Mple Question Paper Mathematics First Term

8/3/2019 Mple Question Paper Mathematics First Term

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mple Question Paper Mathematics First Term (SA-I) Class IX 2010-2011

Time: 3 to 3Y2hoursM.M.: 80

General Instructions

i) All questions are compulsory.

ii) The questions paper consists of 34 questions divided into four sections A, B, C

and D.

Section A comprises of 10 questions of 1 mark each, Section B comprises of 8 questions

of 2 marks each section C comprises of 10 questions of 3 marks each and section D

comprises of 6 questions of 4 marks each.

iii) Question numbers 1 to 10 in section A are multiple choice questions where you

are to

select one correct option out of the given four.

iv) There is no overall choice. However, internal choice has been provided in 1

question of

two marks, 3 questions of three marks each and 2 questions of four marks each. You have

to attempt only one of the alternatives in all such questions.

v) Use of calculators is not permitted.

Section-A

Question numbers 1 to 10 carry 1 mark each.

1. Decimal expresion of a rational number cannot be

(a) non-terminating (B) non-terminating and

recurring

(C) terminating (D) non-terminating and non-

recurring

2. One of the factors of (9x2-1) (1 +3x)2 is

(A) 3+x (B) 3-x (C) 3x-1

(D) 3x+1

3. Which of the following needs a proof?

(A) Theorem (B) Axiom (C) Definition (D) Postulate

4. An exterior angle of a triangle is 110 and the two interior opposite angles are

equal. Eachof these angles is

(A) 70 (B) 55 (C) 35

(D) 110

5. In APQR, if ZR > ZQ, then

(A) QR>PR (B) PQ>PR (C) PQ


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(A) 2.010010001 (B) ^6 (C) 5/2

(D) 4.^/2

8. The coefficient of x2 in (2x2-5) (4+3x2) is

(A) 2 (B) 3 (C)

8 (D) -7

9. In triangles ABC and DEF, ZA = ZD, ZB = ZE and AB=EF, then are the two

triangles

congruent? If yes, by which congruency criterion?

(A) Yes, byAAS (B) No (C) Yes, by ASA (D) Yes.byRHS

10. Two lines are respectively perpendicular to two parallel lines. Then these lines to

each

other are

(A) Perpendicular (B) Parallel

(C) Intersecting (D) incllined at some acute angle

SECTION B

Question numbers 11 to 18 carry 2 marks each.

1. x is an irrational number. What can you say about the number x2? Support youranswer with examples.

2. Let OA, OB, OC and OD be the rays in the anticlock wise direction starting from

OA, such that ZAOB = ZCOD = 100, ZBOC = 82 and ZAOD = 78. Is it true that

AOC and BOD are straight lines? Justify your answer.

OR

In APQR, ZP=70, ZR=30. Which side of this triangle is the longest? Give reasons foryour answer.

13.

14.

In Fig. 2, it is given that Z1 =Z4 and Z3=Z2. By which Euclids axiom, it can be

shown that if Z2 = Z4 then Z1 = Z3.

( 8

~

)3 MY

,

15,

J v3j

How will you justify your answer, without actually calculating the cubes?

2

15. 16.

-13

_8_.

75 Is


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27

In Fig. 3, ifABIICDthen find the measure of x.

88

Q

R.

Fig. 3

1. In an isosceles triangle, prove that the altitude from the vertex bisects the base.

2. Write down the co-ordinates of the points A, B, C and D as shown in Fig. 4.

SECTION C


19. Simplify the following by rationalising the denominators

2^6 6^2

OR

V5+V3 rr=.

If ^ = a_v1 ^b, find the values of a and b.

20.

If a=9-4>/5, find the value of a-.

a

OR

If x = 3+2V2, find the value of x2 + \

21. Represent 73~5 on the number line.

1

22. If(x-3)and x~~rare both factors of ax2+5x+b, show that a=b.

o

23. Find the value of x3+y3+15xy-125 when x+y=5.

OR

If a+b+c=6, find the value of (2-a)3

+(2-b)3

+(2-c)3

-3(2-a)(2-b)(2-c)

24.

25.

26.

27.

28.

a

b/ \ c


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( (-

3,0)

1 0 (3,0)

Fig. 5

In Fig. 8, D and E are points on the base BC of a AABC such that BD=CE and AD=AE.

Prove that AABC = AACD.

rig. o

Find the area of a triangle, two sides of which are 18 cm and 10 cm and the perimeter is

42 cm.

SECTION D


29. Let p and q be the remainders, when the polynomials x3+2x2-5ax-7 and x3+ax2-

12x+6 are

divided by (x+1) and (x-2) respectively. If 2p+q=6, find the value of a.

OR

Without actual division prove that x4-5x3+8x2-10x+12 is divisible by x2-5x+6.

30. Prove that:

(x+y)3 + (y+z)3 + (z+x)3 3(x+y) (y+z) (z+x) = 2(x3+y3+z3-3xyz)

1.

Factorize x12-y12.

2. In Fig. 9, PS is bisector of ZQPR; PT IRQ and ZQ>ZR. Show that

ZTPS = -(ZQ-ZR). R

OR A

In AABC, right angled at A, (Fig. 10), AL is drawn perpendicular to BC. Prove that ZBAL =ZACB.

33. In Fig. 11, AB=AD, AC=AE and

ZBAD = ZCAE. Prove that BC = DE.

34. In Fig. 12, if Zx=Zy and AB = BC, prove that AE = CD.

Answers

Section A


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1.

6.

(D) (C)

2. 7.

(D) (C)

3. 8.

(A) (D)

4. 9.

(B) (B)

5. 10.

(B) (B)

SECTION B

11. x2 may be irrational or may not be.

For example ; ifX=V3 , x2=3 rational; ifx=2+>/3, x

2=7+4^ -> irrational

12. No, AOC and BOD are not straight lines

v i) ZAOC = 182 *180

ii) ZBOD = 178 *180 OR

ZQ=180o-r70o+30o]=80 which is largest

.-. Longest side is PR

1. By Euclids I Axiom, which states. ["Things which are

equal to the same thing are equal to one another"]

2. The LHS can be written as

y2

y2

y2

1

1

2

(8 3 r-i> 3


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+ +

U5, v 3 ,

-(i)

y2

8 1 1 8-5-3 . AS 15-3-5 = ^5=

y2

(1) = 3

_8_ v15y

_8_ 75

RHS

y2

Justification : By the formula: If a+b+c=0, then a3+b3+c3=3abc

y2

1| 2

3

15. K27, v3 J

1 4 = 9

_1V 1

16. Zx=-70+88=18

(v ZQLM=180-110o=70 and ABIICD=*ZPML=88)

17. Let ABC be isosceles a in which AB=AC

Draw AD1BC

A sADB and ADC are congruent by RHS .-. BD=DC(cpct)

i.e, Altitude AD bisects the base BC

18. The coordinates of the points are:

A(2,4), B(0, -3), C(-3, -5) and D(5, 0)

y

y

1/2+1/2+1/2+1/2

SECTON-C

19.

27*5 6>/2 2V6(V2-V3~) + 6V2(V6-V3

6-3


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72 + 73 S + J3~ (2)-(3)

= 2718-2712 + 2^-2^ = 672-276

OR

i+y2 i+y2

LHS =

75 + 73 _(V5 + y3)(75 + 73 75-73 ~

= |^I = 4+715 = a-7l5b

a=4, b=-1

20.

= 9-475 =>- =

a 9-475 81-80

= 9 + 475

a~=9-475-9-475 = -875 a

OR

x=3+2j2 => x2=9+8+12>/2 = 17 +12V2

1= 1 17-12V2=17_^ x2 17 + 12V2 289-288

.-. x2+ =17 + 12V2 +17-12V2 = 34

A respresents V3~5 on the number line

3.5 cm

Let f(x) = ax2+5x+b

f(3) = 0 => 9a+15+b=0 9a+b=-15

0)

(i) = (ii) => a=b

If x+y=5 => x+y+(-5)=0

., (x)3+(y)3+(-5)3 = 3(x)(y)(-5)

=> x3+y3+15xy=125

=> x3+y3+15xy-125=0

OR a+b+c=6 => (2-a)+(2-b)+(2-c)=0

.-. (2-a)3+(2-b)3+(2-c)3 = 3(2-a)(2-b)(2-c) .-. (2-a)3+(2-b)3+(2-c)3-3(2-a)(2-b)(2-c)=0

AB=BC=AC=6 units as AABC is equilateral AO bisects base BC => OB=3 units

.-. OA2=AB2-OB2=62-32 = 27 OA=3y/3

25. DrawADIIPQ, BEIILMIIPQ

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=> ZPAD=15 => ZDAB=20

=> ZDAB=ZABE=20 and ZEBM=ZBML=10C

=> x=30

1. In right triangle QTR, x=90-40=50 Again y is the exterior angle of APSR =>y=30o+x=50+30o=80

2. BD+DE = CE+DE => BE=CD In As ABE and ACD

BE=CD,AE=AD, ZADE=ZAED .-. AABE = AACD (SAS)

Vz

Vz

VA

VA 1

28. s=y=21, let a=18cm, b=10cm, c=42-(28)=14cm

Ar(A) = Vs(s-a)(s4D)(& p=5a-6 q=8+4a-24+6 q=4a-10

2p+q=6 => 10a-12+4a-10=6 => 14a=28 => a=2

OR

x2-5x+6 = (x-2)(x-3)

P(x) = x4-5x3+8x-10x+12

P(2) = 16-40+32-20+12=0

P(3) = 81-135+72-30+12=0

(x-2)(x-3) divides P(x) completely

30. Let x+y=p, y+z=q, z+x=r

LHS = p3+q3+r3-3pqr

= (P+q+r) (p2+q2+r2-pq-qr-rp)

i+y2 i+y2

Yz+Vz

1 1 1

Now p+q+r=2(x+y+z)

p2+q2+r2-pq-qr-rp = (x+y)2+(y+z)2+(z+x)2-(x+y)(y+z)-(y+z)(z+x)-(z+x)(x+y)

x2+y2+2xy+z2+2yz+2zx x2+y2 -xy +z2 -yz -xz

-y2 -xy -z2 -yz -xz x2 -xy -yz -xz x2+y2+z2-xy-yz-zx

.-. (p+q+r) (p2+q2+r2-pq-qr-rp) = 2(x+y+z) (x2+y2+z2-xy-yz-zx)

= 2(x3+y3+z3-3xyz)


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X12.y12 (X6-y6)(x6+y6)

= (x3-y3)(x3+y3)(x2+y2)(x4+y4-x2y2)

= (x-y)(x2+y2+xy)(x+y)(x2+y2-xy)(x2+y2)(x4+y4-x2y2)

ZQ+ZR=180-2ZQPS=180-2 [ZQPT+ZTPS]

= 1800-2[90-Z1 + ZTPS]

=>Z1+Z2 = 2Z1-2ZTPS =>ZTPS=1(Z1-Z2) = 1(ZQ-ZR) OR

ZB+ZC=90 => ZB=90-ZC

ZBAL=90-ZB=90-(90 ZC) = ZC

.-. ZBAL = ZACB

ZBAD+ZDAC = ZCAE+ZCAD => ZBAC=ZDAE

In as ABC and ADE

AB=AD, AC=AE and ZBAC=ZDAE .-. as are congruent

.-. BC = DE (cpct)

Zx=Zy => ZBDC=ZAEB In as ABE and CBD

AB=BC, ZB=ZB, ZBDC=ZAED 1 .-. AABE ACBD [AAS]

.-. AE=CD

Mple Question Paper Mathematics First Term

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