KVS Junior Mathematics Olympiad (JMO) SAMPLE PAPER 1

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Claude McCoy
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1 M.M. 00 Note : Attempt all questions. All questions carry equal marks KVS Junior Mathematics Olympiad (JMO) SAMPLE PAPER Q. If a x b x c x 0 Prove that (a + b + c + 3x) (a + b + c - x) (bc + ca + ab) xy Q. If a x y, b xz yz, and c x z y z a, b and c are other than zero. Find the value of x, y, z in terms of a,b and c. Time : 3 hours Q3. Two clocks showed correct time at noon. After that one started gaining 0 seconds and other started loosing 50 seconds in every hours. After what interval the difference of time shown by the two clock was 6 minutes? What was then the correct time? Q. A triangle has sides of lengths 6, 8 and 0. Find the distance between the center of its inscribed circle and the center of the circumscribed circle. Q5. A pair of poles are s meters apart and is supported by two cables which run from top of each pole to the bottom of other. The poles m and 6m tall. Determine the height of the point T. What happens to this height if s increases. Q6. In the figure square ABCD is having unit area. Find the value of a such that area of wxyz /00 A B C D Prepared by: M. S. KumarSwamy, TGT(Maths) Page - -

2 Q7. Solve for n : 00 /n x 00 /n x 00 3/n x..x /n 000 Q8. A right triangle has base and altitude of b and a. A circle of radius r touches the two sides and has its center on the hypotenuse. Show that a b r Q9. One goes on spiraling walk on cartesian plane. Starting at (0,0). The first five steps are (,0) (,) (0,), (-, ) and (-,0). Find the point on 00 nd step. Q0. Find the ratio of sum of squares of the medians of a triangles to sum of the squares of its sides. Prepared by: M. S. KumarSwamy, TGT(Maths) Page - -

3 Q. a x b x c x 0 a x SOLUTION AND HINTS (SAMPLE PAPER ) b x c x ( a b) (b x) a x b x c x a + b c x - a x b x a + b + c + ab bc ca x (a+b-c) + x {ab x(a+b) + x } (a + b + c) + x (a + b + c ) 3x (ab + bc + ca) (a + b + c + 3x) (a + b + c - x) (bc + ca + ab) Q. xy a x y a x y (i) xz b x z b x z (ii) yz c y z c y z (iii) Now (i) (iii) x z a c and (ii) + (iv) gives x a b c bc ca ab x abc abc x ac bc ab Q3. Two clocks show a difference of 90 secs in hrs. 3 minutes in hrs. two clocks shows a difference of 6 minutes in /3 x x 6 56 hrs. 56 hrs 0 days 6 hrs so the actual time at the moment o clock in the morning The correct time was at noon. Prepared by: M. S. KumarSwamy, TGT(Maths) Page - 3 -

4 A Q. B C Let AB 8 BC 0 AC 6 As AB + AC BC BAC 90 o So area (ABC).AB.AC Let r and R is radii of inscribed and circumscribed circles respectively. d is distance between incentre and circumcentre. Then, d R Rr [Formulae] r / abc R R 5 So, d 5 x 5 x 5 d 5 Q5. h Prepared by: M. S. KumarSwamy, TGT(Maths) Page - -

5 Since, QPR and TOR are similar, a h d d c dh d-c (i) a SRP and TOP are similar, b h d c dh c (ii) b so from (i) and (ii) we get, d dh a b ab h a b Here a, b6 h 5 Height of T meters. 5 Height h is not dependent on d. Q6. Let s construct a coordinate plane A and (,0) B (,) C (0,) and D at (0,0) Then the slope of DR x and slope of AS x Thus DR and AS is and others. Hence PQRS is a rectangle. By symmetry PQRS is a square PQ x sin (<CBP) Now CBP 90 o - PCB. Let CP meet AB at T. Then 90 o - PCB BTC CBP Prepared by: M. S. KumarSwamy, TGT(Maths) Page - 5 -

6 Sin CBP sin BTC then PQ x x x x x x + x Q7. Since, 00 /n x 00 /n x..x /n ( ) n n n x x Q8. B r O r C T In ABC, let T be the point where the circle touches the side AC. Since ABC and AOT are similar, a r ab ar rb ab r (b a) b b r r a b Q9. Prepared by: M. S. KumarSwamy, TGT(Maths) Page - 6 -

7 For the figure, 7 th step is (0,-) th step is (0,-) n th point is (0, - n) at slep no. (n+) (n+) n + 3n Let us see it n + 3n 00 (n-) (n+9) 0 so nd point on y-axis i.e. (0, -) is the 00 nd step. Q0. Find the ratio of sum of squares of the medians of a triangle to sum of the squares of its sides. We know b c a m a ma b c a mb a c b mc a b c b c a a c ma +mb + mc 3 ma +mb + mc (a b c ) ma mb mc a b c 3 b a b c Prepared by: M. S. KumarSwamy, TGT(Maths) Page - 7 -

27 th ARYABHATTA INTER-SCHOOL MATHEMATICS COMPETITION 2010 CLASS = VIII

27 th ARYABHATTA INTER-SCHOOL MATHEMATICS COMPETITION 2010 CLASS = VIII 7 th ARYABHATTA INTER-SCHOOL MATHEMATICS COMPETITION 00 CLASS = VIII Time Allowed: Hours Max. Marks: 00 Roll No. of the Participant: GENERAL INSTRUCTIONS :. Participant should not write his/her name on