UPSC CDS · Solved paper
All 100 questions, textbook-style — tap “Show answer” on any question to reveal the correct option and explanation.
Directions: Read the following frequency distribution for two series of observations and answer the two items that follow:
| Class interval | Series-I | Series-II |
|---|---|---|
| 10–20 | 20 | 4 |
| 20–30 | 15 | 8 |
| 30–40 | 10 | 4 |
| 40–50 | X | 2X |
| 50–60 | Y | Y |
| Total | 100 | 100 |
What is the mean of frequency distribution of Series-I?
Directions: Read the following frequency distribution for two series of observations and answer the two items that follow:
| Class interval | Series-I | Series-II |
|---|---|---|
| 10–20 | 20 | 4 |
| 20–30 | 15 | 8 |
| 30–40 | 10 | 4 |
| 40–50 | X | 2X |
| 50–60 | Y | Y |
| Total | 100 | 100 |
What is the mode of the frequency distribution of Series-II?
Direction: Read the following information and answer the four items that follow: Let the distribution of the number of scooters of companies X and Y sold by 5 showrooms (A, B, C, D and E) in a certain year be denoted S1 and the distribution of the number of scooters of only company X sold by the five showrooms in the same year be denoted by S2.
| Showroom | A | B | C | D | E | Total scooters sold |
|---|---|---|---|---|---|---|
| S1 (both companies X and Y, in %) | 19 | 21 | 15 | 33 | 12 | 6400 |
| S2 (only company X, in %) | 24 | 18 | 20 | 30 | 8 | 3000 |
The number of scooters of company Y sold by showroom E is what percent of the number of scooters of both companies sold by showroom C?
Direction: Read the following information and answer the four items that follow: Let the distribution of the number of scooters of companies X and Y sold by 5 showrooms (A, B, C, D and E) in a certain year be denoted S1 and the distribution of the number of scooters of only company X sold by the five showrooms in the same year be denoted by S2.
| Showroom | A | B | C | D | E | Total scooters sold |
|---|---|---|---|---|---|---|
| S1 (both companies X and Y, in %) | 19 | 21 | 15 | 33 | 12 | 6400 |
| S2 (only company X, in %) | 24 | 18 | 20 | 30 | 8 | 3000 |
The number of scooters of both the companies sold by showroom B is what per cent more than the number of scooters of company X sold by showroom A?
Direction: Read the following information and answer the four items that follow: Let the distribution of the number of scooters of companies X and Y sold by 5 showrooms (A, B, C, D and E) in a certain year be denoted S1 and the distribution of the number of scooters of only company X sold by the five showrooms in the same year be denoted by S2.
| Showroom | A | B | C | D | E | Total scooters sold |
|---|---|---|---|---|---|---|
| S1 (both companies X and Y, in %) | 19 | 21 | 15 | 33 | 12 | 6400 |
| S2 (only company X, in %) | 24 | 18 | 20 | 30 | 8 | 3000 |
What is the average number of scooters of company Y sold by the showrooms A, C and E?
Direction: Read the following information and answer the four items that follow: Let the distribution of the number of scooters of companies X and Y sold by 5 showrooms (A, B, C, D and E) in a certain year be denoted S1 and the distribution of the number of scooters of only company X sold by the five showrooms in the same year be denoted by S2.
| Showroom | A | B | C | D | E | Total scooters sold |
|---|---|---|---|---|---|---|
| S1 (both companies X and Y, in %) | 19 | 21 | 15 | 33 | 12 | 6400 |
| S2 (only company X, in %) | 24 | 18 | 20 | 30 | 8 | 3000 |
What is the difference between the number of scooters of company X and Y sold by showroom A and the total number of scooters of company X sold by showrooms B and E together?
Direction: Study the following table and answer the four items that follow:
| Year | 2014 | 2015 | 2016 | 2017 |
| Number of bikers killed | 40957 | 46070 | 52750 | 48746 |
| Number of pedestrians killed | 12330 | 13894 | 15747 | 20475 |
| Number of cyclists killed | 4037 | 3125 | 2585 | 3559 |
What was the approximate average number of pedestrians killed per day in the year 2017?
Direction: Study the following table and answer the four items that follow:
| Year | 2014 | 2015 | 2016 | 2017 |
| Number of bikers killed | 40957 | 46070 | 52750 | 48746 |
| Number of pedestrians killed | 12330 | 13894 | 15747 | 20475 |
| Number of cyclists killed | 4037 | 3125 | 2585 | 3559 |
What is the approximate percentage change in the pedestrians’ fatalities during the period 2014–17?
Direction: Study the following table and answer the four items that follow:
| Year | 2014 | 2015 | 2016 | 2017 |
| Number of bikers killed | 40957 | 46070 | 52750 | 48746 |
| Number of pedestrians killed | 12330 | 13894 | 15747 | 20475 |
| Number of cyclists killed | 4037 | 3125 | 2585 | 3559 |
What is the approximate average number of bikers killed daily in road accidents in the year 2017?
Direction: Study the following table and answer the four items that follow:
| Year | 2014 | 2015 | 2016 | 2017 |
| Number of bikers killed | 40957 | 46070 | 52750 | 48746 |
| Number of pedestrians killed | 12330 | 13894 | 15747 | 20475 |
| Number of cyclists killed | 4037 | 3125 | 2585 | 3559 |
What is the approximate average number of cyclists killed daily in road accidents in 2017?
Let a and b be two positive real numbers such that a√a + b√b = 32 and a√b + b√a = 31. What is the value of 5(a + b)/7?
If x = (1 + √3)/2 and y = x³, then y satisfies which one of the following equations?
HCF of two numbers is 12. Which one of the following can never be their LCM?
Consider the following statements:
Which of the above statements are correct?
The rate of interest on two different schemes is the same and it is 20%. But in one of the schemes, the interest is compounded half-yearly and in the other, the interest is compounded annually. Equal amounts are invested in the schemes. If the difference of the returns after 2 years is Rs. 482, then what is the principal amount in each scheme?
For what value of k can the expression x³ + kx² - 7x + 6 be resolved into three linear factors?
X, Y and Z start at some point and same time in the same direction to run around a circular stadium. X completes a round in 252 seconds, Y in 308 seconds and Z in 198 seconds. After what time will they meet again at the starting point?
What is the LCM of 1/3, 5/6, 2/9, 4/27?
If the equations x² + 5x + 6 = 0 and x² + kx + 1 = 0 have a common root, then what is the value of k?
A lent Rs. 25000 to B and at the same time lent some amount to C at the same 7% simple interest. After 4 years A received Rs. 11200 as interest from B and C. How much did A lend to C?
A trader sells two computers at the same price, making a profit of 30% on one and a loss of 30% on the other. What is the net loss or profit percentage on the transaction?
The monthly incomes of A and B are in the ratio 4 : 3. Each saves Rs. 600. If their expenditures are in the ratio 3 : 2, then what is the monthly income of A?
The train fare and bus fare between two stations is in the ratio 3 : 4. If the train fare increases by 20% and bus fare increases by 30%, then what is the ratio between revised train fare and revised bus fare?
When N is divided by 17, the quotient is equal to 182. The difference between the quotient and the remainder is 175. What is the value of N?
A stock of food grains is enough for 240 men for 48 days. How long will the same stock last for 160 men?
A hollow right circular cylindrical vessel of volume V whose diameter is equal to its height, is completely filled with water. A heavy sphere of the maximum possible volume is then completely immersed in the vessel. What volume of water remains in the vessel?
Three parallel lines x, y and z are cut by two transversals m and n. Transversal m cuts the lines x, y, z at P, Q, R respectively; and transversal n cuts the lines x, y, z at L, M, N respectively. If PQ = 3 cm, QR = 9 cm and MN = 10.5 cm, then what is the length of LM?
The area of a sector of a circle of radius 4 cm is 25.6 cm². What is the radian measure of the arc of the sector?
Which one of the following is correct in respect of a right-angled triangle?
Let the bisector of the ∠BAC of △ABC meet BC in X. Which one of the following is correct?
What is the value of log₁₀(cos θ) + log₁₀(sin θ) + log₁₀(tan θ) + log₁₀(cot θ) + log₁₀(sec θ) + log₁₀(cosec θ)?
If cos²x + cos x = 1, then what is the value of sin¹²x + 3sin¹⁰x + 3sin⁸x + sin⁶x?
If 0 < θ < 90°, sin θ = 3/5 and x = cot θ, then what is the value of 1 + 3x + 9x² + 27x³ + 81x⁴ + 243x⁵?
The angles of elevation of the tops of two pillars of heights h and 2h from a point P on the line joining the feet of the two pillars are complementary. If the distances of the feet of the pillars from the point P are x and y respectively, then which one of the following is correct?
What is the value of (sin 19°/cos 71°) + (cos 73°/sin 17°)?
The perimeter of a triangle is 22 cm. Through each vertex of the triangle, a straight line parallel to the opposite side is drawn. What is the perimeter of the triangle formed by these lines?
The sides AD, BC of a trapezium ABCD are parallel and the diagonals AC and BD meet at O. If the area of △AOB is 3 cm² and the area of △BDC is 8 cm², then what is the area of △AOD?
A line segment AB is the diameter of a circle with centre at O having radius 6.5 cm. Point P is in the plane of the circle such that AP = x and BP = y. In which one of the following cases does the point P not lie on the circle?
The perimeters of two similar triangles ABC and PQR are 75 cm and 50 cm respectively. If the length of one side of △PQR is 20 cm, then what is the length of the corresponding side of △ABC?
Let PQRS be a parallelogram whose diagonals PR and QS intersect at O. If △QRS is an equilateral triangle having a side of length 10 cm, then what is the length of the diagonal PR?
The areas of three adjacent faces of a cuboid are x, y and z. If V is the volume of the cuboid, then which one of the following is correct?
If l is the length of the median of an equilateral triangle, then what is its area?
A piece of wire is in the form of a sector of a circle of radius 20 cm, subtending an angle 150° at the centre. If it is bent in the form of a circle, then what will be its radius?
Suppose P, Q and R are the midpoints of the sides of a triangle of area 128 cm². If △ABC is drawn by joining the midpoints of the sides of △PQR, then what is the area of △ABC?
Let two lines p and q be parallel. Consider two points B and C on the line p and two points D and E on the line q. The line through B and E intersects the line through C and D at A in between the two lines p and q. If AC : AD = 4 : 9, then what is the ratio of the area of △ABC to that of △ADE?
An equilateral triangle and a square are constructed using metallic wires of equal length. What is the ratio of the area of the triangle to that of the square?
All the four sides of a parallelogram are of equal length. The diagonals are in the ratio 1:2. If the sum of the lengths of the diagonals is 12 cm, then what is the area of the parallelogram?
ABC is a triangle right-angled at B. If AB = 5 cm and BC = 10 cm, then what is the length of the perpendicular drawn from the vertex B to the hypotenuse?
Two cylinders of equal volume have their heights in the ratio 2:3. What is the ratio of their radii?
The length and breadth of a rectangle are increased by 20% and 10% respectively. What is the percentage increase in the area of the rectangle?
If the length of the hypotenuse of a right-angled triangle is 10 cm, then what is the maximum area of such a right-angled triangle?
A square is drawn such that its vertices are lying on a circle of radius 201 mm. What is the ratio of the area of the circle to that of the square?
A right circular cylinder has a diameter of 20 cm, and its curved surface area is 1000 cm². What is the volume of the cylinder?
A piece of wire of length 33 cm is bent into an arc of a circle of radius 14 cm. What is the angle subtended by the arc at the centre of the circle?
What is the ratio of the area of a square inscribed in a semicircle of radius r to the area of a square inscribed in a circle of radius r?
The quotient when x⁴ - x² + 7x + 5 is divided by (x + 2) is ax³ + bx² + cx + d. What are the values of a, b, c and d respectively?
The sides of a triangle are 30 cm, 28 cm and 16 cm respectively. In order to determine its area, the logarithm of which of the quantities are required?
If log₁₀1995 = 3.3000, then what is the value of (0.001995)¹/⁸?
What is (x - a)(x - b)(x - c) equal to?
Let XYZ be an equilateral triangle in which XY = 7 cm. If A denotes the area of the triangle, then what is the value of log₁₀A⁴? (Given that log₁₀1050 = 3.0212 and log₁₀35 = 1.5441)
A hollow sphere of external and internal diameters 6 cm and 4 cm respectively is melted into a cone of base diameter 8 cm. What is the height of the cone?
A solid metallic cylinder of height 10 cm and radius 6 cm is melted to make two cones in the ratio of volume 1:2 and of the same height as 10 cm. What is the percentage increase in the flat surface area?
If one side of a right-angled triangle (with all sides integers) is 15 cm, then what is the maximum perimeter of the triangle?
A thin rod of length 24 feet is cut into rods of equal size and joined so as to form a skeleton cube. What is the area of one of the faces of the largest cube thus constructed?
Consider a trapezium ABCD, in which AB is parallel to CD and AD is perpendicular to AB. If the trapezium has an incircle which touches AB at E and CD at F, where EB = 25 cm and FC = 16 cm, then what is the diameter of the circle?
Three copper spheres of radii 3 cm, 4 cm and 5 cm are melted to form a large sphere. What is its radius?
The volume of a hemisphere is 155232 cm³. What is the radius of the hemisphere?
A bucket is in the form of a truncated cone. The diameters of the base and top of the bucket are 6 cm and 12 cm respectively. If the height of the bucket is 7 cm, what is the capacity of the bucket?
A right circular cone has a height 8 cm. If the radius of its base is 6 cm, then what is its total surface area?
Six cubes, each with 12 cm edge are joined end to end. What is the surface area of the resulting cuboid?
If the sum of a real number and its reciprocal is 26/5, then how many such numbers are possible?
Consider the following statements:
Which of the above statements is/are correct?
Radha and Rani are sisters. Five years back, the age of Radha was three times that of Rani, but one year back the age of Radha was two times that of Rani. What is the age difference between them?
A person carries Rs. 500 and wants to buy apples and oranges out of it. If the cost of one apple is Rs. 5 and the cost of one orange is Rs. 7, then what is the number of ways in which the person can buy both apples and oranges using the total amount?
Given y is inversely proportional to √x, and x = 36 when y = 36. What is the value of x when y = 54?
What is the square root of 16 + 6√7?
What is the number of digits in 7²⁵, 8²³ and 9²⁰ respectively? [Given log₁₀2 = 0.301, log₁₀3 = 0.477, log₁₀7 = 0.845]
Let x be the smallest positive integer such that when 14 divides x, the remainder is 7; and when 15 divides x, the remainder is 5. Which one of the following is correct?
Two taps X and Y are fixed to a water tank. If only X is opened, it drains out the full tank of water in 20 minutes. If both X and Y are opened, then they drain out the full tank of water in 15 minutes. If only Y is opened, how long does it take to drain out the full tank of water?
Consider the following statements:
Which of the above statements are correct?
A library has an average number of 510 visitors on Sunday and 240 on other days. What is the average number of visitors per day in a month of 30 days beginning with Saturday?
If 36/11 = 3 + 1/(x + 1/(y + 1/z)), where x, y and z are natural numbers, then what is (x + y + z) equal to?
A person sells two items each at Rs. 990, one at a profit of 10% and another at a loss of 10%. What is the combined percentage of profit or loss for the two items?
It takes 11 hours for a 600 km journey if 120 km is done by train and the rest by car. It takes 40 minutes more if 200 km are covered by train and the rest by car. What is the ratio of the speed of the car to that of the train?
A real number x is such that (x - x²) is maximum. What is x equal to?
If 10ⁿ divides 6²³ × 75⁹ × 105², then what is the largest value of n?
What is the digit in the unit’s place of the number represented by 3⁹⁸ - 3⁸⁹?
The sum of the squares of four consecutive natural numbers is 294. What is the sum of the numbers?
The equation x² + px + q = 0 has roots equal to p and q where q ≠ 0. What are the values of p and q respectively?
How many pairs of natural numbers are there such that the difference of their squares is 35?
If (b - 6) is one root of the quadratic equation x² - 6x + b = 0, where b is an integer, then what is the maximum value of b²?
If a = √(7 + 4√3), then what is the value of a + 1/a?
What is the maximum value of the expression 1/(x² + 5x + 10)?
If the ratio of the work done by (x + 2) workers in (x - 3) days to the work done by (x + 4) workers in (x - 2) days is 3:4, then what is the value of x?
Which one of the following is not correct?
What is the sum of all integer values of n for which n² + 19n + 92 is a perfect square?
What is the LCM of the polynomials x³ + 3x² + 3x + 1, x³ + 5x² + 5x + 4 and x² + 5x + 4?
What is the value of [(x - y)³ + (y - z)³ + (z - x)³] / [9(x - y)(y - z)(z - x)]?
If X = {{a}, {b}, c}, Y = {{a}, b, c} and Z = {a, b, {c}}, then (X ∩ Y) ∩ Z equals to
Two numbers p and q are such that the quadratic equation px² + 3x + 2q = 0 has -6 as the sum and the product of the roots. What is the value of (p - q)?
