CDS 2018 Maths Question Paper -1

5¹⁷ + 5¹⁸ + 5¹⁹ + 5²⁰ is divisible by

(a) 7

(b) 9

(d) 13

If a + b = 2c, then what is the value of a/a-c +c/b-c?

(a) -1

(b) 0

(d) 2

If x = y^1/a, y = z^1/band z = x^1/cwhere x ≠ 1, y ≠ 1, z ≠ 1, then what is the value of abc ?

(a) -1

(b) 1

If 2b = a + c and y2 = xz, then what is x^b-cy^c-az^a-bequal to?

(a) 3

(b) 2

(d) -1

Which one of the following is correct ?

(a) Decimal expansion of a rational number is terminating.

(b) Decimal expansion of a rational number is non-terminating.

(d) Decimal expansion of an irrational number is non-terminating and non-repeating.

If the roots of the equation px2 + x + r = 0 are reciprocal to each other, then which one of the following is correct ?

(a) P = 2r

(b) P = r

(d) P = 4r

If 65x – 33y = 97 and 33x – 65y = 1, then what is xy equal to ?

(a) 2

(b) 3

(d) -3

8. If b/y + z/c =1 and c/z +x/a =1, then what is ab+xy/bx equal to ?

(a) 1

(b) 2

(d) -1

If a2-1/a =5, then what is the value of a6-1/a3 ?

(a) 125

(b) -125

(d) – 140

If x + y + z = 0, then what is (y + z – x)3 + (z + x – y)3 + (x + y – z)3 equal to?

(a) (x + y + z)3

(b) 3 (x + y) (y + z) (z + x)

(d) – 24xyz

If (x + 3) is a factor of x3 + 3×2 + 4x + k, then what is the value of k ?

(a) 12

(b) 24

(d) 72

The smallest integer with 4 digits which is a perfect square is

(a) 1000

(b) 1024

(d) None of the above

Which one of the following is a zero of the polynomial 3×3 + 4×2 – 7 ?

(a) 0

(b) 1

(d) – 1

There are two numbers which are greater than 21 and their LCM and HCF are 3003 and 21 respectively. What is the sum of these numbers?

(a) 504

(b) 508

(d) 528

If α and β are the roots of the equation ax2 + bx + c = 0, then what is the value of the expression (α + 1) (β + 1) ?

(a) a + b + c/a

(b) b + c-a/a

(d) a b – c/a

The remainder when 3×3 + kx2 + 5x – 6 is divided by (x + 1) is – 7. What is the value of k?

(a) – 14

(b) 14

(d) 7

If fix) and g(x) are polynomials of degree and q respectively, then the degree {(f(x) ± g(x)} (if it is non-zero) is

(a) Greater than min(p, q)

(b) Greater than max(p, q)

(d) Equal to min(p, q)

What is the value of

(a) -2√15

(b) 2√15

(d) -√15

What is the value of where x ≠ 0 ?

(a) -1

(b) 0

(d) 3

The sum of a number and its square is 20. Then the number is

(a) – 5 or 4

(b) 2 or 3

(d) 5 or – 4

If the price of wheat rises by 25%, then by how much percent must a man reduce his consumption in order to keep his budget the same as before ?

(a) 15%

(b) 20%

(d) 30%

1/25 of the students who registered did not appear for the examination, 11/20 of those who appeared passed. If the number of registered students is 2000, the number who passed is

(a) 1920

(b) 1056 .

(d) 864

What is the difference between 0.9 and 0.9 ?

(a) 0

(b) 0.099

(d) 0.09

If A : B = 1 : 2, B : C = 3 : 4, C : D = 2 : 3 and D : E = 3 : 4, then what is B : E equal to ?

(a) 3:2

(b) 1:8

(d) 4:1

A work when done by 10 women is completed in 12 days. The same work can be completed in 8 days when done by 5 men. How many days will it take to complete when 6 women and 3 men are employed to perform the same job ?

(a) 12

(b) 10

(d) 5

A man undertakes to do a certain work in 150 days. He employs 200 men. He finds that only a quarter of the work is done in 50 days. How many additional men should he employ so that the whole work is finished in time ?

(a) 75

(b) 85

(d) 120

A train moving with a speed of 60 km per hour crosses an electric pole in 30 seconds. What is the length of the train in metres ?

(a) 300

(b) 400

(d) 600

Rs.120 is distributed among A, B and C so that A’s share is Rs. 20 more than B’s and Rs. 20 less than C’s. What is B’s share ?

(a) Rs. 10

(b) Rs. 15

(d) Rs. 25

In the following table of inverse variation, what are the values of A, B and C respectively ?

(a) 10, -30,-1

(b) 10,-1,30

(d) -1,-30, 10

A person borrowed Rs. 5,000 at 5% rate of interest per annum and immediately lent it at 5.5%. After two years he collected the amount and settled his loan. What is the amount gained by him in this transaction ?

(a) Rs .25

(b) Rs .50

(d) Rs .200

At present the average of the ages of a father and a son is 25 years. After seven years the son will be 17 years old. What will be the age of the father after 10 years ?

(a) 44 years

(b) 45 years

(d) 52 years

If 5 tractors can plough 5 hectares of land in 5 days, then what is the number of tractors required to plough 100 hectares in 50 days ?

(a) 100

(b) 20

(d) 5

A merchant commences with a certain capital and gains annually at the rate of 25%. At the end of 3 years he has Rs .10,000. What is the original amount that the merchant invested ?

(a) Rs .5,120

(b) Rs .5,210

(d) Rs .5,500

Which one of the following decimal numbers is a rational number with denominator 37 ?

(a) 0.459459459 .. .

(b) 0.459459459

(d) 0.00459459 . ..

The annual income of a person decreases by Rs .64 if the annual rate of interest decreases from 4% to 3.75%. What is his original annual income ?

(a) Rs .24,000

(b) Rs .25,000

(d) Rs .24,600

For 0 < m < 1, which one of the following is correct ?

(a) log₁₀ m < m² < m < m^-1

(b) m < m^-1 < m² < log₁₀ m

(d) log₁₀ m < m^-1 < m < m²

A gentleman left a sum of Rs .39,000 to be distributed after his death among his widow, five sons and four daughters. If each son receives 3 times as much as a daughter receives, and each daughter receives twice as much as their mother receives, then what is the widow’s share ?

(a) Rs .1,000

(b) Rs .1,200

(d) None of the above

Three numbers which are co-prime to each other, are such that the product of the first two is 286 and that of the last two is 770. What is the sum of the three numbers ?

(a) 85

(b) 80

(d) 70

The age of a woman is a two-digit integer. On reversing this integer, the new integer is the age of her husband who is elder to her. The difference between their ages is one-eleventh of their sum. What is the difference between their ages ?

(a) 8 years

(b) 9 years

(d) 11 years

A passenger train and a goods train are running in the same direction on parallel railway tracks. If the passenger train now takes three times as long to pass the goods train, as when they are running in opposite directions, then what is the ratio of the speed of the passenger train to that of the goods train? (Assume that the trains run at uniform speeds)

(a) 2 : : 1

(b) 3 : : 2

(d) 1 : : 1

All odd prime numbers upto 110 are multiplied together. What is the unit digit in this product ?

(a) 0

(b) 3

(d) None of the above

An alloy A contains two elements, copper and tin in the ratio of 2 : 3, whereas an alloy B contains the same elements in the ratio of 3 : 4. If 20 kg of alloy A, 28 kg of alloy B and some more pure copper are mixed to form a third alloy C which now contains copper and tin in the ratio of 6 : 7, then what is the quantity of pure copper mixed in the alloy C ?

(a) 3 kg

(b) 4 kg

(d) 7 kg

A quadratic polynomial ax² + bx + c is such that when it is divided by x, (x – 1) and (x + 1), the remainders are 3, 6 and 4 respectively. What is the value of (a + b) ?

(a) 3

(b) 2

(d) -1

If the average of 9 consecutive positive integers is 55, then what is the largest integer ?

(a) 57

(b) 58

(d) 60

The average of the ages of 15 students in a class is 19 years. When 5 new students are admitted to the class, the average age of the class becomes 18-5 years. What is the average age of the 5 newly admitted students ?

(a) 17 years

(b) 17.5 years

(d) 18.5 years

A man can row at a speed of x km/hr in still water. If in a stream which is flowing at a speed of y km/hr it takes him z hours to row to a place and back, then what is the distance between the two places?

(a) z(x2 -y2)/2y

(b) z (x2 – y2)/2x

(d) z(x2 -y2)/x

A water tank has been fitted with two taps P and Q and a drain pipe R. Taps P and Q fill at the rate of 12 litres per minute and 10 litres per minute respectively.

Consider the following statements S1, S2 and S3 :

S1 : Pipe R drains out at the rate of 6 litres per minute.

S2 : If both the taps and the drain pipe are opened simultaneously, then the tank is filled in 5 hours 45 minutes.

S3 : Pipe R drains out (fully) the filled tank in 15 hours 20 minutes.

To know what is the capacity of the tank, which one of the following is correct ?

(a) S2 is only sufficient

(b) S1, S2 and S3 are necessary

(d) None of the above

A car has an average speed of 60 km per hour while going from Delhi to Agra and has an average speed of y km per hour while returning to Delhi from Agra (by travelling the same distance). If the average speed of the car for the whole journey is 48 km per hour, then what is the value of y ?

(a) 30 km per hour

(b) 35 km per hour

(d) 45 km per hour

An article is sold at a profit of 32%. If the cost price is increased by 20% and the sale price remains the same, then the profit percentage becomes

(a) 10%

(b) 12%

(d) 20%

A, B, C, D and E start a partnership firm. Capital contributed by A is three times that contributed by D. E contributes half of A’s contribution, B contributes one-third of E’s contribution and C contributes two-third of A’s contribution. If the difference between the combined shares of A, D and E and the combined shares of B and C in the total profit of the firm is Rs .13,500, what is the combined share of B, C and E ? (The shares are supposed to be proportional to the contributions)

(a) Rs .13,500

(b) Rs .18,000

(d) Rs .20,250

A Pie Chart is drawn for the following data :

What is the angle (approximately) subtended by the Social Services Sector at the centre of the circle ?

(a) 45°

(b) 46°

(d) 98°

The arithmetic mean of two numbers is 10 and their geometric mean is 8. What are the two numbers ?

(a) 15,5

(b) 12,8

(d) 18,2

The arithmetic mean of 11 observations is 11. The arithmetic mean of the first 6 observations is 10.5 and the arithmetic mean of the last 6 observations is 11.5. What is the sixth observation ?

(a) 10.0

(b) 10.5

(d) 11.5

What is sin4 Ө – cos4 Ө equal to for any real number Ө ?

(a) 1

(b) 1-2 sin2 Ө

(d) 1-2 cos2 Ө

What is cot 1° cot 23° cot 45° cot 67° cot 89° equal to ?

(a) 0

(b) 1

(d) 1/3

What angle does the hour hand of a clock describe in 10 minutes of time ?

(a) 1°

(b) 5°

(d) 10°

57. Consider the following statements :

(sec2 Ө – 1) (1 – cosec2 Ө) = 1
sin Ө (1 + cos Ө)-1 + (1 + cos Ө) (sin Ө)^-1 = 2 cosec Ө

Which of the above is/are correct ?

(a) 1 only

(b) 2 only

(d) Neither 1 nor 2

Each side of a square subtends an angle of 60° at the tip of a tower of height h metres standing at the centre of the square. If l is the length of each side of the square, then what is h2 equal to ?

(a) 2l²

(b) l²/2

(d) 2l²/3

From a height of h units, a man observes the angle of elevation as a and angle of depression as p of the top and the bottom respectively of a tower of height H (> 4h). To what further height should he climb so that the values of angle of elevation and angle of depression get interchanged for the top and bottom of the tower ?

(a) H — h units

(b) H – 2h units

(d) H – 4h units

If sec x cosec x = 2, then what is tanⁿ x + cotⁿ x equal to ?

(a) 2

(b) 2ⁿ⁺¹

(d) 2^n-1

If cos x + cos2 x = 1, then what is sin2 x + sin4 x equal to ?

(a) 1

(b) 1.5

(d) 3

If sin A + cos A = p and sin3 A + cos3 A = q, then which one of the following is correct ?

(a) p3 – 3p + q = 0

(b) q3 – 3q + 2p =0

(d) p3 + 3p + 2q = 0

If x = sec2 Ө – tan Ө / sec2 Ө + tan Ө, then which one of the following is correct ?

ABC is a right angled triangle with base BC and height AB. The hypotenuse AC is four times the length of the perpendicular drawn to it from the opposite vertex. What is tan C equal to ?

(a) 2 – √3

(b) √3-1

(d) √3 + 1

ABC is a triangle right angled at C with BC = a and AC = b. If p is the length of the perpendicular from C on AB, then which one of the following is correct ?

(a) a2 b2 = p2 (a2 + b2)

(b) a2 b2 = p2 (b2 – a2)

(d) a2 b2 = 2p2 (a2 + b2)

The radius and slant height of a right circular cone are 5 cm and 13 cm respectively. What is the volume of the cone ?

(a) 100π cm3

(b) 50 π cm3

(d) 169 π cm3

Two equal circular regions of greatest possible area are cut off from a given circular sheet of area A. What is the remaining area of the sheet ?

(a) A/2

(b) A/3

(d) 2A/5

If the ratio of the radius of the base of a right circular cone to its slant height is 1 : 3, what is the ratio of the total surface area to the curved surface area ?

(a) 5:3

(b) 3:1

(d) 4:3

A right circular cone is sliced into a smaller cone and a frustum of a cone by a plane perpendicular to its axis. The volume of the smaller cone and the frustum of the cone are in the ratio 64 : 61. Then their curved surface areas are in the ratio

(a) 4 :1

(b) 16:9

(d) 81:64

In a room whose floor is a square of side 10 m, an equilateral triangular table of side 2 m is placed. Four book-shelves of size 4mxlmx9m are also placed in the room. If half of the rest of the area in the room is to be carpeted at the rate of Rs.100 per square metre, what is the cost of carpeting (approximately) ?

(a) Rs.7,600

(b) Rs.5,635

(d) Rs. 3.200

A region of area A bounded by a circle C is divided into n regions, each of area A/n, by drawing circles of radii r1, r2, r3, … rn-1 such that r1 < r2 < r3 < … rn-1 concentric with the circle C. If p_m = r_m+1/r_m where m = 1, 2, 3, …(n – 2), then which one of the following is correct ?

(a) p increases as m increases

(b) p decreases as m increases

(d) p increases for some values of m as m increases and then decreases thereafter

What is the volume of a cone of maximum volume cut out from a cube of edge 2a such that their bases are on the same plane ?

(a) πa3

(b) πa3/3

(d) 3πa3/4

The radii of two circles are 4.5 cm and 3.5 cm respectively. The distance between the centres of the circles is 10 cm. What is the length of the transverse common tangent ?

(a) 4 cm

(b) 5 cm

(d) 7 cm

There are as many square centimetres in the surface area of a sphere as there are cubic centimetres in its volume. What is the radius of the sphere ?

(a) 4 cm

(b) 3 cm

(d) 1 cm

The length of a line segment AB is 2 cm. It is divided into two parts at a point C such that AC2 = AB x CB. What is the length of CB ?

(a) 3√5 cm

(b) 3 – √5 cm

(d) √5 – 1 cm

The locus of the mid-points of the radii of length 16 cm of a circle is

(a) A concentric circle of radius 8 cm

(b) A concentric circle of radius 16 cm

(d) A straight line passing through the centre of the circle

The curved surface area of a right circular cone is 1.76 m2 and its base diameter is 140 cm. What is the height of the cone ?

(a) 10 cm

(b) 10 √2 cm

(d) 10√5 cm

Consider the following statements :
The orthocentre of a triangle always lies inside the triangle.
The centroid of a triangle always lies inside the triangle.
The orthocentre of a right angled triangle lies on the triangle.
The centroid of a right angled triangle lies on the triangle.

Which of the above statements are correct ?

(a) 1 and 2

(b) 1 and 4

(d) 2 and 4

The locus of a point equidistant from two intersecting lines is

(a) A straight line

(b) A circle

(d) None of the above

Consider the following statements :

Two triangles are said to be congruent, if

Three angles of one triangle are equal to the corresponding three angles of the other triangle.
Three sides of one triangle are equal to the corresponding three sides of the other triangle.
Two sides and the included angle of one triangle are equal to the corresponding two sides and the included angle of the other triangle.
Two angles and the included side of one triangle are equal to the corresponding two angles and the included side of the other triangle.

Which of the above statements are correct ?

(a) 1, 2 and 3

(b) 1, 3 and 4

(d) 2, 3 and 4

Given that the angles of a polygon are all equal and each angle is a right angle.

Statement-1: The polygon has exactly four sides.

Statement-2: The sum of the angles of a polygon having n sides is (3n – 8) right angles.

Which one of the following is correct in respect of the above statements ? *

(a) Both Statement-1 and Statement-2 are true and Statement-2 is the correct explanation of Statement-1

(b) Both Statement-1 and Statement-2 are true but Statement-2 is not the correct explanation of Statement-1

(d) Statement-1 is false but Statement-2 is true

If the length of a side of a square is increased by 8 cm, its area increases by 120 square cm. What is the length of a side of the square ?

(a) 2.5 cm

(b) 3.5 cm

(d) 5.5 cm

What is the largest power of 10 that divides the product 1x2x3x4…x23x24x25?

(a) 2

(b) 4

(d) None of the above

Walls (excluding their roofs and floors) of 5 identical rooms having length, breadth and height 6 m, 4 m and 2.5 m respectively are to be painted. Paints are available only in cans of 1 L and one litre of paint can be used for painting 20 square metres. What is the number of cans required for painting ?

(a) 10

(b) 12

(d) 14

A rectangular pathway having width 4-5 m and length 10 m will have to be tiled using square tiles of side 50 cm. Each packet of such tiles contains 20 pieces and costs Rs.100. What will be the total cost of tiles for the pathway ?

(a) Rs.1,200

(b) Rs.1,100

(d) Rs.900

A cube of maximum volume (each corner touching the surface from inside) is cut from a sphere. What is the ratio of the volume of the cube to that of the sphere ? .

(a) 3 : 4π

(b) √3 : 2 π

(d) 4 : 3 π

If the ratio of the circumference of the base of a right circular cone of radius r to its height is 3:1, then what is the area of the curved surface of the cone ?

A wire is in the form of a circle of radius 98 cm. A square is formed out of the wire. What is the length of a side of the square ? (Use π = 22/7)

(a) 146 cm

(b) 152 cm

(d) 156 cm

Consider the following for the next two (02) questions:

In a triangle ABC, a, b and c are the lengths of the sides and p, q and r are the lengths of its medians.

Which one of the following is correct ?

(a) 2 (p + q + r) = (a + b + c)

(b) 2 (p + q + r) > 3 (a + b + c)

(d) 11 (p + q + r) > 10 (a + b + c)

Which one of the following is correct ?

(a) (a + b + c) < (p + q + r)

(b) 3 (a + b + c) < 4 (p + q + r)

(d) 3 (a + b + c) > 4 (p + q + r)

What is the area of the largest circular disc cut from a square of side 2/ √π = units ?

(a) π square units

(b) 1 square unit

(d) 2 square units

The product of the lengths of the diagonals of a square is 50 square units. What is the length of a side of the square ?

(a) 5√2 units

(b) 5 units

(d) 2√5 units

The surface area of a closed cylindrical box is 352 square cm. If its height is 10 cm, then what is its diameter ? (Use π = 22/7 )

(a) 4 cm

(b) 8 cm

(d) 19.26 cm

A square and an equilateral triangle have the same perimeter. If the diagonal of the square is 6√2 cm, then what is the area of the triangle ?

(a) 12√2 cm2

(b) 12√3 cm2

(d) 16√3 cm2

What is the area of the region bounded internally by a square of side of length ‘a’ and externally by a circle passing through the four comers of the square ?

(a) (π – 1) a2 square units

(b) (π -1)a2/2 square units

(d) (π – 2)/a2 square units

In the figure given below, XA and XB are two tangents to a circle. If ∠AXB = 50° and AC is parallel to XB, then what is ∠ACB equal to ?

(a) 70°

(b) 65°

(d) 55°

In the figure given below, p, q, r are parallel lines; I and m are two transversals.
AB : AC = DE : DF
AB x EF = BC x DE Which of the above is/are correct ?

(a) 1 only

(b) 2 only

(d) Neither 1 nor 2

98.In the equilateral triangle ABC given below, AD = DB and AE = EC. If I is the length of a side of the triangle, then what is the area of the shaded region ?

In the figure given below, SPT is a tangent to the circle at P and O is the centre of the circle. If ∠QPT = a, then what is ∠POQ equal to ?

(a) a

(b) 2a

(d) 180° – 2a

In the figure given below, two equal chords cut at point P. If AB = CD – 10 cm, OC = 13 cm (O is the centre of the circle) and PB = 3 cm, then what is the length of OP ?

(a) 5 cm

(b) 6 cm

(d) 2√37 cm