Mathematics Design of Sample Question Paper

Mathematics

Design of Sample Question Paper

SA-I Class IX (2010-2011)

Type of Question	Marks per question	Total No. of Questions	Total Marks
M.C.Q.	1	10	10
SA-I	2	8	16
SA-I I	3	10	30
LA	4	6	24
TOTAL		34	80

Blue Print Sample Question Paper-1 SA-1

I Term

Topic / Unit	MCQ	SA(I)	SA(II)	LA	Total
Number System	2(2)	2(4)	3(9)	-	7(15)
Algebra	2(2)	1(2)	2(6)	3(12)	8(22)
Geometry	6(6)	4(8)	3(9)	3(12)	16(35
Coordinate Geometry	-	1(2)	1(3)	-	2(5)
Mensuration	-	-	1(3)	-	1(3)
TOTAL	10(10)	8(16)	10(30)	6(24)	34(80)

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

Time: 3 to 3Y₂ hours M.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. Each
of 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

6.            Two sides of a triangle are of lengths 7 cm and 3.5 cm. The length of the third side of the
triangle cannot be

(A)    3.6 cm        (B)    4.1cm        (C)    3.4 cm        (D)    3.8 cm.

7.           A rational number between 2 and 3 is

(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 your answer 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 for your answer.

13.

14.

_8_. 75

Is

In Fig. 2, it is given that Z1 =Z4 and Z3=Z2. By which Euclid’s 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

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

Question numbers 19 to 28 carry 3 marks each.

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~~r are 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
(    (-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

Question numbers 29 to 34 carry 4 marks each.

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

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 ; if _X=V3 , x²=3    rational; if _x=2+>/3, x²=7+4^ -> irrational

12.         No, AOC and BOD are not straight lines
v i) ZAOC = 182° *180°

ii) ZBOD = 178° *180° OR

ZQ=180^o-r70^o+30^o]=80° which is largest

.-. Longest side is PR

1. By Euclid’s 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
—	+		+
U5,		v 3 ,

-(i)

y₂

8 1 1 8-5-3 . ^AS 15-3-5 ⁼ ^5—⁼ °

y₂

••• (1) = 3

_8_ _v15_y

_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°-110^o=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

72 + 73   S + J3~ (2)-(3)

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

OR

i+y₂ i+y₂

LHS =

5-3

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

=> ZPAD=15° => ZDAB=20°

=> ZDAB=ZABE=20° and ZEBM=ZBML=10^C

=> x=30°

1. In right triangle QTR, x=90°-40°=50° Again y is the exterior angle of APSR => y=30^o+x=50°+30^o=80°
2. BD+DE = CE+DE => BE=CD In A’s 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)(&<:) = ^21(3)(11)(7) = 2lVncm²

SECTION-D

29.          Let P(x) = x³+2x²-5ax-7 and Q(x) = x³+ax²-12x+6
P(-1) = pandQ(2) = q

… p=-l+2+5a-7 => 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)

_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 a’s ABC and ADE

AB=AD, AC=AE and ZBAC=ZDAE .-. a’s are congruent

.-. BC = DE (cpct)

Zx=Zy => ZBDC=ZAEB In a’s ABE and CBD

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