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Mathematics Mock Test - 1

100 Questions MCQ Test CDS ( Combined Defence Services ) Mock Test Series | Mathematics Mock Test - 1

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Attempt Mathematics Mock Test - 1 | 100 questions in 120 minutes | Mock test for Defence preparation | Free important questions MCQ to study CDS ( Combined Defence Services ) Mock Test Series for Defence Exam | Download free PDF with solutions
QUESTION: 1

Solution:

QUESTION: 2

Solution:

QUESTION: 3

In this given figure an equilateral triangle based prism is cut off a cube of side of 3 cm as shown in figure. Find the total surface area (in cm2) of remaining structure.

Solution:

QUESTION: 4

The average weight of 3 men A,B and C is 85 kg another man D joins the group and the average now become 81 kg. If another man E whose weight is 4 kg more than that of D, replaces A then the average weight of B, C, D and E becomes 80 kg. What is the weight of A?

Solution:

QUESTION: 5

If sin θ + cos θ = a and sec θ +cosec θ = b, then value of b(a2 -1) is equal to:

Solution:

QUESTION: 6

If a2 + b+ c2 = 2( 3a - 4b - 6c) - 61. Find value of (a-b-c).

Solution:

QUESTION: 7

Work done by (x + 4) men in (x + 5) days is equal to the work done by (x – 5) men in (x + 20) days. The value of x is

Solution:

QUESTION: 8

What is the number of distinct triangles with integral valued sides and perimeter as 14?

Solution:

QUESTION: 9

If x = , then the value of 5x2 - 5x - 1 is

Solution:

QUESTION: 10

Find the area (in cm2) of given quadrilateral ABDC.

Solution:

QUESTION: 11

The train A left Delhi at noon sharp. Four hours later, another train B started from Delhi in the same direction. The train B overtook the train A at 10 p.m. Find the average speed of the both trains over this journey if the sum of their speed is 80 km/h.

Solution:

QUESTION: 12

The third proportional to and  is

Solution:

QUESTION: 13

is equal to

Solution:

QUESTION: 14

If a secθ + b tanθ =1 and a2 sec2θ - b2 tan2 = 5, then a2b2 + 4a2 is equal to

Solution:

QUESTION: 15

If x varies inversely as (y2 - 1) and is equal to 24 when y = 10, then the value of x when y = 5 is

Solution:

QUESTION: 16

Which term of the AP: 121, 117, 113……… is its first negative term.

Solution:

Here a = 121 and d = -4 The first negative term will be less than zero.
Moreover each term gives 1 as remainder when divided by 4.
So the last positive term should be 5 let us check the value of n for last term as 5

QUESTION: 17

If cot θ + cos θ = p and cot θ - cos θ = q, then (p2 - q2)2 in terms of p and q is-

Solution:

QUESTION: 18

A dishonest dealer defrauds to the extent of 10% in buying and 20% in selling and claims that he earns only 10% profit what will be the gain percent on his outlay.

Solution:

QUESTION: 19

In given figure ABCD is a square in which three circles are drawn touching one another. Radius of two smaller circles is R units and bigger circle is 2R units. Diagonal AC of the square passes through the center of all the circles. What is the ratio of the radius of the smaller circles to the side of the square?

Solution:

QUESTION: 20

The angles of a triangle are in the ratio of 4 : 1 : 1. Then the ratio of the longest side to the perimeter is

Solution:

QUESTION: 21

Three circles of radius a, b, c touch each other externally. The area of the triangle formed by joining their centre is

Solution:

QUESTION: 22

A man makes 80 articles in the 1st hour. His efficiency decreases by 25% in the 2nd hour, increases by 40% in the 3rd hour, decreases by  in the 4th hour and increases by  in the 5th hour (his efficiency increases or decreases with respect to the efficiency in previous hour). If he works for 5 hours, then find the average of total article made by the man in 5 hours.

Solution:

QUESTION: 23

Let A, B, C, D be the angles of a quadrilateral. If they are concyclic then the value of cos A + cos B + cos C + cos D is

Solution:

QUESTION: 24

If x = 5 -  , then the value of is

Solution:

QUESTION: 25

A can do as much work as B and C together can do. A and B can together do a piece of work in 10 hours and C can do it in 50 hrs. Then, time that B needs to do the work alone is –

Solution:

QUESTION: 26

If , then the value of  is:

Solution:

QUESTION: 27

If a right circular cone is cut into three solids of volumes V1,V2, and V3 by two cuts which are parallel to the base and trisects the altitude then  V1,V2, and V3 is

Solution:

QUESTION: 28

A man goes from Delhi to Ghaziabad at v1 km/h, goes back from Ghaziabad to Delhi at v2 km/h and again Delhi to Ghaziabad at v2 km/h. The average speed of man is

Solution:

QUESTION: 29

If a+b+c+d=4 then the value of   is

Solution:

QUESTION: 30

A spiral is made up of successive semicircles with centers alternately at A and B, starting with center at A of radii 0.5 cm, 1.0 cm, 1.5 cm…………as shown in figure. What is the total length of such a spiral made up of thirteen consecutive semicircles?

Solution:

Circumference of first semicircle = πr= .5π
Circumference of IInd semicircle =1π
Circumference of IIIrd semicircle = 1.5π
It is clear that a =.5π, d= .5π
and n = 13
hence, length of spiral can be calculated as follows

QUESTION: 31

IF tan4 θ + tan2 θ = 1 then the value of cos4 θ + cos2 θ is —

Solution:

QUESTION: 32

If  then the value of (p3-q3) is

Solution:

QUESTION: 33

If the sum of all prime numbers is x and all odd prime numbers is y then what will be x -y?

Solution:

​Keep in mind that 2 is the only prime numbers which is also an even number.

QUESTION: 34

If , then the value of cosß in terms of a and b is

Solution:

QUESTION: 35

A cistern measuring 80 cm × 50 cm × 30 cm has 12000 cm3 water in it. Porous bricks are placed in the water until the cistern is full to the brim. Each brick absorb one seventeenth of its own volume of water. How many bricks can be put in without overflowing the water, each brick being 22.5 cm × 7.5 cm × 8.5 cm

Solution:

Volume of cistern = 80 x 50 x 30 = 120000 cm3
Vacant space = volume of cistern - volume of water
= 120000 - 12000 = 108000 cm3 Volume of brick = 22.5 x 7.5 x 8.5 cm
Since the brick absorb one seventeenth its volume hence remaining will be equal to 16/17 the volume of brick.
remaining Volume =22.5 x 7.5 x 8.5 cm3
Number of bricks == 80 bricks

QUESTION: 36

In given figure find the area (in cm2 ) of shaded regions A & B if radius of circle is 1 cm.

Solution:

In ΔAOC
AC = OC = R
Area of ΔAOC =

Required Area =

QUESTION: 37

The houses on ABC lane are numbered consecutively from 1 to 49. If the sum of the numbers of the houses preceding the house numbered n is equal to the sum of the numbers of the houses following it. Then find the value of n?

Solution:

QUESTION: 38

if  are in AP. Find a, and also find a,b & c's progression.

Solution:

a,b,c are in GP

QUESTION: 39

A cubical block of side 7 cm is surmounted by a hemisphere of diameter equal to the side of the cube. Then find the total surface area of the solid.

Solution:

Diameter = side of the cube = 7
ATQ,
Total Surface area of solid = surface area of cube – surface area of base of hemisphere + curved surface area of hemisphere
= 6 X a2— πr2 + 2πr2 = 6a2 + πr2
= 6 X 49 + X 3.5 X 3.5
= 294 sq. cm + 38.5 sq. cm =
332.5 sq. cm.

QUESTION: 40

If  = 0.6, then  is

Solution:

We know thet,

QUESTION: 41

Find the area (in sq. units) of quad ABCD as shown in figure. Where ∠B = ∠D = 90° AB = BC & AD + DC = 1 unit

Solution:

QUESTION: 42

A man purchased 70 story books. But he found that he could get 8 extra books by spending 564 more and then the overall average price per book would be reduced by 2 Rs. The previous average price of each book was.

Solution:

Let the previous average price be Rs x
According to the question
70x + 564 = (x - 2) 78
70x + 564 = 78x - 156
8x = 720
x = 90

QUESTION: 43

If

Solution:

QUESTION: 44

The population of a town is 20,000. If the Males increase by 3% and the females by 4% the population will be 20680. How many females are there?

Solution:

If the Number of woman be x then men = 20000-x=20680-20000
=4x + 60000 - 3x = 68000
x = 8000

QUESTION: 45

A Vendor has 5913 bottles of whiskey, 6059 Bottles of soda and 7081 bottles of water If he wants to pack them in cans so that each can contains same number of bottles and does not want to mix any two kinds of bottles in a can, then find the least number of cans required.

Solution:

Maximum bottles in each can
= HCF of all three Types of liquid bottles
5913, 6059, 7081 = 73
Required least number of cans

= 81 + 83 + 97=261

QUESTION: 46

A canteen requires 98 kgs of rice for fourteen days. The quantity of rice required for the months of February and March together is (not for leap year)

Solution:

Number of days in February and March = 28 + 31 = 59
∴ Required of Rice for 14 days = 98 kg
∴ Requirement of Rice for 59 days

=413 kg

QUESTION: 47

A person spend 1/3rd of his income on food, clothes & study in ratio 3 : 2 : 3 respectively. He spends 2/5 of his income on transport & entertainment in the ratio 3 : 5 & deposit remaining amount of 6464 in a bank. Find total amount spent on foods, clothes & transportation.

Solution:

QUESTION: 48

If ,then:

Solution:

By componendo and dividendo,

QUESTION: 49

A wire is bent in the form of a circle whose area is A cm². If the same wire is bent into the form of an equilateral triangle and next time in the form of a square, then find the ratio of area of all three structures?

Solution:

In Equilateral triangle

QUESTION: 50

A number which when divided by 13 leaves a remainder of 12, when divided by 12 leaves a remainder of 11 and when divided by 11 leaves a reminder of 10 ; is

Solution:

LCM of 13, 12, 11 = 1716
Required number (LCM of 13, 12, 11 -1)
= 1716 - 1 = 1715

QUESTION: 51

Two years ago the value of Arun’s motor bike was Rs. 390625. If the value depreciates at the rate of 8% p.a. per half year, then its present worth is:

Solution:

QUESTION: 52

If x = a sinn θ and y = b cosm θ, then by eliminating θ

Solution:

QUESTION: 53

The HCF and LCM of two numbers are 23 and 460 respectively. If one of the number lies between 93 and 125 then that number is-

Solution:

QUESTION: 54

If = -3, then the value of (x3 + 8x2 + 27z) is

Solution:

QUESTION: 55

If a farmer packs 8 or 11 mangoes in a box, he is left with 5 mangoes. But if he packs 6 or 7 mangoes in each box also he is left with 5 mangoes. Find the no. of mangoes that he had

Solution:

LCM of 8, 11, 6 and 7
= 1848
Required number of mangoes = 1848 + 5 = 1853

QUESTION: 56

Find ∠ODC in the given figure, if  ∠ABC = 30ο & ∠BCO = ∠OCD = 20ο

Solution:

QUESTION: 57

Relation between arithmetic and Geometric mean of two numbers a and b (a > b) is A.M. = 2 G.M. Find the ratio between numbers.

Solution:

QUESTION: 58

A cylinder of height 15 cm and radius 14 cm is wound around with a string of width ‘5’ cm. The string covers the lateral surface area of the cylinder completely without keeping any space between two turns and without overlapping one another. What is the length of the string?

Solution:

QUESTION: 59

The difference between the ages of two brothers is one third of difference between the ages of their parents. The elder brother is 19 yrs of age. Their father’s age was 33 years when the younger brother was born who is now 17 years old. What is their Mother’s age?

Solution:

Father's present age = 33 + 17 = SO yrs Atq,
(SO - m) = 3(19 - 17) = 6
x = 44 years

QUESTION: 60

Two cylinder of length 2.5m and 15m are to be cut into equal pieces without leaving any extra length of cylinder. Find the greatest length of the cylinder pieces of same size which can be cut from these two

Solution:

Maximum length of each piece
= HCF of 2.5m and 15in = 2.5m
Required size = 2.5 m

QUESTION: 61

The factors of (a2 +4b2 +4b - 4ab - 2a -8 ) are

Solution:

QUESTION: 62

A spherical ball was painted black. After getting it painted, it was cut diametrically (two times vertically and one time horizontally) into eight similar pieces. Then what is the ratio of the painted area to the non-painted Area?

Solution:

QUESTION: 63

Ravi covers a certain distance by his car. If he moved 3 km/h faster then he takes 4 hour less & if he had moved 2 km/h slower then he would have taken 4 hours more. Find the distance covered.

Solution:

QUESTION: 64

Find No. of zeroes at the end of the product 1 × 2 × 3 × 4 × 5 ……. × 50

Solution:

​For required zeroes Calculate pair of 2 × 5 = 10 In given question factor 2 is repeated more times than factor 5, so we calculate number of 5’s as factors in 1 × 2 ×3 × …..50
Number of factor 5 = 12 So, required zeros = 12

QUESTION: 65

The equation of the line passing through the point of intersection of the lines 5x – 2y = 3, 4x – 7y + 3 = 0 and parallel to the line 5x – 4y + 5 = 0

Solution:

Intersection point.
5x-2y=3 (1)
4x-7y=-3 (2)
On solving we have x = 1, y= 1
So, intersection points (1, 1)
Now, slope of the required line x slope of the line 5x - 4y + 5 = 0
As both are parallel lines,
So, M1= M2, here M2 =
M1
So, required equation (Y - Y1) = m1 (x - x1) (y - 1) = (x - 1)
4y - 4 = 5x - 5
⇒5x- 4y = 1

QUESTION: 66

If a + b + c= 8 and ab + bc + ca = 13, then value of bc(b + c) + ca(c + a) + ab(a+b) + 3abc is

Solution:

bc(b+c) + ca(c+a) + ab(a+b)+3abc
= bc(b+c) + abc + ca(c+a) + abc + ab(a+b) + abc
= bc (a + b+ c) + ca(c+ a+ b) + ab (a + b +c)
= (a + b +c) (ab + bc +ca) = 8 x 13
= 104

QUESTION: 67

If x = ∛65,y = ∛64, then the value of (x + y) is

Solution:

QUESTION: 68

AD is perpendicular to the internal bisector of ∠ABC of ΔABC. DE is drawn through D and parallel to BC to meet AC at E. If the length of AC is 10 cm, then the length of AE (in cm) is

Solution:

QUESTION: 69

Find the area of a regular octagon inscribed in a circle of radius r?

Solution:

QUESTION: 70

In the expression x2 y the values of both x and y are decreased by 10%. Due to this, the value of the expression is decreased by

Solution:

QUESTION: 71

A right angle triangle with base and height measuring 15 cm and 20 cm is rotated along its hypotenuse and formed a new structure. Find the volume (in cm3) of the structure

Solution:

QUESTION: 72

Rate of interest on a sum of money is 4% pa for the first 2 years. 6% pa for the next 4 years & 8% pa for the period beyond 6 years. SI accrued by the sum for a total period of 9 years is Rs. 1120. What is the sum (in Rs.)?

Solution:

QUESTION: 73

If = 1 and =0 where p, q, r and a, b, c are non-zero, then the value of  is

Solution:

QUESTION: 74

Given area ∆ FHI = 2 unit² & HI = 2 & is perpendicular to line m & BD‖EG‖HI & BC = 4. Also, line l || line m. AC = FC = 2FI & DC = CG then, find ∠BCE?

Solution:

QUESTION: 75

If ( a+ b +7)2 + a2 + b2 + 1 + 2b = 2ab + 2a, then the value of a is

Solution:

QUESTION: 76

If (a - 2) + = -1 then the value of (a + 2 )2 is

Solution:

QUESTION: 77

Given that AB || DG || JI, AC=JH=10 cm and CE=EH=5 cm, then find the sum of perimeter of ∆ABC, ∆CDE, ∆EGH & ∆HIJ? (In cm)

Solution:

QUESTION: 78

In a colored picture of turquoise & magenta, colors are used in the ratio 5 : 3 respectively. If in upper half turquoise : magenta is 5 : 3, then in the lower half magenta : turquoise is,

Solution:

​Clearly, the ratio between magenta & turquoise should be the same in another half also.

QUESTION: 79

Solution:

QUESTION: 80

In ∆ ABC with centroid G, if AG = BC & BG = 9 cm & GC = 12 cm. What is the sum of areas of the circle passing through points B, G & C & ∆BGC? (cm2)

Solution:

QUESTION: 81

Find ∠PQR of the given isosceles ΔAPQ, when PQ = PA & QR = RA?

Solution:

∠RQA = ∠RAQ = 30° & ∠PQA = ∠PAQ = 55°
So, ∠PQR = 55° - 30°= 25°

QUESTION: 82

Given an equilateral ∆ABC inscribing two circles as shown in the figure below. If DE tangent to both the circles. Then find the ratio of perimeters of the two circles.

Solution:

QUESTION: 83

If = 5, then the value of   is

Solution:

QUESTION: 84

Amit plans to buy a scooter for his sister for which he saves Rs. 15625 at the start of every year for 3 year. If the rate of CI is 4% pa. then amount at which he plans to buy the scooter is (in Rs.)

Solution:

4% = 1/25
so,
Principal Amount

16250+16900+17576= Rs.50726

QUESTION: 85

What is the value of a, if (x - a ) is a factor of (x3 - a2x + x +1)?

Solution:

By factor theorem = (x - a) = 0
x = a
atq,
x3 - a2x + x + 1 = 0
by putting value of x,
a3 - a3 + a + 1 = 0
a = -1

QUESTION: 86

Given ∠D = ∠F = 110ο & BD = DC 1.5 & EF = FG = 1.5 & BC ll EG. Then find ∠y.

Solution:

QUESTION: 87

If AE = CE & FE = ED & FD || AC then AC + x = ?

Solution:

QUESTION: 88

Find radius r (in c.m) of the given circle where CH is tangent to circle & HBA is a secant.

Solution:

QUESTION: 89

In the given fig. AB, CD & EF are three towers. The angle of elevation of the top of the tower CD from the top of the tower AB is 60° and that from EF is 30°. If BD = 2√3 m, CD : EF = 5 : 4 & DF = 4 m .Then, find height of tower AB ?

Solution:

QUESTION: 90

Find the value of m if the polynomial x3 - mx2 - 13x + n has (x + 1) and (x - 3) as factors

Solution:

QUESTION: 91

PQR is a Δ with three squares put on the three sides of the Δ. Then ∠a + ∠b + ∠c =?

Solution:

QUESTION: 92

Measure of regular polygon’s interior angle is 4 times its exterior angle. Then no. of diagonals the polygon can have?

Solution:

QUESTION: 93

A cube is inscribed in a hemisphere of radius R. Such that four of its vertices lie on the base of the hemisphere and the other four touch the hemispherical surface of the half sphere. What is the volume of the cube?

Solution:

Let a be the side of cube ABCDEFGH and O be the center of the hemisphere
Then OA = OC = R

QUESTION: 94

A tank has a leak which would empty the completely filled cistern in 20 hours. If the tank is full of water and a tap is opened which admits 2 liters of water per minute in the tank, the leak now takes 30 hours to empty the tank. How many liters of water does the tank hold?

Solution:

QUESTION: 95

A boat running downstream covers a distance of 20 kms in 2 hours. While coming back the boat takes 4 hours to cover the same distance. What is the speed of the boat in still water in kmph?

Solution:

QUESTION: 96

The following pie-chart shows the distribution of total cost per day incurred to a manufacturer. Study the chart to answer the questions based on it.

Q. Which cost is second highest?

Solution:

Machinery cost = 250 is 2nd highest

QUESTION: 97

The following pie-chart shows the distribution of total cost per day incurred to a manufacturer. Study the chart to answer the questions based on it.

Q. Packaging cost is what percent of total cost?

Solution:

QUESTION: 98

The following pie-chart shows the distribution of total cost per day incurred to a manufacturer. Study the chart to answer the questions based on it.

Q. What would be the central angle for material cost?

Solution:

QUESTION: 99

The following pie-chart shows the distribution of total cost per day incurred to a manufacturer. Study the chart to answer the questions based on it.

Q. Machinery cost is what percent more than power cost?

Solution:

QUESTION: 100

If loga (ab) = x, then what is log(ab) equal to?

Solution:

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