(d) none of these. 9. When the curves y log 10 x and y x. x y plane, how many times do they intersect for values

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1 at Solved Paper Leaked Paper No. of Questions : 50 Time : 40 min Each wrong answer carry rd negative mark. irections for Question no. to : nswer the questions independently of each other. NOTE. In the figure given elow, EF is a regular heagon and OF 90. FO is parallel to E. What is the ratio of the area of the triangle OF to that of the heagon EF? (a) () 6 (c) 4 (d) 8. In a 4000 meter race around a circular stadium having a circumference of 000 meters, the fastest runner and the slowest runner reach the same point at the end of the 5th minute, for the first time after the start of the race. ll the runners have the same starting point and each runner maintains a uniform speed throughout the race. If the fastest runner runs at twice the speed of the slowest runner, What is the time taken y the fastest runner to finish the race? (a) 0 min () 5 min (c) 0 min (d) 5 min. Given that v, u 0.5 and z 0.5 and w vz, then which of the following is necessarily true? u (a) 0.5 w () 4 w 4 (c) 4 w (d) w 0.5 irections for Question no. 4 and 5 : nswer the questions on the asis of the information given elow : New age consultants have three consultants Gyani, Medha and uddhi. The sum of the numer of projects handled y Gyani and uddhi individually is equal to the numer of projects in which Medha is involved. ll three consultants are involved together in 6 projects. Gyani works with Medha in 4 projects. uddhi has projects with Medha ut without Gyani and projects with Gyani ut without Medha. The total numer of projects for New age consultants is one less than twice the numer of projects in which more than one consultant is involved. 4. What is the numer of projects in which Gyani alone is involved? (a) Uniquely equal to zero () Uniquely equal to (c) Uniquely equal to 4 (d) annot e determined uniquely 5. What is the numer of projects in which Medha alone is involved? (a) Uniquely equal to zero () Uniquely equal to F E O (c) Uniquely equal to 4 (d) annot e determined uniquely irections for Question no. 6 to 8 : nswer the questions on the asis of the information given elow : city has two perfectly circular and concentric ring roads, the outer ring road (OR) eing twice as long as the inner ring road (IR). There are also four (straight line) chord roads from E, the east end point of OR to N, the north end point of IR, from N, the north end points of OR to W, the west end point of IR; fromw, the west end point of OR, to S, the south end point of IR and from S, the south end point of OR to E, the east end point of IR. Traffic moves at a constant speed of 0 km/h. on the OR road, 0 km/hr on the IR road and 5 5 km/h on all the chord roads. 6. The ratio of the sum of the lengths of all chord roads to the length of the outer ring road is : (a) 5 : () 5 : (c) 5 : (d) none of these 7. mit wants to reach N from S. It would take him 90 minutes if he goes on minor arc S E on OR, and then on the chord road E N. What is the radius of the outer ring road in kms? (a) 60 () 40 (c) 0 (d) 0 8. mit wants to reach E from N using first the chord N W and then the inner ring road. What will e his travel time in minutes on the asis of information given in the aove question? (a) 60 () 45 (c) 90 (d) 05 irections for Question no. 9 to 8 : nswer the questions independently of each other. 9. When the curves y log 0 and y are drawn in the y plane, how many times do they intersect for values? (a) Never () Once (c) Twice (d) More than twice 0. The sum of rd and 5th elements of an arithmetic progression is equal to the sum of 6th, th and th elements of the same progression. Then which element of the series should necessarily e equal to zero? (a) st () 9th (c) th (d) none of these. test has 50 questions. student scores mark for a correct answer, for a wrong answer, and for not attempting a 6 question. If the net score of a student is, the numer of

2 Solved Paper 00 (Leaked Paper) questions answered wrongly y that student cannot e less than : (a) 6 () (c) (d) 9. How many even integers n, where 00 n 00, are divisile neither y seven nor y nine? (a) 40 () 7 (c) 9 (d) 8. Which one of the following conditions must p, q and r satisfy so that the following system of linear simultaneous equations has at least one solution, such that p q r 0? y z p 6y z q y 7z r (a) 5p q r 0 () 5p q r 0 (c) 5p q r 0 (d) 5p q r 0 4. The function f( ).5.6, where is a real numer, attains a minimum at : (a). ().5 (c).7 (d) none of these 5. Let and e two solid spheres such that the surface area of is 00% higher than the surface area of. The volume of is found to e k% lower than the volume of. The value of k must e : (a) 85.5 () 9.5 (c) 90.5 (d) Twenty-seven persons attend a party. Which one of the following statements can never e true? (a) There is a person in the party who is acquainted with all the twenty si others. () Each person in the party has a different numer of acquaintances. (c) There is a person in the party who has an odd numer of acquaintances. (d) In the party, there is no set of three mutual acquaintances. 7. Let g( ) ma( 5, ). The smallest possile value of g( )is : (a) 4.0 () 4.5 (c).5 (d) none of these 8. positive whole numer M is less than 00 is represented in ase notation, ase notation and ase 5 notation. It is found that in all three cases the last digit is, while in eactly two out of the three cases the leading digit is. Then M equals : (a) () 6 (c) 75 (d) 9 irections for Questions no. 9 and 0 : nswer the questions on the asis of the information given elow : certain perfume is availale at a duty free shop at the angkok international airport. It is priced in the Thai currency aht ut other currencies are also acceptale. In particular, the shop accepts Euro and US dollar at the following rates of echange : US ollar 4ahts Euro 46 ahts The perfume is priced at 50 ahts per ottle. fter one ottle is purchased, susequent ottles are availale at a discount of 0%. Three friends S, R and M together purchase three ottles of the perfume, agreeing to share the cost equally. R pays Euros, M pays 4 Euros and 7 Thai ahts and S pays the remaining amount in US ollars. 9. How much does R owe to S in Thai aht? (a) 48 () 46 (c) 4 (d) 4 0. How much does M owe to S in US ollars? (a) () 4 (c) 5 (d) 6 irections for Questions to 8 : nswer the questions independently of each other.. leather factory produces two kinds of ags, standard and delue. The profit margin is Rs. 0 on a standard ag and Rs. 0 on a delue ag. Every ag must e processed on machine and on machine. The processing times per ag on the two machines are as follows : Time Required (Hours/ag) Machine Machine Standard ag 4 6 elue ag 5 0 The total time availale on machine is 700 hours and on machine is 50 hours. mong the following production plans, which one meets the machine availaility constraints and maimizes the profit? (a) Standard 75 ags, delue 80 ags () Standard 00 ags, delue 60 ags (c) Standard 50 ags, delue 00 ags (d) Standard 60 ags, delue 90 ags. t the end of year 998, shepard ought nine dozen goats. Hence forth, every year he added p% of the goats at the eginning of the year and sold q% of the goats at the end of the year where p 0 and q 0. If shepard had nine dozen goats at the end of year 00, after making the sales for the year, which of the following is true? (a) p q () p q q (c) p q (d) p. Each side of a given polygon is parallel to either the X or the Y ais. corner of such a polygon is said to e conve if the internal angle is 90 or concave if the internal angle is 70. If the numer of conve corners in such a polygon is 5, the numer of concave corners must e : (a) 0 () 0 (c) (d) 4. Let p and q e the roots of the quadratic equation ( ) 0. What is the minimum possile value of p q? (a) 0 () (c) 4 (d) 5 5. The 88th term of the series a,,, c, c, c, d, d, d, d, e, e, e, e, e, f, f, f, f, f, f is : (a) u () v

3 Solved Paper 00 (Leaked Paper) (c) w (d) 6. There are two concentric circles such that the area of the outer circle is four times the area of the inner circle. Let, and e three distinct points on the perimeter of ths outer circle such that and are tangents to the inner circle. If the area of the outer circle is square centimeters then the area (in square centimeters) of the triangle would e : (a) (c) 9 () 9 (d) 6 7. Let a,, c, d e four integers such that a c d 4m, where m is a positive integer. Given m, which one of the following is necessarily true? (a) The minimum possile value of a c d is 4m m () The minimum possile value of a c d is 4m m (c) The maimum possile value of a c d is 4m m (d) The maimum possile value of a c d is 4m m 8. The numer of non-negative real roots of 0 equals : (a) 0 () (c) (d) 9. Three horses are grazing within a semi-circular field. In the diagram given elow, is the diameter of the semi-circular field with centre at O. Horses are tied up at P, R and S such that PO and RO are the radii of semi circles S with centres at P and R respectively, and S is the centre of the circle touching the two semi P O R circles with diameters O and O. The horses tied at P and R can graze within the respective semi circles and the horses tied at S can graze within the circle centred at S. The percentage of the area of the semi circle with diameter that cannot e grazed y the horses is nearest to : (a) 0 () 8 (c) 6 (d) In a triangle, 6, 8 and 0. perpendicular dropped from, meets the side at. circle of radius (with centres ) is drawn. If the circle cuts and at P and Q respectively, then P : Q is equal to : (a) : () : P (c) 4 : (d) : 8. In the diagram given elow, PQ 90. If : :, the ratio of : PQ is : (a) : 0.69 () : 0.75 (c) : 0.7 (d) none of these. If 7 log, log ( 5), log are in arithmetic progression, then the value of is equal to : (a) 5 () 4 (c) (d). In the figure given elow, is the chord of a circle with centre O. is etended to such that O. The straight line O is produced to meet the circle at. If y degrees and O degrees such that ky, then the value of k is : (a) () (c) (d) none of these 4. How many three digit positive integers, with digits, y and z in the hundred s ten s, and unit s place respectively, eist such that y, z y and 0 : (a) 45 () 85 (c) 40 (d) 0 5. There are 846 steel alls, each with a radius of centimeter, stacked in a pile, with all on top, alls in the second layer, 6 in the third layer, 0 in the fourth, and so on. The numer of horizontal layers in the pile is : (a) 4 () 8 (c) 6 (d) 6. If the product of n positive real numers is unity, then their sum is necessarily : (a) a multiple of n () equal to n n (c) never less than n (d) a positive integer 7. In the figure elow, the rectangle at the corner measures 0 cm 0 cm. The corner of the rectangle is also a point on the circumference of the circle. What is the radius of the circle in cm? (a) 0 cm () 40 cm (c) 50 cm (d) none of the aove 8. vertical tower OP stands at the centre O of a square. Let h and denote the length OP and respectively. Suppose P 60 the relationship etween h and can e epressed as : (a) h () h (c) h (d) h O Q

4 4 Solved Paper 00 (Leaked Paper) irections for Questions no. 9 to 4: Each question is followed y two statements, and. nswer each question using the following instructions. hoose. If the question can e answered y one of the statements alone ut not y the other. hoose. If the question can e answered y using either statement alone. hoose. If the question can e answered y using oth the statements ut cannot e answered y using either statement alone. hoose 4. If the question cannot e answered even y using oth the statements together Is a, given a and is an integer?. is even. is greater than What are unique values of and c in the equation 4 c 0 if one of the roots of the equation is?. The second root is. The ratio of c and is. 4. is a chord of a circle. 5 cm. tangent parallel to touches the minor arc at E. What is the radius of the circle?. is not a diameter of the circle.. The distance etween and the tangent at E is 5 cm 4. If a a a a a a?. a. One of the roots of the equation is a. 4., E, F are the mid points of the sides, and of triangle respectively. What is the area of EF in square centimeters?. cm, F cm and perimeter of EF cm. Perimeter of 6 cm, cm and cm irections for Question no. 44 to 50 : nswer the questions independently of each other. nswers (a) 5 () 7 (c) (d) If, y, z are distinct positive real numers then ( y z) y ( z) z ( y) would e : yz (a) greater than 4 () greater than 5 (c) greater than 6 (d) none of these 46. In a certain eamination paper, there are n questions. For J,, n, there are n j students who answered jor more questions wrongly. If the total numer of wrong answers is 4095, then the value of n is : (a) () (c) 0 (d) onsider the following two curves in the y plane : y 5 and y 5 Which of the following statements is true for? (a) The two curves intersect once () The two curves intersect twice (c) The two curves do not intersect (d) The two curves intersect thrice 48. Let T e the set of integers {,, 9, 7, 45, 459, 467} and S e a suset oft such that the sum of no two elements of S is 470. The maimum possile numer of elements in S is : (a) () 8 (c) 9 (d) graph may e defined as a set of points connected y lines called edges. Every edge connects a pair of points. Thus, a triangle is a graph with edges and points. The degree of a point is the numer of edges connected to it. For eample, a triangle is a graph with three points of degree each. onsider a graph with points. It is possile to reach any point from any other point through a sequence of edges. The numer of edges, e in the graph must satisfy the condition : (a) e 66 () 0 e 66 (c) e 65 (d) 0 e 50. There are 6 oes numered,, 6. Each o is to e filled up either with a red or a green all in such a way that at least o contains a green all and the oes containing green alls are consecutively numered. The total numer of ways in which this can e done is : (a). (c). () 4. (d) 5. () 6. (c) 7. (c) 8. (d) 9. () 0. (c). (c). (c). (a) 4. () 5. (d) 6. () 7. (d) 8. (d) 9. (d) 0. (c). (a). (c). (c) 4. (d) 5. (d) 6. (c) 7. () 8. (c) 9. (c) 0. (d). (). (d). (a) 4. (c) 5. (c) 6. (c) 7. (c) 8. () 9. (a) 40. () 4. (a) 4. (a) 4. () 44. () 45. (c) 46. (a) 47. (d) 48. (d) 49. (a) 50. () 44. The numer of positive integers n in the range n 40 such that the product ( n )( n ) is not divisile y n is : (a) 5 () (c) (d) 60. rea of PF rea of heagon E 6 Since the whole heagon is made up of 6 congruent equilateral triangles.

5 Solved Paper 00 (Leaked Paper) 5 Hints & Solutions If area of the heagon e S, then the area of an equilateral triangle ( PF ) is S 6. Now since the area of OF is half of the area of PF. Hence 4 w 4 Solutions for question no. 4 and 5 : rea of OF PF ( ) Hence, area of OF area of heagon EF S S 6 lternatively : In the given figure there are triangles congruent to OF i. e., the whole heagon is the comination of triangles eactly similar to OF. Hence, area of OF area of heagon. Let F and S denote the faster and slower runner. Since the ratio of speeds of slower is to faster runner is :, hence when S completes one round, F completes round of the circular track. Thus in 5 minutes (when they meet for the first time) S has completed one round (of 000 meter) and F has completed two rounds and hence traversed 000 m. Thus F needs 5 0 minutes to traverse m.. v, u 0.5 and z 0.5 w vz u For the maimum value of w : Since u is always negative, so ( v. z) must e negative ( ve ve ve) in order to get the positive value of w. lso, to get the maimum positive value of w, numerical value of numerator (i. e., v. z) must e greatest and numerical value of denominator (i. e., u) must e least. Thus u 0.5 and v and z w vz 4 u 0.5 For the minimum value of w : Since u is always negative, so ( v. z) must e positive ( ve ve ve) in order to get the negative value of w. lso, to get the minimum value of w, numerical value of numerator (i.e., v. z) must e greatest and numerical value of denominator (i.e.,u) must e least. Thus u 0.5 and v and z w vz 4 u 0.5 Let a c, c y z, k Given a c ( k z ) k 6, k 4 8 and y and z lso, ( ) ( ) ( 9) ( 9) 8 a c 8 () lso, a c k z 6 a c 6 8 [from ()] 4. We have a c 6 7. Further there is no any equation with which a c can e solved to get the values of a and c individually. Hence we cannot determine the no. of projects a in which Gyani alone is involved. 5., therefore Medha alone is involved in only one project. Solutions for question no. 6 to 8 : Let the radius of inner circle (IR) e rand the radius of the outer circle (OR) e R and O e the centre of concentric circles. W G N W S E Then, R ( r ) R r S Length of N E ( N O ) ( OE ) r ( r) r 5 Hence length of each chord a N EN NW W S SE r 5 Length of all chord roads 6. Length of the outer ring road ( r) y k z E M N O 5 E

6 6 Solved Paper 00 (Leaked Paper) [ Total time 4 ( r)] r r r 0 5 r 5 R r 0 km ( r r 5 8. Total required time ) hrs 05 min 4 4 ( r 5 km) 9. To get the point(s) of intersection of two curves, we equate the two equation as log Hence there is only one possile value of ;( ) Therefore the two graphs intersect each other only once. lternatively : Plotting the rough sketch of two graphs we get the following diagram : s we increases the value of the graph of log 0 diverges ' from -ais and the graph of ( i. e., ) getting closed to -ais. The two graphs are intersecting some where etween and. 0. T T5 T6 T T a 6d a 7d a d T n 0 a ( n ) d 0 d ( n ) d 0 n y y' / log 0 Hence, T i. e., th term will necessarily e zero.. Let the numer of correct answers e a, numer of wrong answers e and numer of questions not attempted e c. Thus a c 50 () c a 6 The second equation can e written as, dding the () and (), we get () 6a c 9 () 7a 4 ( 4 ) a 7 Since a and oth are integers. Thus at,, a is not an integer hence at, we otain a as an integer. lternatively : Go through options and try the least valued option first, then higher valued option since it is eing asked to find the least possile no. of wrong answers. orrect Wrong Not attempted No. of questions 5 Marks scored 5 Net Score Since the net score of is possile when there are wrong questions, hence the choice (c) is correct.. No. of even integers among 00, 0, 00 5 No. of even integers divisile y 7 7 (These numers are :, 6, 40, 96) No. of even integers divisile y 9 6 (These numers are : 08, 6, 44 98) No. of even integers divisile y oth 7 and 9 (The numer is 6) No. of even integers which are divisile y either 7 or 9 ( 7 6 ) Hence, the required no. of even integers 5 9. Given, p y z q 6y z r y 7z 5p 5 0y 5z () q 4 y z () r y 7 z () Now we can see that option (a) is correct. i.e., 5p q r ( 5 0y 5z) 4. t, f( ) at.5, at.6, ( 4 y z) ( y 7z) 0 f( ) f( ) Thus at.5, f( )will e minimum. NOTE If you draw the graphs of,.5 and.6 you will find that at and.6, values of all the three functions certainly increases. 5. Let the radii of spheres and e r and r. Since the surface area of is 00% more than that of the surface area of is 400% (i. e., 4 times) of area of. Hence, r : r : 4 r : r : 4

7 Solved Paper 00 (Leaked Paper) 7 r : r : r : r : 8 r : r : 8 Volume of : Volume of : 8 Volume ( ) Volume( ) Volume( ) % Hence volume of sphere is 87.5% less than the volume of sphere. 6. Statement (a) can e true, for eample it may e no one ut the host or some other person e acquainted with all the rest 6 guests. Statement (c) can e true, for eample there is a very new friend of the host who knows only to the host (an odd numer of acquaintance). Statement (d) can also e true that there is no set of three mutual acquantances. Hence only choice () is the most appropriate answer. 7. Please note that we are required to find out the minimum value of the g( ), ut g( )always prefer to give the maimum of the two values ( i. e., 5, ). Thus there is only one possile condition to get the minimum of g( )that is when oth the values i. e., 5 and are equal. 5 Thus at, we get minimum value of g( ) (.5,.5).5 lternatively : From the graph it is clear that at P 7,, 5 4 O y (5 ) P (/, 7/) ( + ) 4 5 i. e., at, y 7 is minimum. Please note that as per the given function ma ( 5, ) we will consider always greater values out of the two ( 5, ) for every. 8. Since in the ase notation, ase notation and ase 5 notation, last digit (i. e., unit digit) is, hence the required nured numer must leave the remainder in each case when divided y either, or 5. Thus there are only choices left to check choice (a) and choice (d) ( 9) ( 00) 0 ( 9) ( 00) 0 ( 9) ( ) 0 5 Hence in each of the three notations last digit is one and out of cases, there are eactly two cases in which leading digit (i.e., MS) is. Hence choice (d) is correct. Solutions for question no. 9 and 0 : ost of all the three ottles Share of each person 46 mount spent y R Euros 9 ahts mount spent y M 4 Euros 7 ahts ahts Paid amount 9 ahts ahts mount Payale to S 4 ahts 05 ahts ut 05 ahts 5 US ollars. 9. R owes 4 ahts to S. 0. M owes 5 US ollars to S.. Go through options. learly choice (c) and (d) are wrong since to process the ags on machine there is no sufficient time availale. Now consider choice (a) Total profit For choice () Total profit Hence, choice (a) is correct since it gives maimum profit under the given conditions.. p q For the clarification of the concept once again study the chapter : Percentages. It is a very fundamental concept of percentage. + p% +5% e.g., 00 (00 + p) or 00 5 q% In this case p q (always). No. of concave corners ( n 4) 5 4 R 0% M

8 8 Solved Paper 00 (Leaked Paper) where n is the numer of conve corners. 4. p q ( ) and pq ( ) ( p q) p q pq ( ) p q [ ( )] p q 6 ( ) 5 p q ( ) 5 Hence the minimum value of p 5. Sum of first n natural numers q is 5. n( n ) The sum of first natural numers and the sum of first 4 natural numers 00 Hence the 88th term of the sequence will e 4th letter of the English alphaet i. e.,. HINT If we epress the sequence in the susets S, S, S S n then the last element ofnth suset is given y n( n ) i. e.,sum of first n natural numers. S { a } S {, } S { c, c, c } S4 { d, d, d, d } S5 { e, e, e, e, e } S6 { f, f, f, f, f, f } etc. i.e., S, S, S S n { a } {, }{ c, c, c } { d, d, d, d } { e, e, e, e, e } { f, f, f, f, f, f } etc. 6. M r Since the area of outer circle is 4 times than the area of inner circle. Therefore the radius of the outer circle (i. e., circumradius) will e two times the inner circle (i. e., inradius) Since. r lso MO sin r 0 M ( MO ) 60 Hence, the given triangle is an equilateral triangle. lternatively : When the ratio etween inradius and circumradius is : then the triangle lies etween incircle and circumcircle is an equilateral triangle. Now, the radius of circumcircle R ( r) O r area Height of the triangle Each side of the triangle 6 rea of the triangle Given, a c d 4m a c d ( a ac ad c d cd) ( 4m ) Since we know that if a c d k ( constant) then the product of any two or more of them (i. e., a,, c, d) will e maimum. Hence, when ( a ac ad c d cd) is maimum then a c d is minimum. lso, in order to a ac ad c d cd e maimum a c d. Now, a c d 4m 4a 4 4c 4d 4m a c d m m ( a ac ad c d cd) ( 6a ) ( 6 ) ( 6c ) ( 6d ) ( m 0.5) m 6m 0.75 Hence, for the minimum value of a c d, a c d ( a ac ad c d cd) ma ( a c d ) m 6m m 8m a c d 4m m 0.5 ( 4m ) Since a,, c, d are integers, hence m is also an integer. Therefore a c d is an integer and hence 4m m 0.5 must e integer. Thus the minimum value of a c d 4m m. lternatively : onsider some positive value of m, then check the correct option using the logic applied in the aove solution. 8. t 0, the given equation satisfies. Hence there are only two roots possile. See the following graph : y for 0

9 Solved Paper 00 (Leaked Paper) 9 P O R 0 9. Let the radius of largest sphere e O O r Radii of the semicircles with centres P and R P PO r R OR R Let r e the radius of circle with centre S, then S S r gain OS O S r r r and SR S R r ( SR ) ( OS) ( OR ) r r r r r r r ( ) Total area of semicircle with diameter r Total area of two semicircles with diameters O and O r and the area of circle with centre S r r Total area that can e grazed y horses r r 4 9 r Ungrazed area 6 r 6 r 5 r P 6 Percentage area that cannot e grazed 6 r r 00 6% 0. P Q r (radius) P P cm and Q Q P Q cm 8. P ~ P ( P P and P) P P gain ~ PQ ( is common and PQ ) P P S Q Q Q Q 4 PQ : PQ 4 : : Go through options. Rememer, If log a, log y, logc z are in P then, y, z are in GP Hence choice (d) is correct. 7 log, log ( 5), log log, log, log ( 4.5) Since,, 4.5 are in GP Threfore the required value of. ( Q Q). Let 0 (considered aritrarily) O O 0 ( O O ) O 40 O ( O O) O 00 O 80 ( O O ) 60 O ( ) y k lternatively : O O O y O y O ( O O) and (O is the eternal angle) O 80 ( y y) 80 4y O 80 ( O O ) 80 ( 80 4y y) y k 4. 0least possile value of ut z can e equal to zero. least possile value of y ; [ y (, z)] y z y z No. of numers 0,, 0,, 6 4,, 0,,, 4 5,,, 4, 0,,,, ,,, 4, 5 0,,,, 4, ,,, 4, 5, 6 0,,,, 4, 5, ,,, 4, 5, 6, 7 0,,,, 4, 5, 6, ,,, 4, 5, 6, 7, 8 0,,,, 4, 5, 6, 7, Total No. of alls in first, second, third, fourth layers etc. is,, 6, 0, 5,, 8, 6, 45, 55 etc. respectively. Total no. of alls in top, top, top, top 4 layers etc. is, 4, 0, 0, 5, 56, 84, etc. respectively.

10 0 Solved Paper 00 (Leaked Paper) Thus if n is the total numer of alls in top 6, top 6, top 6, top 6, top 46 layers etc, then the unit digit of n is 6. lso if n is the total numer of alls in top, top, top, top, top 4 layers etc, then the unit digit of n is 6. ut as per the given choices only 6 is the suitale value of n. Hence choice (c) is correct. lternatively : No. of alls in first, second, third nth layer is,, 6, n( n ). Sum of all the alls in top first, top second, top n layers is n( n 6 ) n( n ) [ ( n n)] [ n n] n( n )( n ) n( n ) 6 n( n ) n n( n )( n ) 6 n( n )( n ) n 6 6. Let n such that 5 hence choices (a) and (d) are eliminated. gain if n, such that Thus choice () is also eliminated. Hence the correct choice is (c). M 5 N O 6 7. raw the perpendiculars OM and N as shown in figure and join the points and O, where O is the centre of circle. ( O) ( N ) ( ON ) ( O) ( N ) ( MO MN ) r ( r 0) ( r 0) ( M N OM O r) r 50 lternatively : Go through options P 8. O O ( O is a right angled) lso P 60 and P is an equilateral triangle. ( P P) P P Now, OP P O ( PO is right angled) h h h 9. can e either of, 4, 6, 8, etc. So we cannot determine the relation y the first statement itself. Now, since 6; I 7, 8, 9, etc. Thus we can determine the unique relation y the second statement itself. s. a a and and 7 7 or 8 or 9 etc. 44 hence 6 7 or 8 or 9 etc. ( least possile value) Thus the question can e answered y second statement alone ut not y the first statement. Hence choice (a) is correct. 40. Let and c a and a 4 c 4 and a c 0 [from statement ()] Hence, statement () alone is sufficient to determine the unique values of and c. Now, c P 4 0 E O M [ c ; statement ()] 4 0, also c ( c) Q

11 Solved Paper 00 (Leaked Paper) Hence, statement () alone is sufficient to determine the unique values of and c. Hence, choice () is correct. 4. Statement () is not sufficient itself. Now, consider statement () Let OE r O O OM ( r 5 ) ( ME 5 cm) and M M.5 cm O M OM r (.5) ( r 5) r.5 cm OE ut OE cannot e less than OM, hence data is inconsistence. ut the information given y statement () is sufficient to answer. Hence, choice (a) is correct. 4. onsider statement () Sum of the first series a a a Sum of the second series a a a a From the first statement (i. e., a ) the relation cannot e determined uniquely since it is different for the +ve and ve values. Now, consider statement () 4a 4a 0 a E Thus for a (a unique value) the sum of the second series will always e greater than that of the first. Hence, choice (a) is correct. 4. Since, E, F are the mid points of, and respectively. Therefore EF, F, E. Hence the area of EF 4 ( ) onsider statement () cm cm and EF cm F cm cm Perimeter of EF cm E cm, hence cm E EF F F EF is an equilateral triangle of side cm each. Thus area can e calculated. onsider statement () Perimeter of 6 cm,perimeter of EF cm cm EF cm cm E cm F cm ( E EF F cm) Hence area can e calculated. Therefore, choice () is correct. 44. When n is a prime numer than ( n )( n ) is not divisile y n. Now, since there are 7 prime numers (, 7, 9,, 9,, 7) from to 40 therefore n e.g., n, then ( n )( n )! Thus!is not divisile y. Hence, choice () is correct. ( y z) y ( z) z ( y) yz ( y z ) y ( z) z ( y) yz yz yz y y z z y z z y y y z z y z y z Now, since a ; if a and are distinct numers. a y y z z y z y 6 z Hence, choice (c) is correct. 46. No. of students who answered or more questions wrongly n No. of students who answered or more questions wrongly n No. of students who answered or more questions wrongly n n No. of students who answered n questions wrongly Hence total no. of wrong answers n n n n 4095 ( ) n n 4096 n 47. For the intersection of two curves we equate them and get the solution as follows ( ) 0 ( )( ) 0

12 Solved Paper 00 (Leaked Paper) NOTE 0,, Thus there are three solutions each lying in the given range (i.e., ). Hence, the two curves intersect thrice. For more clarification of the concept draw the two graphs on the same plane and you will find that the two graphs intersect each other at, 0and i. e.,the corresponding values of y for each of the two curves are same Thus there are 9 pairs which gives the sum of 470 and a single numer (which is the 0th numer) 4 which never gives the sum of 470 y comining with any other numer of the set T. 4 Therefore we can take 9 numers (one from each pair) alongwith 4. Thus the maimum numers in the suset S can e The least no. of edges will e when points will e connected to a single point through the edges. Hence this comination will give us least possile edges. The maimum no. of edges will e when all the points will e non collinear and connected with each other. Hence the maimum no. of edges No. of ways of filling green all 6 (,,, 4, 5, 6) No. of ways of filling green alls 5 No. of ways of filling green alls 4 No. of ways of filling 4 green alls No. of ways of filling 5 green alls [(, ); (, ); (, 4); ( 4, 5); ( 5, 6)] [(,, );(,, 4);(, 4, 5);( 4, 5, 6)] [(,,, 4);(,, 4, 5);(, 4, 5, 6)] [(,,, 4, 5);(,, 4, 5, 6)] No. of ways of filling 6 green alls [(,,, 4, 5, 6)] Hence, the required no. of ways 6 5 4