CDS (I) 2025 — Elementary Mathematics (Full Question Paper)

A real number M is squared to give the value N. What is the minimum value of (M+N)?

• (a)-0.25
• (b)-0.50
• (c)0
• (d)0.25

What is the sum of all 3-digit numbers that give a remainder of 5 when they are divided by 50?

• (a)9005
• (b)9540
• (c)9600
• (d)9640

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?

• (a)68
• (b)69
• (c)70
• (d)71

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:

• (a)x
• (b)y
• (c)yz
• (d)xz

Let p be the remainder when 7^84 is divided by 342 and q be the remainder when 7^84 is divided by 344. What is (p-q) equal to?

• (a)0
• (b)1
• (c)2
• (d)6

N is the smallest 5-digit number which when divided by 2, 2^2, 2^3, 2^4, …, 2^n leaves a remainder 1. What is the value of n?

• (a)12
• (b)13
• (c)14
• (d)15

What is the minimum value of p for which 1/532900 + p^2/266450 + p^4/523900 is an integer?

• (a)729
• (b)243
• (c)27
• (d)1

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?

• (a)ab+a+b
• (b)ab-a-b
• (c)a+b-ab
• (d)-(ab+a+b)

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)100
• (b)110
• (c)115
• (d)120

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)10 L
• (b)20 L
• (c)25 L
• (d)Not possible

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?

• (a)3 m
• (b)5 m
• (c)6 m
• (d)7 m

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?

• (a)231 square cm
• (b)256 square cm
• (c)272 square cm
• (d)297 square cm

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?

• (a)2.5d
• (b)2.75d
• (c)3d
• (d)3.25d

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?

• (a)5:4
• (b)5:3
• (c)4:3
• (d)3:2

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?

• (a)2.5 cm
• (b)6 cm
• (c)7.5 cm
• (d)8 cm

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?

• (a)3 square cm
• (b)32 square cm
• (c)33 square cm
• (d)3.3 square cm

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?

• (a)40√3 square cm
• (b)40 square cm
• (c)20√3 square cm
• (d)20 square cm

The measure of an 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?

• (a)65°
• (b)75°
• (c)80°
• (d)85°

What is log₁₀2000 + log₁₀400 + 4log₁₀25 + 5log₁₀20 equal to?

• (a)10
• (b)16
• (c)18
• (d)20

If log₁₀(100001-4^x)/(5-x) = 1, then what is x equal to?

• (a)0
• (b)1
• (c)10
• (d)100

If 2sin^4α + 2cos^4α – 1 = 0, where 0 ≤ α < π/2, then what is sin2α + cos2α equal to?

• (a)0
• (b)1
• (c)(√3+1)/2
• (d)(√3-1)/2

Consider the following: I. 1-sin^6α = cos^2α(cos^4α-3cos^2α+3); II. cos^8α-sin^8α = 2sin^2α(1-cos^4α+sin^2αcos^2α). Which of the above is/are identities?

• (a)I only
• (b)II only
• (c)Both I and II
• (d)Neither I nor II

If p = 1/(cosecθ+cotθ) and q = cosecθ, then what is p^2-2pq equal to?

• (a)-1
• (b)0
• (c)1
• (d)2

Consider the following statements: I. (cosecα-secα) is always positive in the first quadrant. II. (tanα-cotα) is always negative in the first quadrant. Which of the statements given above is/are correct?

• (a)I only
• (b)II only
• (c)Both I and II
• (d)Neither I nor II

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?

• (a)2h
• (b)2.5h
• (c)3h
• (d)3.5h

What is sinθ/(1-cotθ) + cosθ/(1-tanθ) (θ≠π/4) equal to?

• (a)sinθ+cosθ
• (b)sinθ-cosθ
• (c)cosθ-sinθ
• (d)-(sinθ+cosθ)

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?

• (a)π
• (b)π/2
• (c)π/3
• (d)π/4

If cot^2θ – 3√3cotθ + 6 = 0, where π/6 ≤ θ < π/2, then what is a value of sinθ+cos2θ?

• (a)0
• (b)1
• (c)√3
• (d)1+√2

Which of the following equations is/are possible? I. sin^2θ = (x+y)^2/4xy, where x, y are positive unequal real quantities. II. sinθ+cosθ = x+1/x, where x is a positive real quantity. Select the correct answer using the code given below.

• (a)I only
• (b)II only
• (c)Both I and II
• (d)Neither I nor II

If m^2(sinθ-1) + n^2(sinθ+1) = 0, where 0 < θ < π/2, then what is (m^2+n^2)cosθ – (m^2-n^2)cotθ equal to?

• (a)4mn
• (b)2mn
• (c)1
• (d)0

If sinα+cosα = √2, where 0 < α < π/2, then what is sin^3α-cos^3α equal to?

• (a)1
• (b)1/2
• (c)1/4
• (d)0

What is (1+cotα-cosecα)(1+tanα+secα) equal to?

• (a)1/2
• (b)1
• (c)2
• (d)4

If tanθ = (sinα-cosα)/(sinα+cosα), where θ and α (α≠π/4) are acute angles, then what is √2sinθ equal to?

• (a)sinα-cosα
• (b)sinα+cosα
• (c)cosα-sinα
• (d)±(sinα-cosα)

For how many values of α does the expression (sinα+2)(sinα+4)(sinα-2)(sinα-4) become zero?

• (a)No value
• (b)One
• (c)Two
• (d)Four

What is the value of x, where 0 ≤ x < 30°, satisfying tan3x tan6x = 1?

• (a)0°
• (b)10°
• (c)12°
• (d)15°

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 a≠b≠c?

• (a)0
• (b)3
• (c)a+b+c
• (d)3(a-b)(b-c)(c-a)

Given that (100×99×98×…×3×2×1)/100^n is an integer. What is the largest value of n for which this is true?

• (a)20
• (b)21
• (c)24
• (d)None of the above

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?

• (a)150 m
• (b)180 m
• (c)200 m
• (d)240 m

If x^2+y^2+z^2 = 3, where x, y and z are integers, then how many values can (xy+yz+zx) have?

• (a)One
• (b)Two
• (c)Three
• (d)Four

If x, y, z are real numbers such that x+y+z = 10 and xy+yz+zx = 18, then what is the value of x^3+y^3+z^3-3xyz?

• (a)400
• (b)440
• (c)460
• (d)500

What is √(17-4√15) + √(8-2√15) equal to?

• (a)√3
• (b)2√3
• (c)2(√5-√3)
• (d)2(√5+√3)

What is the maximum value of the sum of the numbers 36, 33, 30, 27, 24, …?

• (a)240
• (b)237
• (c)234
• (d)231

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? I. The remainder when (m+n) is divided by 12 is 10. II. The remainder when (m-n) is divided by 12 is 10. Select the correct answer using the code given below.

• (a)I only
• (b)II only
• (c)Both I and II
• (d)Neither I nor II

If (x+y):(y+z):(z+x) = 3:5:6 and x+y+z = 14, then what is x^2+y^2+z^2 equal to?

• (a)81
• (b)84
• (c)87
• (d)90

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?

• (a)13:5
• (b)25:1
• (c)9:4
• (d)16:1

Travelling at 3/5th of his usual speed, a man is late by 20 minutes. What is the usual time if he travels with his usual speed?

• (a)25 minutes
• (b)30 minutes
• (c)32 minutes
• (d)35 minutes

What is the remainder when 2^p-1 is divided by p, where p>5 is a prime number?

• (a)1
• (b)2
• (c)3
• (d)4

What is the number of factors of 24^3-16^3-8^3?

• (a)33
• (b)30
• (c)28
• (d)24

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)2
• (b)3
• (c)4
• (d)5

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)?

• (a)(2yt+xt)/(y-x)
• (b)(yt+xt)/(y-x)
• (c)(3yt-xt)/(2y-x)
• (d)(2yt-xt)/(y-x)

What is the whole surface area of the frustum?

• (a)39πk^2/8
• (b)41πk^2/8
• (c)43πk^2/8
• (d)45πk^2/8

What is the volume of the frustum?

• (a)61πk^3/48
• (b)59πk^3/48
• (c)57πk^3/48
• (d)53πk^3/48

What is the area of the triangle ABC?

• (a)18 square cm
• (b)24 square cm
• (c)36 square cm
• (d)48 square cm

A circle is inscribed in the triangle. What is its radius?

• (a)1 cm
• (b)1.5 cm
• (c)2 cm
• (d)2.5 cm

What is the length of the common tangent PQ?

• (a)16 cm
• (b)16√2 cm
• (c)24 cm
• (d)24√2 cm

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?

• (a)192√2 square cm
• (b)192 square cm
• (c)96√2 square cm
• (d)96 square cm

What is AB:BC:CA equal to?

• (a)10:6:5
• (b)5:10:6
• (c)6:5:3
• (d)3:5:6

What is the approximate area of the triangle ABC?

• (a)175 square cm
• (b)190 square cm
• (c)205 square cm
• (d)285 square cm

What is the inner radius of the circular opening of the pot so formed?

• (a)10√2 cm
• (b)15 cm
• (c)10√3 cm
• (d)12 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?

• (a)π/3
• (b)π/4
• (c)π/6
• (d)π/12

What is h equal to?

• (a)3.5 m
• (b)4 m
• (c)4.5 m
• (d)5 m

If α is the angle between the line joining P and Q, and the line joining P and R, then what is cosα equal to?

• (a)2√5/7
• (b)3√5/7
• (c)1/3
• (d)2/3

If the material weighs 2000 kg per cubic metre, then what is the weight of the material used in the construction of the box?

• (a)360 gm
• (b)365 gm
• (c)720 gm
• (d)730 gm

If water weighs 1000 kg per cubic metre, then what is the weight of the box with water?

• (a)1.850 kg
• (b)1.900 kg
• (c)2.050 kg
• (d)2.100 kg

What is AD equal to?

• (a)4.8 cm
• (b)5.0 cm
• (c)5.2 cm
• (d)5.4 cm

What is ratio of area of triangle ADC to area of triangle ADB?

• (a)7:15
• (b)9:16
• (c)2:3
• (d)3:4

What is the volume of rainfall in cubic meter per hectare?

• (a)40
• (b)400
• (c)4000
• (d)40000

What is the weight of water (in metric tonnes) of annual rainfall falling there on a hectare of land?

• (a)40
• (b)400
• (c)4000
• (d)40000

What is the ratio of the radius of the conical body to its slant height?

• (a)1:2
• (b)√3:1
• (c)√3:2
• (d)√2:1

If the sum of slant height, height and radius is (9+3√3) cm, then what is the volume of the cone?

• (a)27π cubic cm
• (b)18√3π cubic cm
• (c)24π cubic cm
• (d)27√3π cubic cm

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?

• (a)₹40,000
• (b)₹40,500
• (c)₹41,000
• (d)₹51,500

100 quintals is what percent of 10 metric tonnes?

• (a)1%
• (b)10%
• (c)100%
• (d)1000%

A circle is inscribed in an equilateral triangle. The radius of the circle is 2 cm. What is the area of the triangle?

• (a)12√3 square cm
• (b)12 square cm
• (c)9√3 square cm
• (d)9 square cm

The sides of a triangle are k, 1.5k and 2.25k. What is the sum of the squares of its medians?

• (a)359k^2/64
• (b)379k^2/64
• (c)389k^2/64
• (d)399k^2/64

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] equal to?

• (a)abc
• (b)2abc
• (c)4abc
• (d)ab+bc+ca

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)

• (a)₹10,406
• (b)₹10,416
• (c)₹10,426
• (d)₹10,436

If p = 0.0̄9 (recurring), then what is the value of 70p^2+43p-5?

• (a)-1
• (b)0
• (c)1
• (d)10

What is the remainder when 2^101 is divided by 101?

• (a)1
• (b)2
• (c)5
• (d)7

If p(≠0) and q(≠0) are the roots of the equation x^2+px+q = 0, then what is p^2+q^2 equal to?

• (a)2
• (b)3
• (c)4
• (d)5

The equations x^2+px+q = 0 and x^2+qx+p = 0 (p≠q) have a common root. What is the value of (p+q)?

• (a)-1
• (b)0
• (c)1
• (d)2

If x^2-5x+4 is a factor of x^4-px^2+q, then what are the values of p and q respectively?

• (a)17, 16
• (b)16, 17
• (c)15, 16
• (d)16, 15

If two quadratic equations px^2+px+4 = 0 and x^2+qx+q = 0 have a common root 2, then what is p+q equal to?

• (a)-3
• (b)-2
• (c)0
• (d)3

What is the HCF of the polynomials x^8+x^4+1 and x^4+x^2+1?

• (a)1
• (b)x^4-x^2+1
• (c)x^4+x^2+1
• (d)x^4-x^2-1

An arc AB of a circle subtends an angle x radian at the centre O. If the area of the sector AOB is equal to half of the square of length of arc AB, then what is x equal to?

• (a)1/4
• (b)1/2
• (c)1
• (d)2

Consider the following statements in respect of prime numbers p and q: I. Their LCM is always an odd number. II. Sum of their LCM and HCF is always an even number. Which of the statements given above is/are correct?

• (a)I only
• (b)II only
• (c)Both I and II
• (d)Neither I nor II

What is the value of f?

• (a)50
• (b)60
• (c)70
• (d)80

What is the median of the frequency distribution?

• (a)3
• (b)5
• (c)6
• (d)It cannot be determined from the given data

What is the mode of the frequency distribution?

• (a)5
• (b)6
• (c)7
• (d)It cannot be determined from the given data

What is the most appropriate graphical representation of the given frequency distribution of X?

• (a)Bar diagram
• (b)Histogram
• (c)Frequency polygon
• (d)Pie Chart

What is the mean of the frequency distribution?

• (a)3.29
• (b)4.29
• (c)5.29
• (d)6.29

What is the number of students who scored between 60 and 70 marks?

• (a)105
• (b)110
• (c)205
• (d)210

What is the number of students who scored more than 50 marks?

• (a)100
• (b)125
• (c)200
• (d)225

In which of the given years was the circulation of the newspaper D close to its average circulation over all the years?

• (a)2020 and 2021
• (b)2022 and 2023
• (c)2022 only
• (d)2020 only

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?

• (a)2019
• (b)2020
• (c)2021
• (d)2022

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?

• (a)One
• (b)Two
• (c)Three
• (d)Four

On an average, how many persons live in every pucca house?

• (a)8
• (b)8.54
• (c)9.54
• (d)7.54

If 5000 more Kutcha houses are built, then what will be approximate change in angle for Kutcha houses in Pie Chart-I?

• (a)11°
• (b)13°
• (c)15°
• (d)23°

If 300 families from the “Houseless” category shift into Kutcha houses, what will be the average number of families in every Kutcha house?

• (a)15
• (b)13.9
• (c)12.9
• (d)10.5

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?

• (a)25.0
• (b)26.7
• (c)28.3
• (d)30.0

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?

• (a)50.1
• (b)51.5
• (c)52.3
• (d)53.4