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  • UPSC Prelims PYQs
  • Mathematics
    • Number System

      • How many 3-digit natural numbers (without repetition of digits) are there such that each digit is odd and the number is divisible by 5?

      • Question: Is x an integer?
         Statements: 1. x/3 is not an integer. 
         Statements: 2. 3x is an integer.
         Which one of the following is correct in respect of the Question and the Statements?

      • 1. A has some coins. He gives half of the coins and 2 more to B.
         2. B gives half of the coins and 2 more to C.
         3. C gives half of the coins and 2 more to D.
         4. The number of coins D has now, is the smallest two-digit number.
         5. How many coins does A have in the beginning?

      • Let p be a two-digit number and q be the number consisting of same digits written in reverse order. If pq = 2430 then what is the difference between p and q?

      • Consider the following statements in respect of two natural numbers p and q such that p is a prime number and q is a composite number:
         1. pq can be an odd number.
         2. q/p can be a prime number.
         3. p + q can be a prime number.
         Which of the above statements are correct?

      • The sum of three consecutive integers is equal to their product. How many such possibilities are there?

      • What is the number of numbers of the form 0-XY, where X and Y are distinct non-zero digits?

      • Consider all 3-digit numbers (without repetition of digits) obtained using three non-zero digits which are multiples of 3. Let S be their sum. Which of the following is/are correct?
         1. S is always divisible by 74.
         2. S is always divisible by 9.
         Select the correct answer using the code given below:

      • Consider the following statements:
         1. The sum of 5 consecutive integers can be 100.
         2. The product of three consecutive natural numbers can be equal to their sum.
         Which of the above statements is/are correct?

      • The difference between a 2-digit number and the number obtained by interchanging the positions of the digits is 54. Consider the following statements:
         1. The sum of the two digits of the number can be determined only if the product of the two digits is known.
         2. The difference between the two digits of the number can be determined.
         Which of the above statements is/are correct?

      • When a certain number is multiplied by 7, the product entirely comprises ones only (1111...). What is the smallest such number?

      • Let p, q, r and s be natural numbers such that p - 2016 = q + 2017 = r - 2018 = s + 2019. Which one of the following is the largest natural number?

      • How many five-digit prime numbers can be obtained by using all the digits 1, 2, 3, 4 and 5 without repetition of digits?

      • In the sum +1+58+8+1=188 for which digit does the symbol & stand?

      • Let A3BC and DE2F be four-digit numbers where each letter represents a different digit greater than 3. If the sum of the numbers is 15902, then what is the difference between the values of A and D?

      • How many integers are there between 1 and 100 which have 4 as a digit but are not divisible by 4?

      • What is the remainder when 51x27x35x62 x 75 is divided by 100?

      • For what value of n, the sum of digits in the number (10n + 1) is 2?

      • The number of times the digit 5 will appear while writing the integers from 1 to 1000 is

      • The ratio of a two-digit natural number to a number formed by reversing its digits is 4: 7. The number of such pairs is

      • A printer numbers the pages of a book starting with 1 and uses 3089 digits in all. How many pages does the book have?

      • Consider two statements S1 and S2 followed by a question: 
            S1: p and q both are prime numbers.
            S2: p + q is an odd integer.

        Question: Is pq an odd integer? 

        Which one of the following is correct?

      • Number 136 is added to 5B7 and the sum obtained is 7A3, where A and B are integers. It is given that 7A3 is exactly divisible by 3. The only possible value of B is

      • How many triplets (x, y, z) satisfy the equation x+y+z=6, where x, y and z are natural numbers?

      • An 8-digit number 4252746B leaves remainder 0 when divided by 3. How many values of B are possible ?

      • Consider the following sum: +1+2+3+1=21. In the above sum, stands for

      • X and Y are natural numbers other than 1, and Y is greater than X. Which of the following represents the largest number?

      • A number consists of three digits of which the middle one is zero and their sum is 4. If the number formed by interchanging the first and last digits is greater than the number itself by 198, then the difference between the first and last digits is

      • While writing all the numbers from 700 to 1000, how many numbers occur in which the digit at hundred's place is greater than the digit at ten's place, and the digit at ten's place is greater than the digit at unit's place?

      • If xy=8, then which of the following must be true?
         1. Both x and y must be positive for any value of x and y.
         2. If x is positive, y must be negative for any value of x and y
         3. If x is negative, y must be positive for any value of x and y
         Select the correct answer using the code given below.

      • There are thirteen 2-digit consecutive odd numbers. If 39 is the mean of the first five such numbers, then what is the mean of all the thirteen numbers?

      • Certain 3-digit numbers have the following characteristics: 
             a. All the three digits are different.
             b. The number is divisible by 7.
             c. The number on reversing the digits is also divisible by 7.
        How many such 3-digit numbers are there?

      • If R and S are different integers both divisible by 5, then which of the following is not necessarily true?

      • The letters L, M, N, O, P, Q, R, S, and T in their order are substituted by nine integers 1 to 9 but not in that order. 
        1. 4 is assigned to P. 
        2. The difference between P and T is 5. 
        3. The difference between N and T is 3. 
        4. What is the integer assigned to N?

      • A gardener has 1000 plants. He wants to plant them in such a way that the number of rows and the number of columns remains the same. What is the minimum number of plants that he needs more for this purpose?

    • LCM AND HCF

      • What is the smallest number greater than 1000 that when divided by any one of the numbers 6, 9, 12, 15, 18 leaves a remainder of 3?

      • If you have two straight sticks of length 7.5 feet and 3.25 feet, what is the minimum length can you measure?

      • What is the greatest length x such that 3 1/2 m and 8 3/4 m are integral multiples of x?

      • What is the least four-digit number when divided by 3, 4, 5 and 6 leaves a remainder 2 in each case?

      • In a school every student is assigned a unique identification number. A student is a football player if and only if the identification number is divisible by 4, whereas a student is a cricketer if and only if the identification number is divisible by 6. If every number from 1 to 100 is assigned to a student, then how many of them play cricket as well as football?

      • Five persons fire bullets at a target at an interval of 6, 7, 8, 9 and 12 seconds respectively. The number of times they would fire the bullets together at the target in an hour is

      • A bell rings every 18 minutes. A second bell rings every 24 minutes. A third bell rings every 32 minutes. If all the three bells ring at the same time at 8 o'clock in the morning, at what other time will they all ring together?

      • Three persons start walking together and their steps measure 40 cm, 42 cm and 45 cm respectively. What is the minimum distance each should walk so that each can cover the same distance in complete steps?

    • Fraction
    • Elementary Algebra

      • Which number amongst 240, 321, 418 and 812 is the smallest?

      • What is the remainder when 91 × 92 × 93 × 94 × 95 × 96 × 97 × 98 × 99 is divided by 1261?

      • If 15 × 14 × 13 × … × 3 × 2 × 1 = 3m × n where m and n are positive integers, then what is the maximum value of m?

      • If 32019 is divided by 10, then what is the remainder?

      • The number 3798125P369 is divisible by 7. What is the value of the digit P?

      • Integers are listed from 700 to 1000. In how many integers is the sum of the digits 10?

      • Consider the following addition problem: 3P+4P+PP+PP = RQ2; where P, Q and R are different digits. What is the arithmetic mean of all such possible sums?

      • Consider the following multiplication problem: (PQ) x 3 = RQQ, where P, Q and R are different digits and R ≠ 0. What is the value of (P+R)  ÷ Q?

      • What is the largest number among the following?

      • There are three pillars X, Y and Z of different heights. Three spiders A, B and C start to climb on these pillars simultaneously. In one chance, A climbs on X by 6 cm but slips down 1 cm. B climbs on Y by 7 cm but slips down 3 cm. C climbs on Z by 6.5 cm but slips down 2 cm. If each of them requires 40 chances to reach the top of the pillars, what is the height of the shortest pillar?

      • If for a sample data Mean < Median < Mode then the distribution is

      • What is the total number of digits printed, if a book containing 150 pages is to be numbered from 1 to 150?

      • There are some nectar-filled flowers on a tree and some bees are hovering on it. If one bee lands on each flower, one bee will be left out. If two bees land on each flower, one flower will be left out. The number of flowers and bees respectively are:

      • A person is standing on the first step from the bottom of a ladder. If he has to climb 4 more steps to reach exactly the middle step, how many steps does the ladder have?

      • If ABC x DEED = ABCABC; where A, B, C, D and E are different digits, what are the values of D and E?

    • Average

      • The average weight of A, B, C is 40 kg. The average weight of B, D, E is 42 kg and the weight of F is equal to that of B. What is the average weight of A, B, C, D, E and F?

      • There are two Classes A and B having 25 and 30 students respectively. In Class-A the highest score is 21 and lowest score is 17. In Class-B the highest score is 30 and lowest score is 22. Four students are shifted from Class-A to Class-B.

        Consider the following statements: 
         1. The average score of Class-B will definitely decrease.
         2. The average score of Class-A will definitely increase.
         Which of the above statements is/are correct?

      • Consider the following data:

          Average marks in English Average marks in Hindi
        Girls 9 8
        Boys 8 7
        Overall average marks 8.8 x

        What is the value of x in the above table?

      • In a class, there are three groups A, B and C. If one student from group A and two students from group B are shifted to group C, then what happens to the average weight of the students of the class?

      • The average score of a batsman after his 50th innings was 46.4. After 60th innings, his average score increases by 2.6. What was his average score in the last ten innings?

      • The average marks of 100 students are given to be 40. It was found later that marks of one student were 53 which were misread as 83. The corrected mean marks are

      • A family has two children along with their parents. The average of the weights of the children and their mother is 50 kg. The average of the weights of the children and their father is 52 kg. If the weight of the father is 60 kg, then what is the weight of the mother?

      • Sunita cuts a sheet of paper into three pieces. Length of first piece is equal to the average of the three single digit odd prime numbers. Length of the second piece is equal to that of the first plus one-third the length of the third. The third piece is as long as the other two pieces together. The length of the original sheet of paper is

      • Suppose the average weight of 9 persons is 50 kg. The average weight of the first 5 persons is 45 kg, whereas the average weight of the last 5 persons is 55 kg. Then the weight of the 5th person will be

      • The average rainfall in a city for the first four days was recorded to be 0.40 inch. The rainfall on the last two days was in the ratio of 4:3. The average of six days was 0.50 inch. What was the rainfall on the fifth day?

      • The monthly average salary paid to all the employees of a company was Rs. 5000. The monthly average salary paid to male and female employees was Rs. 5200 and Rs. 4200 respectively. Then the percentage of males employed in the company is:

      • The average monthly income of a person in a certain family of 5 is Rs. 10,000. What will be the average monthly income of a person in the same family if the income of one person increased by Rs. 1,20,000 per year?

      • Consider the following information regarding the performance of a class of 1000 students in four different tests:

        Tests

        I

        II

        III

        IV

        Average marks

        60

        60

        70

        80

        Range of marks

        30

        To

        90

        45

        To

        75

        20

        To

        100

        0

        To

        100

        If a student scores 74 marks in one of the four tests, in which one of the following tests is her performance the best comparatively?

      • A student on her first 3 tests received an average score of N points. If she exceeds her previous average score by 20 points on her fourth test, then what is the average score for the first 4 tests?

    • Percentage

      • Two candidates X and Y contested an election. 80% of voters cast their vote and there were no invalid votes. There was no NOTA (None of the above) option. X got 56% of the votes cast and won by 1440 votes. What is the total number of voters in the voters list?

      • In a class, 60% of students are from India and 50% of the students are girls. If 30% of the Indian students are girls, then what percentage of foreign students are boys?

      • A student appeared in 6 papers. The maximum marks are the same for each paper. His marks in these papers are in the proportion of 5 : 6 : 7 : 8 : 9 : 10. Overall he scored 60%. In how many number of papers did he score less than 60% of the maximum marks?

      • Half of the villagers of a certain village have their own houses. One-fifth of the villagers cultivate paddy. One-third of the villagers are literate. Four-fifth of the villagers are under 25 years of age. Which one of the following statements is certainly correct?

      • P scored 40 marks more than Q in an examination. If Q scored 10% less marks than P, then how much did Q score?

      • A person bought a car and sold it for ₹3,00,000. If he incurred a loss of 20%, then how much did he spend to buy the car?

      • In adult population of a city, 40% men and 30% women are married. What is the percentage of married adult population if no man marries more than one woman and no woman marriєв more than one man; and there are no widows and widowers?

      • A and B are two heavy steel blocks. If B is placed on the top of A, the weight increases by 60%. How much weight will reduce with respect to the total weight of A and B, if B is removed from the top of A?

      • Raju has ₹9000 with him and he wants to buy a mobile handset; but he finds that he has only 75% of the amount required to buy the handset. Therefore, he borrows ₹2000 from a friend. Then

      • All members of a club went to Mumbai and stayed in a hotel. On the first day, 80% went for shopping and 50% went for sightseeing, whereas 10% took rest in the hotel. Which of the following conclusion(s) can be drawn from the above data?
         1. 40% members went for shopping as well as sightseeing.
         2. 20% members went for only shopping.

        Select the correct anser using the code given below:

      • In an examination, A has scored 20 marks more than B. If B has scored 5% less marks than A, how much has B scored?

      • If the numerator and denominator of a proper fraction are increased by the same positive quantity which is greater than zero, the resulting fraction is

      • A student has to get 40% marks to pass in an examination. Suppose he gets 30 marks and fails by 30 marks, then what are the maximum marks in the examination?

      • P (40% of A) + (65% of B) and Q = (50% of A) + (50% of B), where A is greater than B. In this context, which of the following statements is correct?

      • In a city, 12% of households earn less than ₹ 30,000 per year, 6% households earn more than ₹ 2,00,000 per year, 22% households earn more than ₹ 1,00,000 per year and 990 households earn between ₹ 30,000 and ₹ 1,00,000 per year. How many households earn between ₹ 1,00,000 and ₹ 2,00,000 per year?

      • Anita's mathematics test had 70 problems carrying equal marks i.e., 10 arithmetic, 30 algebra and 30 geometry. Although she answered 70% of the arithmetic, 40% of the algebra and 60% of the geometry problems correctly, she did not pass the test because she got less than 60% marks. The number of more questions she would have to answer correctly to earn a 60% passing marks is:

      • Two numbers X and Y are respectively 20% and 28% less than a third number Z. By what percentage is the number Y less than the number X?

      • In a test, a candidate attempted only 8 questions and secured 50% marks in each of the questions. If he obtained a total of 40% in the test and all questions in the test carried equal marks, how many questions were there in the test?

      • In a town, 45% population read magazine A, 55% read magazine B, 40% read magazine C, 30% read magazines A and B, 15% read magazines B and C, 25% read magazines A and C; and 10% read all the three magazines. What percentage do not read any magazine?

      • Candidates in a competitive examination consisted of 60% men and 40% women. 70% men and 75% women cleared the qualifying test and entered the final test where 80% men and 70% women were successful. Which of the following statements is correct?

      • A and B decide to travel from place X to place Y by bus. A has ₹10 with him and he finds that it is 80% of the bus fare for two persons. B finds that he has ₹3 with him and hands it over to A. In this context, which one of the following statements is correct?

      • As per agreement with a bank, a businessman had to refund a loan in some equal instalments without interest. After paying 18 instalments he found that 60 percent of his loan was refunded. How many instalments were there in the agreement?

      • There are 100 students in a particular class. 60% students play cricket, 30% student play football and 10% students play both the games. What is the number of students who play neither cricket nor football?

      • In a group of persons, 70% of the persons are male and 30% of the persons are married. If two-sevenths of the males are married, what fraction of the females is single?

    • Order of Magnitude
    • Profit, Loss and Discount

      • The increase in the price of a certain item was 25%. Then the price was decreased by 20% and then again increased by 10%. What is the resultant increase in the price?

      • If the price of an article is decreased by 20% and then the new price is increased by 25%, then what is the net change in the price?

      • A shop owner offers the following discount options on an article to a customer:
         1. Successive discounts of 10% and 20%, and then pay a service tax of 10%
         2. Successive discounts of 20% and 10%, and then pay a service tax of 10%
         3. Pay a service tax of 10% first, then successive discounts of 20% and 10%
         Which one of the following is correct?

      • Rakesh had money to buy 8 mobile handsets of a specific company. But the retailer offered very good discount on that particular handset. Rakesh could buy 10 mobile handsets with the amount he had. What was the discount the retailer offered?

      • A bookseller sold 'a' number of Geography textbooks at the rate of ₹x per book, 'a + 2' number of History textbooks at the rate of ₹(x - 2) per book and 'a - 2' number of Mathematics textbooks at the rate of ₹(x - 2) per book. What is his total sale in ₹?

      • A shopkeeper sells an article at ₹40 and gets X% profit. However, when he sells it at ₹20, he faces same percentage of loss. What is the original cost of the article?

      • A person allows a 10% discount for cash payment from the marked price of a toy and still he makes a 10% gain. What is the cost price of the toy which is marked Rs. 770?

      • What will be the expected monthly consumption (in litres) if the price goes up to ₹ 80 per litre, given the following price-consumption relationship? 
        Price (₹ per litre):                           40, 50, 60, 75 
        Monthly consumption (in litres):   60, 48, 40, 32 
        If the price goes up to ₹ 80 per litre, his expected consumption (in litres) will be: 

      • A person ordered 5 pairs of black socks and some pairs of brown socks. The price of a black pair was thrice that of a brown pair. While preparing the bill, the bill clerk interchanged the number of black and brown pairs by mistake which increased the bill by 100%. What was the number of pairs of brown socks in the original order?

      • A cow costs more than 4 goats but less than 5 goats. If a goat costs between ₹600 and ₹800, which of the following is a most valid conclusion?

      • If Sohan, while selling two goats at the same price, makes a profit of 10% on one goat and suffers a loss of 10% on the other

      • For a charity show, the total tickets sold were 420. Half of these tickets were sold at the rate of ₹5 each, one-third at the rate of ₹3 each and the rest for ₹2 each. What was the total amount received ?

      • A sum of ₹700 has to be used to give seven cash prizes to the students of a school for their overall academic performance. If each prize is ₹20 less than its preceding prize, what is the least value of the prize ?

      • A person can walk a certain distance and drive back in six hours. He can also walk both ways in 10 hours. How much time will he take to drive both ways?

    • Ratio, Proportion, Partnership and Mixture

      • When 70% of a number x is added to another number y, the sum becomes 165% of the value of y. When 60% of the number x is added to another number z, then the sum becomes 165% of the value of z. Which one of the following is correct?

      • There are two containers X and Y. X contains 100 ml of milk and Y contains 100 ml of water. 20 ml of milk from X is transferred to Y. After mixing well, 20 ml of the mixture in Y is transferred back to X. If m denotes the proportion of milk in X and n denotes the proportion of water in Y, then which one of the following is correct?

      • Jay and Vijay spent an equal amount of money to buy some pens and special pencils of the same quality from the same store. If Jay bought 3 pens and 5 pencils, and Vijay bought 2 pens and 7 pencils, then which one of the following is correct?

      • An amount of money was distributed among A, B and C in the ratio p: q: r. Consider the following statements:
         1. A gets the maximum share if p is greater than (q+r).
         2. C gets the minimum share if r is less than (p+q).
         Which of the above statements is/are correct?

      • A sum of ₹2,500 is distributed among X, Y and Z in the ratio1/2 : 3/4 : 5/6. What is the difference between the maximum share and the minimum share?

      • A bottle contains 20 litres of liquid A. 4 litres of liquid A is taken out of it and replaced by same quantity of liquid B. Again 4 litres of the mixture is taken out and replaced by same quantity of liquid B. What is the ratio of quantity of liquid A to that of liquid B in the final mixture?

      • There is a milk sample with 50% water in it. If 1/3rd of this milk is added to equal amount of pure milk, then water in the new mixture will fall down to

      • P works thrice as fast as Q, whereas P and Q together can work four times as fast as R. If P, Q and R together work on a job, in what ratio should they share the earnings?

      • The monthly incomes of X and Y are in the ratio of 4:3 and their monthly expenses are in the ratio of 3:2. However, each saves 6,000 per month. What is their total monthly income?

      • 30 g of sugar was mixed in 180 ml water in a vessel A, 40 g of sugar was mixed in 280 ml of water in vessel B and 20 g of sugar was mixed in 100 ml of water in vessel C. The solution in vessel B is:

      • In aid of charity, every student in a class contributes as many rupees as the number of students in that class. With the additional contribution of Rs. 2 by one student only, the total collection is Rs. 443. Then how many students are there in the class ?

      • The total emoluments of two persons are the same, but one gets allowances to the extent of 65% of his basic pay and the other gets allowances to the extent of 80% of his basic pay. The ratio of the basic pay of the former to the basic pay of the latter is:

      • Two equal glasses of same type are respectively 1/3 and 1/4 full of milk. They are then filled up with water and the contents are mixed in a pot. What is the ratio of milk and water in the pot?

      • The monthly incomes of Peter and Paul are in the ratio of 4: 3. Their expenses are in the ratio of 3: 2. If each saves ₹6,000 at the end of the month, their monthly incomes respectively are (in ₹)

      • In a rare coin collection, there is one gold coin for every three non-gold coins. 10 more gold coins are added to the collection and the ratio of gold coins to non-gold coins would be 1:2. Based on the information, the total number of coins in the collection now becomes

      • Out of 120 applications for a post, 70 are male and 80 have a driver's license. What is the ratio between the minimum to maximum number of males having driver's license ?

      • Two glasses of equal volume are respectively half and three-fourths filled with milk. They are then filled to the brim by adding water. Their contents are then poured into another vessel. What will be the ratio of milk to water in this vessel?

    • Time and Work

      • 24 men and 12 women can do a piece of work in 30 days. In how many days can 12 men and 24 women do the same piece of work?

      • A man completes 7/8 of a job in 21 days. How many more days will it take him to finish the job if quantum of work is further increased by 50%?

      • A person X can complete 20% of work in 8 days and another person Y can complete 25% of the same work in 6 days. If they work together, in how many days will 40% of the work be completed?

      • A lift has the capacity of 18 adults or 30 children. How many children can board the lift with 12 adults?

      • Ram and Shyam work on a job together for four days and complete 60% of it. Ram takes leave then and Shyam works for eight more days to complete the job. How long would Ram take to complete the entire job alone ?

      • W can do 25% of a work in 30 days, X can do 1/4 of the work in 10 days, Y can do 40% of the work in 40 days and Z can do 1/3 of the work in 13 days. Who will complete the work first ?

      • Two pipes A and B can independently fill a tank completely in 20 and 30 minutes respectively. If both the pipes are opened simultaneously, how much time will they take to fill the tank completely?

      • In a garrison, there was food for 1000 soldiers for one month. After 10 days, 1000 more soldiers joined the garrison. How long would the soldiers be able to carry on with the remaining food?

      • The tank-full petrol in Arun's motor-cycle lasts for 10 days. If he starts using 25% more everyday, how many days will the tank-full petrol last?

      • A contract on construction job specifies a penalty for delay in completion of the work beyond a certain date is as follows: ₹200 for the first day, ₹250 for the second day, ₹300 for the third day etc., the penalty for each succeeding day being ₹50 more than that of the preceding day. How much penalty should the contractor pay if he delays the work by 10 days?

      • A village having a population of 4000 requires 150 litres of water per head per day. It has a tank measuring 20 m × 15 m × 6 m. The water of this tank will last for

    • Speed, Time and Distance

      • X and Y run a 3 km race along a circular course of length 300 m. Their speeds are in the ratio 3:2. If they start together in the same direction, how many times would the first one pass the other (the start-off is not counted as passing)?

      • On one side of a 1.01 km long road, 101 plants are planted at equal distance from each other. What is the total distance between 5 consecutive plants?

      • A man started from home at 14:30 hours and drove to village, arriving there when the village clock indicated 15:15 hours. After staying for 25 minutes, he drove back by a different route of length 1.25 times the first route at a rate twice as fast reaching home at 16:00 hours. As compared to the clock at home, the village clock is

      • A person X from a place A and another person Y from a place B set out at the same time to walk towards each other. The places are separated by a distance of 15 km. X walks with a uniform speed of 1.5 km/hr and Y walks with a uniform speed of 1 km/hr in the first hour, with a uniform speed of 1-25 km/hr in the second hour and with a uniform speed of 1-5 km/hr in the third hour and so on. Which of the following is/are correct?
         1. They take 5 hours to meet.
         2. They meet midway between A and B.
         Select the correct answer using the code given below:

      • A car travels from a place X to place Y at an average speed of v km / hr from Y to X at an average speed of 2v km / hr again from X to Y at an average speed of 3v km / hr and again from Y to X at an average speed of 4v km / hr Then the average speed of the car for the entire journey

      • A man takes half time in rowing a certain distance downstream than upstream. What is the ratio of the speed in still water to the speed of current?

      • When a runner was crossing the 12 km mark in a race, she was told that she had completed only 80% of the race. What was the total distance of the race?

      • X, Y and Z are three contestants in a race of 1000 m. Assume that all run with different uniform speeds. X gives Y a start of 40 m and X gives Z a start of 64 m. If Y and Z were to compete in a race of 1000 m, how many metres start will Y give to Z?

      • The figure drawn below gives the velocity graphs of two vehicles A and B. The straight line OKP represents the velocity of vehicle A at any instant, whereas the horizontal straight line CKD represents the velocity of vehicle B at any instant. In the figure, D is the point where perpendicular from P meets the horizontal line CKD such that 1 PD=LD: 2 What is the ratio between the distances covered by vehicles A and B in the time interval OL?

      • A train 200 metres long is moving at the rate of 40 kmph. In how many seconds will it cross a man standing near the railway line?

      • Two persons, A and B are running on a circular track. At the start, B is ahead of A and their positions make an angle of 30 deg at the centre of the circle. When A reaches the point diametrically opposite to his starting point, he meets B. What is the ratio of speeds of A and B, if they are running with uniform speeds?

      • A freight train left Delhi for Mumbai at an average speed of 40 km/hr. Two hours later, an express train left Delhi for Mumbai, following the freight train on a parallel track at an average speed of 60 km/hr. How far from Delhi would the express train meet the freight train?

      • A daily train is to be introduced between station A and station B starting from each end at 6 AM and the journey is to be completed in 42 hours. What is the number of trains needed in order to maintain the shuttle service?

      • A and B walk around a circular park. They start at 8 a.m. from the same point in the opposite directions. A and B walk at a speed of 2 rounds per hour and 3 rounds per hour respectively. How many times shall they cross each other after 8.00 a.m. and before 9.30 a.m. ?

      • In a 500 metres race, B starts 45 metres ahead of A, but A wins the race while B is still 35 metres behind. What is the ratio of the speeds of A to B assuming that both start at the same time?

      • Two cities A and B are 360 km apart. A car goes from A to B with a speed of 40 km/hr and returns to A with a speed of 60 km/hr. What is the average speed of the car?

      • A worker reaches his factory 3 minutes late if his speed from his house to the factory is 5 km/hr. If he walks at a speed of 6 km/hr, then he reaches the factory 7 minutes early. The distance of the factory from his house is

      • Location of B is north of A and location of C is east of A. The distances AB and AC are 5 km and 12 km respectively. The shortest distance (in km) between the locations B and C is

      • Two cars start towards each other, from two places A and B which are at a distance of 160 km. They start at the same time 08:10 AM. If the speeds of the cars are 50 km and 30 km per hour respectively, they will meet each other at

      • A thief running at 8 km/hr is chased by a policeman whose speed is 10 km/hr. If the thief is 100 m ahead of the policeman, then the time required for the policeman to catch the thief will be

      • A train travels at a certain average speed for a distance of 63 km and then travels a distance of 72 km at an average speed of 6 km/hr more than its original speed. If it takes 3 hours to complete the total journey, what is the original speed of the train in km/hr?

      • Four cars are hired at the rate of 6 per km plus the cost of diesel at 40 a litre. In this context, consider the details given in the following table:

        Car

        Mileage (km/l)

        Hours

        Total Payment (₹)

        A

        8

        20

        2120

        B

        10

        25

        1950

        C

        9

        24

        2064

        D

        11

        22

        1812

        Which car maintained the maximum average speed?

      • Mr. Kumar drives to work at an average speed of 48 km per hour. The time taken to cover the first 60% of the distance is 10 minutes more than the time taken to cover the remaining distance. How far is his office?

      • Consider the following Velocity - Time graph. It shows two trains starting simultaneously on parallel tracks. [Image of Velocity-Time graph] With reference to the above graph, which one of the following statements is not correct?

      • Consider the following distance time graph. The graph shows three athletes A, B and C running side by side for a 30 km race. With reference to the above graph, consider the following statements:
         1. The race was won by A.
         2. B was ahead of A up to 25 km mark.
         3. C ran very slowly from the beginning.
         Which of the statements given above is/are correct?

      • If a bus travels 160 km in 4 hours and a train travels 320 km in 6 hours at uniform speeds, then what is the ratio of the distances travelled by them in one hour

    • A person bought a refrigerator worth ₹22,800 with 12.5% interest compounded yearly. At the end of first year he paid ₹8,650 and at the end of second year ₹9,125. How much will he have to pay at the end of third year to clear the debt?

    • Sets and Functions

      • In a group of 120 persons, 80 are Indians and rest are foreigners. Further, 70 persons in the group can speak English. The number of Indians who can speak English is

      • In a group of 15 people; 7 can read French, 8 can read English while 3 of them can read neither of these two languages. The number of people who can read exactly one language is

      • 19 boys turn out for playing hockey. Of these, 11 are wearing hockey shirts and 14 are wearing hockey pants. There are no boys without shirts and/or pants. What is the number of boys wearing full uniform?

      • Out of 130 students appearing in an examination, 62 failed in English, 52 failed in Mathematics, whereas 24 failed in both English and Mathematics. The number of students who passed finally is

      • In a group of persons travelling in a bus, 6 persons can speak Tamil, 15 can speak Hindi and 6 can speak Gujarati. In that group none can speak any other language. If 2 persons in the group can speak two languages only and one person can speak all the three languages, then how many persons are there in the group?

      • There are 50 students admitted to a nursery class. Some students can speak only English and some can speak only Hindi. 10 students can speak both English and Hindi. If the number of students who can speak English is 21, then how many students can speak Hindi, how many can speak only Hindi and how many can speak only English?

      • Out of a total of 120 musicians in a club, 5% can play all the three instruments, guitar, violin and flute. It so happens that the number of musicians who can play any two and only two of the above instruments is 30. The number of musicians who can play the guitar alone is 40. What is the total number of those who can play violin alone or flute alone?

    • Clocks and Calendars

      • Which date of June 2099 among the following is Sunday?

      • How many seconds in total are there in x weeks, x days, x hours, x minutes and x seconds?

      • Consider the following statements:
         1. Between 3:16 p.m. and 3:17 p.m., both hour hand and minute hand coincide.
         2. Between 4:58 p.m. and 4:59 p.m., both minute hand and second hand coincide.
         Which of the above statements is/are correct?

      • From January 1, 2021, the price of petrol (in Rupees per litre) on m ^ (th) day of the year is 80 + 0 * 1m where m = 1, 2, 3 ,..., 100 and thereafter remains constant. On the other hand, the price of diesel (in Rupees per litre) on n ^ (th) day of 2021 is 69 + 0 * 15n for any n. On which date in the year 2021 are the prices of these two fuels equal?

      • At which one of the following times, do the hour hand and the minute hand of the clock make an angle of 180° with each other?

      • If in a particular year, 12 January is a Sunday, then which one of the following is correct?

      • A wall clock moves 10 minutes fast in every 24 hours. The clock was set right to show the correct time at 8:00 a.m. on Monday. When the clock shows the time 6:00 p.m. on Wednesday, what is the correct time?

      • Which year has the same calendar as that of 2009?

      • A watch loses 2 minutes in every 24 hours while another watch gains 2 minutes in every 24 hours. At a particular instant, the two watches showed an identical time. Which of the following statements is correct if 24-hour clock is followed?

      • A clock strikes once at 1 o'clock, twice at 2 o'clock and thrice at 3 o'clock, and so on. If it takes 12 seconds to strike at 5 o'clock, what is the time taken by it to strike at 10 o'clock?

      • A class starts at 11:00 am and lasts till 2:27 pm. Four periods of equal duration are held during this interval. After every period, a rest of 5 minutes is given to the students. The exact duration of each period is:

      • Between 6 PM and 7 PM the minute hand of a clock will be ahead of the hour hand by 3 minutes at

      • Assume that
         1. the hour and minute hands of a clock move without jerking.
         2. the clock shows a time between 8 o'clock and 9 o'clock.
         3. the two hands of the clock are one above the other.
         After how many minutes (nearest integer) will the two hands be again lying one above the other?

    • Area and Perimeter

      • Consider the following statements in respect of a rectangular sheet of length 20 cm and breadth 8 cm:
         1. It is possible to cut the sheet exactly into 4 square sheets.
         2. It is possible to cut the sheet into 10 triangular sheets of equal area.
         Which of the above statements is/are correct?

      • Twelve equal squares are placed to fit in a rectangle of diagonal 5 cm. There are three rows containing four squares each. No gaps are left between adjacent squares. What is the area of each square?

      • An agricultural field is in the form of a rectangle having length X1 meters and breadth X2 meters (X1 and X2 are variable). If X1 + X2 = 40 meters, then the area of the agricultural field will not exceed which one of the following values ?

      • A round archery target of diameter 1 m is marked with four scoring regions from the centre outwards as red, blue, yellow and white. The radius of the red band is 0.20 m. The width of all the remaining bands is equal. If archers throw arrows towards the target, what is the probability that the arrows fall in the red region of the archery target ?

      • A gardener increased the area of his rectangular garden by increasing its length by 40% and decreasing its width by 20%. The area of the new garden

      • Consider the following figure and answer the item that follows:

        15

         

         

        48

        A square is divided into four rectangles as shown above. The lengths of the sides of rectangles are natural numbers. The areas of two rectangles are indicated in the figure. What is the length of each side of the square?

    • Volume and Surface Area

      • Let x, y be the volumes; m, n be the masses of two metallic cubes P and Q respectively. Each side of Q is two times that of P and mass of Q is two times that of P. Let u = m / x and v = n / y. Which one of the following is correct?

      • A cylindrical overhead tank of radius 2 m and height 7 m is to be filled from an underground tank of size 5.5 m x 4 m x 6 m. How much portion of the underground tank is still filled with water after filling the overhead tank completely?

    • Sequences and Series

      • What is the value of X in the sequence 20, 10, 10, 15, 30, 75, X?

      • What is the value of X in the sequence 2, 12, 36, 80, 150, X?

      • Replace the incorrect term by the correct term in the given sequence 3, 2, 7, 4, 13, 10, 21, 18, 31, 28, 43, 40 where odd terms and even terms follow the same pattern.

      • What is the value of 'X' in the sequence 2, 7, 22, 67, 202, X, 1822?

      • A simple mathematical operation in each number of the sequence 14, 18, 20, 24, 30, 32, ... results in a sequence with respect to prime numbers. Which one of the following is the next number in the sequence?

      • Consider the following sequence of numbers:
         51473985726315 863852243496
         How many odd numbers are followed by the odd number in the above sequence?

      • What is X in the sequence 132, 129, 124, 117, 106, 93, X?

      • In the sequence 1, 5, 7, 3, 5, 7, 4, 3, 5, 7, how many such 5s are there which are not immediately preceded by 3 but are immediately followed by 7?

      • Consider the sequence given below: 4/12/95, 1/1/96, 29/1/96, 26/2/96, .... What is the next term of the series?

      • What is the missing number 'X' of the series 7, X, 21, 31, 43?

    • Plane Geometry

      • There are eight equidistant points on a circle. How many right-angled triangles can be drawn using these points as vertices and taking the diameter as one side of the triangle?

      • The number of parallelograms that can be formed from a set of four parallel lines intersecting another set of four parallel lines, is

      • There are 24 equally spaced points lying on the circumference of a circle. What is the maximum number of equilateral triangles that can be drawn by taking sets of three points as the vertices?

      • There are 4 horizontal and 4 vertical lines, parallel and equidistant to one another on a board. What is the maximum number of rectangles and squares that can be formed?

      • Two walls and a ceiling of a room meet at right angles at a point P. A fly is in the air 1 m from one wall, 8 m from the other wall and 9 m from the point P. How many meters is the fly from the ceiling?

      • AB is a vertical trunk of a huge tree with A being the point where the base of the trunk touches the ground. Due to a cyclone, the trunk has been broken at C which is at a height of 12 meters, broken part is partially attached to the vertical portion of the trunk at C. If the end of the broken part B touches the ground at D which is at a distance of 5 meters from A, then the original height of the trunk is:

      • In a plane, line X is perpendicular to line Y and parallel to line Z; line U is perpendicular to both lines V and W; line X is perpendicular to line V. Which one of the following statements is correct?

      • Consider the figure given below and answer the item that follows:
         

        In the figure shown above, OP₁ and OP₂ are two plane mirrors kept perpendicular to each other. S is the direction of a beam of light falling on the mirror OP₁.  The direction of the reflected beam of light from the mirror OP₂ will be \n.

    • Probability

      • Raj has ten pairs of red, nine pairs of white and eight pairs of black shoes in a box. If he randomly picks shoes one by one (without replacement) from the box to get a red pair of shoes to wear, what is the maximum number of attempts he has to make?

      • A bag contains 15 red balls and 20 black balls. Each ball is numbered either 1 or 2 or 3. 20% of the red balls are numbered 1 and 40% of them are numbered 3. Similarly, among the black balls, 45% are numbered 2 and 30% are numbered 3. A boy picks a ball at random. He wins if the ball is red and numbered 3 or if it is black and numbered 1 or 2. What are the chances of his winning?

      • A bag contains 20 balls. 8 balls are green, 7 are white and 5 are red. What is the minimum number of balls that must be picked up from the bag blindfolded (without replacing any of it) to be assured of picking at least one ball of each colour?

      • Six identical cards are placed on a table. Each card has number '1' marked on one side and number '2' marked on its other side. All the six cards are placed in such a manner that the number '1' is on the upper side. In one try, exactly four (neither more nor less) cards are turned upside down. In how many least number of tries can the cards be turned upside down such that all the six cards show number "2' on the upper side?

    • Permutation and Combination

      • In how many ways can a batsman score exactly 25 runs by scoring single runs, fours and sixes only, irrespective of the sequence of scoring shots?

      • The digits 1 to 9 are arranged in three rows in such a way that each row contains three digits, and the number formed in the second row is twice the number formed in the first row; and the number formed in the third row is thrice the number formed in the first row. Repetition of digits is not allowed. If only three of the four digits 2, 3, 7 and 9 are allowed to use in the first row, how many such combinations are possible to be arranged in the three rows?

      • There is a numeric lock which has a 3-digit PIN. The PIN contains digits 1 to 7. There is no repetition of digits. The digits in the PIN from left to right are in decreasing order. Any two digits in the PIN differ by at least 2. How many maximum attempts does one need to find out the PIN with certainty?

      • One non-zero digit, one vowel and one consonant from English alphabet (in capital) are to be used in forming passwords, such that each password has to start with a vowel and end with a consonant. How many such passwords can be generated?

      • There are 9 cups placed on a table arranged in equal number of rows and columns out of which 6 cups contain coffee and 3 cups contain tea. In how many ways can they be arranged so that each row should contain at least one cup of coffee?

      • There are three points P, Q and R on a straight line such that PQ QR = 3: 5. If n is the number of possible values of PQ: PR, then what is n equal to ?

      • On a chess board, in how many different ways can 6 consecutive squares be chosen on the diagonals along a straight path?

      • Using 2, 2, 3, 3, 3 as digits, how many distinct numbers greater than 30000 can be formed?

      • How many different sums can be formed with the denominations ₹50, ₹100, ₹200, ₹500 and ₹2,000 taking at least three denominations at a time?

      • How many different 5-letter words (with or without meaning) can be constructed using all the letters of the word 'DELHI' so that each word has to start with D and end with I?

      • Suppose you have sufficient amount of rupee currency in three denominations: ₹1,₹10 and ₹50. In how many different ways can you pay a bill of ₹107?

      • How many numbers are there between 99 and 1000 such that the digit 8 occupies the units place?

      • If 2 boys and 2 girls are to be arranged in a row so that the girls are not next to each other, how many possible arrangements are there?

      • How many numbers are there between 100 and 300 which either begin with or end with.2?

      • Four-digit numbers are to be formed using the digits 1, 2, 3, and 4; and none of these four digits are repeated in any manner. Further,
         1. 2 and 3 are not to immediately follow each other
         2. 1 is not to be immediately followed by 3
         3. 4 is not to appear at the last place
         4. 1 is not to appear at the first place
         How many different numbers can be formed?

      • A selection is to be made for one post of Principal and two posts of Vice-Principal. Amongst the six candidates called for the interview, only two are eligible for the post of Principal while they all are eligible for the post of Vice-Principal. The number of possible combinations of selectees is

      • A student has to opt for 2 subjects out of 5 subjects for a course, namely, Commerce, Economics, Statistics, Mathematics I and Mathematics II. Mathematics II can be offered only if Mathematics I is also opted. The number of different combinations of two subjects which can be opted is

      • There are 5 tasks and 5 persons. Task-1 cannot be assigned to either person-1 or person-2. Task-2 must be assigned to either person-3 or person-4. Every person is to be assigned one task. In how many ways can the assignment be done?

    • Statistics

      • An Identity Card has the number ABCDEFG, not necessarily in that order, where each letter represents a distinct digit (1, 2, 4, 5, 7, 8, 9 only). The number is divisible by 9. After deleting the first digit from the right, the resulting number is divisible by 6. After deleting two digits from the right of original number, the resulting number is divisible by 5. After deleting three digits from the right of original number, the resulting number is divisible by 4. After deleting four digits from the right of original number, the resulting number is divisible by 3. After deleting five digits from the right of original number, the resulting number is divisible by 2. Which of the following is a possible value for the sum of the middle three digits of the number?

      • Five friends P, Q, X, Y and Z purchased some notebooks. The relevant information is given below:
         1. Z purchased 8 notebooks more than X did.
         2. P and Q together purchased 21 notebooks.
         3. Q purchased 5 notebooks less than P did.
         4. X and Y together purchased 28 notebooks.
         5. P purchased 5 notebooks more than X did.
         If each notebook is priced 40, then what is the total cost of all the notebooks?

      • A person X wants to distribute some pens among six children A, B, C, D, E and F. Suppose A gets twice the number of pens received by B, three times that of C, four times that of D, five times that of E and six times that of F. What is the minimum number of pens X should buy so that the number of pens each one gets is an even number?

      • Let A, B and C represent distinct non-zero digits. Suppose x is the sum of all possible 3-digit numbers formed by A, B and C without repetition. Consider the following statements:
         1. The 4-digit least value of x is 1332.
         2. The 3-digit greatest value of x is 888.
         Which of the above statements is/are correct?

      • A biology class at high school predicted that a local population of animals will double in size every 12 years. The population at the beginning of the year 2021 was estimated to be 50 animals. If P represents the population after n years, then which one of the following equations represents the model of the class for the population?

      • A boy plays with a ball and he drops it from a height of 1.5 m. Every time the ball hits the ground, it bounces back to attain a height of 4/5th of the previous height. The ball does not bounce further if the previous height is less than 50 cm. What is the number of times the ball hits the ground before the ball stops bouncing?

      • In an objective type test of 90 questions, 5 marks are allotted for every correct answer and 2 marks are deducted for every wrong answer. After attempting all the 90 questions, a student got a total of 387 marks. What is the number of incorrect responses?

      • As a result of 25% hike in the price of rice per kg, a person is able to purchase 6 kg less rice for ₹1,200. What was the original price of rice per kg?

      • Rakesh and Rajesh together bought 10 balls and 10 rackets. Rakesh spent 1300 and Rajesh spent 1500. If each racket costs three times a ball does, then what is the price of a racket?

      • If x is greater than or equal to 25 and y is less than or equal to 40, then which one of the following is always correct?

      • If X is between -3 and 1, and Y is between -1 and 1, then X² - Y² is in between which of the following?

      • If there is a policy that 1/3rd of a population of community has migrated every year from one place to some other place, what is the leftover population of that community after the sixth year, if there is no further growth in the population during this period?

      • In a class, there are 18 very tall boys. If these constitute three-fourths of the boys and the total number of boys is two-thirds of the total number of students in the class, what is the number of girls in the class?

      • A person has only ₹1 and ₹2 coins with her. If the total number of coins that she has is 50 and the amount of money with her is ₹75, then the number of ₹1 and ₹2 coins are, respectively

    • Vein Diagram
  2. PYQs and Practice Questions
  3. UPSC IAS (CSE)
  4. UPSC Prelims
  5. Mathematics
  6. Order of Magnitude
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