THE ASSOCIATION OF MATHEMATICS

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0 th NMT-08 THE SSOITION OF MTHEMTIS TEHERS OF INDI (Regd.

Standards TEST PPER WITH SOLUTION & NSWER KEY Date: st September,

1 0 th NMT-08 THE SSOITION OF MTHEMTIS TEHERS OF INDI (Regd.) Screening Test Ramanujan ontest NMT at INTER LEVEL XI & XII Standards TEST PPER WITH SOLUTION & NSWER KEY Date: st September, 08 Duration: Hours National Mathematics Talent ompetitions (NMT) 08 Resonance Eduventures Ltd. ORPORTE OFFIE : G Tower, -46 &, IPI, Near ity Mall, Jhalawar Road, Kota (Raj.) Reg. Office : J-, Jawahar Nagar, Main Road, Kota (Raj.)-3400 Ph. No.: FX No. : Ph.No. : , 663 To Know more : sms RESO at 6677 IN : U8030RJ007PL0409

2 National Mathematics Talent ompetitions (NMT) 08 XI & XII 0-Sep.-08 Instructions to andidates. Do not open he Question booklet until you are told to do so.. a) This is a Multiple choice and fill in the blanks test. In multiple choices each Question is followed by four answer (), (), () and (D) Only one of them it correct. When you have decided on your choice, shade the corresponding letter in the bo against the question number. For eample, if you decide that for a question, the correct response is (), shade of the bo marked (Eg.[] on in the bo shading that bo. b) In the case of fill in the blanks questions write the correct answer in the space provided. 3. Each question carries marks. 4. a) If you are unable to solve a problem, it is better not to answer the question. void guessing. Since you are penalized for each wrong answer. b) ½ mark will be deducted for wrong answers in part c) ½ mark will be deducted for wrong answers in part. e certain that you understand thoroughly the coding system for your answer sheet. If you are not sure, ask your supervisor to clarify it. 6. You are permitted to use rough paper. No other aid, like the instrument bo, calculator, etc, are permitted. 7. Diagrams are not drawn to scale. They are intended as aids only. 8. efore commencing to write your answers. Fill in the details of your bio-data (Name, School/address etc.,) in the appropriate places in the response sheet. If you are writing in a centre other than your school write the agency through whom you write the eam. Eg,., MTI for open quota. 9. fter completion, return only the response sheet. The question-booklet and the roughworksheets may be retained by you or as directed by your supervisor. 0. When your supervisor instructs you to begin, you have 0 minutes if working time. NOTE:. Fill in the response sheet with your Name, lass and the institution through which you appear in the specified places.. Diagrams are only visual aids, they are NOT drawn to scale. 3. You are free to do rough work on separate sheets. 4. Duration of the test: pm to 4 pm ( hours).. In the addition shown, each of the letters T,H,,S represents a non zero digit. What is T + H + + S? T H S S 0 8 ns. () 34 () 3 () 4 (D) () orp. / Reg. Office : G Tower, -46 &, IPI, Near ity Mall, Jhalawar Road, Kota (Raj.)-3400 Toll Free : IN: U8030RJ007PL0409 NMT08 -

3 National Mathematics Talent ompetitions (NMT) 08 XI & XII 0-Sep.-08 T H S S 0 8 learly S 4 or S 9 If S 4 can not take any value If S 9 can take value only Hence T H 9 9 T(H+) 8 08 T and H 9 So the number T H S 99 Hence T + H + + S We have four sets S, S, S 3, S 4 each containing a number of parallel lines. The set S i contains i + parallel lines i,,3,4. line in S i is not parallel to lines in S j when i j. In how many points do these lines intersect? () 4 () 63 () 7 (D) 9 ns. () Number of point of intersection ( ) + 3 ( 4 + ) + 4 () + 3(9) n old tanker is 00 km due north of a cruise liner. The tanker sails Southeast at a speed of 0 kilometers per hour and the liner sail Northwest at a speed of 0 kilometres per hour. What is the shortest distance between the two boats during the subsequent motion? ns. () 0 km () 60 km () 80 km (D) 00 km () From relative motion Let initially oil tanker at and d cruise liner at 0 oth move parallely opposite directions Shortest distance O 0 0 km/hr 0 km/hr O orp. / Reg. Office : G Tower, -46 &, IPI, Near ity Mall, Jhalawar Road, Kota (Raj.)-3400 NMT08-3 Toll Free : IN: U8030RJ007PL0409

4 National Mathematics Talent ompetitions (NMT) 08 XI & XII 0-Sep Volume equals one fourth of the sum of the volumes and, while volume equals one sith of the sum of the volumes and. The ratio of volume to the sum of volumes of and is () : 3 () 9 : 0 () 7 : (D) : 3 ns. (D) V 4 ( + V )... () V 6 (V + V )... () To find V V V from () and () 4V 6V V V 7V Put in () 4V 7 V + V 3V 7V Hence V V V 3 V 7 V V In the ninety-nine shop every item costs some whole number of rupees plus 99 paise. Rhea spent sity five rupees and seventy si paise in buying some items from the shop. How many items did she buy? () 3 () 4 () 6 (D) 66 ns. () Let p items cost Rs. 99, Rs. 99 and so on p p 6.76 p The diagram shows a rectangle D where : D :. Point E on is such that DE is perpendicular to. What is the ratio of the area of the triangle DE to the rectangle D? E 90 D ns. () : 4 () : 6 () : 8 (D) : 0 (D) orp. / Reg. Office : G Tower, -46 &, IPI, Near ity Mall, Jhalawar Road, Kota (Raj.)-3400 NMT08-4 Toll Free : IN: U8030RJ007PL0409

5 National Mathematics Talent ompetitions (NMT) 08 XI & XII 0-Sep.-08 E p D ED p 4 ( p) p p rea of DE rea of rec tangle D p p 0 7. The numbers, 3,, 4,, 0, may be divided into two sets so that the product of the numbers in each set is the same. What is this product? () 40 () 60 () 0 (D) 670 ns. () P P 0 8. D is a trapezium with D D D 0 units and units. Semi circles are drawn as shown in the figure. The area of the region bounded by these semi circles in square units is D ns. () () () (D) () S P P P S P Required rea area of trapezium ( + P ) + S + S P area of trapezium + S + S ( + P + P ) area of trapezium + S + S (0 + ) 8 + ( ) orp. / Reg. Office : G Tower, -46 &, IPI, Near ity Mall, Jhalawar Road, Kota (Raj.)-3400 NMT08 - Toll Free : IN: U8030RJ007PL0409

6 National Mathematics Talent ompetitions (NMT) 08 XI & XII 0-Sep onsider the number of ways in which five girls and five boys sit in ten seats that are equally ns. spaced around a circle. The proportion of the seating arrangements in which no two girls sit at the ends of a diameter is () () () 8 63 () 63 (D) None of the above Proportion!! 0 8 (0 )! Let , the sum of reciprocals of fourth powers of integers and ns , the sum of reciprocals of fourth powers of odd positive integers. The value of / as a fraction is () 6 () () () () 6 () (D) 8 7. The number ns. 7 (6 ) is written on the board (in base 0). Gia takes two of the digits at a time, erases them but appends the sum of those digits at the end. She repeats this till she ends up with one digit on the board. What is the digit that remains on the board? () () () 6 (D) 7 () obviously. Seven points are marked on the circumference of a circle and all pairs of points are joined by straight lines. No three of these lines have a common point and any two intersect at a point inside the circle. Into how many regions is the interior of the circle divided by these lines? () 64 () 63 () 7 (D) 6 ns. () Point s Section orp. / Reg. Office : G Tower, -46 &, IPI, Near ity Mall, Jhalawar Road, Kota (Raj.)-3400 NMT08-6 Toll Free : IN: U8030RJ007PL0409

7 National Mathematics Talent ompetitions (NMT) 08 XI & XII 0-Sep The diagram below shows a regular heagon with side length, insceibed in a square. Two of the vertices lie on the diagonal of the square and the remaining vertices lie on its sides. What is the area of the square? () 7 () 4 () + 3 (D) 3 + ns. () cos sin sin Length + cos + sin + sin ns. 4. is a diameter of a semicircle of centre O. is the midpoint of the arc. and the tangent at ns. to the semicircle meet at P. D is the midpoint of P. If DO is a parallelogram and then sin is () () () O O O PD D r OD r 0 () 0 (D) 3 0 PD, D r orp. / Reg. Office : G Tower, -46 &, IPI, Near ity Mall, Jhalawar Road, Kota (Raj.)-3400 NMT08-7 Toll Free : IN: U8030RJ007PL0409

8 National Mathematics Talent ompetitions (NMT) 08 XI & XII 0-Sep.-08 D 3 sin sin3 r r sin 0 r 3 r r r P r D r r O r. The real valued function f() satisfies the equation f( ) + f() for all. Then ( + 4) f() equals () () () + (D) 3 ns. (D) f( ) + f() Replace by f() + ( ) f( ) f() f() + ( ) 4f() + ( ) f() ( ) ( + 4) f() 3 PRT Note : Write the correct answer in the space provided in the response sheet. For each correct response you get mark. For each incorrect response you lose 4 Mark. 6. The number of ways in which 6 identical chocolates be distributed between my, ob, athy ns. (078) and Daniel so that each receives at least one chocolate and my receives more chocolates than ob is. Total number of natural solution of equation is total number of natural solution of equation Total number of natural solution of equation when > is orp. / Reg. Office : G Tower, -46 &, IPI, Near ity Mall, Jhalawar Road, Kota (Raj.)-3400 NMT08-8 Toll Free : IN: U8030RJ007PL0409

9 National Mathematics Talent ompetitions (NMT) 08 XI & XII 0-Sep set S contains numbers. The average of the numbers in S is 30. The average of the si smallest numbers of S is 00 and the average of the si largest of the numbers is 300. What is the median of the numbers in S. ns. (360) onus (wrong question) 8. The sum of the angles,,3,4,,6,7,8 in degrees shows in the following figure is ns. (360) G F 7 H + S 6 R 8 U E V T Q GST + HIV HUV + 4 P O TVJ UVI In quadrilateral UVH sum of angle D N M 3 9. The number of positive integers less than 08 that are divisible by 6 but are not divisible by at least one of the numbers 4 or 9 is. ns. (80) Total number divisible by 6 but less than 08 are 336 Total number divisible by 36 but less than 08 are 6 Required number equals to orp. / Reg. Office : G Tower, -46 &, IPI, Near ity Mall, Jhalawar Road, Kota (Raj.)-3400 NMT08-9 Toll Free : IN: U8030RJ007PL0409

10 0. If ( + )( + )... ( + 3) ns. () of 3 is. In elements of set {0,,,...3} 8 elements are of the form 3 8 elements are of the form 3+ 8 elements are of the form 3 National Mathematics Talent ompetitions (NMT) 08 XI & XII 0-Sep.-08 oefficient of 4 is (not divisible by 3) oefficient of 3 is ( ) oefficient of is not multiple of 3 oefficient of is multiple of 3 oefficient of 8 is not multiple of 3 oefficient of 7 is multiple of 3 4 n an the number of coefficients a n that are multiples n 4 3 multiple of 3 Now all coefficient of power of less than 7 are multiple of 3 oefficient of 4, oefficient of,... oefficient of 8 are not multiple of 3 Number of coefficient which are multiple of 3 are 4 9. square is cut into 37 squares of which 36 have area square cms. The length of the side of the original square is. ns. (0) Let a square is of side length a is taken which cuts in 36 square of side length and one of side length b. a 36() + b a b 36 (a b)(a + b) 36 clearly a + b and a b both even a b 8 a 0 and b 8 a b Hence length of original square 0.. There are 4 coins in a row and all are showing heads to start with. The coins can be flipped with the following rules : (a) The fourth coin (from the left) can be flipped any time (b) n intermediate coin can be changed to tail only if its immediate neighbor on the right is heads and all other coins (if any) to its right are tails. (c) Only one coin can be flipped in one step. The minimum number of steps required to bring all coins to show tails is. ns. (8) HHHH, HHTH, HHTT, THTT, THHT, TTHT, TTHH, TTTH, TTTT orp. / Reg. Office : G Tower, -46 &, IPI, Near ity Mall, Jhalawar Road, Kota (Raj.)-3400 NMT08-0 Toll Free : IN: U8030RJ007PL0409

11 National Mathematics Talent ompetitions (NMT) 08 XI & XII 0-Sep poet met a tortoise sitting under a tree. When the tortoise was the poet's age, the poet was only a quarter of his current age. When the tree was the tortoise's age, the tortoise was only a seventh of its current age. If all the ages are in whole number of years, and the sum of their ages is now 64, the age of the tree in years is, ns. (43) p poet, t tortoise, T tree p, t, T p (t p) p 4 and t (T t) t 7 p 4t 7 T + t + p 64 and t 7T 3 T + 7T 3 + 4T 7 64 T The sum of all real value of satisfying is. ns. (3) 7 7 Let + 7 t t t + 34 t t 34 0 t 36 t 37 or or or 3 37 ( ) + 0 and (3 37 ) ns. 3 orp. / Reg. Office : G Tower, -46 &, IPI, Near ity Mall, Jhalawar Road, Kota (Raj.)-3400 NMT08 - Toll Free : IN: U8030RJ007PL0409

12 National Mathematics Talent ompetitions (NMT) 08 XI & XII 0-Sep.-08. On the inside of a square with side length 6, construct four congruent isosceles triangles each with base 6 and height, and each having one side coinciding with a different side of the square. The area of the octagonal region common to the interiors of all four triangles is. ns. (3.6) D F G 4 E L H J 9 N V M 6 tan PM M 3 GN N 3 3 N 3 N 9 tan y 3 y 9 y y + y 3y Now FL y 6 y rea area of square + 4(area of EFL) 4y + (y)() In a triangle with integer side lengths, one side is thrice the other. The third side is cm. The ns. (43) greatest possible perimeter of the triangle is (in cm). 3 3 < orp. / Reg. Office : G Tower, -46 &, IPI, Near ity Mall, Jhalawar Road, Kota (Raj.)-3400 NMT08 - Toll Free : IN: U8030RJ007PL0409

13 National Mathematics Talent ompetitions (NMT) 08 XI & XII 0-Sep.-08 and 3 + > < and > 4 greatest value of can be 7 perimeter cube has edge length (an integer). three faces meeting at a corner are painted blue. The cube is then cut into smaller cubes of unit length. If eactly 343 of these cubes have no faces painted blue, then the value of is. ns. (8) ( ) If f() a 4 b + + and f(3) 8, the value of f( 3) is. ns. () f(3) f( 3) 6 f( 3) 9. rchana has to choose a three-digit code for her bike lock. The digits can be chosen from to 9. ns. (84) To help her remember them, she decides to choose three different digits in increasing order, for eample 78. The number of such codes she can choose is. Number of ways ! 30. Let S be a set of five different positive integers, the largest of which is n. It is impossible to construct a quadrilateral with non-zero area, whose side-lengths are all distinct elements of S. The smallest possible value of n is. ns. () S {,,3,6,} In quadrilateral with side a,b,c,d we have a + b + c > d etc which is not possible in set S. orp. / Reg. Office : G Tower, -46 &, IPI, Near ity Mall, Jhalawar Road, Kota (Raj.)-3400 NMT08-3 Toll Free : IN: U8030RJ007PL0409

CAREER POINT THE ASSOCIATION OF MATHEMATICS TEACHERS OF INDIA

CAREER POINT THE ASSOCIATION OF MATHEMATICS TEACHERS OF INDIA REER POINT THE SSOITION OF MTHEMTIS TEHERS OF INI Screening Test Gauss ontest NMT at PRIMRY LEVEL V & VI Standards Saturday, 6 th ugust, 017 P Note : 1. Fill in the response sheet with your Name, lass