Q1
What is √((0.064× 6.25)/(0.081× 4.84)) equal to ?
Elementary MathematicsNumber System
UPSC CDS · Solved paper
All 100 questions, textbook-style — tap “Show answer” on any question to reveal the correct option and explanation.
What is √((0.064× 6.25)/(0.081× 4.84)) equal to ?
(x + 4) is a factor of which one of the following expressions ?
If α and β are the roots of the quadratic equation 2x²+6x+k=0, where k < 0, then what is the maximum value of α/β + β/α ?
Consider the following statements :
Which of the above statements is/are correct ?
What is the square root of ((0.35)²+0.70+1)/2.25+0.19 ?
What is the number of digits in 2⁴⁰ ? (Given that log₁₀ 2 = 0.301)
If one root of (a²-5a+3)x²+(3a-1)x+2=0 is twice the other, then what is the value of‘a’?
What is the remainder when the number (4444)⁴⁴⁴⁴ is divided by 9 ?
If x=(√(a+b)-√(a-b))/(√(a+b)+√(a-b)), then what is bx²-2ax+b equal to (b≠ a) ?
What is the value of ((443+547)²+(443-547)²)/(443 × 443 + 547 × 547) ?
If x = t^(1/(t-1)) and y = t^(t/(t-1)), t>0, t ≠ 1, then what is the relation between x and y ?
If A : B = 3 : 4, then what is the value of the expression ((3A²+4B)/(3A-4B²)) ?
If A = (x : x is a multiple of 7}, B = x : x is a multiple of 5) and C = x : x is a multiple of 35, then which one of the following is a null set ?
If x=2+2^(2/3)+2^(1/3), then what is the value of x³-6x²+6x ?
If √(x/y) = 24/5 + √(y/x) and x+y=26,then what is the value of xy ?
What is the solution of the equation x log₁₀ (10/3)+log₁₀3=log₁₀(2+3ˣ)+x?
If α and β are the roots of the equation x²+px+q=0, then what is α²+β² equal to ?
If a³=335+b³ and a=5+b, then what is the value of a+b (given that a > 0 and b > 0) ?
If 9ˣ 3ʸ = 2187 and 2³ˣ2²ʸ - 4ˣʸ = 0, then what can be the value of (x + y) ?
The pair of linear equations kx + 3y + 1 = 0 and 2x + y + 3 = 0 intersect each other, if
The number of prime numbers which are less than 100 is
The cost of a diamond varies directly as the square of its weight. A diamond broke into four pieces with their weights in the ratio of 1: 2: 3: 4. If the loss in total value of the diamond was ₹ 70,000 , what was the price of the original diamond?
In a 100 m race, A runs at a speed of 5/3 m/s. If A gives a start of 4 m to B and still beats him by 12 seconds, what is the speed of B ?
If 15 men take 21 days of 8 hours each to do a piece of work, then what is the number of days of 6 hours each that 21 women would take, if 3 women would do as much work as 2 men ?
What number must be subtracted from both the numerator and the denominator of the fraction 27/35 so that it becomes 2/3 ?
A sum of ₹ 8,400 was taken as a loan. This is to be paid in two equal instalments. If the rate of interest is 10% per annum, compounded annually, then the value of each instalment is
Leela got married 6 years ago. Today her age is 1 1/4 times her age at the time of her marriage. Her son’s age is 1/10 times her age. What is the present age of her son ?
A and B working together can finish a piece of work in 12 days while B alone can finish it in 30 days. In how many days can A alone finish the work ?
The values of x which satisfy the equation 5¹⁺ˣ+5¹⁻ˣ=26 are
If 5 men can do a piece of work in 10 days and 12 women can do the same work in 15 days, the number of days required to complete the work by 5 men and 6 women is
A passenger train departs from Delhi at 6 p.m. for Mumbai. At 9 p.m., an express train, whose average speed exceeds that of the passenger train by 15 km/hour leaves Mumbai for Delhi. Two trains meet each other mid-route. At what time do they meet, given that the distance between the cities is 1080 km ?
In a class of 49 students, the ratio of girls to boys is 4 : 3. If 4 girls leave the class, the ratio of girls to boys would be
If a + b = 5 and ab=6, then what is the value of a³+b³ ?
Rajendra bought a mobile with 25% discount on the selling price. If the mobile cost him ₹ 4,875, what is the original selling price of the mobile ?
A 225 m long train is running at a speed of 30 km/hour. How much time does it take to cross a man running at 3 km/hour in the same direction ?
Which one among the following is the largest ?
The difference between the simple and the compound interest on a certain sum of money at 4% per annum in 2 years is ₹ 10. What is the sum ?
If a% of a + b% of b = 2% of ab, then what percent of a is b ?
5/9 th part of the population in a village are males. If 30% of the males are married, the percentage of unmarried females in the total population is
Sunil wants to spend ₹ 200 on two types of sweets, costing ₹ 7 and ₹ 10 respectively. What is the maximum number of sweets he can get so that no money is left over ?
What is the LCM of x³+8,x²+5x+6 and x³+4x²+4x ?
The HCF of two expressions p and q is 1. What is the reciprocal of their LCM ?
What is the value of ∛(4 12/125) ?
A thief is spotted by a policeman from a distance of 100 m. When the policeman starts the chase, the thief also starts running. If the speed of the thief is 8 km/hour and that of the policeman is 10 km/hour, then how far will the thief have to run before he is overtaken ?
Aman and Alok attempted to solve a quadratic equation. Aman made a mistake in writing down the constant term and ended up in roots (4, 3). Alok made a mistake in writing down the coefficient of x to get roots (3, 2). The correct roots of the equation are
Consider the following statements :
Which of the above statements is/are correct ?
The system of equations 2x + 4y = 6 and 4x + 8y = 8 is
(Nᵖ⁻¹-1) is a multiple of p, if N is prime to p and p is a
The ratio of two numbers is 1 : 5 and their product is 320. What is the difference between the squares of these two numbers ?
25 kg of alloy X is mixed with 125 kg of alloy Y. If the amount of lead and tin in the alloy X is in the ratio 1 : 2 and the amount of lead and tin in the alloy Y is in the ratio 2 : 3, then what is the ratio of lead to tin in the mixture ?
The mean of 5 numbers is 15. If one more number is included, the mean of the 6 numbers becomes 17. What is the included number ?
The mean marks obtained by 300 students in a subject are 60. The mean of top 100 students was found to be 80 and the mean of last 100 students was found to be 50. The mean marks of the remaining 100 students are
Consider the following distribution:
| Class | Frequency |
|---|---|
| 0–20 | 17 |
| 20–40 | 28 |
| 40–60 | 32 |
| 60–80 | f |
| 80–100 | 19 |
If the mean of the above distribution is 50, what is the value of f ?
In a pie diagram, there are four slices with angles 150°, 90°, 60° and 60°. A new pie diagram is formed by deleting one of the slices having angle 60° in the given pie diagram. In the new pie diagram
In an asymmetrical distribution, if the mean and median of the distribution are 270 and 220 respectively, then the mode of the data is
Let a, b, c, d, e, f, g be consecutive even numbers and j, k, I, m, n be consecutive odd numbers. What is the average of all the numbers ?
An individual purchases three qualities of pencils. The relevant data is given below : Quality Price perpencil (in ₹) Money spent (in ₹) A 1.00 50 B 1.50 x C 2.00 20 It is known that the average price per pencil is ₹ 1.25. What is the value of x ?
Consider the following frequency distribution:
| X | Frequency | Cumulative frequency |
| 1 | 8 | 8 |
| 2 | 10 | 18 |
| 3 | f₁ | 29 |
| 4 | f₂ | 45 |
What are the values of f₁ and f₂ respectively ?
If D is the number of degrees and R is the number of radians in an angle θ , then which one of the following is correct ?
What is the minimum value of 9 tan²θ+4 cot²θ ?
If x sinθ = y cosθ = (2z tanθ)/(1-tan²θ), then what is 4z²(x²+y²) equal to ?
If cosθ₁+cosθ₂+cosθ₃=3, then what is sinθ₁+sinθ₂+sinθ₃ equal to ?
What is the value of θ which satisfies the equation cos θ + tan θ=1 ?
What is the value of sin x √(1/(1+cos x) + 1/(1−cos x)) ?
What is (cos⁴A - sin⁴A)/(cos²A - sin²A) equal to ?
If 7 sin²x + 3 cos²x = 4, 0 < x < 90°, then what is the value of tan x ?
An aeroplane flying at a height of 300 m above the ground passes vertically above another plane at an instant when the angles of elevation of the two planes from the same point on the ground are 60° and 45° respectively. What is the height of the lower plane from the ground ?
If x=a cosθ+b sinθ and y=a sinθ-b cosθ, then what is x²+y² equal to ?
From the top of a building 90 m high, the angles of depression of the top and the bottom of a tree are 30° and 45° respectively. What is the height of the tree ?
Which one of the following triples does not represent the sides of a triangle ?
If the perimeter of a rectangle is 10 cm and the area is 4 cm², then its length is
The angles of a triangle are in the ratio 2:4:3. The smallest angle of the triangle is
A ball of radius 1 cm is put into a cylindrical pipe so that it fits inside the pipe. If the length of the pipe is 14 m, what is the surface area of the pipe ?
The areas of two circular fields are in the ratio 16 : 49. If the radius of the bigger field is 14 m, then what is the radius of the smaller field ?
Let ABCD be a rectangle. Let P, Q, R, S be the mid-points of sides AB, BC, CD, DA respectively. Then the quadrilateral PQRS is a
Let P, Q, R be the mid-points of sides AB, BC, CA respectively of a triangle ABC. If the area of the triangle ABC is 5 square units, then the area of the triangle PQR is
If each of the dimensions of a rectangle is increased by 200%, the area is increased by
Three circles each of radius 3.5 cm touch one another. The area subtended between them is
The area of a regular hexagon of side ‘a’ is equal to
ABCDEF is a regular polygon. Two poles at C and D are standing vertically and subtend angles of elevation 30° and 60° at A respectively. What is the ratio of the height of the pole at C to that of the pole at D ?
Two parallel chords of a circle whose diameter is 13 cm are respectively 5 cm and 12 cm in length. If both the chords are on the same side of the diameter, then the distance between these chords is
If the radius of a right circular cone is increased by p% without increasing its height, then what is the percentage increase in the volume of the cone ?
A copper wire when bent in the form of a square encloses an area of 121 cm². If the same wire is bent in the form of a circle, it encloses an area equal to
ABC is a triangle and D is a point on the side BC. If BC = 12 cm, BD = 9 cm and ∠ADC = ∠BAC, then the length of AC is equal to
If the surface area of a sphere is reduced to one-ninth of the area, its radius reduces to
In a trapezium ABCD, AB is parallel to CD and the diagonals intersect each other at O. What is the ratio of OA to OC equal to ?
Ice-cream, completely filled in a cylinder of diameter 35 cm and height 32 cm, is to be served by completely filling identical disposable cones of diameter 4 cm and height 7 cm. The maximum number of persons that can be served in this way is
The radius of a circle is increased so that its circumference increases by 15%. The area of the circle will increase by
ABCD is a rectangle. The diagonals AC and BD intersect at O. If AB = 32 cm and AD = 24 cm, then what is the length of OD ?
A field is divided into four regions as shown in the given figure. What is the area of the field in square metres ?
In the figure given below, D is the diameter of each circle. What is the diameter of the shaded circle ?
In the figure given below, AC is parallel to ED and AB = DE = 5 cm and BC = 7 cm. What is the area ABDE : area BDE : area BCD equal to?
In the figure given below, PQRS is a parallelogram. PA bisects ∠ P and SA bisects ∠ S. What is ∠ PAS equal to ?
In the figure given below, ∠ A = 80° and ∠ ABC = 60°. BD and CD bisect angles B and C respectively. What are the values of x and y respectively ?
In the figure given below, PQR is a non-isosceles right-angled triangle, right angled at Q. If LM and QT are parallel and QT = PT, then what is ∠ RLM equal to ?
In the figure given below, PQ is parallel to RS and PR is parallel to QS. If ∠ LPR = 35° and ∠ UST = 70°, then what is ∠ MPQ equal to ?
In the figure given below, ABC is a triangle with AB = BC and D is an interior point of the triangle ABC such that ∠ DAC = ∠ DCA.Consider the following statements :
Which of the above statements are correct ?
In the figure given below, M is the mid-point of AB and ∠ DAB = ∠ CBA and ∠ AMC = ∠ BMD. Then the △ ADM is congruent to the △ BCM by
ABCD is a square. X is the mid-point of AB and Y is the mid-point of BC. Consider the following statements :
Which of the above statements are correct ?
From an aeroplane vertically over a straight horizontal road, the angles of depression of two consecutive kilometre-stones on the opposite sides of the aeroplane are observed to be a and p. The height of the aeroplane above the road is