simo_ix_2010

Standard – IX(SIMO) - 2010

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SIMO EDUCATION SOUTH INDIAN MATHEMATICS OLYMPIAD 2010

SCREENING TESTSTANDARD IX

Time : 2 Hours. Max. Marks : 60

General Instructions: This Question Paper booklet contains 20 Questions. Each question carries THREE marks. ONE mark will be deducted for every wrong answer.

No marks are deducted for un-attempted questions. Few questions may have more than one correct option. They are not given separately at one

place/section.

SECTION – ACOMPREHENSIONS

Comprehension – I

A 2010 sided regular polygon G has vertices V1, V2, V3… ,V2009 and V2010. Each side of G measures 0.008 cms and area of G is approximately 20.5 sq cms. Let P be midpoint of diagonal V2V4 and O be the center of the polygon G. Also, OV3 measures 2.57 cms approximately.

01. From point P, perpendiculars are dropped to all 2010 sides. Let length of perpendicular dropped from P on to side VrVr+1 be Hr (1 ≤ r ≤ 2009). H2010 is perpendicular dropped on to side V2010V1. Then, value of H1+H2+H3+………+H2009+H2010 is (A) 5025 (B) 6225 (C) 5125 (D) 6025

02. The value of V7VrVi is 60o. It is also known that if we know the value of r, the value of i can be determined uniquely. Area of triangle V7VrVi is (approximately) …….. (in sq cms)(A) 8.5 (B) 12.5 (C) 15.5 (D) 16.5

03. A circle C is circumscribed over the polygon G. A regular polygon of n sides is doubled to give

another regular polygon of 2n sides through the process below. The midpoint of arc �1 2

VV is marked

as P1. Similarly midpoint of arc �1r r

V V

is marked as Pr. A new polygon of 4020 sides is formed by

joining Pr to vertices Vr and Vr+1. The doubled polygon is V1P1V2P2V3P3…….V2009P2009V2010P2010. This process is repeated 2010 times to give a polygon M with 22010.2010 sides. The area of polygon M is (approximately) equal to …….. (in sq cms) (A) 19.6 (B) 20.75 (C) 25.6 (D) 28.9



Comprehension – II

Let Sn (n >1) represent set of all natural numbers which can be expressed as sum of n consecutive natural numbers. Consider the set = S4 (S2 S3) and set = S4 (S2 S3). Now, answer the following questions.

04. Which of the following statements is/are true?(A) (B) 2011 (C) - = (D) 2009

05. Let Set P = {x2 / x }. Number of elements of set (P ) which are less than 2010 is(A) 6 (B) 16 (C) 26 (D) None

06. Let elements of be arranged in ascending order. Let xi represents ith element in ascending order. If xi = 2010, the value of i is(A) 148 (B) 124 (C) 167 (D) None

07. The number of values of n for which 2010 Sn is(A) 19 (B) 7 (C) 17 (D) None

Comprehension – III

A(2,-4) and B(7,3) are two points on Coordinate Plane. Point C lies on the line x-y=2 such that area of triangle ABC, i.e., [ABC] = 3 sq units. Also, point C lies in the half plane x+y≤20. Point D lies on axis of X such that line BD divides triangle ABC into two halves in area. Points K, L and M are marked on lines AB, BC and AD respectively such that AK: KB =1:3, BL: LC =5:3 and DM: MA=4:1. Now, answer the following questions.

08. P is a point on boundary of quadrilateral ABCD such that we have exactly three different positions of P for which [KPM] = Δ. The value of Δ (in sq units) is

(A) 3

5(B)

3

10(C)

3

15(D)

3

20

09. KLQM is a parallelogram whose area is

(A) 7

3(B)

27

80(C)

49

160(D)

37

160

10. AB cuts X-axis at R. Slope of line passing through R such that the line cuts quadrilateral ABCD into two halves in area is

(A) 7

3(B) -

5

7(C)

7

5(D)

5

7

11. Area of triangle COD is (O is origin)(A) 11.5 (B) 14 (C) 17 (D) None



SECTION – B

12. x and y are real numbers that satisfy x3 = y3+6xy+8. Possible value(s) of (x-y) is/are(A) -4 (B) 2 (C) 0 (D) 6

13. ABC is an integer sided triangle with lengths 4x+5, x+3 and 10-3x for some integral value of x.Possible value(s) of area of triangle ABC is(A) 8 (B) 10√3 (C) 6√5 (D) 8√3

14. Consider a sequence of numbers given by an = 1 1+ 5 1- 5

2 25

n n

. Which of the

following is/are true with respect to the sequence {an}?(A) a5 = 5 (B) a10 = 1+2+3+……+ 10 (C) a12 =122 (D) a2009 + a2010 = 2.a2011

15. ABCD is a pyramid such that AB=4cms, AC=5cms, AD=3cms. Maximum possible volume of the pyramid is(A) 10 cm3 (B) 20 cm3 (C) 30 cm3 (D) 6 cm3

16. A sequence {an} is given by an+1 = an7. If a1=2007, unit’s digit of a2010 is

(A) 3 (B) 1 (C) 9 (D) 7

17. A three digit number N leaves a remainder of 6 when divided by 7 and a remainder of 7 when divided by 11. Number of possible values of N is(A) 11 (B) 12 (C) 15 (D) 16

SECTION – C

ABC is a triangle such that I, O, S are Incenter, Orthocenter and Circumcenter respectively. B is greater than C. Lines AI and OS cut at P. Also, OP =√3 PS. Also, OAS = 10o. Coordinates of O and S are (6,4) and (3,-2) Now, answer the following questions

18. Ratio of area of triangle AOI to that of SAI is(A) √3 (B) 3 (C) √6 (D) 3√2

19. Centroid of the triangle ABC is located at(A) (5, 2) (B) (4.5, 1) (C) (5, -2) (D) (4, 0)

20. D is midpoint of AB and E is foot of perpendicular of B on to AC. Ratio of area of triangle BEC to that of ADS is(A) 1:2 (B) 2: 1 (C) 3:2 (D) 1: 1

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