CDS (II) 2022 — Elementary Mathematics (Full Question Paper)

If x² − 20 = √(20 + √(20 + √(20 + √(20 + …)))) infinite terms, then what is x equal to ?

• (a)4
• (b)5
• (c)√5
• (d)2√5

If a + bb + c = c + dd + a where a ≠ c, then which one of the following is correct ?

• (a)a + b = c + d
• (b)a + c = b + d
• (c)a − b − c + d = 0
• (d)a + b + c + d = 0

If (a³ + b³) is proportional to (a² − b²), then (a² − ab + b²) is proportional to

• (a)(a − b)
• (b)(a + b)
• (c)(a + ab + b)
• (d)(a³ − b³)

If √(x+20) + √(x-1)√(x+20) – √(x-1) = 73, then what is the value of √((x+20)(x-1)) ?

• (a)8
• (b)9
• (c)10
• (d)12

What is the HCF of (x⁸ − y⁸) and (x⁷ − y⁷ + x⁵y² − x²y⁵) ?

• (a)(x² + y²)
• (b)(x² − y²)
• (c)(x³ − y³ − x²y + xy²)
• (d)(x³ − y³ + x²y − xy²)

If the sum of the squares of the roots of the equation x² − 14x + k = 0 is 100, then what is the value of k ?

• (a)42
• (b)48
• (c)52
• (d)56

If (x + k) is the HCF of x² + px + q and x² + qx + p, where p ≠ q, then what is the value of k ?

• (a)−1
• (b)0
• (c)12
• (d)1

If three times the greater of two numbers is divided by the smaller number, the quotient will be 6 and the remainder will be 6. If five times the smaller number is divided by the greater number, the quotient will be 2 and the remainder will be 3. What is the difference between the numbers ?

• (a)8
• (b)9
• (c)10
• (d)12

Consider the following statements: 1. (ab + bc + ca) is a factor of a²(b−c)³ + b²(c−a)³ + c²(a−b)³. 2. (a + b + c) is a factor of a²(b−c)³ + b²(c−a)³ + c²(a−b)³. Which of the statements given above is/are correct ?

• (a)Only 1
• (b)Only 2
• (c)Both 1 and 2
• (d)Neither 1 nor 2

What is 1x(x-y)(x-z) + 1y(y-z)(y-x) + 1z(z-x)(z-y) equal to ?

• (a)0
• (b)1
• (c)1xyz
• (d)−1xyz

Consider the following statements: 1. x⁴(y−z) + y⁴(z−x) + z⁴(x−y) is positive if x > y > z. 2. x⁴(y−z) + y⁴(z−x) + z⁴(x−y) is negative if x < y < z. Which of the above statements is/are correct ?

• (a)Only 1
• (b)Only 2
• (c)Both 1 and 2
• (d)Neither 1 nor 2

What is x⁶ − 24x⁴ + 144x²(x² + 4√3x + 12)(x – 2√3)² equal to ?

• (a)x²
• (b)x² − 2
• (c)x² + 2√3
• (d)x² − 2√3

X and Y can do a piece of work in 45 days and 40 days respectively. They begin to work together, but X leaves after n days and then Y completes the remaining work in 23 days. What is n equal to ?

• (a)8
• (b)9
• (c)10
• (d)12

480 persons working 10 hours per day complete one-fourth of a work in 10 days. How many additional persons are to be employed in order to complete the remaining work in 20 days, working 8 hours per day ?

• (a)400
• (b)420
• (c)480
• (d)500

12 women and 16 men can do a piece of work in 5 days. 13 women and 24 men can do it in 4 days. How long will 25 women and 50 men take to do it ?

• (a)1 day
• (b)2 days
• (c)3 days
• (d)4 days

There are two stations X and Y, 1320 km apart. A train starts from station X at 6 a.m. and moves at an average speed of 60 km/hr. At 2 p.m. another train starts from Y towards X and moves at an average speed of 80 km/hr. When do they meet ?

• (a)6 p.m.
• (b)7 p.m.
• (c)8 p.m.
• (d)9 p.m.

Two glasses of equal volume are filled with a mixture of alcohol and water in the ratio 3 : 2 and 4 : 1, respectively. These glasses are emptied into a third glass. What is the ratio of alcohol and water in the third glass ?

• (a)5 : 4
• (b)7 : 2
• (c)7 : 3
• (d)7 : 4

If x is the HCF and y is the LCM of 35, 625, 920, 2750, then which one of the following is correct ?

• (a)y = 90x
• (b)y = 180x
• (c)y = 270x
• (d)y = 360x

There are two natural numbers x and y, where x > y. When x is divided by 6, it leaves the remainder 2 and; when y is divided by 6, it leaves the remainder 3. What is the remainder when (x − y) is divided by 6 ?

• (a)1
• (b)3
• (c)5
• (d)Remainder cannot be determined

The value of a 2-digit number is 5 times the sum of the digits. What is the product of the digits ?

• (a)15
• (b)18
• (c)20
• (d)27

Consider the number N = 12⁶ × 3⁸ × 5³. Which of the following statements is/are correct ? 1. The number of odd factors of N is 60. 2. The number of even factors of N is 720. Select the correct answer using the code given below :

• (a)Only 1
• (b)Only 2
• (c)Both 1 and 2
• (d)Neither 1 nor 2

If log₁₀ ab – c = log₁₀ bc – a = log₁₀ ca – b, (a ≠ b ≠ c), then what is the value of abc ?

• (a)−1
• (b)0
• (c)1
• (d)3

What is the angle between the hour hand and the minute hand of a clock when the clock shows 4 hours 40 minutes ?

• (a)80°
• (b)100°
• (c)120°
• (d)220°

If log₁₀[995 + √(x² − 12x + 60)] = 3, then what is the sum of the roots of the equation ?

• (a)12
• (b)11
• (c)10
• (d)9

1. x = √(p+q) + √(p-q)√(p+q) – √(p-q), where p > q
2. x = p + √(p²-q²)q, where p > q
3. x = √(p+q)√(p-q), where p > q
Select the correct answer using the code given below:

How many of the following values of x would satisfy the equation qx² − 2px + q = 0 ?

• (a)Only one value
• (b)Only two values
• (c)All three values
• (d)None

Let p(x) = x⁴ + x² + 1, q(x) = x⁴ − 2x³ + 3x² − 2x + 1. If GCD of p(x) and q(x) is x² − x + 1, then what is their LCM ?

• (a)(x²+x+1)(x²-x+1)²
• (b)(x⁴+x²+1)²(x²-x+1)
• (c)(x⁴+x²+1)(x²+x+1)²
• (d)(x⁴+x²+1)(x²-x+1)²

Question: What is the 2-digit number ?
Statement-1: The sum is divisible by 9.
Statement-2: The sum is divisible by 2.
Which one of the following is correct in respect of the question and the statements ?

A 2-digit number is added to the number formed by reversing the digits of the 2-digit number.

• (a)Statement-1 alone is sufficient to answer the question
• (b)Statement-2 alone is sufficient to answer the question
• (c)Both Statement-1 and Statement-2 are sufficient to answer the question
• (d)Both Statement-1 and Statement-2 are not sufficient to answer the question

Question: What is the value of (3x − 2y) ?
Statement-1: y > 1.
Statement-2: HCF of x and y is 1.
Which one of the following is correct in respect of the question and the statements ?

LCM of two numbers x and y is 481 where x > y.

• (a)Statement-1 alone is sufficient to answer the question
• (b)Statement-2 alone is sufficient to answer the question
• (c)Both Statement-1 and Statement-2 are sufficient to answer the question
• (d)Both Statement-1 and Statement-2 are not sufficient to answer the question

Statement-1: n is a natural number.
Statement-2: n is an even natural number.
Which one of the following is correct in respect of the question and the statements ?

Is (xⁿ + yⁿ) divisible by (x + y) ?

• (a)Statement-1 alone is sufficient to answer the question
• (b)Statement-2 alone is sufficient to answer the question
• (c)Both Statement-1 and Statement-2 are sufficient to answer the question
• (d)Both Statement-1 and Statement-2 are not sufficient to answer the question

Question: Is xy > 0 ?
Statement-1: x⁸y⁹ < 0.
Statement-2: x⁹y¹⁰ < 0.
Which one of the following is correct in respect of the question and the statements ?

Let x and y be two real numbers.

• (a)Statement-1 alone is sufficient to answer the question
• (b)Statement-2 alone is sufficient to answer the question
• (c)Both Statement-1 and Statement-2 are sufficient to answer the question
• (d)Both Statement-1 and Statement-2 are not sufficient to answer the question

If a = 3b, 4b = 5c, 6c = 7d, then what is d + ad – a equal to ?

• (a)2743
• (b)4327
• (c)−2743
• (d)−4327

What are the values of k for which the polynomial (k − 3)x² − kx − 1 has no real linear factors ?

• (a)k < −6
• (b)−6 < k < 2
• (c)2 < k < 6
• (d)k > 6

If bc + cd = 2bd and a + c = 2b, then which one of the following is correct ?

• (a)ab − cd = 0
• (b)ac − bd = 0
• (c)ad − bc = 0
• (d)ad + bc = 0

In a flight of 2800 km, an aircraft was slowed down due to bad weather. Its average speed for the trip was reduced by 100 km/hr and time of flight increased by 30 minutes. What was the original average speed of the aircraft ?

• (a)700 km/hr
• (b)750 km/hr
• (c)800 km/hr
• (d)900 km/hr

If x = b + c, y = c + a, z = a + b, then what is (x + y + z)³ − 24xyz equal to ?

• (a)a³ + b³ + c³
• (b)2(a³ + b³ + c³)
• (c)8(a³ + b³ + c³)
• (d)None of the above

A person X starts from a place A and another person Y starts simultaneously from another place B which is d km away from A. They walk in the same direction. X walks at an average speed of u km/hr and Y walks at an average speed of v km/hr. How far will X have walked before he overtakes Y ?

• (a)ud(u – v)
• (b)vd(u – v)
• (c)(ud – vd)(u – v)
• (d)(ud + vd)(u + v)

If p is the difference between a number and its reciprocal and q is the difference between the square of the same number and the square of its reciprocal, then what is p⁴ + 4p² equal to ?

• (a)4q
• (b)8q
• (c)4q²
• (d)q²

If x = 2 + 2¹⁄², then what is the value of x⁴ + 16x⁻⁴ ?

• (a)152
• (b)144
• (c)136
• (d)132

X takes 3 hours longer than Y to walk 30 km. If X doubles his speed, he takes 2 hours less than Y. What is the speed of Y ?

• (a)3 km/hr
• (b)4 km/hr
• (c)427 km/hr
• (d)437 km/hr

If C is the compound interest on ₹ 10,000 for one year at 4% per annum when compounded quarterly, then which one of the following is correct ?

• (a)C < ₹ 100
• (b)₹ 100 < C < ₹ 200
• (c)₹ 200 < C < ₹ 400
• (d)C > ₹ 400

Consider the equation 6x² − 25x + 6x² + 25x + 12 = 0. What is one of the possible values of x − 1x ?

• (a)12
• (b)32
• (c)2
• (d)52

What is one of the possible values of x² + 1x² ?

• (a)6
• (b)629
• (c)8
• (d)829

A triangle ABC with sides AB = 15 cm, BC = 9 cm, CA = 12 cm is inscribed in a circle. What is cos²A + cos²B + cos²C equal to ?

• (a)34
• (b)1
• (c)54
• (d)2

What is sin²A + sin²B + sin²C equal to ?

• (a)2
• (b)54
• (c)1
• (d)34

What is the radius of the circle ?

• (a)4·5 cm
• (b)6 cm
• (c)7·5 cm
• (d)15 cm

The angle of elevation of a cloud at C from a point (P), H metres above the surface of a lake is 30°. The height of the cloud from the surface of the lake is 2H metres. Let θ be the angle of depression of the reflection of the cloud in the lake from the point P. What is the value of θ ?

• (a)30°
• (b)45°
• (c)60°
• (d)Cannot be determined due to insufficient data

What is PC equal to ?

• (a)H metres
• (b)√2H metres
• (c)√3H metres
• (d)2H metres

A frequency distribution table (x: 0,1,2,3,4,5; f: 46,p,q,25,10,5) is given. Total frequency is 200 and mean of the distribution is 1·46. What is the value of p ?

• (a)70
• (b)72
• (c)76
• (d)78

What is the value of q ?

• (a)32
• (b)34
• (c)36
• (d)38

What is the median of the distribution ?

• (a)1
• (b)2
• (c)3
• (d)4

If tan θ = √(q² – p²)p; 0 < θ < 90°, then what is sec θ + cos θ + 2 equal to ?

• (a)p² + q²pq
• (b)(p + q)²pq
• (c)(p + q)²2pq
• (d)(p – q)²pq

If 11 sin θ + 60 cos θ = 61; 0 < θ < 90°, then what is the value of √(660(tan θ + cot θ)) ?

• (a)61
• (b)61√2
• (c)122
• (d)122√2

What is the diameter of a circle inscribed in a regular polygon of 15 sides with side length unity ?

• (a)0·5 cot 12°
• (b)cot 12°
• (c)0·5 tan 12°
• (d)tan 12°

If x = 1 + sin θcos θ, then what is tan θ + sec θ – 1tan θ – sec θ + 1 equal to ?

• (a)−x
• (b)x
• (c)2x
• (d)x2

Which of the above is/are identities ?

Consider the following: 1. tan θ + sin θtan θ – sin θ = sec θ + 1sec θ – 1, where 0 < θ < π/2. 2. cos²θ – sin²θcos²θ + sin²θ = 2tan θtan²θ + 1, where 0 < θ < π/2.

• (a)Only 1
• (b)Only 2
• (c)Both 1 and 2
• (d)Neither 1 nor 2

If sin θ + cos θsin θ – cos θ = 5, where 0 < θ < π/2, θ ≠ π/4, then what is the value of 2sin θ + 3cos θ3sin θ – 2cos θ ?

• (a)85
• (b)2
• (c)125
• (d)3

If 1cosec θ – cot θ − 1sin θ = x, then what is 1cosec θ + cot θ − 1sin θ equal to, where 0 < θ < π/2 ?

• (a)−x
• (b)x
• (c)1x
• (d)−1x

What is 2sin⁶θ + 2cos⁶θ − 3sin⁴θ − 3cos⁴θ equal to ?

• (a)−1
• (b)0
• (c)1
• (d)2

What is the minimum value of 3600sec²θ + 121cosec²θ, where 0 < θ < π/2 ?

• (a)1320
• (b)2401
• (c)3721
• (d)5041

What is √[(1 + sin θcos θ)² + (cos θ1 + sin θ)² − 2] equal to, where 0 < θ < π/2 ?

• (a)tan θ
• (b)cot θ
• (c)2tan θ
• (d)2cot θ

If x is the harmonic mean between y and z, then which one of the following is correct ?

• (a)xy + xz − yz = 0
• (b)xy + xz − 2yz = 0
• (c)xy + xz + yz = 0
• (d)xy + xz − 4yz = 0

What is the median of all possible factors of 120 ?

• (a)10
• (b)11
• (c)12
• (d)13·5

The sum of deviations of a set of n values measured from 50 is −10 and the sum of deviations of the values measured from 46 is 70. What is the mean of the values ?

• (a)48·5
• (b)49·0
• (c)49·5
• (d)50·0

A table shows 4 schools (I: 60 students, avg 60; II: 50 students, avg 80; III: 50 students, avg 40; IV: x students, avg 50). If the average marks of the students of all four schools are 58, then how many students appeared from School-IV ?

• (a)38
• (b)40
• (c)42
• (d)44

The data of different natural numbers 4, 7, 10, 14, 2x+3, 2x+5, 22, 23, 30, 50 are in ascending order. How many possible values are there for the median of the data for various values of x ?

• (a)Only one value
• (b)Only two values
• (c)Only three values
• (d)Five values

If M is the mean of the first 100 even natural numbers, then what is the mean of the first 100 odd natural numbers ?

• (a)M − 1
• (b)M
• (c)M + 1
• (d)M + 2

Mean marks of 50 students were found to be 78·4. But later it was detected that 95 was misread as 59 and 25 was misread as 52. What is the difference between correct mean and incorrect mean ?

• (a)0·04
• (b)0·08
• (c)0·12
• (d)0·18

In a triangle ABC, ∠A = 2θ, ∠B = ∠C = 4θ and θ satisfies 4sin²θ + 2sinθ − 1 = 0. What is the ratio of BC to AB ?

• (a)√5 − 1
• (b)(√5 – 1)2
• (c)(√5 – 1)4
• (d)2(√5 − 1)

A circle is inscribed in a triangle ABC. It touches the sides BC, CA, AB at D, E, F respectively. What is ∠EDF equal to ?

• (a)90° − A
• (b)90° − (B + C)2
• (c)90° − 2A
• (d)90° − A2

The length of a room is 2116 times its breadth and breadth is 43 times its height. If H is the height of the room and L is the longest rod that can be placed in the room, then which one of the following is correct ?

• (a)12L = 29H
• (b)9L = 25H
• (c)7L = 23H
• (d)5L = 13H

A conical vessel of radius 12 cm and height 16 cm is filled with water. A sphere is lowered into water and its size is such that it touches the sides of the vessel and it is just immersed. What is the radius of the sphere ?

• (a)5 cm
• (b)6 cm
• (c)6·5 cm
• (d)7 cm

How much water will remain in the vessel after the overflow ?

• (a)288π mL
• (b)360π mL
• (c)480π mL
• (d)500π mL

What is the ratio of lateral surface area of the vessel to the surface area of the sphere ?

• (a)43
• (b)32
• (c)53
• (d)2

A chord of length l of a circle makes an angle 90° at the centre of the circle. What is the area of the minor segment ?

• (a)l²2(π − 12)
• (b)l²4(π − 12)
• (c)l²4(π2 − 1)
• (d)l²2(π2 − 12)

What is the area of the major segment ?

• (a)l²4(3π2 + 1)
• (b)l²4(3π2 − 1)
• (c)l²2(3π2 + 1)
• (d)l²2(3π2 − 1)

Triangle ABC is right-angled at A and AD is perpendicular to BC. If BD = 7·5 cm and DC = 10 cm, then what is AD equal to ?

• (a)5 cm
• (b)5√2 cm
• (c)5√3 cm
• (d)10 cm

The perpendicular AD on the base BC of a triangle ABC intersects BC at D so that DB = 3CD. Which one of the following is correct ?

• (a)2(AB + AC)(AB − AC) = BC²
• (b)3(AB + AC)(AB − AC) = 2BC²
• (c)4(AB + AC)(AB − AC) = 3BC²
• (d)5(AB + AC)(AB − AC) = 4BC²

Triangle ABC is right-angled at C and AC = √3 BC. What is ∠ABC equal to ?

• (a)30°
• (b)45°
• (c)60°
• (d)75°

In a triangle ABC, ∠A = 60°. What is AB² + AC² − BC² equal to ?

• (a)AB.AC
• (b)AB.BC
• (c)AC.BC
• (d)2 AB.AC

The diameter of a sphere made of copper is 3 cm. The sphere is melted and recast into a wire. If the length of the wire is half metre, then what is the diameter of the wire ?

• (a)0·3 cm
• (b)0·45 cm
• (c)0·6 cm
• (d)0·75 cm

The height of a cone is 30 cm. A small cone is cut off at the top by a plane parallel to its base. If its volume is 127 of the volume of the given cone, then what is the height of the frustum of the cone ?

• (a)10 cm
• (b)12 cm
• (c)18 cm
• (d)20 cm

A rectangular metal sheet is of length 24 cm and breadth 18 cm. From each of its corners a square of side x cm is cut off and an open box is made of the remaining sheet. If the volume of the box is 640 cubic cm, then what is the value of x ?

• (a)2
• (b)3
• (c)4
• (d)6

Let x, y, z be the length, breadth, height of a cuboid. If its volume is 400 cubic cm and total surface area is 340 square cm, then what is x⁻¹ + y⁻¹ + z⁻¹ equal to ?

• (a)1720
• (b)1740
• (c)710
• (d)910

Two parallel sides of a trapezium are 29 cm and 21 cm. Non-parallel sides are equal and each is of length 8·5 cm. What is the area of the trapezium ?

• (a)187·5 square cm
• (b)227·5 square cm
• (c)375 square cm
• (d)455 square cm

A square copper plate of side 16 cm weighs 128 gm. A circular disc of diameter 14 cm is cut off from the plate. What is the weight of the remaining part ? (π = 22/7)

• (a)48 gm
• (b)49 gm
• (c)50 gm
• (d)51 gm

The area of a rhombus is 96 square cm and one of its diagonals is of length 12 cm. What is the perimeter of the rhombus ?

• (a)36 cm
• (b)40 cm
• (c)44 cm
• (d)48 cm

A right-angled triangle ABC is inscribed in a circle of radius 10 cm. The altitude drawn to the hypotenuse AC is of length 8 cm. If AB = x cm and BC = y cm, then what is the value of xy ?

• (a)60
• (b)80
• (c)120
• (d)160

Two circles touch externally. The sum of their areas is 89π square cm and the distance between their centres is 13 cm. What is the difference in their radii ?

• (a)2 cm
• (b)2·5 cm
• (c)3 cm
• (d)3·5 cm

A square and a rectangle have same perimeter. They differ in areas by 1 square cm. The length of the rectangle exceeds its breadth by

• (a)1 cm
• (b)2 cm
• (c)3 cm
• (d)4 cm

Two rectangles are of same area equal to 480 square cm. They differ in lengths by 6 cm and breadths by 4 cm. What is the difference in their perimeters ?

• (a)2 cm
• (b)4 cm
• (c)6 cm
• (d)10 cm

A closed box is built of wood of uniform thickness. Its external dimensions are 12 cm, 10 cm and 8 cm. If the inner surface area is 376 square cm, then what is the thickness of the wood ?

• (a)0·5 cm
• (b)1 cm
• (c)1·5 cm
• (d)2 cm

X, Y and Z are three equilateral triangles. The sum of the areas of X and Y is equal to the area of Z. If the side lengths of X and Y are 6 cm and 8 cm respectively, then what is the side length of Z ?

• (a)9 cm
• (b)9·5 cm
• (c)10 cm
• (d)10·5 cm

What is the time taken by a person to cover one round of a circular park of diameter 210 m if he walks at a speed of 6 km/hr ? (π = 22/7)

• (a)6·6 minutes
• (b)5·5 minutes
• (c)4·4 minutes
• (d)3·3 minutes

In an equilateral triangle of side 2√3 cm, a circle is inscribed touching the sides. What is the area of the remaining portion of the triangle ?

• (a)(2√3 − π) square cm
• (b)(3√3 − π) square cm
• (c)(4√3 − 2π) square cm
• (d)(4√3 − π) square cm

A triangle has side lengths x cm, x + 13 cm and x + 26 cm. If its area is 126 square cm, then what is the value of x ?

• (a)18
• (b)17
• (c)16
• (d)15

ABCD is a parallelogram. A circle through A, B and C intersects CD (produced) at E. Which of the following is/are correct ? 1. AE = AD 2. CD = DE. Select the correct answer using the code given below :

• (a)Only 1
• (b)Only 2
• (c)Both 1 and 2
• (d)Neither 1 nor 2

Consider the following statements: 1. In an equilateral triangle, the centroid and centre of circumcircle coincide. 2. Angle bisectors of a cyclic quadrilateral form another cyclic quadrilateral. 3. Every cyclic parallelogram is a rectangle. Which of the statements given above are correct ?

• (a)Only 1 and 2
• (b)Only 2 and 3
• (c)Only 1 and 3
• (d)1, 2 and 3

ABCD is a trapezium in which AB is parallel to DC. The vertices A, B, C and D pass through a circle. Which of the following are correct ? 1. AD = BC 2. ∠A + ∠C = 180° 3. ∠A + ∠D = 180°. Select the correct answer using the code given below :

• (a)Only 1 and 2
• (b)Only 2 and 3
• (c)Only 1 and 3
• (d)1, 2 and 3

ABCD is a cyclic quadrilateral. AB and DC when produced, meet in E. Which of the following statements is/are correct ? 1. ΔEBC is similar to ΔEAD. 2. ∠CBE + ∠DAE = 180°. Select the correct answer using the code given below :

• (a)Only 1
• (b)Only 2
• (c)Both 1 and 2
• (d)Neither 1 nor 2

In a triangle ABC, DE is a line segment which intersects AB at D and AC at E such that DE is parallel to BC. The line segment divides the triangle in two parts of equal area. What is BDAB equal to ?

• (a)√2 – 12
• (b)√2 – 1√2
• (c)√3 – 1√3
• (d)√2