Latest TCS Aptitude Question SOLUTION: 222^222 when divided by 7 gives a remainder of?

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Previous Question: 2-Given the word PARROT. Find the 20th word in dictionary order? Ans-APRTOR 3- 1(1!)+2(2!)+3(3!)+....+2010(2010!)+2011(2011!)+2012(2012!)=? 1(1!)+2(2!)=5 ie 3!-1 1(1!)+2(2!)+3(3!)=23 ie 4!-1 1(1!)+2(2!)+3(3!)+.............+2012(2012!)=2013!-1 4- how many polynomial functions f of degree greater than 1 satisfy f(x^2)=[f(x)]^2=f(x)? a)more than 2 b)2 c)0 d)1 ans- Let f(x)=x;then f(x^2)=x^2,[f(x)]^2=x^2,will b equal when put"0".tk any polynomial it's true so ans is 0 5-what will be the next 3-digit in the series 1123783131.. ANS=574 11*2+1=23 23*3+9=78 78*4+1=313 313*5+9=1574 watch carefully 6-Prime factorisaton of int N is A*A*B*C. a,b,c all are distinct prime integers. how many factors does N have ?? ans- (2+1)(1+1)(1+1)=12 7- In how MANY WAYS can we distribute 10 identical pencil into 4 students so that each student get at least 1 pencil ?? Ans- 9C3 Let us distribute 1 pencil to each. Then pencils left =10-4=6 Now ways of distributing 6 pencils among 4 such that each gets 0 or more =(n+r-1)C(r-1) - this is a standard formula =(6+4-1)C(4-1) =9C3 =84 8-All faces of a cube with an 8meter edge are painted red.if the cube is cut into smaller cubes with a 2 meter edge. how many of two meter cubes have paint on exacly one face ?? please explain in detail with ol the logics used. . always confuesd in such type of cube questions ! ans exactly one face painted cube =24 two side painted= 24 3 side painted=8 not painted= 8 .just draw cube in copy u can found answer 9- what the formula must be applied to find no of RECTANGULAR in m*n grid ?? Ans- (m+1)c2*(n+1)c2 10- x^2-16>0 in what range x must lie ?? x^2-16>0 =>x^2>16, x>+or-4 therefore, -4>x>4 11- 3y+x>2, x+2y>=3 find range?? Ans- y > -1 How many ways can Jason sit with his five friends in a row of six seats with an aisle on either side of the row, if Jason insists on sitting one of the aisles? a) 120 b) 240 c) 360 d) 540 Ans- 2*5c4*4!=240 or ans is 240 : Because Jason must sit on either aisle, there are two places he can sit. And for each of those two seats, there are five friends left for five seats, making 5! ways to arrange them. So the answer is 2 * 5!, which is 240. Florence and Jackie are running laps on a circular quarter-mile track. If Florence completes each lap in 75 seconds and Jackie completes each lap in 90 seconds, how many laps will Jackie have run before sheâ€™s passed by Florence, assuming they start at the same time. 1) 5 2) 6 3) 7 4) 8 5) 9 lcm of 75 and 90 is 450 and 450/90(time taken by jackie)=5 Grass in lawn grows equally thick in uniform rate. It takes 24 days for 70 cows and 60 days for 30 cows to eat the whole grass. how many cows are needed to eat the grass in 96 days? Ans- 20 I think this approach is very long to solve this problem i try to reduce some step g= grass initially, r= rate at which grass grow/day, c= cow eat grass/day g+24r=70*24c=1680c-----------1 g+60r=60*30c=1800c--------->g=1800c-60r----------------2 by solving this 2 equation we have relation c=(3/10)r-------------3 g+96r=96nc =>96nc=1800c-60r+96r=1800c+36r=1800c+120c=1920c =>n=20 find the 25 th word if the word RANKED is arranged in dictionary order...... AD_ _ _ _ 4*3*2*1 =24 next will be AEDKNR A,B ,C can do some work in 36days.A and B together do twice as much work as c alone, and A and C together can do thrice as much work as B alone..find the time taken by C to do whole work? a)96 b)108 c)120 d)72 ans:108. . Ans 108 1 day work for (A+B+C) = 1/36 ............(ii) Now, (A+B) = 2C ..........................(i) From (i)and(ii): 3C = 1/36 C = 1/108 Thus, C takes 108 days to complete the whole work alone. >At 12:00 hours jake starts to walk from his house at 6kms an hour. At 13:30 hours, paul follows him from jake's house on his bicycle at 8 kmph. When will jake be 3 kms behind paul?,ans:19.30. Ans- distance travelled y jake= 6*1.5= 9 Relativespeed=8-6=2kmph Time= (9 + 3 )/2= 6hr Time= 1:30+ 6:00= 7:30 hr 3>There are 5 men and 11 women. How many ways can a panel of 11 be formed such that the number of men is not more than 3,ans:2256. Ans- possible combination((0,11) or (1,10) or (2,9) or (3,8) ) Aplly permutation combination to get nswer 6 different chocolates 10 boxes how they can be arranged ,,Ans : 10^6. . if a^x=b, b^y=c, c^z=a ..then what is the value of x*y*z?ans:1. Ans- aplly log to all individual equation and find value of x, y,z and then multiply, get =1 SENIOR 36th word ans:ENOSRI. there are 60%male and 70% female are allowed for vote. 70% of male voter and 60% of female voter give vote to a particular person .what percentage of vote for him ,ans: 42. . ans- let 100 men and 100 women therefore 60 male 70 female now 70% of 60=42 60%of70=42 Therefore reuired %=(42+42)/200= .42 = 42 % when m+n is divided by 12 we get reminder 8.when m-n is divided by 12 we get reminder 6.if m>n then what will be the reminder when mn is divided by 12 ANS- 7 ans is 7 m+n/12=12x+8 m-n/12=12x+6 if x=1 then m+n=20 and m-n=18 as m>n so m=19 n=1 so mn/12=7 How many 6 digit even numbers can be formed from digits 1 2 3 4 5 6 7 so that the digit should not repeat and the second last digit is even?,ans:720 Logical reasoning hw many ways can 4 couple can cross a river,ans: 16. Ans- select 1 girl out of 4 select 1 boy out of 4 Multiply them i.e. 4x4=16 1.2.3 are 3 numbers maximum way it can be arranged,u ll get 1 point if 1 is at 1st positin or 2 is at 2nd position or 3 is at 3rd position(atleast) how many points ll u get,ans:4. Ans – possible ways= 123, 132, 321, 213 A series of book was published at 7 year intervals.When the 7th book was issued the sum of publication year is 13524.When was the 1st book published ? ,ans:1911. sol: sum of A.P. series=(n/2)(2*a+(n-1)*d) here sum=13524,n=7,d=7 substituting these values in above eq. n solving we get a=1911 which was the 1st book published year. George can do some work in 8hrs. paul can do the same work in 10hours while hari can do the same work in 12hrs. all the 3 of them start working at 9am while George stops work at 11am and the remaining two complete the work. Approximately at what time will the work be finished?,ans:2pm Ans efficiency of George= 15 efficiency of paul= 12 efficiency of hari= 10 total work= 120 work left after 2 hr= 120- ((12+15+10))x2=66 time=66/(10+12)= 3hr A and B are running around a circular track of length 120 meters with speeds 12 m/s and 6 m/s in the same direction. When will they meet for the first time? a)25 b)30 c)10 d)20 Ans whenever two people r running in same direction and they meet for first time=total distance/relative speed; same direction s1-s2;opposite direction s1+s2; 120/(12-6)=20 In a class there are 60% of girls of which 25% poor. What is the probability that a poor girl is selected is leader? a)15% b)50% c)80% d)30% Ans a) 15% probability for a girl to be leader is 60% becuase there are 60% girls in class. after this probability that tht girl will be poor is 25% of 60 ie..15% so answer is a) 15% x=(sin nx/n)=? a)90 b)0 c)1 d)2 ans:-b.) 0 Raj writes a number.He sees that athe number of two digits exceeds four times the sum of its digits by 3. If the number is increased by 18,the result is the same as the number formed by reversing the digits. Find the number Ans let the unit digit is y and tens digit is x now 10x+y=4(x+y)+3 .....(1) and 10y+x=10x+y+18 ......(2) by solving the eq.x=3 and y=5 so the ans is 35 15 students join a summer camp. every day 3 students are on duty after the school to clean classrooms. After the course it was found that every pair has been on duty exactly once. How many days does the course last for ? Anc 15C2 Raj invested in Indigo, HUL and SBI shares at Rs. 300, Rs. 200 and Rs. 5 per share, 100 shares for Rs. 10000. The number of Indigo and HUL shares he bought are respectively 1.15, 25 2.23, 17 3.17, 23 4.17, 60 Ans 17 and 23 would be the answer.as we have two equations 300x+200y+5z=10000 and x+y+z=100 if we solve these two equations we will get 295x+195y=9500 by puting x=17 and y=23,we will get the result at 9500 There is a tank,and two pipes A and B.A can fill the tank in 25 minutes and B can empty the tank in 20 minutes.If both pipes are opened at same time.how much time required the taank can fill? Ans Never will fill…. As B have more efficiency tha that of A [Q-11]If X^Y denotes X raised to the power of Y, find out last two digits of (2957^3661)+(3081^3643) Options o 42 o 38 o 98 o 22 Ans- 98 6 different chocolates 10 boxes how they can be arranged Ans : 10^6 ADEORV find the 45th word in dictionary order Ans- AEVODR if f(x) is a function for all real numbers x holds good for maximum of 2x+4 and 12+3x..then wat is the value of x so that f(x) is equal to 2x+4..Ans is -9 ans- let f1(x)=12+3x f1(-8)=-12 and f2(x)=2x+4 f2(-8)=-12 so ans is -8 2^74+2^2048+2^2n what is the value of n so that sum will be perfect square since, (a+b)^2=a^2+b^2+2ab 2^74+2^2n+2^2048=(2^37)^2+(2^n)^2+2^2048 2*(2^37)*(2^n)=2^2048 2^(1+37+n)=2^2048 since base are same ,equating powers,we get 1+37+n=2048 38+n=2048 n=2010 15 women or 10 men complete in 55 days..how many days 4 women and 5 men vil take Ans 10 men= 15 women 5 men= 15/2 women now,4 women and 5men 4women + 15/2 women 23/2 women (4women &5men = 23/2women) can complete a work in = (55*15*2)/23=71.739 days #1.A series of book was published at 7 seven year intervals. When the seventh book was published the total some of publication year was 13,524. First book was published in?? -sum of A.P ->Sn=n/2(2a+(n-1)d) where Sn=13524(given),n=7,d=7 13524=7/2(2*a+(7-1)7) a=1911 #2.How many motor vehicles registration number plates can be formed with the digits 1, 2, 3, 4, and 5(no digits being repeated) if it is given that registration number can have 1 to 5 digits? -1 digit = 5 ways 2 digit = (5*4)ways 3 digit = (5*4*3)ways 4 digit = (5*4*3*2)ways 5 digit = (5*4*3*2*1)ways total ways = 5+20+60+120+120= 325 ways. #3.Five rivers Ya-Fen,Volga,Yamuna,Hudson and Siene flow through the five cities v-z. New York,Paris,Moscow,Tokyo and Delhi,but not necessarily in same order.No two rivers flow in the same city and no two cities have same river flowing through them.Vogla flows through neither Moscow nor Delhi.Ya-Fen is neither in New York nor in Paris. Yamuna is neither in Delhi nor in Moscow.Hudson is in Moscow while new York has Yamuna flowing through it. Which of following is definitely FALSE? 1. If Volga flows through Tokyo,then Ya-Fen flows through Delhi. 2. If Tokyo has Volga,then Siene must flow through Delhi. 3. If Ya-Fen flows through Tokyo, then Siene must flow through Delhi. 4. Ya-Fen may flow through Tokyo or Delhi. -we have moscow=hudson ny=yamuna so if tokyo=volga nd siene= delhi den paris= yafen which is false option 2 is the FALSE statement! #4.If 9/12/2001 is sunday then what day of week will be on 9/12/9071???? -as it seems it is not a leap year so their r 365 days, so 9071-2001=7070 yrs r left and 368650 days r their in 7070 yrs. when 368650 is divided by 7 we get 0,that means sunday. sunday is ans. #5.Five people Pradeep,Qutub,Raj,Sandy and Tej are in a row. Raj sits to the immediate right of Tej. If Pradeep and Qutub have only two persons sitting between them, which of following must be true? 1. Pradeep is at one extreme end of row. 2. Qutub is sitting beside Pradeep. 3. Tej or Raj must be at one extreme end of the row. 4. Sandy must be at one of ends of the row. -The sitting order may be sandy,pradeep,tej,raj,qutub or pradeep,tej,raj,qutub,sandy or sandy,qutub,tej,raj,pradeep or qutub,tej,raj,pradeep,sandy in all the cases sandy is at one end of the row hence ANS:4 #6.find the number of zeroes in 1^1*2^2*3^3...48^48*49^49? -5^5+10^10+15^15....45^45 will have zeroes total=5+10+15+20+50+30+35+40+45=250. #7.A person has to go both Northwards & Southwards in search of a job. He decides to go by the first train he encounters.There are trains for every 15 min both southwards and northwards. First train towards south is at 6:00 A.M. and that towards North is at 6:10. If the person arrives at any random time, what is the probability that he gets into a train towards North.? -In a frequency of 15 mins if the person comes in first 10 mins he will catch northwards train.so the probability that he gets into a train towards North is 10/15=0.66 If he comes in the next 5 mins he will catch southwards train.so the probability of getting into a train towards southwards is 5/15=0.33 #8.Banu plays with two boxes seperately containing 25 red cards and 33 block cards and she allowed to move the cards between the boxes. She won the game by choose a box at random and a card at random from the chosen box the probability of getting a black card is minimized. Now,the question is to find that minimum probability. -to minimise probability of black card, put 1 red card in one box and 24 red cards and 33 black cards in second box. then min prob of getting black card = 0*1/2 + 0.5 * 33/57 = 11/38. #9.A completes a work in 20 days,B completes in 60 days and C in 45 days. all 3 prsn working 2gthr on a project got a profit of Rs 26000.what is profit of B?? -Profit will be divided in ratio of 1/20 : 1/60 : 1/45 or 9:3:4 profit of B = 26000*3/16 = 78000/16 = Rs. 4875. #9.My flight takes off at 2am from a place at 18N 10E and landed 10 Hrs later at a place with coordinates 36N70W.What is the local time when my plane landed? a) 6:00 am b) 6:40 am c) 7:40 am d) 7:00 am e) 8:00 am. -Latitudianal difference=(10+70)=80 So, time difference=80X4=320mins=5 hrs 20 mins Since the place is located towards west Therefore local time of that place during take off=8:40 p.m. So, local time during landing=8:40p.m.+10hrs=6:40a.m. #10.trucks A,B,C// A loads 10 kg/min B loads 13 1/3 kg/min C unloads 5kg/min If three simultaneously works then what is the time taken to load 2.4 tonnes?? -a loaded in 1 min =10kg// work day me diya hota to reciprocal lete but ye time me hain isliye direct jod //denge b loaded in 1 min=40/3kg c unloaded in 1 min=5kg so a,b,c are loaded in 1 min=55/3kg so 1 kg loaded by a,b,c in time =3/55min now, 2.4 tons loaded by a,b,c in time=(3/55)*2.4*1000=131min . #11.I was talking to a friend Tammy, the middle child of five. she has a younger sister Tracey, and three brothers Tommy (oldest child), Timmy and Tony (youngest child). I asked her how old she was, but she hardly ever gives a straight answer. The following is the conversation we had. "How old are you Tammy?" "I am three times as old as Timmy was born. Actually, it's a funny, but Timmy's,tracey's, Tommy's and my ages were all factors of Mum's age when Tony was born. The only other year where the ages of more than two of us were factors of Mum's age was when Tracey was half as old as I am now." "How old is your Mum now?" "There's another funny thing. when Mum turns 50, tony will be the same age Mum was when she gave birth to Tommy" Assuming Tammy was talking in terms of whole years, and with my mathematical and reasoning abilities i was able to quickly deduce not just Tammy's age, but all of thier ages. so how old are Tommy,Timmy,Tammy,Tracey,Tony and Mum Now? a) Tommy is 26, Timmy is 17, Tammy is 16,Tracey is 13, tony is 12 and Mum is 43 b) Tommy is 23, Timmy is 19, Tammy is 18,Tracey is 16, tony is 13 and Mum is 43 c) Tommy is 28, Timmy is 21, Tammy is 23,Tracey is 19, tony is 17 and Mum is 43 d) Tommy is 21, Timmy is 20, Tammy is 19,Tracey is 17, tony is 15 and Mum is 43. -the ans is b..based on the options given.because.in this opetion when we subtract 13 from each age then all the ages are facotros of moms age. #12.Seven Applicants for a Job - F, G, H, I, J, K, and L are each to be interviewed on monday or Tuesday . Each Applicant is to be Interviewed exactly once, and the interview schedule must conform to the following conditions: There must be at least two interviews each day I cannot have the first interview on Monday F must have the first or last interview on either Monday or Tuesday H and L must interview on the same day, with no other applicant interviewing between them I and K cannot interview on the same day. At least one interview must be scheduled between the interviews of G and L If G has very First interview , and if exactly four people are interviewed on Tuesday, which of the following could be the schedule for the two interviews immediately following G's? A) First H, then L B) First L, then F C) First I, Then L D) First F, Then I -Answer C First I than L Monday Tuesday G F I K L H=L. #13.A,B and C can do a piece of work alone in 15 , 25 and 40 days respectively.They start together but B left the work as soon as the work was half completed and c left the work when it was 3/4th completed. A doing the remaining work alone. In how many days the work completed? -ans 300/79+30/11+15/4=10.2747 days. #14.there is a 14*8 grid.what is the total number of squares and rectangles? -square:- 14*8+(14-1)(8-1)+(14-2)(8-2)+(14-3)(8-3)+(14-4)(8-4)+(14-5)(8-5)+(14-6)(8-6)+(14-7)(8-7)=420 rectangle:- (1+2+3+4+5+6+7+8+9+10+11+12+13+14)(1+2+3+4+5+6+7+8)=105*36=3780 total: 420+3780= 4200. #15.what is the remainder when 1!+2!+3!.....100! is divided by 9? all the numbers after 6! are divisible by 9. remainder will be same as obtained when 1!+2!+3!.....5! is divided by 9 which is when (1+2+6+24+120)is divided by 9 or 153 is divided by 9. so zero(0) is the remainder as 153/9 =17 //zero is theanswer -9! is 241920,which is multiple of 9 also 9! divides n! for all n>9 so, we can compute remainder just by looking 1!+2!+3!+4!+5!+6!+7!+8!=30873 which has remainder 3 on dividing by 9. #16.what is the ramainder when 50! is divided by 47*47? first understand dis (6*10*55)/9 remainder of this is 6. there r two ways to calculate this 1--- 6*10*55 ie 3300 mod 9 = 6 2--- 6 mod 9 =6 10 mod 9 =1 55 mod 9=1 thn multiply all the rem n u get the ans ie 6 in case of 47^2 50*49*48*47*46! / 47^2 47 will b cancel out 50*49*48*46 ! / 47 nw find out remainder individually 50 % 47 =3 49 % 47 =2 48 % 47=1 and acc to wilson theorem if p is a prime no thn (p-1)! mod p =-1 here 47 is a prime no so 46!% 47 = -1 or 46 so multiply all the rem 3 * 2*1* 46 mod 47 = 276 mod 47 = 41 nw multiply 41 with 47 so we get 41*47= 1927...ans -the ans will be 1927... 50! can be written as 50*48*47*46!*49/47*47 50*48*49*(47-1)!/47 and (n-1)!/n=-1 so reminder when 47 is divided by 50 is 3,by 48 is 1 and by 49 is 2 and by (47-1)! is -1 so total is -6 and +ve reminder is 41 and we have to multiply it by 47 becoz first we cut it so final ans is 41*47=1927. #17.the first 44 positive integers are written in the order to form the largest number n=123456789101111213..424344. what will be the remainder when n is divided by 45? -no will be div by 45 when it is divided y both 9 and 5 a no is div by 9 if sum of digits is div by 9 now sum of digits of given no is 270 so the no is div by 9 and since last digit is 4 when div by 5 it leaves rem 4 let us say numbers such as 9,54... we see they are div by 9 but leaves rem 4 when div by 5.. so when div by 45 they leave rem 9 so req ans=9 . #18.Given 2^74 + 2^2074 + 2^2n. For what value of(among the options)n will the sum be a perfect square? -we know, (a+b)^2= a*a+b*b+2a*b so 2^74+2^2074+2^2n= (2^37)^2 + 2*2^37*2^n + (2^n)^2 So, we get 38+n=2074 therefoe n = 2036. #19. 50!/(16^15) Then What is the reminder? -there should be 47 2s in 50! and we can write 16^15 as 2^47 then there will remain 2^(60-47)in denominator now in numerator the last digit will be 5 because all the 2s has removed now 2^13=8192 when numerator is divided by 8192 the remainder would be 1. because the last digit of denominator is 2.......... #20.There was two problem on rolling of 3dice and what is the probability if the sum of outcome is 13,10? -for sum as 10 p= 27/216 13 p= 21/216 total p=49/216. #21.if a divided by 5,3,2 respectively give the remainder 1,2,0.what is the no when is divided by 2 3, 5? when the no is devided by 5 it gives remainder 1 so the possible no can be determined by (5*1)+1=6 (5*2)+1=11 (5*3)+1=16 (5*4)+1=21 (5*5)+1=26----so on among them only 26 gives the remainder 2 and 0 when it is devided by 3 and 2 so the no is 26. -11 is the no. 0*2+0*3+2*5+1=11 rem= 1,2,1 11/2= rem 1 5/3= 2 1/5 = 1. its not what is the no. its what is the remainder when the same no. is divided by 2,3,5 respectively. so let the no. be x then x=5y+1 y=3z+2 z=2*1+0 thus x=41 now on dividing 41by 2,3,5 respectively we get 1,2,1 as remainder.... #22.From a circle of radius 10 cm a sector of 40% area is removed. a cone is formed with the remaining part of the circle. find the ratio of the radius and height of the cone? > 1/2 >2/3 >3/4 >4/3. -major sector length= 160pi/9. it is = to base of cone i.e bottom circle. =>2 pi r= 160pi/9 => r=80/9 slant height of cone= 10 h=sqrt(10^2-(80/9)^2)) h=(10sqrt17)/4 r/h= (9sqrt17)/32 r/h= 1.154> 3/4 . #23.A, B and C are friends and went on a picnic. A bought 5 sandwiches and B bought 3 for some amount. C paid Rs.4 as his share. Find the amount B vl get in return > 1.5 >1.4 >0.5 >1.0 -total sandwiches = 8 total money invested = 3*4 ( since each of a,b and c should pay equally) price of 1 sandwich = 3*4/8 = 1.5 so, A invested 7.5 rs, B invested 4.5, so he should get 0.5 back. #24.how many 2's are there between 223 to 2222? - from 201-222 there are 28 2's and from 201-300 there are 120 2's so from 223-300 there are 120-28=92 2's now from 301-1000 there are 7*20=140 2's now from 1000-2000 there are 300 2's now we left with 2001-2222 now consider only thousands place there are 222 2's and from 000 to 222 there are 20+21+28=69 2's so total=92+140+300+222+69=823ans. #25.In a staircase, there are 10 steps. A child is attempting to climb the staircase. Each time, she can either make 1 step or 2 steps. In how many different ways can she climb the stair case? a. 10 b. 21 c. 36 d. None of these. -ans d none of these.\ #26.what is the value of 1(1!)+2(2!)+3(3!)+.......+2012(2012!)? -1(1!)+2(2!)=5(3!-1) 1(1!)+2(2!)+3(3!)=23(4!-1) ... ... 1(1!)+2(2!)+3(3!)+.....+2012(2012!)=2013!-1. #27.Probability of 1 ace which is red and one spade? -total no of ace=4 two red(1diamond,1artin),two black(1spade,1cluver) prob getting red ace=2/4 prob getting spade ace=1/4 prob 1 ace which is red and one spade(and means multiplication)=2/4*1/4 1/2*1/4=1/8. #28.find the average speed if a man travles at speed 24kmph up and 36kmph down at an altitude of 200m? a) 28.8 mph b)27.8mph c)27.5mph d)30mph. -a=24kmph b=36kmph avegage speed=2ab/(a+b) altitude has nothing to do with average speed,.28.5. PAGE NO 5 : 1. 1! + 2! + â€¦â€¦. + 50!=? a)3.1035*10^64 b)2.1021*10^65 c)3.1035*10^63 d)3.1035*10^62 ans- (a) what is the reminder when 6^17+17^6 is divided by 7 ? a) 3 b) 0 c) 6 d) 1 ans --- (B) 6^2=1 mod 7 [(6^2)=36,divided by 7 gives remainder 1] =>(6)^2*8=1^8 mod 7 =>6^16=1 mod 7 [all even power also gives 1 mod 7] =>(6^16)*6=1*6 mod 7 So, 6^17=6 mod 7. Likewise, 17^2=2 mod 7 =>(17)^2*3=2^3 mod 7 =>17^6=8 mod 7 [simplify: 8 is one time divisible by 7 & gives remainder 1] So, 17^6=1 mod 7. ((6^17+17^6)/7)= ((6 mod 7)+(1 mod 7))/7 =>((6+1) mod 7)/7 =>(7 mod 7)/7 =>Hence the remainder is 0 Next Question: what is the remainder of 100!/(97^2)?

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Previous Question: 2-Given the word PARROT. Find the 20th word in dictionary order? Ans-APRTOR 3- 1(1!)+2(2!)+3(3!)+....+2010(2010!)+2011(2011!)+2012(2012!)=? 1(1!)+2(2!)=5 ie 3!-1 1(1!)+2(2!)+3(3!)=23 ie 4!-1 1(1!)+2(2!)+3(3!)+.............+2012(2012!)=2013!-1 4- how many polynomial functions f of degree greater than 1 satisfy f(x^2)=[f(x)]^2=f(x)? a)more than 2 b)2 c)0 d)1 ans- Let f(x)=x;then f(x^2)=x^2,[f(x)]^2=x^2,will b equal when put"0".tk any polynomial it's true so ans is 0 5-what will be the next 3-digit in the series 1123783131.. ANS=574 11*2+1=23 23*3+9=78 78*4+1=313 313*5+9=1574 watch carefully 6-Prime factorisaton of int N is A*A*B*C. a,b,c all are distinct prime integers. how many factors does N have ?? ans- (2+1)(1+1)(1+1)=12 7- In how MANY WAYS can we distribute 10 identical pencil into 4 students so that each student get at least 1 pencil ?? Ans- 9C3 Let us distribute 1 pencil to each. Then pencils left =10-4=6 Now ways of distributing 6 pencils among 4 such that each gets 0 or more =(n+r-1)C(r-1) - this is a standard formula =(6+4-1)C(4-1) =9C3 =84 8-All faces of a cube with an 8meter edge are painted red.if the cube is cut into smaller cubes with a 2 meter edge. how many of two meter cubes have paint on exacly one face ?? please explain in detail with ol the logics used. . always confuesd in such type of cube questions ! ans exactly one face painted cube =24 two side painted= 24 3 side painted=8 not painted= 8 .just draw cube in copy u can found answer 9- what the formula must be applied to find no of RECTANGULAR in m*n grid ?? Ans- (m+1)c2*(n+1)c2 10- x^2-16>0 in what range x must lie ?? x^2-16>0 =>x^2>16, x>+or-4 therefore, -4>x>4 11- 3y+x>2, x+2y>=3 find range?? Ans- y > -1 How many ways can Jason sit with his five friends in a row of six seats with an aisle on either side of the row, if Jason insists on sitting one of the aisles? a) 120 b) 240 c) 360 d) 540 Ans- 2*5c4*4!=240 or ans is 240 : Because Jason must sit on either aisle, there are two places he can sit. And for each of those two seats, there are five friends left for five seats, making 5! ways to arrange them. So the answer is 2 * 5!, which is 240. Florence and Jackie are running laps on a circular quarter-mile track. If Florence completes each lap in 75 seconds and Jackie completes each lap in 90 seconds, how many laps will Jackie have run before sheâ€™s passed by Florence, assuming they start at the same time. 1) 5 2) 6 3) 7 4) 8 5) 9 lcm of 75 and 90 is 450 and 450/90(time taken by jackie)=5 Grass in lawn grows equally thick in uniform rate. It takes 24 days for 70 cows and 60 days for 30 cows to eat the whole grass. how many cows are needed to eat the grass in 96 days? Ans- 20 I think this approach is very long to solve this problem i try to reduce some step g= grass initially, r= rate at which grass grow/day, c= cow eat grass/day g+24r=70*24c=1680c-----------1 g+60r=60*30c=1800c--------->g=1800c-60r----------------2 by solving this 2 equation we have relation c=(3/10)r-------------3 g+96r=96nc =>96nc=1800c-60r+96r=1800c+36r=1800c+120c=1920c =>n=20 find the 25 th word if the word RANKED is arranged in dictionary order...... AD_ _ _ _ 4*3*2*1 =24 next will be AEDKNR A,B ,C can do some work in 36days.A and B together do twice as much work as c alone, and A and C together can do thrice as much work as B alone..find the time taken by C to do whole work? a)96 b)108 c)120 d)72 ans:108. . Ans 108 1 day work for (A+B+C) = 1/36 ............(ii) Now, (A+B) = 2C ..........................(i) From (i)and(ii): 3C = 1/36 C = 1/108 Thus, C takes 108 days to complete the whole work alone. >At 12:00 hours jake starts to walk from his house at 6kms an hour. At 13:30 hours, paul follows him from jake's house on his bicycle at 8 kmph. When will jake be 3 kms behind paul?,ans:19.30. Ans- distance travelled y jake= 6*1.5= 9 Relativespeed=8-6=2kmph Time= (9 + 3 )/2= 6hr Time= 1:30+ 6:00= 7:30 hr 3>There are 5 men and 11 women. How many ways can a panel of 11 be formed such that the number of men is not more than 3,ans:2256. Ans- possible combination((0,11) or (1,10) or (2,9) or (3,8) ) Aplly permutation combination to get nswer 6 different chocolates 10 boxes how they can be arranged ,,Ans : 10^6. . if a^x=b, b^y=c, c^z=a ..then what is the value of x*y*z?ans:1. Ans- aplly log to all individual equation and find value of x, y,z and then multiply, get =1 SENIOR 36th word ans:ENOSRI. there are 60%male and 70% female are allowed for vote. 70% of male voter and 60% of female voter give vote to a particular person .what percentage of vote for him ,ans: 42. . ans- let 100 men and 100 women therefore 60 male 70 female now 70% of 60=42 60%of70=42 Therefore reuired %=(42+42)/200= .42 = 42 % when m+n is divided by 12 we get reminder 8.when m-n is divided by 12 we get reminder 6.if m>n then what will be the reminder when mn is divided by 12 ANS- 7 ans is 7 m+n/12=12x+8 m-n/12=12x+6 if x=1 then m+n=20 and m-n=18 as m>n so m=19 n=1 so mn/12=7 How many 6 digit even numbers can be formed from digits 1 2 3 4 5 6 7 so that the digit should not repeat and the second last digit is even?,ans:720 Logical reasoning hw many ways can 4 couple can cross a river,ans: 16. Ans- select 1 girl out of 4 select 1 boy out of 4 Multiply them i.e. 4x4=16 1.2.3 are 3 numbers maximum way it can be arranged,u ll get 1 point if 1 is at 1st positin or 2 is at 2nd position or 3 is at 3rd position(atleast) how many points ll u get,ans:4. Ans – possible ways= 123, 132, 321, 213 A series of book was published at 7 year intervals.When the 7th book was issued the sum of publication year is 13524.When was the 1st book published ? ,ans:1911. sol: sum of A.P. series=(n/2)(2*a+(n-1)*d) here sum=13524,n=7,d=7 substituting these values in above eq. n solving we get a=1911 which was the 1st book published year. George can do some work in 8hrs. paul can do the same work in 10hours while hari can do the same work in 12hrs. all the 3 of them start working at 9am while George stops work at 11am and the remaining two complete the work. Approximately at what time will the work be finished?,ans:2pm Ans efficiency of George= 15 efficiency of paul= 12 efficiency of hari= 10 total work= 120 work left after 2 hr= 120- ((12+15+10))x2=66 time=66/(10+12)= 3hr A and B are running around a circular track of length 120 meters with speeds 12 m/s and 6 m/s in the same direction. When will they meet for the first time? a)25 b)30 c)10 d)20 Ans whenever two people r running in same direction and they meet for first time=total distance/relative speed; same direction s1-s2;opposite direction s1+s2; 120/(12-6)=20 In a class there are 60% of girls of which 25% poor. What is the probability that a poor girl is selected is leader? a)15% b)50% c)80% d)30% Ans a) 15% probability for a girl to be leader is 60% becuase there are 60% girls in class. after this probability that tht girl will be poor is 25% of 60 ie..15% so answer is a) 15% x=(sin nx/n)=? a)90 b)0 c)1 d)2 ans:-b.) 0 Raj writes a number.He sees that athe number of two digits exceeds four times the sum of its digits by 3. If the number is increased by 18,the result is the same as the number formed by reversing the digits. Find the number Ans let the unit digit is y and tens digit is x now 10x+y=4(x+y)+3 .....(1) and 10y+x=10x+y+18 ......(2) by solving the eq.x=3 and y=5 so the ans is 35 15 students join a summer camp. every day 3 students are on duty after the school to clean classrooms. After the course it was found that every pair has been on duty exactly once. How many days does the course last for ? Anc 15C2 Raj invested in Indigo, HUL and SBI shares at Rs. 300, Rs. 200 and Rs. 5 per share, 100 shares for Rs. 10000. The number of Indigo and HUL shares he bought are respectively 1.15, 25 2.23, 17 3.17, 23 4.17, 60 Ans 17 and 23 would be the answer.as we have two equations 300x+200y+5z=10000 and x+y+z=100 if we solve these two equations we will get 295x+195y=9500 by puting x=17 and y=23,we will get the result at 9500 There is a tank,and two pipes A and B.A can fill the tank in 25 minutes and B can empty the tank in 20 minutes.If both pipes are opened at same time.how much time required the taank can fill? Ans Never will fill…. As B have more efficiency tha that of A [Q-11]If X^Y denotes X raised to the power of Y, find out last two digits of (2957^3661)+(3081^3643) Options o 42 o 38 o 98 o 22 Ans- 98 6 different chocolates 10 boxes how they can be arranged Ans : 10^6 ADEORV find the 45th word in dictionary order Ans- AEVODR if f(x) is a function for all real numbers x holds good for maximum of 2x+4 and 12+3x..then wat is the value of x so that f(x) is equal to 2x+4..Ans is -9 ans- let f1(x)=12+3x f1(-8)=-12 and f2(x)=2x+4 f2(-8)=-12 so ans is -8 2^74+2^2048+2^2n what is the value of n so that sum will be perfect square since, (a+b)^2=a^2+b^2+2ab 2^74+2^2n+2^2048=(2^37)^2+(2^n)^2+2^2048 2*(2^37)*(2^n)=2^2048 2^(1+37+n)=2^2048 since base are same ,equating powers,we get 1+37+n=2048 38+n=2048 n=2010 15 women or 10 men complete in 55 days..how many days 4 women and 5 men vil take Ans 10 men= 15 women 5 men= 15/2 women now,4 women and 5men 4women + 15/2 women 23/2 women (4women &5men = 23/2women) can complete a work in = (55*15*2)/23=71.739 days #1.A series of book was published at 7 seven year intervals. When the seventh book was published the total some of publication year was 13,524. First book was published in?? -sum of A.P ->Sn=n/2(2a+(n-1)d) where Sn=13524(given),n=7,d=7 13524=7/2(2*a+(7-1)7) a=1911 #2.How many motor vehicles registration number plates can be formed with the digits 1, 2, 3, 4, and 5(no digits being repeated) if it is given that registration number can have 1 to 5 digits? -1 digit = 5 ways 2 digit = (5*4)ways 3 digit = (5*4*3)ways 4 digit = (5*4*3*2)ways 5 digit = (5*4*3*2*1)ways total ways = 5+20+60+120+120= 325 ways. #3.Five rivers Ya-Fen,Volga,Yamuna,Hudson and Siene flow through the five cities v-z. New York,Paris,Moscow,Tokyo and Delhi,but not necessarily in same order.No two rivers flow in the same city and no two cities have same river flowing through them.Vogla flows through neither Moscow nor Delhi.Ya-Fen is neither in New York nor in Paris. Yamuna is neither in Delhi nor in Moscow.Hudson is in Moscow while new York has Yamuna flowing through it. Which of following is definitely FALSE? 1. If Volga flows through Tokyo,then Ya-Fen flows through Delhi. 2. If Tokyo has Volga,then Siene must flow through Delhi. 3. If Ya-Fen flows through Tokyo, then Siene must flow through Delhi. 4. Ya-Fen may flow through Tokyo or Delhi. -we have moscow=hudson ny=yamuna so if tokyo=volga nd siene= delhi den paris= yafen which is false option 2 is the FALSE statement! #4.If 9/12/2001 is sunday then what day of week will be on 9/12/9071???? -as it seems it is not a leap year so their r 365 days, so 9071-2001=7070 yrs r left and 368650 days r their in 7070 yrs. when 368650 is divided by 7 we get 0,that means sunday. sunday is ans. #5.Five people Pradeep,Qutub,Raj,Sandy and Tej are in a row. Raj sits to the immediate right of Tej. If Pradeep and Qutub have only two persons sitting between them, which of following must be true? 1. Pradeep is at one extreme end of row. 2. Qutub is sitting beside Pradeep. 3. Tej or Raj must be at one extreme end of the row. 4. Sandy must be at one of ends of the row. -The sitting order may be sandy,pradeep,tej,raj,qutub or pradeep,tej,raj,qutub,sandy or sandy,qutub,tej,raj,pradeep or qutub,tej,raj,pradeep,sandy in all the cases sandy is at one end of the row hence ANS:4 #6.find the number of zeroes in 1^1*2^2*3^3...48^48*49^49? -5^5+10^10+15^15....45^45 will have zeroes total=5+10+15+20+50+30+35+40+45=250. #7.A person has to go both Northwards & Southwards in search of a job. He decides to go by the first train he encounters.There are trains for every 15 min both southwards and northwards. First train towards south is at 6:00 A.M. and that towards North is at 6:10. If the person arrives at any random time, what is the probability that he gets into a train towards North.? -In a frequency of 15 mins if the person comes in first 10 mins he will catch northwards train.so the probability that he gets into a train towards North is 10/15=0.66 If he comes in the next 5 mins he will catch southwards train.so the probability of getting into a train towards southwards is 5/15=0.33 #8.Banu plays with two boxes seperately containing 25 red cards and 33 block cards and she allowed to move the cards between the boxes. She won the game by choose a box at random and a card at random from the chosen box the probability of getting a black card is minimized. Now,the question is to find that minimum probability. -to minimise probability of black card, put 1 red card in one box and 24 red cards and 33 black cards in second box. then min prob of getting black card = 0*1/2 + 0.5 * 33/57 = 11/38. #9.A completes a work in 20 days,B completes in 60 days and C in 45 days. all 3 prsn working 2gthr on a project got a profit of Rs 26000.what is profit of B?? -Profit will be divided in ratio of 1/20 : 1/60 : 1/45 or 9:3:4 profit of B = 26000*3/16 = 78000/16 = Rs. 4875. #9.My flight takes off at 2am from a place at 18N 10E and landed 10 Hrs later at a place with coordinates 36N70W.What is the local time when my plane landed? a) 6:00 am b) 6:40 am c) 7:40 am d) 7:00 am e) 8:00 am. -Latitudianal difference=(10+70)=80 So, time difference=80X4=320mins=5 hrs 20 mins Since the place is located towards west Therefore local time of that place during take off=8:40 p.m. So, local time during landing=8:40p.m.+10hrs=6:40a.m. #10.trucks A,B,C// A loads 10 kg/min B loads 13 1/3 kg/min C unloads 5kg/min If three simultaneously works then what is the time taken to load 2.4 tonnes?? -a loaded in 1 min =10kg// work day me diya hota to reciprocal lete but ye time me hain isliye direct jod //denge b loaded in 1 min=40/3kg c unloaded in 1 min=5kg so a,b,c are loaded in 1 min=55/3kg so 1 kg loaded by a,b,c in time =3/55min now, 2.4 tons loaded by a,b,c in time=(3/55)*2.4*1000=131min . #11.I was talking to a friend Tammy, the middle child of five. she has a younger sister Tracey, and three brothers Tommy (oldest child), Timmy and Tony (youngest child). I asked her how old she was, but she hardly ever gives a straight answer. The following is the conversation we had. "How old are you Tammy?" "I am three times as old as Timmy was born. Actually, it's a funny, but Timmy's,tracey's, Tommy's and my ages were all factors of Mum's age when Tony was born. The only other year where the ages of more than two of us were factors of Mum's age was when Tracey was half as old as I am now." "How old is your Mum now?" "There's another funny thing. when Mum turns 50, tony will be the same age Mum was when she gave birth to Tommy" Assuming Tammy was talking in terms of whole years, and with my mathematical and reasoning abilities i was able to quickly deduce not just Tammy's age, but all of thier ages. so how old are Tommy,Timmy,Tammy,Tracey,Tony and Mum Now? a) Tommy is 26, Timmy is 17, Tammy is 16,Tracey is 13, tony is 12 and Mum is 43 b) Tommy is 23, Timmy is 19, Tammy is 18,Tracey is 16, tony is 13 and Mum is 43 c) Tommy is 28, Timmy is 21, Tammy is 23,Tracey is 19, tony is 17 and Mum is 43 d) Tommy is 21, Timmy is 20, Tammy is 19,Tracey is 17, tony is 15 and Mum is 43. -the ans is b..based on the options given.because.in this opetion when we subtract 13 from each age then all the ages are facotros of moms age. #12.Seven Applicants for a Job - F, G, H, I, J, K, and L are each to be interviewed on monday or Tuesday . Each Applicant is to be Interviewed exactly once, and the interview schedule must conform to the following conditions: There must be at least two interviews each day I cannot have the first interview on Monday F must have the first or last interview on either Monday or Tuesday H and L must interview on the same day, with no other applicant interviewing between them I and K cannot interview on the same day. At least one interview must be scheduled between the interviews of G and L If G has very First interview , and if exactly four people are interviewed on Tuesday, which of the following could be the schedule for the two interviews immediately following G's? A) First H, then L B) First L, then F C) First I, Then L D) First F, Then I -Answer C First I than L Monday Tuesday G F I K L H=L. #13.A,B and C can do a piece of work alone in 15 , 25 and 40 days respectively.They start together but B left the work as soon as the work was half completed and c left the work when it was 3/4th completed. A doing the remaining work alone. In how many days the work completed? -ans 300/79+30/11+15/4=10.2747 days. #14.there is a 14*8 grid.what is the total number of squares and rectangles? -square:- 14*8+(14-1)(8-1)+(14-2)(8-2)+(14-3)(8-3)+(14-4)(8-4)+(14-5)(8-5)+(14-6)(8-6)+(14-7)(8-7)=420 rectangle:- (1+2+3+4+5+6+7+8+9+10+11+12+13+14)(1+2+3+4+5+6+7+8)=105*36=3780 total: 420+3780= 4200. #15.what is the remainder when 1!+2!+3!.....100! is divided by 9? all the numbers after 6! are divisible by 9. remainder will be same as obtained when 1!+2!+3!.....5! is divided by 9 which is when (1+2+6+24+120)is divided by 9 or 153 is divided by 9. so zero(0) is the remainder as 153/9 =17 //zero is theanswer -9! is 241920,which is multiple of 9 also 9! divides n! for all n>9 so, we can compute remainder just by looking 1!+2!+3!+4!+5!+6!+7!+8!=30873 which has remainder 3 on dividing by 9. #16.what is the ramainder when 50! is divided by 47*47? first understand dis (6*10*55)/9 remainder of this is 6. there r two ways to calculate this 1--- 6*10*55 ie 3300 mod 9 = 6 2--- 6 mod 9 =6 10 mod 9 =1 55 mod 9=1 thn multiply all the rem n u get the ans ie 6 in case of 47^2 50*49*48*47*46! / 47^2 47 will b cancel out 50*49*48*46 ! / 47 nw find out remainder individually 50 % 47 =3 49 % 47 =2 48 % 47=1 and acc to wilson theorem if p is a prime no thn (p-1)! mod p =-1 here 47 is a prime no so 46!% 47 = -1 or 46 so multiply all the rem 3 * 2*1* 46 mod 47 = 276 mod 47 = 41 nw multiply 41 with 47 so we get 41*47= 1927...ans -the ans will be 1927... 50! can be written as 50*48*47*46!*49/47*47 50*48*49*(47-1)!/47 and (n-1)!/n=-1 so reminder when 47 is divided by 50 is 3,by 48 is 1 and by 49 is 2 and by (47-1)! is -1 so total is -6 and +ve reminder is 41 and we have to multiply it by 47 becoz first we cut it so final ans is 41*47=1927. #17.the first 44 positive integers are written in the order to form the largest number n=123456789101111213..424344. what will be the remainder when n is divided by 45? -no will be div by 45 when it is divided y both 9 and 5 a no is div by 9 if sum of digits is div by 9 now sum of digits of given no is 270 so the no is div by 9 and since last digit is 4 when div by 5 it leaves rem 4 let us say numbers such as 9,54... we see they are div by 9 but leaves rem 4 when div by 5.. so when div by 45 they leave rem 9 so req ans=9 . #18.Given 2^74 + 2^2074 + 2^2n. For what value of(among the options)n will the sum be a perfect square? -we know, (a+b)^2= a*a+b*b+2a*b so 2^74+2^2074+2^2n= (2^37)^2 + 2*2^37*2^n + (2^n)^2 So, we get 38+n=2074 therefoe n = 2036. #19. 50!/(16^15) Then What is the reminder? -there should be 47 2s in 50! and we can write 16^15 as 2^47 then there will remain 2^(60-47)in denominator now in numerator the last digit will be 5 because all the 2s has removed now 2^13=8192 when numerator is divided by 8192 the remainder would be 1. because the last digit of denominator is 2.......... #20.There was two problem on rolling of 3dice and what is the probability if the sum of outcome is 13,10? -for sum as 10 p= 27/216 13 p= 21/216 total p=49/216. #21.if a divided by 5,3,2 respectively give the remainder 1,2,0.what is the no when is divided by 2 3, 5? when the no is devided by 5 it gives remainder 1 so the possible no can be determined by (5*1)+1=6 (5*2)+1=11 (5*3)+1=16 (5*4)+1=21 (5*5)+1=26----so on among them only 26 gives the remainder 2 and 0 when it is devided by 3 and 2 so the no is 26. -11 is the no. 0*2+0*3+2*5+1=11 rem= 1,2,1 11/2= rem 1 5/3= 2 1/5 = 1. its not what is the no. its what is the remainder when the same no. is divided by 2,3,5 respectively. so let the no. be x then x=5y+1 y=3z+2 z=2*1+0 thus x=41 now on dividing 41by 2,3,5 respectively we get 1,2,1 as remainder.... #22.From a circle of radius 10 cm a sector of 40% area is removed. a cone is formed with the remaining part of the circle. find the ratio of the radius and height of the cone? > 1/2 >2/3 >3/4 >4/3. -major sector length= 160pi/9. it is = to base of cone i.e bottom circle. =>2 pi r= 160pi/9 => r=80/9 slant height of cone= 10 h=sqrt(10^2-(80/9)^2)) h=(10sqrt17)/4 r/h= (9sqrt17)/32 r/h= 1.154> 3/4 . #23.A, B and C are friends and went on a picnic. A bought 5 sandwiches and B bought 3 for some amount. C paid Rs.4 as his share. Find the amount B vl get in return > 1.5 >1.4 >0.5 >1.0 -total sandwiches = 8 total money invested = 3*4 ( since each of a,b and c should pay equally) price of 1 sandwich = 3*4/8 = 1.5 so, A invested 7.5 rs, B invested 4.5, so he should get 0.5 back. #24.how many 2's are there between 223 to 2222? - from 201-222 there are 28 2's and from 201-300 there are 120 2's so from 223-300 there are 120-28=92 2's now from 301-1000 there are 7*20=140 2's now from 1000-2000 there are 300 2's now we left with 2001-2222 now consider only thousands place there are 222 2's and from 000 to 222 there are 20+21+28=69 2's so total=92+140+300+222+69=823ans. #25.In a staircase, there are 10 steps. A child is attempting to climb the staircase. Each time, she can either make 1 step or 2 steps. In how many different ways can she climb the stair case? a. 10 b. 21 c. 36 d. None of these. -ans d none of these.\ #26.what is the value of 1(1!)+2(2!)+3(3!)+.......+2012(2012!)? -1(1!)+2(2!)=5(3!-1) 1(1!)+2(2!)+3(3!)=23(4!-1) ... ... 1(1!)+2(2!)+3(3!)+.....+2012(2012!)=2013!-1. #27.Probability of 1 ace which is red and one spade? -total no of ace=4 two red(1diamond,1artin),two black(1spade,1cluver) prob getting red ace=2/4 prob getting spade ace=1/4 prob 1 ace which is red and one spade(and means multiplication)=2/4*1/4 1/2*1/4=1/8. #28.find the average speed if a man travles at speed 24kmph up and 36kmph down at an altitude of 200m? a) 28.8 mph b)27.8mph c)27.5mph d)30mph. -a=24kmph b=36kmph avegage speed=2ab/(a+b) altitude has nothing to do with average speed,.28.5. PAGE NO 5 : 1. 1! + 2! + â€¦â€¦. + 50!=? a)3.1035*10^64 b)2.1021*10^65 c)3.1035*10^63 d)3.1035*10^62 ans- (a) what is the reminder when 6^17+17^6 is divided by 7 ? a) 3 b) 0 c) 6 d) 1 ans --- (B) 6^2=1 mod 7 [(6^2)=36,divided by 7 gives remainder 1] =>(6)^2*8=1^8 mod 7 =>6^16=1 mod 7 [all even power also gives 1 mod 7] =>(6^16)*6=1*6 mod 7 So, 6^17=6 mod 7. Likewise, 17^2=2 mod 7 =>(17)^2*3=2^3 mod 7 =>17^6=8 mod 7 [simplify: 8 is one time divisible by 7 & gives remainder 1] So, 17^6=1 mod 7. ((6^17+17^6)/7)= ((6 mod 7)+(1 mod 7))/7 =>((6+1) mod 7)/7 =>(7 mod 7)/7 =>Hence the remainder is 0 Next Question: what is the remainder of 100!/(97^2)?

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