Q1
The number of values of x satisfying x + 100/x > 50,where x is a natural number less than or equal to 100 is
Elementary MathematicsNumber System
UPSC CDS · Solved paper
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The number of values of x satisfying x + 100/x > 50,where x is a natural number less than or equal to 100 is
What is the sum of digits of the least multiple of 13, which when dividedby 6, 8 and 12 leaves 5, 7 and 11 respectively as the remainders ?
A milkman claims to sell milk at its cost price only. Still he is making a profit of 20% since he has mixed some amount of water in the milk. What is the percentage of milk in the mixture ?
What is √(4 + √(4 − √(4 + √(4 − …)))) equal to?
The largest natural number which divides every natural number of the form (n³-n)(n-2),where n is a natural number greater than 2 is
The average of m numbers is n^4 and the average of n number is m^4.The average of (m + n) numbers is
The digit in the units place of the resulting number of the expression (234)¹00+(234)¹01 is
If x={3}+{2}, then the value of x³+x+1/x+1/x³ is
The value of 1/1× 4+1/4× 7+1/7× 10++1/16× 19 is
Two trains are moving in the same direction at 1.5 km/minute and 60 km/ hour respectively. A man in the faster train observes that it takes 27 seconds to cross the slower train. The length of the slower train is
A tin of oil was 4/5 full.When 6 bottles of oil were taken out from this tin and 4 bottles of oil were poured into it, it was 3/4 full.Oil of how many bottles can the tin contain ?(All bottles are of equal volume)
A number when divided by 7 leaves a remainder 3 and the resulting quotient when divided by 11 leaves a remainder 6. If the same number when divided by 11 leaves a remainder m and the resulting quotient when divided by 7 leaves a remainder n. What are the values of m and n respectively ?
Consider the following in respect of the equation y = √((x − 1)²)/(x − 1):
Which of the above statements is/are correct?
If (x+1) i the HCF of Ax²+Bx+C and Bx²+Ax+C where A ≠ B, then the value of C is
The seven digit number 876p37q is divisible by 225. The values of p and qcan be respectively
If a, b and c satisfy the equation x³ - 3x² + 2x + 1 = 0 then what is the value of 1/a + 1/b + 1/c ?
A and B are two taps which can fill a tank individually in 10 minutes and 20 minutes respectively. However, there is a leakage at the bottom, which can empty a filled tank in 40 minutes. If the tank is empty initially, how much time will both the taps take to fill the tank with leakage ?
Which one of the following is correct ?
The value of a single discount on some amount which is equivalent to a series of discounts of 10%, 20% and 40% on the same amount, is equal to
If k=x-y+2z where -2 ≤ x ≤ 1, -1 ≤ y ≤ 2 and 3 ≤ z ≤ 6, then which one of the following is correct ?
The number of pairs (x, y) where x, y are integers satisfying the equation 21+48y=5 is
Let x and y be positive integers such that x is prime and y is composite.Which of the following statements are correct ?
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Let x and y be positive integers such that x > y. The expressions 3x+2y and 2x+3y when divided by 5 leave remainders 2 and 3 respectively. What is the remainder when (x-y) is divided by 5 ?
In a race A, B and C take part. A beats B by 30 m, B beats C by 20 m and A beats C by 48 m. Which of the following is/are correct ?
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If the roots of the quadratic equation x² - 4x - ₁0} N = 0 are all real then the minimum value of N is
A motor boat, whose speed is 15 km/hour in still water goes 30 km downstream and comes back in a total of 4 hour and 30 minutes. The speed of the stream is
If 4 men working 4 hours per day for 4 days complete 4 units of work, then how many units of work will be completed by 2 men working for 2 hours per day in 2 days ?
By increasing the speed of his car by 15 km/hour, a person covers 300 km distance by taking an hour less than before. The original speed of the car was
If m persons can paint a house in d days, how many days will it take for(m + 2) persons to paint the same house?
If a:b=3:5 and b:c=7:8, then 2a:3b:7c is equal to
Two trains, one is of 121 m in length at the speed of 40 km/hour and the other is of 99 m in length at the speed of 32 km/hour are running in opposite directions. In how much time will they be completely clear from each other from the moment they meet ?
Three athletes run a 4 km race. Their speeds are in the ratio 16 : 15 : 11. When the winner wins the race, then the distance between the athlete in the second position to the athlete in the third position is
The sum of first 47 terms of the series 1/4+1/5-1/6-1/4-1/5+1/6+1/4+1/5-1/6.... is
Which one of the following is correct?
20% of a number when added to 20 becomes the number itself, then thenumber is
Consider the following statements :
Which of the above statements is/arecorrect ?
If x = a θ and y = b θ , then (ax^{-1}-by^{-1})(ax^{-1}+by^{-1}) is equal to
θ/1-θ is equal to (where θ ≠ π/2)
If (x+40)^{}(x+20)^{}(3x)^{}(70-x)^{}(50-x)^{}=1 then the value of x is equal to
If θ is an acute angle and sin θ cos θ = 2 cos³ θ − 1.5 cos θ, then what is sin θ equal to?
Consider the following statements :
Which of the above statements is/arecorrect ?
If a and b are positive, then the relation θ=2a+3b/3b is
An aeroplane flying at a height of 3000 m passes vertically above another aeroplane at an instant when the angles of elevation of the two planes from some point on the ground are 60° and 45° respectively. Then the vertical distance between the two planes is
A pole is standing erect on the ground which is horizontal. The tip of the pole is tied tight with a rope of length {12} m to a point on the ground. If the rope is making 30° with the horizontal, then the height of the pole is
If θ+θ=2, then θ is equal to
A’s salary was increased by 40% and then decreased by 20%. On the whole A’s salary is increased by
A number consists of two digits, whose sum is 7. If the digits are reversed, the number is increased by 27. The product of digits of the number is
In a race of 100 m, A beats B by 4 m and A beats C by 2 m. By how many metres (approximately) would C beat B in another 100 m race assuming C and B run with their respective speeds as in the earlier race ?
In an election 10% of the voters on the voter list did not cast their vote and 60 voters cast their ballot papers blank. There were only two candidates. The winner was supported by 47% of total voters m the voter list, and he got 308 votes more than his rival. The number of voters on the voter list is
Consider all positive two digit numbers each of which when divided by 7 leaves a remainder 3. What is their sum ?
A pipe with square cross-section is supplying water to a cistern which was initially empty. The area of cross-section is 4 {cm}² and the nozzle velocity of water is 40 m/s. The dimensions, of the cistern are 10 m x 8 m x 6 m. Then the cistern will be full in
A hollow cylindrical drum has internal diameter of 30 cm and a height of 1 m. What is the maximum number of cylindrical boxes of diameter 10 cm and height 10 cm each that can be packed in the drum?
The speeds of three buses are in the ratio 2:3:4. The time taken by these buses to travel the same distance will be in the ratio
A square and an equilateral triangle have equal perimeter. If the diagonal of the square is 12{2} cm, then the area of the triangle is
A boy is cycling such that the wheels of the cycle are making 140 revolutions per minute. If the radius of the wheel is 30 cm, the speed of the cycle is
There are 437 fruit plants in an orchard planted in rows. The distance between any two adjacent rows is 2 m and the distance between any two adjacent plants is 2 m. Each row has the same number of plants.- There is 1 m clearance on all sides of the orchard. What is the cost of fencing the area at the rate of Rs. 100 per metre ?
ABCD is a parallelogram with AB and AD as adjacent sides. If ∠A=60° and AB=2AD, then the diagonal BD will be equal to
The point O is equidistant from die three sides of a triangle ABC. Consider the following statements:
Which of the above statements are correct ?
Consider the following statements :
Which of the above statements is/are correct ?
A large water tank has the shape of a cube. If 128{m}³ of water is pumped out, the water level goes down by 2 m. Then the maximum capacity of the tank is
The circumference of a circle is 100 cm. The side of the square inscribed in the circle is
From the solid gold in the form of a cube of side length 1 cm, spherical solid balls each having the surface area π¹/3}\, ext{cm}² are to be made. Assuming that there is no loss of the material in the process of making the balls, the maximum number of balls made will be
Water is filled in a container in such a manner that its volume doubles every 5 minutes. If it takes 30 minutes for the container to be full, in how much time will it be one-fourth full ?
30 metallic cylinders of same size are melted and cast in the form of cones having the same radius and height as those of the cylinders Consider the following statements :
Which one of the following is correct in respect of the above ?
An equilateral triangle BOC is drawn inside a square ABCD. If angle AOD=2θ, what is tanθ equal to ?
The diameter of a wheel- that makes 452 revolutions to move 2 km and 26 decametre is equal to
If X is any point within a square ABCD and on AX a square AXYZ is described, which of the following is/are correct ?
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The two adjacent sides of a cyclic quadrilateral are 2 cm and 5 cm and the angle between them is 60°. If the third side is 3 cm, then the fourth side is of length
Let A denote the set of quadrilaterals having two diagonals equal and bisecting each other. Let B denote the set of quadrilaterals having diagonals bisecting each other at 90°. Then A∩ B denotes
A square is inscribed in a right triangle with legs x and y and has common right angle with the triangle. The perimeter of the square is given by
From a rectangular sheet of sides 18 cm and 14 cm, a semicircular portion with smaller side as diameter is taken out. Then the area of the remaining sheet will be
ABCD is a square. If the sides AB and CD are increased by 30%, sides BC and AD are increased by 20%, then the area of the resulting rectangle exceeds the area of the square by
A water tank, open at the top, is hemispherical at the bottom and cylindrical above it. The radius is 12 m and the capacity is 3312π m³. The ratio of the surface areas of the spherical and cylindrical portions is
The areas of three mutually perpendicular faces of a cuboid are x, y, z. If F is the volume, then xyz is equal to
Let F be the volume of ah inverted cone with vertex at origin and the axis of the cone is along positive y-axis. The cone is filled with water up to half of its height. The volume of water is
Three rectangle R₁,R₂ and R₃ same area.Their lengths x₁,x₂ and x₃ respectively are such that x₁
If the surface area of a cube is 13254 {cm}², then the length of its diagonal is
The area of a trapezium is 336 {cm}² . If its parallel sides are in the ratio 5 : 7 and the perpendicular distance between them is 14 cm, then the smaller of the parallel sides is
How many spherical bullets each of 4 cm in diameter can be made out of a cube of lead whose edge is 44 cm ?
A river 2.5 m deep and 45 m wide is flowing at the speed of 3.6 km/hour. The amount of water that runs into the sea per minute is
AB is a line segment of length 2a, with M as mid point. Semicircles are drawn on one side with AM, MB and AB as diameter as shown in the above figure. A circle.with centre O and radius r is drawn such that this circle touches all the three semicircles. The value of r is
From an external point P tangents PA and PB are drawn to the circle as shown in the above figure. CD is the tangent to the circle at E. If AP = 16 cm, then the perimeter of the triangle PCD is equal to
In the above figure, ABCD is a parallelogram. P is a point on BC such that PB : PC = 1 : 2 . DP and AB when both* produced meet at Q. If area of triangle BPQ is 20 square unit, the area of triangle DCP is
Chord CD intersects the diameter AB of a circle at right angle at a point P in the ratio 1 : 2. If diameter of circle is d, then CD is equal to
A circle of radius r is inscribed in a regular polygon with n sides (the circle touches all sides of the polygon). If the perimeter of the polygon is p, then the area of the polygon is
The LCM of two numbers is 12 times their HCF. The sum of HCF and LCM is 403. If one of the numbers is 93, then the other number is
The value of ≤ft(0.{-}{63})+0.{-}{37} is
A sum of Rs. 10,000 is deposited for 1 year at the rate of interest 10% compounded half yearly. What will be the interest at the end of one year ?
In a mixture of milk and water of volume 30 litre, the ratio of milk and water is 7:3. The quantity of water to be added to the mixture to make the ratio of milk and water 1:2 is
The difference of maximum values of the expressions (6+5x-x²) and (y-6-y²) for any real values of x and
A clock is started at noon. By 10 minutes past 5, through what angle, the hour hand moves ?
The minimum value of ² x + ² y - ² z is
The value of 32cot²(π/4) − 8sec²(π/3) + 8cos³(π/6) is equal to
If x = 91/216, then the value of 3 − 1/∛(1 − x) is
How many right angled triangles can be formed by joining the vertices of a cuboid ?
The average weight of students in a class is 43 kg. Four new students are admitted to the class whose weights are 42 kg, 36-5 kg, 39 kg and 42-5 kg respectively. Now the average weight of the students of the class is 42-5 kg. The number of students in the beginning was
Four years ago, the average age of A and B was 18 years. Now the average age of A, B and C is 24 years. After 8 years, the age of C will be
If a variable takes discrete values a+4, a-3.5, a-2.5, a-3, a-2, a+0.5, a+5 and a-0-5 where a > 0, then the median of the data set is
If each of n numbers x_i = i (i = 1,2,3,,n) is replaced by (i+1)x_i, then the new mean is
The weighted arithmetic mean of first 10 natural numbers whose weights are equal to the. corresponding numbers is equal to