Standard – X(SIMO) X(SIMO) - 2012
SIMO EDUCATION INDIAN MATHEMATICS OLYMPIAD (SIMO) 2012 SCREENING TEST STANDARD X
Time : 90 mins
Max. Marks : 120
Instructions: The question paper contains 30 questions to be answered in 90 minutes All questions have ONLY ONE correct answer. Each question carries 4 marks. One mark would be deducted for every wrong answer. No marks would be deducted for unattempted questions. You can retain the Question paper with you after the Olympiad. Fill the OMR sheet very carefully. Your name in the participation certificate will be the same as the one you give here. Please leave space after initials.
Comprehension-I (Questions 01 to 03) [ANGELS] O, S and I are orthcenter, circumcenter and incenter respectively of a triangle ABC. Let AOB AOB , ASB and AIB be represented by α, β and γ (not necessarily in that respective order). Also, we 3 2 have f(x)= 2x -(α+β+γ)x +αβγ. Now, answer the following questions.
01. It is found that for all angles of triangle ABC, f(α)≤0. Possible value(s) of β if o
(A) 105
(B) 60
o
(C) Either A or B
o
C =30
is
(D) Neither A nor B
02. Which of the following statements is/are correct?
(A) f(α)=0
=0 3
(B) f(α)=0
f
(C) f(α) f(β) f(γ)≥0 for all angles of the triangle ABC.
ABC is equilateral
(D) All the above
03. The points A1(f(α), f(β)), A2(f(β),f(γ)) and A3(f(γ),f(α)) are marked in the 2D Cartesian plane. Consider the following statements I. None of the three points lie in the III Quadrant II. No two points lie in the same Quadrant Which of the above statements is/are true? (A) Only I (B) Only II (C) Both I and II (D) Neither I nor II Comprehension-II (Questions 04 to 06) [THE RING]
∑ is a circle of radius 3 and center at O. P is a point on circumference of circle. Q is a point on the ray OP , outside the circle such that line segment QB is bisected by point A (A is different from P) with A and B on circumference of ∑. Now, answer the following questions. 04. If |PQ| is of integral length (i.e., length of PQ is integer), then number of possible integral values of |AB| is (A) 1 (B) 3 (C) 5 (D) None of these
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Standard – X(SIMO) X(SIMO) - 2012 05. If |PQ| is of integral length, the maximum possible area of triangle OAB is (A)
8 5 3 2
(B)
9 15 8
(C)
9 5
(D) None of these
4 2
06. Angles of triangles PAB and QOB are equal. However, they need to be similar in respective order. Area of triangle APQ is (A)
9(3 3 ) 8
(B)
3(2 2 1)
(C) Either A or B
5
(D) Neither A nor B
Comprehension- III (Questions 07 and 08) [REM(A)INDER!] f(x) is a polynomial such that f(x)=0 has exactly one root between 0 and 1. Also, f(x) when divided 2 p 1 3 by x -x leaves a remainder of g(x)= x 2 px , p R 4
07. Consider the following statements. I. f(x)=0 has a root between -1 and 0. II. g(x)=0 has a root between 0 and 1 Which of the above statements is/are true? (A) Only I (B) Only II (C) Both I and II 08. Number of values of p for which g(x)≥ (A) 1
(B) 2
1 50
(D) Neither I nor II
for all
x R is
(C) infinitely many
(D) None
Comprehension-IV (Questions 09 and 10) [LADDERS] Consider the function f(x) defined below f(x) =
| x 1 | if | x 3 | 1 | x | if | x 3 | 1
Let A be set of lines such that they pass through (-16,0) and intersects f(x) at three distinct points. Let B be set of all points of intersection of lines of A with f(x). Now, answer the following questions. 09. Maximum possible area of quadrilateral formed by an y four points in set B is (A) 32/5 (B) 36/5 (C) 38/5 (D) None 10. A line l 1 makes an angle θ with X-axis such that it intersects f(x) in exactly two points and cot θ N . Number of possible values of θ such that cot θ≤2012 is (A) 2007 (B) 2004 (C) 2005 (D) None Questions 11 to 13 [GOOD BYE 2012] m+n
n
11. Consider set S = {2 – m.2 / m,n N }. The elements s1, s2, s3.,…. such that s1
of the set S are ordered
12. S is arithmetic progression of positive integers such that sum of first 8 terms, ie., S 8=2012. If a and d are first term and common difference of S, number of possible pairs (a,d) is (A) 34 (B) 35 (C) 36 (D) None of these
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Standard – X(SIMO) X(SIMO) - 2012 13. A and B are 2x2 matrices defined as below.
1 1 3a 5b , B 7c 20 where a,b,c N 0 2 2 0 1 2
A
Number of possible matrices B such that AB=BA is (A) 134 (B) 135 (C) 136 (D) None of these Comprehension-V (Questions 14 and 15) 2n
Let Sn(α)=sin 3
α+cos2n α where α 0, . If S1(α), S2(α), S3(α) are roots of the equation
2
2
x -ax +bx-c=0. Now answer the following questions. 14. Consider the following statements I. b>c II. a+c=1+b Which of the above statements is/are true? (A) Only I (B) Only II (C) Both I and II
(D) Neither I nor II
o
15. If 0< α<45 , which of the below is enough to know the value of α? (A) a is an integer (B) b is an integer (C) c is an integer (D) Any two of the above 16. ABC is an isosceles triangle with then (A)
BG DG 7 2
o
A =120
. If G is centroid of the triangle and AD is a median,
=
(B)
7
(C)
2 7
(D) None of these
17. Consider three series given below.
S : 3,7,11,15,….. P: 2,7,12,17,….. Q: 3x2, 7x7, 11x12, 15x17,……
Let Sn, Pn and Qn represent sum of first n terms of respective series. If value of n is (A) 672
(B) 675
(C) 669
6Qn
n
10 Sn
8P n
2012 ,
(D) None
18. Consider the following statements. I.We can find an irrational number between an y two rational numbers II. We can find a rational number between any two irrational numbers Which of the above statements is/are true? (A) Only I (B) Only II (C) Both I and II (D) Neither I nor II 19. f(x) and g(x) are functions that satisfy below f(x)=2x+3, g(0)=1, fog=gof. Value of g(9) is (A) 13 (B) 14 (C) 15 (D) None 20. x and y are reals that satisfy x+y≤10, y-x≤2, 0≤x≤6 and 0≤y≤3. Maximum value of 2x+5y subject to above constraints is (A) 22 (B) 27 (C) 32 (D) None
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Standard – X(SIMO) X(SIMO) - 2012 2
21. Consider the inequality 15≥8x-x value of | α- β| is (A) 2 (B) 1 (C) 4
≥12. If α, β satisfy above inequality, then maximum possible (D) None 2
2
2
22. Consider the quadratic equation (x+a) +(x+b) =(x+c) where x
R
. The above equation have 2
2
2
two real roots whose difference is 28 . Then minimum value of (x+a) +(x+b) -(x+c) is (A) -3.5 (B) -7 (C) can not be determined (D) None 23. A, B, C are sets such that A B A C ; Then it is necessary that (A) B=C (B) B C (C) Either A or B (D) None 2
24. A and B are 2x2 square matrices such that AB+B =4I. If det(A+B)=2, value of debt is (A) 2 (B) 4 (C) 8 (D) None o
o
o
o
o
o
25. Area formed by points P(cos10 , sin10 ), Q(cos40 , sin40 ), R(cos70 , sin70 ) is (A)
2 4
3
(B)
5
1
4
(C) 4
2 2
7
3
3
(D) None
2
26. Remainder obtained when x +4x +3x +2x-1 is divided by (x+1-√2) is (B) 4√2-5 (C) 5√2-4 (A) -1 (D) None 27. ABC ids a right angled triangle with hypotenuse AC=13cms. Also, AB=5cms. ∑ is a circle with diameter AB. ∑ cuts AC again at D. Also, tangent at D cuts BC at E and AB extended at F. Area of the triangle BEF is approximately (A) 18 (B) 16 (C) 17 (D) None Comprehension VI (Questions 28 to 30) Let f(x) =
x 1 x 1
(x≠1)
28. f(x) : R-{1}R is (A) only onto (B) only one-one
(C) bijection
(D) Neither onto nor one-one
29. The line y=2x+3 cuts f(x) at points p oints P and Q. Area of triangle OPQ is (O is origin) (A) 6√2 (B) 4√2 (C) 3√2 (D) None 30. If f:[3,4]A is a bijection, then A is (A) [0, 5/3] (B) [0,2] (C) [5/3, 2]
(D) None
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