Test: Elementary Mathematics - 10 - CDS MCQ

Given :

Curved surface area of a circular cone is 1100 cm²

Base diameter is 14 cm

Formula used :

The curved surface area of cone = πrl

Volume of cone

Where r is radius and l is slant height

Calculations:

Let l and h be slant height & height of cone respectively.

⇒ 1100 = (22/7) × 7 × l

⇒ 1100 = 22 × l

⇒ l = 50 cm

We know that, in a right-angle triangle,

hypotenuse² = base² + perpendicular²

⇒ 50² = 7² + h²

⇒ 2500 = 49 + h²

⇒ h² = 2451

⇒ h ≈ 49.5 cm

∴ The approximate height of the cone is 49.5 cm.

Given:

Total Rs. = Rs.550

Calculation:

Let the number of denominations of 50p = 2x

the number of denominations of 25p = 3x

the number of denominations of 20p = 5x

Total paisa = 550 × 100 = 55000 paise

According to the question:

⇒ (50 × 2x) + (25 × 3x) + (20 × 5x) = 55000

⇒ 100x + 75x + 100x = 55000

⇒ 275x = 55000

⇒ x = 55000/275 = 200

Amount in (Rs.) of 50p = 100x = (100 × 200) = 20000 paise = Rs.200

Amount in (Rs.) of 20p = 100x = (100 × 200) = 20000 paise = Rs.200

Required difference = 200 - 200 = 0

∴ The correct answer is 0.

Formula used:

Circumference of the circle = 2πr

Calculation:

Let Radius of first, second & third circles be R₁, R₂ & R respectively.

It is given that,

⇒ R₁ = 0.17 m = 17 cm

⇒ R₂ = 9 cm

Circumference of third circle = Circumference of first circle + Circumference of second circle

⇒ 2πR = 2πR₁ + 2πR₂

⇒ R = R₁ + R₂

⇒ R = 17 + 9 = 26 cm

∴ The correct answer is 26 cm.

Calculation:

According to the question

Side of ABCD = radius of sector CBXD = 20 cm

⇒ Area of square = (20)² = 400 sq.cm

Area of the shaded region inside square = Area of ABCD – Area of sector CBXD

⇒ Area of the shaded region inside square =

⇒ Area of the shaded region inside square = 86 sq.cm.

Radius of bigger sector = Length of diagonal of ABCD = 20√2 cm

Now,

Area of the shaded region outside the square = Area of sector APCQ – Area of square ABCD

⇒ Area of the shaded region outside the square = 228 sq.cm.

∴ Total area of shaded region = 86 + 228 = 314 sq.cm

Calculation:

(1 + cot A + tan A)(sin A - cos A)

⇒ (1 + + )(sin A - cos A)

⇒ ()

Given:

Annual depreciation of car value = 20%, 15% and 10%

Cost = 7,00,000

Concept used:

If P is the initial price of the commodity, then after ‘n’ years the value of the commodity will become P × [1 – (n/100)]

Calculation:

Percentage fall in price after 3 years = 700000 × (100 - 20)/100 × (100 - 15)/100 × (100 - 10)/100

⇒ 700000 × 8/10 × 85/100 × 9/10

⇒ 70 × 8 × 85 × 9

⇒ 428400

The answer is 428400

Concept used:

Efficiency = Total Work/ Total Time

Calculation:

Let the total capacity of the tank = 72 units [LCM of 12, 18 and 36]

According to the question

Efficiency of Pipe A = 72/12 = 6

Efficiency of Pipe B = 72/18 = 4

Efficiency of Pipe C = 72/36 = -2 (outlet pipe)

Combined efficiency of (A + B + C) = 6 + 4 + (-2) = 8

So, Time taken by (A + B + C) to fill the tank = 72/8 = 9 minutes

The answer is 9 minutes.

Formula used:

(a - b)(a + b) = a² - b²

Calculation:

(sec A - tan A + 1)(sec A - tan A - 1)

((sec A - tan A)² - 1²)

sec² A + tan² A - 2sec A tan A - 1

tan2 A + 1 + tan2 A - 2sec A tan A - 1

2tan2 A - 2sec A tan A

2 tan A (tan A - sec A)

∴ Option 2 is the correct answer.

Calculation:

Total production of wheat by 6 states = 360° = 3600

⇒ 1° = 10

Production of state P = 110° = 110 × 10 = 1100

Production of state Q = 70° = 70 × 10 = 700

Average production by state P & Q = (1100 + 700)/2 = 900

Production of state R = 80° = 80 × 10 = 800

Production of state T = 20° = 20 × 10 = 200

Average production by state R & T = (800 + 200)/2 = 500

Difference between the average production of P and Q and R & T = 900 - 500 = 400

∴ The correct answer is 400.

Calculation:

We have

Put a = 1, b = 1, c = - 2

Alternate Method

Formula used:

If a + b + c = 0 then

a⁴ + b⁴ + c⁴ = 2(a²b² + b²c² + c²a²)

Calculation:

We have

By using the above identity

= 2

Concept -

The formula for the mean deviation (also called the mean absolute deviation) is given by:

where:
- n is the number of observations,
- x_i are the individual observations,
- is the mean of the observations,
- is the absolute deviation of each observation from the mean.

Explanation -

Mean deviation:

Largest six observations: 34, 39, 40, 45, 48, 50

Mean = = 42.67

Deviations from the Mean:
|34 - 42.67| = 8.67
|39 - 42.67| = 3.67
|40 - 42.67| = 2.67
|45 - 42.67| = 2.33
|48 - 42.67| = 5.33
|50 - 42.67| = 7.33

Mean deviation =

Hence Option(4) is correct.

Concept:

According to the rule of a linear function, we know that, if, f(x) = x

where a is the coefficient of x.

Formula used:

• a³ + b³ = (a + b) (a² + b² - ab) ----(1)

• (a + b)² = a² + b² + 2ab ----(2)​

• a3 + b3 = (a + b) {(a + b)² - 3ab} [using (1) & (2)] ----(3)

Calculation:

We have,

Using equation (3), we get

Let

⇒ f(t) = t (t² - 3)

On putting t = √3, we get

⇒ f(√3) = √3 {(√3)2 - 3}

⇒ f(√3) = √3 (3 - 3)

⇒ f(√3) = √3 (0)

⇒ f(√3) = 0

Given:

D and E are points on the sides AB and AC, respectively, of ΔABC such that DE is parallel to BC and AD ∶ DB = 7 ∶ 9.

CD and BE intersect each other at F

Calculation:

AD / AB = DE / BC

= 7 : (7+9) 7 : 16

So areas of ΔDEF and ΔCBF = 7² : 16²

= 49 : 256

∴ The correct option is 3.

Given:

The ratio of the two numbers = 9 : 16

After adding 112 to both the numbers the ratio becomes 37 : 44

Calculation:

Let the two numbers be 9x and 16x

Now, according to the question,

(9x + 112)/(16x + 112) = 37/44

⇒ 44(9x + 112) = 37(16x + 112)

⇒ 396x + 4928 = 592x + 4144

⇒ 592x - 396x = 4928 - 4144

⇒ 196x = 784

⇒ x = 4

∴ The two numbers are 9 × 4 = 36 and 16 × 4 = 64

If 12 is subtracted from both the numbers then, their ratio will be

(36 - 12) : (64 - 12)

⇒ 24 : 52

⇒ 6 : 13

∴ The required ratio is 6 : 13

Given:

Edge of the large cube = 48 cm.

Volume of each small cube = 64 cm3.

Formula Used:

Volume of the cube = side³

Calculation:

Volume of the large cube = 48³

⇒ Volume of the large cube = 48 × 48 × 48

⇒ Volume of the large cube = 110592 cm³

Number of small cubes = Volume of large cube / Volume of one small cube

⇒ Number of small cubes = 110592 / 64

⇒ Number of small cubes = 1728

∴ The number of small cubes obtained is 1728.

Formula Used:

1. (a + b)² = a² + b² + 2ab
2. (a - b)2 = a2 + b2 - 2ab

Calculation:

We have (3x - 2y)2(x2 + y3)2

By using the above formula

(9x² + 4y² - 6xy)(x⁴ + y⁶ + 2x²y³)

We will focus on those terms which will give x⁴y³ after multiplication.

We can see,

9x² × 2x²y³ = 18x⁴y³

∴ The correct answer is option 1.

Concept -

Variance =

Explanation -

Variance:

Mean = 42.67

Squared deviations:

(34 - 42.67)² = (-8.67)² = 75.1689
(39 - 42.67)² = (-3.67)² = 13.4689
(40 - 42.67)² = (-2.67)² = 7.1289
(45 - 42.67)² = (2.33)² = 5.4289
(48 - 42.67)² = (5.33)² = 28.4089
(50 - 42.67)² = (7.33)² = 53.7289

Sum of Squared Deviations:
75.1689 + 13.4689 + 7.1289 + 5.4289 + 28.4089 + 53.7289 = 183.3334

Variance =

Hence Option(2) is correct.

Given

Total sum paid by x and y = Rs 770

Calculation

According to question

y + x = 770

⇒ y + 120% of Y = 770

⇒ 11y = 770 × 5

⇒ y = 350

∴ The required answer is Rs 350

Given:

Diameter of sphere = 6 cm

Diameter of cylinder = 18 cm

Height of cylinder rises by = 40 cm

Formula Used:

Volume of sphere =

Volume of cylinder =

Calculation:

Radius of sphere = 6/2 = 3 cm

Radius of cylinder = 18/2 = 9 cm

Number of spherical ice pieces dropped in cylinder is,

x = (Volume of cylinder) / (Volume of sphere)

x = [take height of cyclinder, h = 40 cm]

⇒ 90

∴ Number of ice pieces are dropped in the container is 90.

Given:

The equation 2x2 + 5x + 5 = 0

Concept Used:

The quadratic equation Ax² + Bx + C = 0

If roots are imaginary, B² - 4ac < 0

Calculation:

Comparing of given equation

Now, A = 2, B = 5 & C = 5

⇒ (5)² - 4 × 2 × 5 < 0

⇒ 25 - 40 < 0

⇒ -15 < 0

∴ The roots are imaginary and distinct.

Given:

The Captain of the warship observes that the submarine makes an angle of depression of 30°

The distance between them from the point of observation is 50 km.

After 30 minutes, the angle of depression becomes 60°

If both moving same direction

Formula Used:

cot 60 = 1/√3

Calculation:

25 cot 60° = 75/√3 − a/2

a = 100√3 km.

So the required speed = 2 ⋅ kmph

• Basic pay per hour of normal work = 200/40 = Rs.5
• Over time is paid at 25% more = (125/100) × 5 =Rs.6.25 per hour
• Man was paid Rs.200 for his regular work and Rs.100 for over time (total Rs.300).
• Number of regular hours = 40
• Number of overtime hours at 6.25 per hour = 100 ÷ 6.25 = 16 hours.
• Total number of hours worked = 40+16 = 56 hours.

Calculation:

Average income is ,

⇒ (55 + 50 + 35 + 40 + 60) / 5

⇒ 240 / 5 = 48

Average expenditure is ,

⇒ (50 + 40 + 32 + 35 + 50) / 5

⇒ 207 / 5 = 41.4

So the difference is 48 - 41.4 = 6.6

∴ The correct option is 3.

Formula used:

1 + cot2 θ = cosec² θ

1 + tan² θ = sec² θ

cosec θ = 1/sin θ and sec θ = 1/cos θ

Calculation:

(1 + cot2 θ) (1 + cos θ) (1 - cos θ) - (1 + tan2 θ) (1 + sin θ) (1 - sin θ)

⇒ cosec2 θ (1 - cos² θ) - sec² θ (1 - sin² θ)

⇒ cosec2 θ . sin² θ - sec2θ . cos² θ

⇒ 1 - 1 = 0

Given:

If P is subtracted from 8, 6, 2, and 9 then these numbers are in proportion.

Concept used:

If a, b, c and d are in proportion then

⇒ a/b = c/d

Mean proportion = √(a × b)

Calculation:

According to the question:

⇒ (8 - P)/(6 - P) = (2 - P)/(9 - P)

⇒ (8 - P) × (9 - P) = (2 - P) × (6 - P)

⇒ 72 - 8P - 9P + P² = 12 - 2P - 6P + P2

⇒ 17P - 8P = 72 - 12

⇒ 9P = 60

⇒ P = 20/3

Mean proportion = √{(3P - 6) × (9P - 4)}

Now, Putting the value of P in the equation:

⇒ √{(3 × (20/3) - 6) × (9 × (20/3) - 4)}

⇒ √{(20 - 6) × (60 - 4)}

⇒ √{14 × 56}

⇒ 14 × 2 = 28

∴ The correct answer is 28.

We have,

|x - 1| ≤ 5 and |x| ≥ 2

⇒ -5 ≤ x - 1 ≤ 5 and (x ≤ -2 or x ≥ 2)

⇒ -4 ≤ x ≤ 6 and ( x ≤ -2 or x ≥ 2)

⇒ x ∈ [-4, 6] and x ∈ (-∞, -2] ∪ [2, ∞)

⇒ x ∈ [-4, -2] ∪ [2, 6]

Concept Used:

Substitution method is one of the method to solve the linear equations of two variable. In this, the value of one variable in terms of other value is putted in the second equation to find the values of variables.

Explanation:-

Suppose first number is x while second number is y.

Now, the product of two digits is 40. Thus,

⇒ xy = 40

⇒ x = (40/y) ..(1)

Difference between digits is 3. Thus,

⇒ x - y = 3

Put the value of x,

Neglecting negative value, we get y = 5. Put this value in equation (1),

⇒ x = 40/5 = 8

Thus, the number will be 85.

Hence, the correct option is 2.

Given:

Let a = rd of 15 = 5

b = th of 25 = 20

c = th of 35 = 15

Concept used:

If a : b :: c : d then,

fourth proportional (d) = (b × c)/a

Calculation:

5 : 20 : : 15 : x

⇒ x = (20 × 15)/5

⇒ x = 60

∴ The correct answer is 60.

Let the number of Security Staff be "a"

Number of male workers = 3a

Number of female workers = 2a

Therefore,

a + 2a + 3a = 1800

6a = 1800

a = 300

Employees retired

1/5 th of female workers = 600/5 = 120

1/4 th of male workers = 900/4 = 225

1/6 th of security staff = 300/6 = 50

the remaining number of employees = 1800 - 120 - 225 - 50

the remaining number of employees = 1405

Given:

EC = 15 m

DG = 5 m

Concept used:

Area of Trapezium = 1/2 × (a + b) × h,

Calculation:

GC = AC - AG = 25 - 12 = 13 m

⇒ Area = 1/2 × (15 + 5) × 13

⇒ 1/2 × 20 × 13

⇒ 130 m²

∴ The area of Trapezum DGCE is 130 m²