Downloaded from - NCERT Solutions, CBSE ...€¦ · / 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:

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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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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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Downloaded from - NCERT Solutions, CBSE ...€¦ · / 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: