (x3)
(2x1) (x3)
2
7 ROS18080100
Reflex ROS360100260
2
8 In a circle with centre at P, PRPQradius In a circle with centre at Q, QRQPradius Euclid’s first axiom : Things which are equal to the same things are equal to one another.
PRPQQR
2
9 (1, 2) and (1, 2)
2
7 3 3 2 2 7 (3 3 2 2)
193 3 2 2 3 3 2 2
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10
Altitude AD
2
/ SECTION-C
11 20 3
Question numbers 11 to 20 carry three marks each.
11 L H S
R H S.
3
12
1
411
323
5 8 27
1
41 1
3 33
3 35 2 3
1
435 2 3
3
a b cs 12 cm
2
Area ABC 12 12 8 12 8 12 8216 3 cm
2 28 4 4 3 cm
30 29 282 2 2
31 30 292 2 2
28 2 28 1 282 2 228 3 28 2 28 12 2 2
28 2 28 28
28 3 28 2 28
2 2 2 2 2
2 2 2 2 2 .2
28
28
2 4 2 1 7
102 8 4 2
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1 1
4 43 45 5 5 5
13 (xy2z) (x2y2
4z2xy2yz2zx)
(xy2z) [x2y2
(2z)2xyy(2z)2z(x)]
By the identity
(abc) (a2b2
c2abbcca)a3
b3c3
3abc
(xy2z) (x2y2
(2z)2xyy(2z)2z(x))
x3y3
(2z)33xy (2z)
x3y3
8z36xyz
3
14 p(x)x313x2
32x20
p(1)11332200
x1 is a factor of p(x)
x212x20x2
10x2420
(x10)(x2) p(x)(x1)(x10)(x2)
3
15 Given, To Prove, Figure 1
Proof :
AXCY
2AX2CY
Q If equals are multiplied with equals, then they are also equal 1
So we have
ACBC 1
3
2p12 20
1
xx x
x
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16
150
23
12503180
502502180
2280
240
3
17
z9050180
z40
y90
xz 40
u50
3
18 EG is the bisector of AEF.
AEGGEF a
3
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Similarly EFHHFDb
GEFEFH (As ab)
EGFH
Again AEF2a and EFD2b
AEFEFD2a or 2 b But these are alternate angles
ABCD
19 Plotting and joining points Measuring sides and diagonals
3
20 For ABD, s 105 m
Area (ABD) m2
525 m2
For BCD, s 80
Area of BCD 400 m2
Area of quad. (525 400 ) m2
3
/ SECTION-D
21 31 4
Question numbers 21 to 31 carry four marks each.
21 Any example and verification of example
1111
4
105 55 15 35
11
80 10 50 20 5
11 5
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22
4
23 Let pm(n2p2)n(p2
m2)p(m2n2)
p(mn)n(n2p2)n(p2
n2)p(n2n2)
n(n2p2)n(n2
p2)0
0
mn is a factor of p Similarly p(np)0 & p(pm)
np is a factor of p. and pm is a factor of p.
4
24 (2x3y)38x3
27y33.2x.3y(2x3y)
8x327y3
18(2x2y3xy2)
4
2 2
2 2
( 7 6) 131 1
7 6 13 ( 7 6) 13 ( 7 6) 13
( 7 6) 13
( 7 6) ( 13)
7 6 13
( 7 ) ( 6) 2 . 7 6 13
7 6 13
7 6 2 42 13
7 6 13 7 6 13 2 42
2 42 2 42 2 42
2 7 42 2 6 42 2 13 42
4 42
.
. . . . .
.
2 7 7 6 2 6 6 7 2 13 42
84
. . . . . . . .
14 6 12 7 2 13 42
84
2 (7 6 6 7 546)
84
(7 6 6 7 546)
42
.
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7301815 1000
2x3y10 x&y are positive
25 p(x)4x411x3
2x211x6
Remainder 7x18 Verification by remainder theorem
4
26 LHS
b c a
c a b
given
and abc
a3b3
c33abc
1RHS
4
27 Things which are equal to same thing are equal to one another.
Donating land area for village school.
Spread of education
Community welfare
Feeling for society
4
28 Let x kg be the weight of Sunil and Shyam. Gain of weight by each
5 kg. Weight of Sunil and Shyam (x5) kg each When equals are added to equals, the wholes
4
Q
22
p( ) 4 3 12
2 2
xx x
x x
2 22b c a bc a
3bc 3ac 3ab
22 2ba c
3bc 3ac 3ab
3 3 3a b c
3abc
3abc
3abcQ
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Page 9 of 10
are equal.
One more axioms
29 (z45)90180
(Sum of co-interior angles on same side of transversal is supplementary )
z 180135
z45
Now y45180
(Sum of co-interior angles on same side of transversal is supplementary )
y18045
y135
Now xy (corresponding angles)
x135
4
30 Given, to prove, figure 1
Proof
ABAC>2 AD 1½
Similarly
ABBC>2BE ½
BCCA>2CF ½
Add above three equations
ABBCCA>ADBECF ½
4
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31 (x40)(x20) 10
2
x
180
5 70 180
2
x
5 250
2
x
x100
So, angles are 60, 80 and 40
4
-o0o0o0o-
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