Q1
What are distinct prime factors of the number 26381 ?
Elementary MathematicsNumber System
UPSC CDS · Solved paper
CDS 2021 — Elementary Mathematics (CDS-II)
All 100 questions, textbook-style — tap “Show answer” on any question to reveal the correct option and explanation.
- Q2
Which one of the following is a factor of the polynomial (x − 1)(x − 2)(x − 4) − 90 ?
Elementary MathematicsAlgebra - Q3
What is the square root of 23 − 4√15 ?
Elementary MathematicsNumber System - Q4
What is the remainder after dividing the number 37^1000 by 9 ?
Elementary MathematicsNumber System - Q5
The sum of LCM and HCF of two numbers is 536 and the difference between LCM and HCF is 296. If one of the numbers is 104, then what is the other number?
Elementary MathematicsHCF and LCM - Q6
20 men are supposed to complete a work in 10 days. After working for 5 days, they realise that only one-fourth of the work is done. How many more men they need to employ to finish the work on time?
Elementary MathematicsTime and Work - Q7
If x is a negative real number, then which of the following are not correct ? 1. There is some natural number k such that kx > 0. 2. x² + x > 0 always. 3. 2x < x < −x. 4. x² is always a rational number. Select the correct answer using the code given below:
Elementary MathematicsNumber System - Q8
What is the sum of the linear factors (in x and y) of the expression 2x² + xy − 3y² ?
Elementary MathematicsAlgebra - Q9
Which one of the following equations does not have real roots ?
Elementary MathematicsAlgebra - Q10
The sum and the product of the roots of a quadratic equation are 7 and 12 respectively. If the bigger root is halved and the smaller root is doubled, then what is the resulting quadratic equation ?
Elementary MathematicsAlgebra - Q11
For which values of k, does the equation x² − kx + 2 = 0 have real and distinct solutions ?
Elementary MathematicsAlgebra - Q12
If α + β + γ = αβ + βγ + γα, then what is (1 − α)(1 − β)(1 − γ) equal to ?
Elementary MathematicsAlgebra - Q13
If log₁₀x + log₁₀x² = 2log₁₀x + 1, then what is the value of x ?
Elementary MathematicsAlgebra - Q14
The LCM of two prime numbers p and q is 2231, where p > q. What is the value of p − q ?
Elementary MathematicsHCF and LCM - Q15
Three runners are running in a circular track, and they complete one round in 20, 30 and 35 minutes respectively. When will they next meet at the starting point ?
Elementary MathematicsHCF and LCM - Q16
5 pencils, 6 notebooks and 7 erasers cost Rs. 250; whereas 6 pencils, 4 notebooks and 2 erasers cost Rs. 180. What is the cost of 2 notebooks and 4 erasers ?
Elementary MathematicsAlgebra - Q17
How many zeros are there in the product 1⁵⁰ × 2⁴⁹ × 3⁴⁸ × … × 50¹ ?
Elementary MathematicsNumber System - Q18
If p and q (p > q) are the roots of the equation x² − 60x + 899 = 0, then which one of the following is correct ?
Elementary MathematicsAlgebra - Q19
If the roots of the equation x² − 4x − log₁₀N = 0 are real, then what is the minimum value of N ?
Elementary MathematicsAlgebra - Q20
If 5^(x−1) = (2.5)^(log₁₀5), then what is the value of x ?
Elementary MathematicsAlgebra - Q21
If 96 − 64a³ + 8/a⁶ − 48/a³ − t³ = 0, then what is a²t + 4a³ equal to ?
Elementary MathematicsAlgebra - Q22
If A + B = (x² − 8)/(x + 2) and A − B = (−x² + 2x + 4)/(x + 2), then what is B equal to ?
Elementary MathematicsAlgebra - Q23
What is x⁴/((x² − y²)(x² − z²)) + y⁴/((y² − x²)(y² − z²)) + z⁴/((z² − x²)(z² − y²)) equal to ?
Elementary MathematicsAlgebra - Q24
If (2ab − b²) : (6a² − ab) = 1 : 6, then what is (a + b) : (a − b) equal to ?
Elementary MathematicsRatio and Proportion - Q25
If x/(b + c − a) = y/(b − c − a) = z/(a + b − c) = k, then what is x² + y² + z² − 2xy − 2yz + 2zx equal to ?
Elementary MathematicsRatio and Proportion - Q26
If three positive numbers are in the ratio 2 : 3 : 5 and the sum of their squares is 1368, then what is sum of the numbers ?
Elementary MathematicsRatio and Proportion - Q27
Consider the following inequalities :
- 1(a² − b²)/(a² + b²) > (a − b)/(a + b) where a > b > 0. 2. (a³ + b³)/(a² + b²) > (a² + b²)/(a + b) only when a > b > 0.
Which of the above is/are correct ?
Elementary MathematicsAlgebra - Q28
Let work done by (3n − 1) men in (2n + 1) days be x and work done by (3n + 1) men in (4n − 3) days be y. If x : y = 6 : 11, then what is the value of n ?
Elementary MathematicsTime and Work - Q29
If (ay − bx)/c = (cx − az)/b = (bz − cy)/a, then which of the following is/are correct ?
- 1x/a = y/b.
- 2(x + y + z)/(a + b + c) = z/c.
Select the correct answer using the code given below:
Elementary MathematicsRatio and Proportion - Q30
A person wishes to fence 375 m² rectangular garden. He has 65 m of barbed wire and is able to fence only three sides of the garden. What is the perimeter of the garden ?
Elementary MathematicsMensuration - Q31
If one of the roots of the equation ax² − 4ax + 15 = 0 is 3/2, then what is the sum of the squares of the roots ?
Elementary MathematicsAlgebra - Q32
A two-digit number is such that the product of the digits is 8. If 63 is added to this number, the digits interchange their places. What is the sum of the digits in the number ?
Elementary MathematicsNumber System - Q33
A motor boat has speed 30 km/hr in still water. It goes 60 km down stream and comes back in 9/2 hours. What is the speed of the stream ?
Elementary MathematicsTime, Speed and Distance - Q34
The present age of a father is equal to sum of the ages of his 4 children. After ten years the sum of the ages of the children will be 1.6 times the age of their father. What is the present age of father ?
Elementary MathematicsAlgebra - Q35
The sum of numerator and denominator of a fraction is 10. If the numerator is increased by 3 and denominator is decreased by 1, the fraction becomes 1. What is the difference between numerator and denominator of the fraction ?
Elementary MathematicsAlgebra - Q36
If the system of equations 7x + ky = 21 and kx + 7y = 19 have unique solution, then which one of the following is correct ?
Elementary MathematicsAlgebra - Q37
A sum of money compounded annually doubles itself in 5 years. In how many years will it become four times of itself ?
Elementary MathematicsSimple and Compound Interest - Q38
Between 3 and 4 O’clock, both hour hand and minute hand will coincide past 3 O’clock between
Elementary MathematicsTime, Speed and Distance - Q39
Sheela can stitch a suit in 2 days, while Meena can stitch a suit in 1 1/2 days. How many days will both take in stitching 35 suits ?
Elementary MathematicsTime and Work - Q40
If 2x − 3y − 7 = 0, then what is the value of 8x³ − 36x²y + 54xy² − 27y³ − 340 ?
Elementary MathematicsAlgebra - Q41
If p varies directly as q and inversely as square of r, what is the percentage increase in p due to an increase in q by 20% and a decrease in r by 20% ?
Elementary MathematicsPercentages - Q42
A person agrees to work for 30 days, on a condition that for every day’s work he should receive Rs. 500, and that for every day’s absence from work he should forfeit Rs. 100. At the end of the time he received Rs. 11,400. How many days did he work ?
Elementary MathematicsAlgebra - Q43
A person bought a chair and a table for Rs. 750. He sold the chair at a gain of 5% and the table at a gain of 20%. He gained 16% on the whole. What is original cost of table ?
Elementary MathematicsProfit and Loss - Q44
A person rode one-third of a journey at 60 km/hr, one-third at 50 km/hr and the rest at 40 km/hr. Had the person ridden half of the journey at 60 km/hr and the rest at 40 km/hr, he would have taken 4 minutes longer to complete the journey. What distance did the person ride ?
Elementary MathematicsTime, Speed and Distance - Q45
A person saves Rs. 1000 more than he did the previous year. If he saves Rs. 2000 in the first year, in how many years will he save Rs. 170000 ?
Elementary MathematicsAlgebra - Q46
What is the minimum value of cos³θ + sec³θ where 0° ≤ θ < 90° ?
Elementary MathematicsTrigonometry - Q47
If 14sin²θ + 10cos²θ = 11 where 0° < θ < 90°, then what is the value of tanθ + cotθ ?
Elementary MathematicsTrigonometry - Q48
What is (sin³θ + cos³θ)/(sinθ + cosθ) + (sin³θ − cos³θ)/(sinθ − cosθ) equal to ?
Elementary MathematicsTrigonometry - Q49
A ladder 10 m long reaches a point 10 m below the top of a vertical flagstaff. From foot of the ladder, the elevation of top of the flagstaff is 60°. What is the height of flagstaff ?
Elementary MathematicsTrigonometry - Q50
What is the maximum value of 1 + 2sin²θcos²θ − sin⁴θ − cos⁴θ where 0° < θ < 90° ?
Elementary MathematicsTrigonometry - Q51
From an aeroplane flying above a river at an altitude of 1200 m, it is observed that the angles of depression of opposite points on the two banks of a river are 30° and θ. If the width of the river is 3000 m, then which one of the following is correct ?
Elementary MathematicsTrigonometry - Q52
If (cos²θ − 3cosθ + 2)/sinθ = 1 where 0° < θ < 90°, then what is sin²θ + cosθ equal to ?
Elementary MathematicsTrigonometry - Q53
Consider the following :
- 1sin⁴θ − sin²θ = cos⁴θ − cos²θ.
- 2sin⁴θ + cos⁴θ = 1 + 2sin²θcos²θ.
- 3tan⁴θ + tan²θ = sec⁴θ − sec²θ.
Which of the above are identities ?
Elementary MathematicsTrigonometry - Q54
What is the value of sin24°sin66° − cos24°cos66° + tan24°tan66° − cot24°cot66° ?
Elementary MathematicsTrigonometry - Q55
If x = p sinA cosB, y = p sinA sinB and z = p cosA, then what is the value of x² + y² + z² ?
Elementary MathematicsTrigonometry - Q56
If x = m secA + n tanA and y = m tanA + n secA, then what is x² − y² equal to ?
Elementary MathematicsTrigonometry - Q57
If for some θ lying between 0° and 90°, tanθ = 1, then what is the value of sin²θ − 2sinθcosθ ?
Elementary MathematicsTrigonometry - Q58
What is (2sin³θ − sinθ)/(cosθ − 2cos³θ), (0° < θ < 90°) equal to ?
Elementary MathematicsTrigonometry - Q59
If A, B and C are interior angles of a triangle ABC, then what is tan((B + C)/2) + sin((B + C)/2) − cot(A/2) − cos(A/2) equal to ?
Elementary MathematicsTrigonometry - Q60
In a triangle ABC, right-angled at B, AB + BC = 10(1 + √3) cm and length of the hypotenuse is 20 cm. What is the value of tanA + tanC ?
Elementary MathematicsGeometry: Lines, Angles and Triangles - Q61
The radius of circum-circle of a right-angled triangle is 10 cm and the altitude drawn to the hypotenuse is 8 cm. What is the area of the triangle ?
Elementary MathematicsGeometry: Lines, Angles and Triangles - Q62
Two circles touch externally. The sum of their areas is 41π square cm. If the distance between their centres is 9 cm, then what is difference between their diameters ?
Elementary MathematicsCircles - Q63
Let p be the area of a square X and q be the area of the square formed on the diagonal of the square X. What is the value of p/q ?
Elementary MathematicsQuadrilaterals and Polygons - Q64
The area of a rhombus is 336 square cm. If the length of one of its diagonals is 48 cm, then what is the perimeter of the rhombus ?
Elementary MathematicsQuadrilaterals and Polygons - Q65
The minute hand of a clock is 21 cm long. What is the area on the face of the clock described by the minute hand between 10.10 a.m and 10.30 a.m ? (take π = 22/7)
Elementary MathematicsCircles - Q66
The length and breadth of a room are 21 m and 16 m respectively. If the length of the longest rod that can be placed in the room is 29 m, then what is the height of the room ?
Elementary MathematicsMensuration - Q67
A hemispherical bowl of internal radius 18 cm contains a liquid. The liquid is filled in small cylindrical bottles of internal radius 3 cm and internal height 4 cm. What is the number of bottles used to empty the bowl ?
Elementary MathematicsMensuration - Q68
A hollow spherical shell is made of a metal of density 7 g/cm³. If its internal and external radii are 3 cm and 6 cm respectively, then what is the mass of the shell ? (take π = 22/7)
Elementary MathematicsMensuration - Q69
A cone of height 16 cm and diameter 14 cm is mounted on a hemisphere of same diameter. What is the volume of the solid thus formed ? (take π = 22/7)
Elementary MathematicsMensuration - Q70
3 cubes each of volume 343 cm³ are joined end to end. What is the total surface area of the resulting cuboid ?
Elementary MathematicsMensuration - Q71
A cubical block of side 14 cm is surmounted by a hemisphere of radius 7 cm. What is the total surface area of the solid thus formed ? (take π = 22/7)
Elementary MathematicsMensuration - Q72
How many silver coins, 3.5 cm in diameter and of thickness 4 mm, must be melted to form a cuboid of dimensions 21 cm x 11 cm x 7 cm ? (take π = 22/7)
Elementary MathematicsMensuration - Q73
A tub is in the shape of a frustum of a cone. The radii of two circular ends of the tub are 105 cm and 42 cm. If the vertical height of the tub is 16 cm, what is its slant height ?
Elementary MathematicsMensuration - Q74
ABC is a triangle right-angled at B with AB = 8 cm and BC = 6 cm. It is made to revolve about its side BC. What is the approximate total surface area of the cone so formed ? (take π = 22/7)
Elementary MathematicsMensuration - Q75
A solid rod consists of a cylinder of height 20 cm and radius 7 cm. It is surmounted by another solid cylinder of height 10 cm and radius 3.5 cm. If 1 cubic metre of rod weighs 10000 kg, what is the mass of the rod ? (take π = 22/7)
Elementary MathematicsMensuration - Q76
Consider the following for the next three (03) items that follow : A solid consisting of a right circular cone of radius x and height 2x standing on a hemisphere of radius x (take π = 22/7). The volume of the solid is equal to that of a
Elementary MathematicsMensuration - Q77
Consider the following for the next three (03) items that follow : A solid consisting of a right circular cone of radius x and height 2x standing on a hemisphere of radius x (take π = 22/7). What is the approximate total surface area of the solid ?
Elementary MathematicsMensuration - Q78
Consider the following for the next three (03) items that follow : A solid consisting of a right circular cone of radius x and height 2x standing on a hemisphere of radius x (take π = 22/7). The solid is placed upright in a right circular cylinder full of water such that it touches the bottom. If the internal radius of cylinder is x and height is 3x, what is the approximate volume of water left in the cylinder ?
Elementary MathematicsMensuration - Q79
Consider the following for the next two (02) items that follow : A chord of a circle of radius 2.1 cm subtends an angle of 120° at the centre. (take π = 22/7 and √3 = 1.732). What is the approximate area of minor segment of the circle ?
Elementary MathematicsCircles - Q80
Consider the following for the next two (02) items that follow : A chord of a circle of radius 2.1 cm subtends an angle of 120° at the centre. (take π = 22/7 and √3 = 1.732). What is the approximate area of major segment of the circle ?
Elementary MathematicsCircles - Q81
Consider the following for the next three (03) items that follow : ABC is a triangle with sides AB = 6 cm, BC = 10 cm and CA = 8 cm. With vertices A, B and C as centres, three circles are drawn each touching the other two externally. What is the sum of the radii of the circles ?
Elementary MathematicsGeometry: Lines, Angles and Triangles - Q82
Consider the following for the next three (03) items that follow : ABC is a triangle with sides AB = 6 cm, BC = 10 cm and CA = 8 cm. With vertices A, B and C as centres, three circles are drawn each touching the other two externally. What is the length of the altitude of the triangle drawn from vertex A on BC ?
Elementary MathematicsGeometry: Lines, Angles and Triangles - Q83
Consider the following for the next three (03) items that follow : ABC is a triangle with sides AB = 6 cm, BC = 10 cm and CA = 8 cm. With vertices A, B and C as centres, three circles are drawn each touching the other two externally. If P, Q and R are the areas of sectors at A, B and C within the triangle respectively, then which of the following is/are correct ?
- 1P = π cm².
- 29Q + 4R = 36π cm².
Select the correct answer using the code given below:
Elementary MathematicsGeometry: Lines, Angles and Triangles - Q84
Consider the following for the next two (02) items that follow : ABC is triangle in which AB = AC and D is any point on BC. Which one of the following is correct ?
Elementary MathematicsGeometry: Lines, Angles and Triangles - Q85
Consider the following for the next two (02) items that follow : ABC is triangle in which AB = AC and D is any point on BC. If AD = 5 cm, BD = 4 cm and CD = 6 cm, then what is AB equal to ?
Elementary MathematicsGeometry: Lines, Angles and Triangles - Q86
Consider the following for the next three (03) items that follow : ABC is a triangle with AB = 1.6 cm, BC = 6.3 cm and CA = 6.5 cm. Let P and Q be the mid-points of AB and BC respectively. What is AB² + 4BQ² equal to ?
Elementary MathematicsGeometry: Lines, Angles and Triangles - Q87
Consider the following for the next three (03) items that follow : ABC is a triangle with AB = 1.6 cm, BC = 6.3 cm and CA = 6.5 cm. Let P and Q be the mid-points of AB and BC respectively. What is AQ² + CP² equal to ?
Elementary MathematicsGeometry: Lines, Angles and Triangles - Q88
Consider the following for the next three (03) items that follow : ABC is a triangle with AB = 1.6 cm, BC = 6.3 cm and CA = 6.5 cm. Let P and Q be the mid-points of AB and BC respectively. What is 4(CP² − AQ²) equal to ?
Elementary MathematicsGeometry: Lines, Angles and Triangles - Q89
Consider the following for the next two (02) items that follow : AB is a diameter of a circle with centre O. Radius OP is perpendicular to AB. Let Q be any point on arc PB. What is ∠BAP equal to ?
Elementary MathematicsCircles - Q90
Consider the following for the next two (02) items that follow : AB is a diameter of a circle with centre O. Radius OP is perpendicular to AB. Let Q be any point on arc PB. What is ∠AQP equal to ?
Elementary MathematicsCircles - Q91
Consider the following for the next four (04) items that follow : The following Pie-Charts show the percentage of different categories (A, B, C, D, E and F) of employees in a company in the year 2019 and 2020. The total number of employees was 4000 in 2019 and 5000 in 2020. (2019: A 20%, B 6%, C 10%, D 15%, E 27%, F 22%. 2020: A 21%, B 10%, C 11%, D 9%, E 27%, F 22%.) There are two categories of employees in 2019 whose total strength remained the same in 2020. What are these two categories ?
Elementary MathematicsStatistics - Q92
Consider the following for the next four (04) items that follow : The following Pie-Charts show the percentage of different categories (A, B, C, D, E and F) of employees in a company in the year 2019 and 2020. The total number of employees was 4000 in 2019 and 5000 in 2020. (2019: A 20%, B 6%, C 10%, D 15%, E 27%, F 22%. 2020: A 21%, B 10%, C 11%, D 9%, E 27%, F 22%.) In which one of the following categories of employees, the percentage change in number of employees in 2020 was maximum as compared to 2019 ?
Elementary MathematicsStatistics - Q93
Consider the following for the next four (04) items that follow : The following Pie-Charts show the percentage of different categories (A, B, C, D, E and F) of employees in a company in the year 2019 and 2020. The total number of employees was 4000 in 2019 and 5000 in 2020. (2019: A 20%, B 6%, C 10%, D 15%, E 27%, F 22%. 2020: A 21%, B 10%, C 11%, D 9%, E 27%, F 22%.) What was the percentage increase of category-E employees in 2020 as compared to that in 2019 ?
Elementary MathematicsStatistics - Q94
Consider the following for the next four (04) items that follow : The following Pie-Charts show the percentage of different categories (A, B, C, D, E and F) of employees in a company in the year 2019 and 2020. The total number of employees was 4000 in 2019 and 5000 in 2020. (2019: A 20%, B 6%, C 10%, D 15%, E 27%, F 22%. 2020: A 21%, B 10%, C 11%, D 9%, E 27%, F 22%.) There are two categories of employees whose strength increased by same percentage in 2020. What are the two categories ?
Elementary MathematicsStatistics - Q95
Consider the following for the next four (04) items that follow : The following Bar-Chart gives the production of cars by three different companies X, Y and Z in different years. (2014: X 300, Y 200, Z 500. 2015: X 400, Y 300, Z 500. 2016: X 500, Y 400, Z 600. 2017: X 600, Y 500, Z 700. 2018: X 700, Y 500, Z 800.) The percentage increase in the total production of cars from 2014 to 2018 was
Elementary MathematicsStatistics - Q96
Consider the following for the next four (04) items that follow : The following Bar-Chart gives the production of cars by three different companies X, Y and Z in different years. (2014: X 300, Y 200, Z 500. 2015: X 400, Y 300, Z 500. 2016: X 500, Y 400, Z 600. 2017: X 600, Y 500, Z 700. 2018: X 700, Y 500, Z 800.) Consider the production of each company separately. In how many instances was the percentage increase in production of cars over the previous year’s production greater than 20% ?
Elementary MathematicsStatistics - Q97
Consider the following for the next four (04) items that follow : The following Bar-Chart gives the production of cars by three different companies X, Y and Z in different years. (2014: X 300, Y 200, Z 500. 2015: X 400, Y 300, Z 500. 2016: X 500, Y 400, Z 600. 2017: X 600, Y 500, Z 700. 2018: X 700, Y 500, Z 800.) In which year was the percentage increase in production of cars minimum as compared to its previous year ?
Elementary MathematicsStatistics - Q98
Consider the following for the next four (04) items that follow : The following Bar-Chart gives the production of cars by three different companies X, Y and Z in different years. (2014: X 300, Y 200, Z 500. 2015: X 400, Y 300, Z 500. 2016: X 500, Y 400, Z 600. 2017: X 600, Y 500, Z 700. 2018: X 700, Y 500, Z 800.) In one of the years the percentage increase in production of cars was minimum as compared to its previous year. What was the minimum percentage ?
Elementary MathematicsStatistics - Q99
Consider the following for the next two (02) items that follow : Collect all the sequences of five consecutive integers such that their product is equal to one of these integers. Let X be the collection of all possible such sequences. Let P be the smallest integer and Q be the largest integer occurring in these sequences. How many such sequences of five consecutive integers are possible ?
Elementary MathematicsNumber System - Q100
Consider the following for the next two (02) items that follow : Collect all the sequences of five consecutive integers such that their product is equal to one of these integers. Let X be the collection of all possible such sequences. Let P be the smallest integer and Q be the largest integer occurring in these sequences. What is the arithmetic mean of P and Q ?
Elementary MathematicsNumber System