# CDS 2021 Maths Question Paper -2

1. What are distinct prime factors of the number 26381 ?

(a)          29, 17, 37

(b)          31, 17, 47

(c)           19, 37, 13

(d)          23, 31, 37

1. Which one of the following is a factor of the polynomial (x – 1)(x- 2)(x – 4) – 90 ?

(a)          x+14

(b)          x-14

(c)           x – 6

(d)          x-1

1. What is the square root of 23-4√15 ?

(a)          √6-3√2

(b)          7-3√5

(c)           √3-2√5

(d)          √5-4-√3

1. What is the remainder after dividing the number 371000 by 9 ?

(a)          1

(b)          3

(c)           7

(d)          9

1. 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?

(a)          420

(b)          480

(c)           484

(d)          506

1. 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?

(a)          40

(b)          30

(c)           20

(d)          15

1. If x is a negative real number, then which of the following are not correct ?
2. There is some natural number k such that kx > 0
3. x2 + x > 0 always
4. 2x<x<-x
5. x2, is always a rational number Select the correct answer using the code given below:

(a)          1, 2 and 3

(b)          1, 2 and 4

(c)           1, 3 and 4

(d)          2, 3 and 4

1. What is the sum of the linear factors (in x and y) of the expression 2X2 +xy- 3-y2 ?

(a)          2x-3y

(b)          3x-2y

(c)           3x + 2y

(d)          2x + 3y

1. Which one of the following equations does not have real roots ?

(a)          2×2 + 16x + 3=0

(b)          2X2 + 10x- 1=0

(c)           x2 – 8x + 1 = 0

(d)          4×2+9x + 6 = 0

1. 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 ?

(a)          x2-6x+12 = 0

(b)          x2 – 8x+ 12 = 0

(c)           x2 + 8x+12 = 0

(d)          x2 – 10x+ 12 = 0

1. For which values of k, does the equation x2 – kx + 2 = 0 have real and distinct solutions ?

(a)          -2√2 <k <2√2

(b)          k<-2√2 only

(c)           k>2√2 only

(d)          k<-2√2 or k> 2√2

1. If α + β + γ = α β + βγ + γα, then what is (1- α)(1- β)(1- γ) equal to?

(a)          1- α β γ

(b)          1 + α β γ

(c)           α 2+2+ γ 2

(d)          (α – β)( β – γ)( γ – α)

1. If log10x + log10x2 = 21og10x + 1, then what is the value of or ?

(a)          1

(b)          2

(c)           5

(d)          10

1. The LCM of two prime numbers p and q is 2231, where p>q. What is the value ofp-ql

(a)          67

(b)          70

(c)           74

(d)          81

1. 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?

(a)          After 3 hour 30 minutes

(b)          After 4 hour 30 minutes

(c)           After 3 hours

(d)          After 7 hours

1. 5 pencils, 6 notebooks and 7 erasers cost Rs.250; whereas 6 pencils, 4 note-books and 2 erasers cost Rs.180. What is the cost of 2 notebooks and 4 erasers ?

(a)          Rs.90

(b)          Rs.75

(c)           Rs.60

(d)          Rs.40

1. How many zeros are there in the product 150 x 249 x 348 x … x 501 ?

(a)          262

(b)          261

(c)           246

(d)          235

1. If p and q (p > q) are the roots of the equation x2 – 60x + 899 = 0, then which one of the following is correct ?

(a)          p-q- 1=0

(b)          p – 2q + 21 = 0

(c)           2p – q – 30 = 0

(d)          3/? – 2<7 – 43 = 0

1. If the roots of the equation x2-4x – log10N = 0 are real, then what is the minimum value of N?

(a)          0.1

(b)          0.01

(c)           0.001

(d)          0.0001

1. If 5x-1 = (2.5)log105, then what is the value of x?

(a)          1

(b)          log102

(c)           log105

(d)          21og105

1. If then what is a2t + 4a3 equal to?

(a)          0

(b)          1

(c)           2

(d)          3

1. If A + B = x2 -8/x+2 and A-B = -x2+2x+4/x+2, then What is B equal to? 1. What is (a)          -1

(b)          0

(c)           1

(d)          x2+y2+z2

1. If (2ab – b2): (6a2 – ab) =1:6, then what is (a + b): (a – b) equal to ?

(a)          3 only

(b)          5 only

(c)           -3 or 3

(d)          -5 or 5

1. 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?

(a)          k2(a2+b2+c2)

(b)          k2(a2-b2 + c2)

(c)           k2(a + b + c)2

(d)          k2(a-b + c)2

1. 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 ?

(a)          30

(b)          45

(c)           60

(d)          75

1. Consider the following inequalities :
2. a2-b2/a2+b2 > a-b/a+b where a > b> 0
3. a3 +b3/a2 + b2 > a2+b2 /a+b only when a> b> 0

Which of the above is/are correct ?

(a)          1 only

(b)          2 only

(c)           Both 1 and 2

(d)          Neither 1 nor 2

1. Let work done by (3/1-1) 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?

(a)          6

(b)          7

(c)           8

(d)          9

1. If ay-bx/c = cx-az/b = bz-cy/a, then which of the following is/are correct ?
2. x/a = y/b
3. x+y + z/a+b+c = z/c

Select the correct answer using the code given below:

(a)          1 only

(b)          2 only

(c)           Both 1 and 2

(d)          Neither 1 nor 2

1. 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?

(a)          80 m

(b)          84 m

(c)           90 m

(d)          100 m

1. 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 ?

(a) 15/2

(b) 17/2

(c) 19/2

(d) 21/2

1. 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?

(a)          6

(b)          7

(c)           8

(d)          9

1. 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?

(a)          5 km/hr

(b)          8 km/hr

(c)           10 km/hr

(d)          12 km/hr

1. 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 ?

(a)          36 years

(b)          40 years

(c)           42 years

(d)          45 year

1. 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 numera-tor and denominator of the fraction ?

(a)          2

(b)          3

(c)           4

(d)          5

1. If the system of equations 7x + ky = 21 and kx + 7y = 19 have unique solution, then which one of the following is correct ?

(a)          k ≠ 7

(b)          k ≠ 13

(c)           k =7

(d)          k =13

1. A sum of money compounded annually doubles itself in 5 years. In how many years will it become four times of itself ?

(a)          10 years

(b)          12 years

(c)           15 years

(d)          20 years

1. Between 3 and 4 O’clock, both hour hand and minute hand will coincide past 3 O’clock between

(a)          15-16 minutes

(b)          16-17 minutes

(c)           17-18 minutes

(d)          18-19 minutes

1. 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 30 suits?

(a)          32 days

(b)          33 days

(c)           35 days

(d)          40 days

1. If 2x – 3y – 7 = 0, then what is the value of 8×3 – 36x2y + 54xy2 – 27y3 – 340 ?

(a)          -1

(b)          0

(c)           1

(d)          3

1. 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% ?

(a)          87.5%

(b)          85%

(c)           82.5%

(d)          80%

1. 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?

(a)          20

(b)          21

(c)           24

(d)          25

1. 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?

(a)          Rs.400

(b)          Rs.450

(c)           Rs.550

(d)          Rs.600

1. 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 ?

(a)          180 km

(b)          210 km

(c)           240 km

(d)          300 km

1. 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?

(a)          16 years

(b)          17 years

(c)           18 years

(d)          19 years

1. What is the minimum value of cos3 Ө + sec3 Ө where 0°≤ Ө <90° ?

(a)          0

(b)          1

(c)           2

(d)          None of the above

1. If 14 sin2 Ө + 10cos2 Ө = 11 where 0° < Ө < 90°, then what is the value of tan Ө + cot Ө ?

(a) 4/√3

(b) 2/√3

(c) √3

(d) 2√3

1. What is sin3 Ө + cos3 Ө/ sin Ө + cos Ө + sin3 Ө – cos3 Ө/sin Ө -cos Ө equal to ?

(a)          0

(b)          1

(c)           2

(d)          4

1. 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?

(a)          12 m

(b)          15 m

(c)           16 m

(d)          20 m

1. What is the maximum value of 1 + 2 sin2 Ө cos2 Ө – sin4 Ө – cos4 Ө where 0°< Ө < 90° ?

(a)          1

(b)          2

(c)           3

(d)          4

1. 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 Ө. If the width of the river is 3000 m, then which one of the following is correct ?

(a)          Ө < 30°

(b)          3O°< Ө <45°

(c)           45° < Ө < 60°

(d)          60° < Ө < 90°

1. If cos2 Ө -3cos Ө + 2/sin Ө = 1 where 0° < Ө < 90° then what is sin2 Ө + cos Ө equal to?

(a) 5/4

(b) 3/2

(c) 7/4

(d) 2

1. Consider the following :
2. sin4 Ө – sin2 Ө = cos4 Ө – cos2 Ө
3. sin4 Ө +cos4 Ө = 1 + 2sin2 Ө cos2 Ө
4. tan4 Ө +tan2 Ө = sec4 Ө – sec2 Ө

Which of the above are identities ?

(a)          1 and 2 only

(b)          2 and 3 only

(c)           1 and 3 only

(d)          1, 2 and 3

1. What is the value of sin 24° sin 66°-cos 24° cos 66° + tan 24° tan 66° – cot 24° cot 66° ?

(a)          0

(b)          1

(c)           2

(d)          3

1. If x = p sin A cos B, y = p sin A sinB and z = p cos A, then what is the value of x2+y2 + z2?

(a)          -p2

(b)          0

(c)           P2

(d)          2p2

1. If x = m sec A + n tan A and y = m tanA + n sec A, then what is x2-y2 equal to ?

(a)          m2-n2

(b)          m2 + n2

(c)           m2 + n2 – mn

(d)          m2 -n2 + mn

1. If for some Ө lying between 0° and 90°, tan Ө = 1, then what is the value of sin2 Ө -2sin Ө cos Ө?

(a)          -1

(b)          0

(c)           1/2

(d)          -1/2

1. What is 2sin3 Ө- sin Ө,/cos Ө- 2cos3 Ө, (0° < Ө < 90°) equal to?

(a)          sin Ө

(b)          cos Ө

(c)           tan Ө

(d)          cot Ө

1. If A, B and C are interior angles of a triangle ABC, then what is equal to ?

(a)          0

(b)          1/2

(c) sin(A + B + C/4)

(d) tan (A + B + C/4)

1. 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 tan A + tan C ?

(a) 4/√3

(b) 2/√3

(c) √3

(d) 2√3

1. 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 ?

(a)          60 cm2

(b)          80 cm2

(c)           100 cm2

(d)          120 cm2

1. 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 ?

(a)          1 cm

(b)          1.5 cm

(c)           2 cm

(d)          4 cm

1. 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 ?

(a) 1/8

(b) 1/4

(c) 1/3

(d) ½

1. 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?

(a)          200 cm

(b)          120 cm

(c)           100 cm

(d)          90 cm

1. 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)

(a)          231 cm2

(b)          331 cm2

(c)           462 cm2

(d)          492 cm2

1. The length and breadth of a room are 21m 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?

(a)          10 m

(b)          11m

(c)           12 m

(d)          13 m

1. 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 ?

(a)          54

(b)          81

(c)           108

(d)          135

1. 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)

(a)          2772 g

(b)          3322g

(c)           4433 g

(d)          5544 g

1. 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)

(a)          1540 cm3

(b)          1078 cm3

(c)           1048 cm3

(d)          770 cm3

1. 3 cubes each of volume 343 cm3 are joined end to end. What is the total surface area of the resulting cuboid ?

(a)          343 cm2

(b)          350 cm2

(c)           686 cm2

(d)          700 cm2

1. A cubical block of side 14 cm is sur-mounted by a hemisphere of radius 7 cm. What is the total surface area of the solid thus formed ? (take π = 22/7)

(a)          1330 cm2

(b)          1306 cm2

(c)           1296 cm2

(d)          1256 cm2

1. 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)

(a)          420

(b)          210

(c)           200

(d)          168

1. 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?

(a)          63.5 cm

(b)          65 cm

(c)           73.5 cm

(d)          75 cm

1. 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)

(a)          452 cm2

(b)          440 cm2

(c)           432 cm2

(d)          420 cm2

1. 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)

(a)          34.65 kg

(b)          31.56 kg

(c)           3.465 kg

(d)          3.156 kg

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)

1. The volume of the solid is equal to that of a

(b)          sphere of diameter x

(d)          cylinder of radius √2 x

1. What is the approximate total surface area of the solid ?

(a)          11.2 x2

(b)          12.5 x2

(c)           13.3 x2

(d)          15.1 x2

1. 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 ?

(a)          5.04 x3

(b)          5.09 x3

(c)           5.14X3

(d)          5.24 x3

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)

1. What is the approximate area of minor segment of the circle?

(a)          2.71 cm2

(b)          2.42 cm2

(c)           1.91 cm2

(d)          1.71 cm2

1. What is the approximate area of major segment of the circle ?

(a)          10.05 cm2

(b)          10.15 cm2

(c)           11.05 cm2

(d)          11.15 cm2

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.

1. What is the sum of the radii of the circles ?

(a)          10.4 cm

(b)          11.2 cm

(c)           12 cm

(d)          13 cm

1. What is the length of the altitude of the triangle drawn from vertex A on BC ?

(a)          2.4 cm

(b)          3 cm

(c)           4 cm

(d)          4.8 cm

1. 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 ?
2. P = π cm2
3. 9Q + 4R = 36 π cm2

Select the correct answer using the code given below:

(a)          1 only

(b)          2 only

(c)           Both 1 and 2

(d)          Neither 1 nor 2

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.

1. Which one of the following is correct ?

1. If AD = 5 cm, BD = 4 cm and CD = 6 cm, then what is AB equal to ?

(a)          7 cm

(b)          6.5 cm

(c)           6 cm

(d)          5.5 cm

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.

1. What is AB2 + 4BQ2 equal to ?

(a)          41.25 cm2

(b)          42.25 cm2

(c)           43.75 cm2

(d)          44.25 cm2

1. What is AQ2 + CP2 equal to ?

(a)          AC2

(b)          1.2 AC2

(c)           1.25 AC2

(d)          1.5 AC2

1. What is 4(CP2-AQ2) equal to?

(a)          101.39 cm2

(b)          111.39 cm2

(c)           121.39 cm2

(d)          131.39 cm2

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.

1. What is ∠BAP equal to ?

(a)          30°

(b)          40°

(c)           45°

(d)          60°

1. What is ∠AQP equal to ?

(a)          30°

(b)          40°

(c)           45°

(d)          60°

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. 1. There are two categories of employees in 2019 whose total strength remained the same in 2020. What are these two categories ?

(a)          C and B

(b)          E and A

(c)           E and F

(d)          C and D

1. In which one of the following categories of employees, the percentage change in number of employees in 2020 was maximum as compared to 2019?

(a)          A

(b)          B

(c)           C

(d)          D

1. What was the percentage increase of category-E employees in 2020 as compared to that in 2019?

(a)          20%

(b)          25%

(c)           30%

(d)          40%

1. There are two categories of employees whose strength increased by same percentage in 2020. What are the two categories ?

(a)          E and F

(b)          C and E

(c)           A and E

(d)          A and C

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 : 1. The percentage increase in the total production of cars from 2014 to 2018 was

(a)          50%

(b)          100%

(c)           150%

(d)          200%

1. 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% ?

(a) 5

(b) 6

(c) 7

(d) 8

1. In which year was the percentage in-crease in production of cars minimum as compared to its previous year ?

(a)          2015

(b)          2016

(c)           2017

(d)          2018

1. 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 ?

(a)          11 1/9%

(b)          11 2/9%

(c)           13 1/3%

(d)          20%

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.

1. How many such sequences of five consecutive integers are possible ?

(a)          One

(b)          Two

(c)           Three

(d)          Five

1. What is the arithmetic mean of P and Q1

(a)          0

(b)          1

(c)           2

(d)          Cannot be determined due to in-sufficient data