Q1
A real number M is squared to give the value N. What is the minimum value of (M + N) ?
Elementary MathematicsAlgebra
UPSC CDS · Solved paper
All 100 questions, textbook-style — tap “Show answer” on any question to reveal the correct option and explanation.
A real number M is squared to give the value N. What is the minimum value of (M + N) ?
What is the sum of all 3-digit numbers that give a remainder of 5 when they are divided by 50 ?
If the average of 64, 69, 72, 75, x lies between 62 and 76 (excluding 62 and 76), then what is the number of possible integer values of x ?
Let x, y, z be variables such that (x + y + z) = k, where k is a constant. If (x + z – y) × (x – z + y) is proportional to yz, then (y + z – x) is proportional to :
Let p be the remainder when 7⁸⁴ is divided by 342 and q be the remainder when 7⁸⁴ is divided by 344. What is (p – q) equal to ?
N is the smallest 5-digit number which when divided by 2, 2², 2³, 2⁴, ..., 2ⁿ leaves a remainder 1. What is the value of n ?
What is the minimum value of p for which (1)/(532900)+(p²)/(266450)+(p⁴)/(532900) is an integer ?
If α and β are the roots of the equation x+a+b=abx/ab+ax+bx,then what is (αβ + α + β) equal to ?
Consider a 2-digit number N. Let P be the product of the digits of the number. If P is added to square of the digit in the tens place of N, we get 84. If P is added to the square of the digit in the unit place of N, we get 60. What is the value of P + N ?
A mixture of 100 L contains kerosene and turpentine oil in the ratio 3 : 2. What is the minimum quantity of kerosene in litres (whole number) that should be mixed in the mixture so that the resulting mixture has 20% of kerosene ?
A lamp is kept on a vertical pole. The height of the top of the lamp above the ground is 5√3/2 m. The perpendicular distances of the bottom of the pole from two adjacent walls meeting perpendicularly are 0.7 m and 2.4 m. What is the distance of the top of the lamp from the corner point of the walls on the ground?
C is the centre of a circle of radius 20 cm. AB is a chord of length 32 cm. E is a point on AB such that CE = 13 cm. What is AE × EB equal to?
The inside of a bowl is part of a sphere. When water is put into the bowl to a depth d, the water surface becomes a circle of radius 2d. What is the radius of the sphere?
In a triangle ABC, AB = 2 cm, BC = 4 cm and AC = 3 cm. The bisector of angle A meets BC at D and the bisector of angle B meets AD at E. What is AE : ED equal to?
In a triangle ABC, the bisector of angle A cuts BC at D. If AB + AC = 10 cm and BD : DC = 3 : 1, then what is the length of AC?
In a triangle ABC, AB + BC = 7.1 cm, BC + CA = 12.1 cm and CA + AB = 7.2 cm. What is the area of the triangle?
The adjacent sides of a parallelogram are 10 cm and 8 cm and the angle between them is 150° . What is the area of the parallelogram?
The measure of and angle formed by the bisectors of the angles A and C of the triangle ABC is 130°. What is the measure of the angle B?
What is log₁₀2000 + log₁₀400 + 4log₁₀25 + 5log₁₀20 equal to?
If (log₁₀(100001 − 4ˣ))/(5 − x) = 1, then what is x equal to?
If 2 sin⁴ α + 2 cos⁴ α - 1 = 0, where 0 < α < (π)/(2), then what is sin 2α + cos 2α equal to ?
Consider the following: (I): 1 - sin⁶α = cos²α ( cos⁴α − 3 cos²α + 3) (II): cos⁸α−sin⁸α = 2sin²α(1 − cos⁴α + sin²α cos²α) Which of the above is/are identities?
If p = (1)/(cscθ + cotθ) and q = cscθ, then what is p² - 2pq equal to?
Consider the following statements:
Which of the statements given above is/are correct?"
A tower subtends an angle 60° at a point A on the same level as the foot of the tower. B is a point vertically above A and AB = h. The angle of depression of the foot of the tower, measured from B is 30°. What is the height of the tower ?
What is (sinθ )/(1-cotθ )+(cosθ )/(1-tanθ ) (θ ≠ π/4) equal to?
The length of an arc of a circle of radius 4 cm is π cm. What is the magnitude of the angle subtended by the arc at the centre?
If cot²θ - 3√3 cotθ + 6 = 0, where (π)/(6) < θ < (π)/(2), then what is the value of sinθ + cos 2θ?
Which of the following equations is/are possible? I. (x+y)²4xy, where x, y are positive unequal real quantities. II. 1/x, where x is a positive real quantity. Select the correct answer using the code given below :
If m²(sin θ−1) + n^(2 )(sin θ + 1)=0 , where (π )/(2), then what is (m^(2 )+ n²) cosθ−(m^(2 )− n²) cot θ equal to?
If sin α + cos α = √2, where 0 < α < (π)/(2), then what is sin³ α - cos³ α equal to?
What is (1 + cot α−cosec α)(1 + tan α + sec α) equal to?
If (sinα -cosα )/(sinα +cosα ) where θ and α are (α ≠(π )/(4)) acute angles, then what is √2 sin θ equal to ?
For how many values of α does the expression (sin α + 2)(sin α + 4)(sin α − 2)(sin α − 4) become zero?
What is the value of x, where 0 ≤ x ≤ 30^(∘), satisfying tan3x tan 6x = 1 ?
What is ((a-b)²)/((b-c)(c-a))+((b-c)²)/((c-a)(a-b))+((c-a)²)/((a-b)(b-c)) - 3 equal to, where ≠?
Given that (100× 99× 98× …× 3× 2× 1)/(100ⁿ) is an integer. What is the largest value of n for which this is true?
A man starting from a place P went x metre (x > 120 m) East before turning South. He went 40 m straight before turning to West. He went 60 m to reach a place Q. From Q he went 200 m North and reached a place R. If PR = 200 m, then what is x equal to?
If x² + y² + z² = 3, where x, y and z are integers, then how many values can (xy + yz + zx) have?
If x, y, z are real numbers such that x + y + z=10 and xy + yz + zx = 18, then what is the value of x³+y³+z³− 3xyz?
What is √(17-4√(15))+√(8-2√(15)) equal to?
What is the maximum value of the sum of the numbers 36, 33, 30, 27, 24, ...?
There are two natural numbers m and n (m > n). When m is divided by 12, it leaves a remainder 4. When n is divided by 12, it leaves a remainder 6. Which of the following statements is/are correct?
Select the correct answer using the code given below :
If (x + y) : (y + z) : (z + x) = 3 : 5 : 6 and x + y + z = 14, then what is x² + y² + z² equal to ?
The ratio of sum of two numbers to their difference is 5 : 1. What is the ratio of the sum of their squares to the difference of their squares?
Travelling at 3/5^(th) of his usual speed, a man is late by 20 minutes. What is the usual time if he travels with his usual speed ?
What is the remainder when 2ᵖ-1 is divided by p, where p > 5 is a prime number?
What is the number of factors of 24³ - 16³ - 8³ ?
What is the least number of complete years in which a sum of money put out at 20% compound interest (compounded annually) will be more than doubled?
A train of certain length takes time t to pass completely through a station of length x. The same train with same speed takes time 2t to pass completely through another station of length y. What is the time taken by the train to pass completely through a station of length (x + y)?
Consider the following for the next two (02) items that follow: A frustum of a right cone has a top of diameter 2k, bottom of diameter 2.5k and height k. What is the whole surface area of the frustum?
Consider the following for the next two (02) items that follow: A frustum of a right cone has a top of diameter 2k, bottom of diameter 2.5k and height k. What is the volume of the frustum?
Consider the following for the next two (02) items that follow: ABC is a triangle right-angled at B. The perimeter of the triangle is 24 cm, and the difference between the sum of the perpendicular sides and the hypotenuse is 4 cm. What is the area of the triangle ABC?
Consider the following for the next two (02) items that follow: ABC is a triangle right-angled at B. The perimeter of the triangle is 24 cm, and the difference between the sum of the perpendicular sides and the hypotenuse is 4 cm. A circle is inscribed in the triangle. What is its radius?
Consider the following for the next two (02) items that follow: A circle M of radius 8 cm touches externally another circle N of radius 16 cm. Let P and Q be the points where the common tangent touches the circles M and N, respectively. What is the length of the common tangent PQ?
Consider the following for the next two (02) items that follow: A circle M of radius 8 cm touches externally another circle N of radius 16 cm. Let P and Q be the points where the common tangent touches the circles M and N, respectively. If U, V are the centres of the circles M and N respectively, then what is the area of the quadrilateral formed by the points P, Q, V and U?
Consider the following for the next two (02) items that follow: The perimeter of a triangle ABC is 105 cm. The altitudes AD, BE and CF are in the ratio 3 : 5 : 6. What is AB : BC : CA equal to?
Consider the following for the next two (02) items that follow: The perimeter of a triangle ABC is 105 cm. The altitudes AD, BE and CF are in the ratio 3 : 5 : 6. What is the approximate area of the triangle ABC?
Consider the following for the next two (02) items that follow: A pot is made from a hollow sphere of inner radius 20 cm by cutting its upper portion horizontally. The height of the pot is 30 cm. What is the inner radius of the circular opening of the pot so formed?
Consider the following for the next two (02) items that follow: A pot is made from a hollow sphere of inner radius 20 cm by cutting its upper portion horizontally. The height of the pot is 30 cm. What is the angle made by the line joining the centre of the sphere and any point on the rim of the circular opening with a vertical line passing through the centre?
Consider the following for the next two (02) items that follow: A hall is of length l, breadth b and height h. The maximum distance between any two points (say P and Q) inside the hall is 14 m, whereas the maximum distance between any two points (say P and R) on the floor is 6 m. What is h equal to?
Consider the following for the next two (02) items that follow: A hall is of length l, breadth b and height h. The maximum distance between any two points (say P and Q) inside the hall is 14 m, whereas the maximum distance between any two points (say P and R) on the floor is 6 m. If α is the angle between the line joining P and Q, and the line joining P and R, then what is cos α equal to?
Consider the following for the next two (02) items that follow: The sides of an open box are 0.5 cm thick and bottom is 1 cm thick. The internal length, breadth and depth are, respectively, 14 cm, 10 cm and 8 cm. It is completely filled with water. If the material weighs 2000 kg per cubic metre, then what is the weight of the material used in the construction of the box?
Consider the following for the next two (02) items that follow: The sides of an open box are 0.5 cm thick and bottom is 1 cm thick. The internal length, breadth and depth are, respectively, 14 cm, 10 cm and 8 cm. It is completely filled with water. If water weighs 1000 kg per cubic metre, then what is the weight of the box with water?
Consider the following for the next two (02) items that follow: ABC is a triangle right-angled at A. Further AB = 8 cm and BC = 10 cm. D is the point on BC such that AD is perpendicular to BC. What is AD equal to?
Consider the following for the next two (02) items that follow: ABC is a triangle right-angled at A. Further AB = 8 cm and BC = 10 cm. D is the point on BC such that AD is perpendicular to BC. What is ratio of area of triangle ADC to area of triangle ADB?
What is the volume of rainfall in cubic meter per hectare?
What is the weight of water (in metric tonnes) of annual rainfall falling there on a hectare of land?
Consider the following for the next two items that follow: The angle at the vertex of a conical body is 120°. What is the ratio of the radius of the conical body to its slant height?
Consider the following for the next two items that follow: The angle at the vertex of a conical body is 120°. If the sum of slant height, height and radius is (9 + 3√3) cm, then what is the volume of the cone?
A person sells article X for ₹34,500 and makes a profit of 15%. He sells article Y at a loss of 10%. He neither loses nor gains on the whole because of these two transactions. What is the selling price of article Y?
100 quintals is what percent of 10 metric tonnes?
A circle is inscribed in an equilateral triangle. The radius of the circle is 2 cm. What is the area of the triangle?
The sides of a triangle are k, 1·5k and 2·25k. What is the sum of the squares of its medians?
if 2s = a + b + c then what is:s(s - a)(s - b)(s - c) [(1)/(s-a)+(1)/(s-b)+(1)/(s-c)-(1)/(s)]
How much will ₹10,000 amount to in one year's time at 4% rate of interest per annum if the interest is compounded once in every three months? (take approximate value)
If p = 0.09̅, then what is the value of 70p² + 43p - 5:
What is the remainder when 2¹⁰¹ is divided by 101 ?
If p ≠ 0 and q ≠ 0 are the roots of the equation x² + px + q = 0, then what is p² + q²?
The equation x² + px +q = 0 and x² + qx +p = 0 ( p ≠ q ) have a common root. what is the value of (p + q) ?
"If x² - 5x + 4 is a factor of x⁴ - px² + q, then what are the values of p and q respectively?"
"If two quadratic equations px² + px + 4 = 0 and x² + qx + q = 0 have a common root 2, then what is p + q equal to?"
what is the Hcf of the polynomials x⁸ + x⁴ + 1 and x⁴ + x² + 1 ?
An arc AB of a circle subtends an angle x radians at the centre O of the circle. If the area of sector AOB is equal to half the square of the length of arc AB, then what is x equal to?
Consider the following statements in respect of prime numbers p and q:
Which of the statements given above is/are correct?
Consider the following for the next five (05) items that follow: The frequency distribution of 205 observations on X is given below: X: 3, 5, 6, 7; Frequency: f, f+2, f−3, f+6. What is the value of f?
Consider the following for the next five (05) items that follow: The frequency distribution of 205 observations on X is given below: X: 3, 5, 6, 7; Frequency: f, f+2, f−3, f+6. What is the median of the frequency distribution?
Consider the following for the next five (05) items that follow: The frequency distribution of 205 observations on X is given below: X: 3, 5, 6, 7; Frequency: f, f+2, f−3, f+6. What is the mode of the frequency distribution?
Consider the following for the next five (05) items that follow: The frequency distribution of 205 observations on X is given below: X: 3, 5, 6, 7; Frequency: f, f+2, f−3, f+6. What is the most appropriate graphical representation of the given frequency distribution of X?
Consider the following for the next five (05) items that follow: The frequency distribution of 205 observations on X is given below: X: 3, 5, 6, 7; Frequency: f, f+2, f−3, f+6. What is the mean of the frequency distribution?
What is the number of students who scored between 60 and 70 marks?
What is the number of students who scored more than 50 marks?
In which of the given years was the circulation of the newspaper D close to its average circulation over all the years?
In which of the years from 2019 to 2022 was the circulation of the newspaper D close to the average circulation of all the newspapers in that year?
How many cases are there in which average of the circulation for an individual newspaper was more than the average of the circulation of all the newspapers?
Consider the following for the next three (03) items that follow: Study the following Pie charts: Chart-I (Number of houses: 1,20,000): Pucca 110°, Semi-pucca 210°, Kutcha 40°. Chart-II (Population: 21,00,000): Living in kutcha houses 20°, Houseless 70°, Living in pucca houses 60°, Living in semi-pucca houses 210°. Chart-III (Number of families: 3,60,000): Living in pucca houses 70°, Living in kutcha houses 185°, Living in semi-pucca houses 85°, Houseless 20°. On an average, how many persons live in every pucca house?
Consider the following for the next three (03) items that follow: Study the following Pie charts: Chart-I (Number of houses: 1,20,000): Pucca 110°, Semi-pucca 210°, Kutcha 40°. Chart-II (Population: 21,00,000): Living in kutcha houses 20°, Houseless 70°, Living in pucca houses 60°, Living in semi-pucca houses 210°. Chart-III (Number of families: 3,60,000): Living in pucca houses 70°, Living in kutcha houses 185°, Living in semi-pucca houses 85°, Houseless 20°. If 5000 more Kutcha houses are built, then what will be approximate change in angle for Kutcha houses in Pie Chart-I?
Consider the following for the next three (03) items that follow: Study the following Pie charts: Chart-I (Number of houses: 1,20,000): Pucca 110°, Semi-pucca 210°, Kutcha 40°. Chart-II (Population: 21,00,000): Living in kutcha houses 20°, Houseless 70°, Living in pucca houses 60°, Living in semi-pucca houses 210°. Chart-III (Number of families: 3,60,000): Living in pucca houses 70°, Living in kutcha houses 185°, Living in semi-pucca houses 85°, Houseless 20°. If 300 families from the "Houseless" category shift into Kutcha houses, what will be the average number of families in every Kutcha house?
Marks obtained by 60 students in a test (Maximum Marks-100) are given below : 60, 82, 63, 40, 88, 80, 24, 60, 09, 58, 44, 74 57, 60, 63, 30, 31, 64, 00, 62, 37, 54, 10, 99 35, 83, 58, 33, 92, 43, 66, 41, 78, 72, 32, 33 65, 11, 60, 17, 53, 93, 57, 03, 29, 31, 82, 50 63, 15, 41, 55, 72, 42, 32, 23, 30, 48, 67, 34 What is the percentage (approximate) of students getting marks higher than or equal to 60 but less than 80 ?
The arithmetic mean of 100 observations was 50. Later on, it was found that two observations were misread as 82 and 6 instead of 182 and 56. What is the value of the corrected mean?