PRMO EXAM-07 (Paper & Solution) Q. How many positive integers less than 000 have the property that the sum of the digits of each such number is divisible by 7 and the number itself is divisible by 3? Sum of digit divisible by 7 S and number itself divisible by 3 So numbers are () 399 () 498 (3) 489 (4) 79 () 97 (6) 88 (7) 669 (8) 678 (9) 687 (0) 696 () 79 () 768 (3) 777 (4) 786 () 79 (6) 849 (7) 88 (8) 867 (9) 876 (0) 88 () 894 () 939 (3) 948 (4) 97 () 966 (6) 97 (7) 984 (8) 993 So total number of integer are 8 Sum of number should be Q. Suppose a, b are positive real numbers such that a a + b b 83, a b + a a a b b 83 a 3 b 3 83 () a b 3 3 3 a b 3 ab a b a b 3 83 + ab a b Given From () & (3) From (3) a b b a 8 3 () a b ab 8 (3) a b 3 83 + 3(8) 79 a b 9 (4) ab (9) 8 8 ab 9 CAREER POINT () 9 b 8. Find a b. CAREER POINT Ltd., CP Tower, IPIA, Road No., Kota (Raj.), Ph: 0744-00 www.careerpoint.ac.in
From (4) a b (9) a + b + ab 8 a + b 8 ab 8 a + b 8 9 a + b 79 364 9 9 9 (a + b) 36 9 36 73 Ans. 9 from () 8 ab 9 Q.3 A contractor has two teams of workers: team A and team B. Team A can complete a job in days and team B can do the same job in 36 days. Team A starts working on the job and team B joins A after four days. The team A withdraws after two more days. For how many more days should team B work to complete the job? Team A completes job in days Team B completes job in 36 days day work of team A day work of team B 36 day work of team A and team B (when they both work together) Now according to question 3 36 36 36 4 9 Let more number of days should team B works to complete the job be x days So, 4 + + x 9 36 x 3 9 36 + 8 + x 36 x 6 days Q.4 Let a, b be integers such that all the roots of the equation (x + ax + 0) (x + 7x + b) 0 are negative integers. What is the smallest possible value of a + b? (x + ax + 0) (x + 7x + b) 0 ; a,b are integer CAREER POINT Ltd., CP Tower, IPIA, Road No., Kota (Raj.), Ph: 0744-00 www.careerpoint.ac.in
According to question a > 0 and b > 0 ( sum of roots < 0 and product > 0) Since + a 0 Since 0 ( 0) ( 0) or (4, ) Min a 9 () and 7 + (, ) (, 6), (, ).. ( 8, 9) Min b 6 (a + b) min a min + b min 9 + 6 Q. Let u, v, w be real numbers in geometric progression such that u > v > w. Suppose u 40 v n w 60. Find the value of n. u, v, w G.P. u > v > w Also given u 40 v n w 60 Let u a v ar w ar u 40 v n w 60 a 40 (ar) n a 60 r 0 a 40 a 60 r 0 a 0 r 0 (ar ) 0 ar 6 a r 6 Now (ar) n (ar ) 60 (r ) n (r 4 ) 60 n 40 n 48 Ans. Q.6 Let the sum n(n )(n 9 n(n )(n ) n..3 9 ) n.3.4 3.4.........3.3.4 9.0. () written in its lowest terms be q p. Find the value of q p. 9.0......3.3 3.4 9.0 0. CAREER POINT Ltd., CP Tower, IPIA, Road No., Kota (Raj.), Ph: 0744-00 www.careerpoint.ac.in 3
. 0. 0 4 7 p 0 0 q q p 0 7 83 Q.7 Find the number of positive integers n, such that n + n <. n + n < Let n + n () n On Rationalisation From Eq. () + () n n n n + n 6 n () n + 6 37 3600 n 9.7 9 values. Q.8 A pen costs ` and a notebook costs `3. Find the number of ways in which a person can spend exactly `000 to buy pens and notebooks. Cost of a pen ` Cost of a Note book ` 3 Cost of x pen x Cost of y Note book 3 y x + 3y 000 () x 8 7 9 46 33 0 7 y 6 7 38 49 60 7 So total ways 7 Q.9 There are five cities A, B, C, D, E on a certain island, Each city is connected to every other city by road. In how many ways can a person starting from city A come back to A after visiting some cities without visiting a city more than once and without taking the same road more than once? (Th e order in which he visits the cities also matters : e.g., the routes A B C A and A C B A are different.) CAREER POINT Ltd., CP Tower, IPIA, Road No., Kota (Raj.), Ph: 0744-00 www.careerpoint.ac.in 4
A B C D E Five cities A BC DE BD BE CD CE 4 C.! A BCD CDE BCE BDE 4 C 3. 3! 4 A BCDE 4 C 4. 4! 4 Total + 4 + 4 60 Q.0 There are eight rooms on the first floor of a hotel, with four rooms on each side of the corridor, symmetrically situated (that is each room is exactly opposite to one other room). Four guests have to be accommodated in four of the eight rooms (that is, one in each) such that no two guests are in adjacent rooms or in opposite rooms. In how many ways can the guests be accommodated? 4! 48 Ans. x 3x Q. Let f(x) sin + cos 3 0 for all real x. Find the least natural number n such that f(n + x) f(x) for all real x. x 3x f(x) sin + cos 3 0 Period of sin x Period of sin 3 x 6 3x Period of cos 0 0 3 period of f(x) (LCM of 6 and 80 60 3 n 60 0 ) 3 Q. In a class, the total numbers of boys and girls are in the ratio 4 : 3. On one day it was found that 8 boys and 4 girls were absent from the class, and that the number of boys was the square of the number of girls. What is the total number of students in the class? Let number of boys 4x and number of girls 3x Given (4x 8) (3x 4) 4x 8 9x + 96 84 x 9x 88x + 04 0 9x 4x 34x + 04 0 9x(x 6) 34(x 6) 0 34 x 6, x (not possible) 9 no. of students 4x + 3x 7x 7(6) 4 Ans. CAREER POINT Ltd., CP Tower, IPIA, Road No., Kota (Raj.), Ph: 0744-00 www.careerpoint.ac.in
Q.3 In a rectangle ABCD, E is the midpoint of AB; F is a point on AC such that BF is perpendicular to AC; and FE perpendicular to BD. Suppose BC 8 3. Find AB. Let In AEF D F k a 8 3 8 3 A x BC AD 8 3 AB CD x AE EB x 90º+ + 90º + + a 80º E a 90º 90º x C B Explanations But a In AFB, F 90º 90º and E is mid point on hypotenue In ABC tan 3 90º 30º 8 3 x 8 3 3 x AB x 4 Ans. Q.4 Suppose x is a positive real number such that {x}, [x] and x are in a geometric progression. Find the least positive integer n such that x n > 00. (Here [x] denotes the integer part of x and {x} x [x].) Given {x} x [x] Let {x} a [x] ar x ar a ar ar ar ar a 0 r r 0 4 r r x Positive real number CAREER POINT Ltd., CP Tower, IPIA, Road No., Kota (Raj.), Ph: 0744-00 www.careerpoint.ac.in 6
[x] Integer (I) ar Integer (I) r I a and 0 < {x} < I 0 < < 0 < I( 0 < I < I ar ) < a and x ar x n > 00 n > 00 n log 0 > n > 9. n 0 Ans. Q. Integers,, 3., n, where n >, are written on a board. Two numbers m, k such that < m < n, < k < n are removed and the average of the remaining numbers is found to be 7. What is the maximum sum of the two removed numbers? Given < m < n, < k < n n(n ) S n + +.. + n Now two number are removed m and k n(n ) then S n (m + k) According to question n(n ) (n ) < 7 < n n(n ) 3 n CAREER POINT Ltd., CP Tower, IPIA, Road No., Kota (Raj.), Ph: 0744-00 www.careerpoint.ac.in 7
n n 4n n n 6 7 (n ) (n ) n n 4n (n ) < 7 < (n 3)(n ) (n ) n 3n (n 3)(n ) 7 (n ) (n ) ( n )(n ) n 3 7 (n ) n n 3 < 7 < n < 3 & n > 3 n 3, 33, 34 Case () n 3 Case () n(n ) P 7 (n ) n(n ) 7(n ) P P 8 When n 33 P 34 Case (3) When n 34 P Maximum sum Q.6 Five distinct -digit numbers are in a geometric progression. Find the middle term. Let the numbers be a, ar, ar, ar 3, ar 4 Since all are digit number. r 3 or 3 Hence five numbers are (6, 4, 36, 4, 8) Middle term 36 ( 4 th power of integer greater than 3 are 3 digit number) Q.7 Suppose the altitudes of a triangle are 0, an. What is its semi-perimeter? Let us suppose 3 altitudes are h, h, h 3 h : h : h 3 0 : : a : a : a 3 0 : : 6 : : 4 a, a & a 3 are sides of triangle a 6k, a k, a 3 4k CAREER POINT Ltd., CP Tower, IPIA, Road No., Kota (Raj.), Ph: 0744-00 www.careerpoint.ac.in 8
S a a a 3 k Area (s a )(S a )(S a ) S 3 k Area 7 4 3k k k Also area h 6k 7 4 k h 7 4 Given h 0 0 7 4 k 8 7 k S k k 6kh 60 7 7 k Q.8 If the real numbers x, y, z are such that x + 4y + 6z 48 and xy + 4yz + zx 4, What is the value of x + y + z? x + 4y + 6z 48 () and xy + 4zy + zx 4 () from () x + (y) + (4z) 48 (3) from () Eq.() 4 4xy + 6zy + 8zx 96 (4) Eq.(3) Eq.(4) [x + (y) + (4z) ] (xy + 8zy + 4zx] 0 (x y) + (y 4z) + (x 4z) 0 x y, y 4z, x 4z x y 4z Now xy + 4zy + zx 4 x. x x x + x. + x. 4 3x 48 x 6 x 4 y and z x + y + z 6 + 4 + Ans. CAREER POINT Ltd., CP Tower, IPIA, Road No., Kota (Raj.), Ph: 0744-00 www.careerpoint.ac.in 9
Q.9 Suppose,, 3 are the roots of the equation x 4 + ax + bx c. Find the value of c. x 4 + ax + bx C 0 Roots are,, 3, Now sum of roots 0 6 + 0 6 and product ()()(3)() C 36 C C 36 Ans. Q.0 What is the number of triples (a, b, c) of positive integers such that (i) a < b < c < 0 and (ii) a, b, c, 0 form the sides of a quadrilateral? a + b + c > 0 (a, b, c) can be Total cases 73 a b c 8,9 3 7,8,9 4 6,7,8,9 6,7,8,9 6 7,8,9 7 8,9 8 9 3 6,7,8,9 4,6,7,8,9 6,7,8,9 6 7,8,9 7 8,9 8 9 3 4,6,7,8,9 3 6,7,8,9 3 6 7,8,9 3 7 8,9 3 8 9 4 6,7,8,9 4 6 7,8,9 4 7 8,9 4 8 9 6 7,8,9 7 8,9 8 9 6 7 8,9 6 8 9 7 8 9 CAREER POINT Ltd., CP Tower, IPIA, Road No., Kota (Raj.), Ph: 0744-00 www.careerpoint.ac.in 0
Q. Find the number of ordered triples (a, b, c) of positive integers such that abc 08. abc 08 abc 3 3 x 3 y x y x a. 3 ; b y 3.3, c.3 x + x + x 3 and y + y + y 3 3 Cases (0,0,), (0,,) (0,0,3), (0,,), (0,,) (,0,), (,,0) (,0,), (,,), (,,0) (0,,0), (,0,0) (,0,), (,,0), (0,3,0) (3,0,0) 6 cases 0 cases Total 6 0 60 Q. Suppose in the plane 0 pair wise nonparallel lines intersect one another. What is the maximum possible number of polygons (with finite areas) that can be formed? No. of non-overlapping polygon 6 0 36 Ans. Explanation line divide plane into regions line divide plane into 4 regions 3 line divide plane into 7 regions 4 line divide plane into regions line divide plane into 6 regions 6 line divide plane into regions 7 line divide plane into 9 regions 8 line divide plane into 37 regions 9 line divide plane into 46 regions 0 line divide plane into 6 regions Now open regions for 3 lines are 6 Similarity for 0 lines are 0 number of polygons (with finite areas) 6 0 36 Q.3 Suppose an integer x, a natural number n and a prime number p satisfy the equation 7x 44x + p n. Find the largest value of p. 7x 44x + p n 7x 4x x + p n (7x ) (x 6) p n 7x p a and x 6 p b 7x 7(x 6) p a 7p b 40 p a 7p b Now if a, b N, p is divisors of 40 p or CAREER POINT Ltd., CP Tower, IPIA, Road No., Kota (Raj.), Ph: 0744-00 www.careerpoint.ac.in
if p, 40 a 7. b 3 a 7. b if b 3 a 40 + 6 a z hence not possible if p then 40 a 7. b 3 a 7. b if b a 40 + 3 so b 0 p a 47 a z (not possible) p 47 and a Q.4 Let P be an interior point of a triangle ABC whose side lengths are 6, 6, 78. The line through P parallel to BC meets AB in K and AC in L. The line through P parallel to CA meets BC in M and BA in N. The line through P parallel to AB meets CA in S and CB in T. If KL, MN, ST are of equal lengths, find this common length. 6 A N ab S K a p b L ac c bc 6 Given B T M C 78 KL MN ST (say) (b c) 6 (a b c) b c 6 a b c a c Similarity 6 a b c 78 6 + + 6 6 78 30, which is not possible as has to be less than 6. CAREER POINT Ltd., CP Tower, IPIA, Road No., Kota (Raj.), Ph: 0744-00 www.careerpoint.ac.in
Q. Let ABCD be a rectangle and let E and F be points on CD and BC respectively such that area (ADE) 6, area (CEF) 9 and area (ABF). What is the area of triangle AEF? a E y a D C 9 6 b x A y x.a 6 xa 3 F x b B b(y a) 9 b(y a) 8 y(x b) y(x b) 0 xa 3 by ab 8 & xy yb 0 3b 8x by 8 b x xy 3 8xy xy 0 xy 3 Let xy t 3t 8t 0 t 00t + 600 0 t 3 t 80, 0 0 not possible xy 0 xy 80 Area of AEF 80 (6 + 9 + ) 30 Ans. t Q.6 Let AB and CD be two parallel chords in a circle with radius such that the centre O lines between these chords. Suppose AB 6, CD 8. Suppose further that the area of the part of the circle lying between the chords AB and CD is (m + n)/k, where m, n, k are positive integers with gcd(m, n, k). What is the value of m + n + k? C 4 N 3 O D A 3 4 M B 6 CAREER POINT Ltd., CP Tower, IPIA, Road No., Kota (Raj.), Ph: 0744-00 www.careerpoint.ac.in 3
Radius AB 6 CD 8 ON 3 and OM 4 A + 3 8 + 4 6 4 3 [ ( + )] tan tan 3 4 A 4 + 48 A given A m n k m, n 48, k (m + n + k) + 48 + 7 Ans. Q.7 Let be a circle with centre O and let AB a diameter of. Let P be a point on the segment OB different from O. Suppose another circle with centre P lies in the interior of. Tangents are drawn from A and B to the circle intersecting again at A and B respectively such that A and B are on the opposite sides of AB. Given that A B, AB and OP 0, find the radius of. A A r O r p 0 r 0 B r r 0 r r 0 B r r r 0 r 0 3 3r 30 r + 0 r 40 r 0 () () CAREER POINT Ltd., CP Tower, IPIA, Road No., Kota (Raj.), Ph: 0744-00 www.careerpoint.ac.in 4
Q.8 Let p, q be prime numbers such that n 3pq n is a multiple of 3pq for all positive integers n. Find the least possible value of p + q. Using Euler's totient function a (n) (mod n) 3pq n 3pq n 3pq n (n 3pq ) It is easy to see that p, q have got to be both odd and neither can be 3 3, p, q are pairwise relatively prive (3pq) (p ) (q ) 3pq n (n 3pq ) n I + if (3pq) (3pq ) (p ) (q ) (3pq ) Since p, q are odd (3pq ) (p ) (3pq ) (q ) (3pq ) (p ) (3q ) Both should hold simultaneously (q ) (3p ) the least p + q is + 7 8 (for p, q 7) Q.9 For each positive integer n, consider the highest common factor h n of the two numbers n! + and (n + )!. For n < 00, find the largest value of h n. n! + is not divisible by,,.., n (n + )! is divisible by,,., n so HCF n + also (n + )! is not divisible by n +, n + 3,. so HCF can be (n + ) only let us start by taking n 99 99! + and 00! HCF 00 is not possible as 00 divides 99! composite number will not be able to make it so let us take prime number i.e. n 97 now 96! + and 97! are both divisible by 97 so HCF 97 by Coilson theorem ((p )! + ) is divisible by p. Q.30 Consider the areas of the four triangles obtained by drawing the diagonals AC and BD of a trapezium ABCD. The product of these areas, taken two at time, are computed. If among the six products so obtained, two products are 96 and 76, determine the square root of the maximum possible area of the trapezium to the nearest integer. A A B A A A 3 D C CAREER POINT Ltd., CP Tower, IPIA, Road No., Kota (Raj.), Ph: 0744-00 www.careerpoint.ac.in
Clearly A 3 A A Now 6 products are A, Reduces to 3 cases only Case () Case () Case (3) When A A A A 3 A, A A, A A, A A 3, A A 3 A 96 and A A 76 A + A + A 3 69 When A 76 and A A 96 A + A + A 3.66 < 69 When A A 96 and A A 3 76 A + A + A 3 4 < 69 Hence Max. value of A A A3 3 Ans. CAREER POINT Ltd., CP Tower, IPIA, Road No., Kota (Raj.), Ph: 0744-00 www.careerpoint.ac.in 6