Given:

The ratio of two positive numbers is 11 ∶ 17 and their HCF is 4.

Concept used:

HCF of two or more numbers is the greatest factor that divides the numbers.

Calculation:

Since the HCF of the two numbers is 4, the two numbers are as follows:

11 × 4 = 44 and 17 × 4 = 68

LCM (44, 68) = 4 × 11 × 17 = 748

∴ Their LCM is 748.

Given:

Concept used:

(H)Hypotenuse² = (P)Perpendicular² + (B)Base²

Calculation:

sinθ = P/H

So, H² = P² + B²

⇒ 17² = 8² + B²

⇒ 289 = 64 + B2

⇒ 225 = B²

⇒ 15 = B

Now,

tanθ = P/B

Shortcut Trick

sinθ = 8/17 = P/H, we know the triplet P = 8, B = 15, H = 17

Now, tanθ = P/B So, tanθ = 8/15

Given:

Ratio of the rates of Simple Interests in two banks A and B = 5 : 4

Half-yearly interests from both the banks are equal.

Concept:

Simple Interest (SI) = (P × R × T)/100;

where P = principal, R = Rate of Interest, T = Time period.

Calculation:

Let the principal amounts invested in banks A and B be A and B, respectively.

Let the respective rates of simple interest for banks A and B be 5x and 4x

Using the formula for SI,

we get: SI for bank A = (A × 5x × 0.5)/100

Also, SI for bank B = (B × 4x × 0.5)/100

Since the SI is the same for both the banks,

we get: (A × 5x × 0.5)/100 = (B × 4x × 0.5)/100

⇒ 2.5A = 2B

⇒ A/B = 4/5

∴ The ratio in which deposit should be made in the banks A and B should be 4 : 5.

Alternate Method

Let the principal amounts deposited in banks A and B be P₁ and P₂ respectively.

Given that the interest rates are in the ratio 5 : 4, let's assume:

Rate for bank A, R₁ = 5x

Rate for bank B, R₂ = 4x

Time, T = 1/2 year (6 months)

As the interests received are equal: I₁ = I₂ 

⇒ P₁ × R₁ × T = P₂ × R₂ × T

⇒ P₁ × 5x × 1/2 = P₂ × 4x × 1/2

⇒ P₁ × 5x = P₂ × 4x

⇒ P₁/P₂ = 4 / 5

∴ The ratio in which he should deposit the savings in banks A and B is 4 : 5.

Given:

cot²A - cos²A

Concept used:

cosec²θ - cot²θ = 1

Calculation:

cot²A - cos²A

GIVEN:

Mean proportional between a number and 20 = 50.

FORMULA USED:

⇒ x × 20 = 2500.

⇒ x = 125.

∴ The required number is 125.

Given:

Principal = Rs 20,000

Rate = 8%

Compound Interest = Rs 3,328

Formula used:

Where, P = Principal , C.I. = Compound Interest , R = Rate of Interest , t = time in years,  A = Amount 

Calculation:

Here, we have P = Rs 20,000 , R = 8% , C.I. = Rs 3,328 

⇒ A = ( P + C.I.) = ( 20,000 + 3,328 ) = Rs23,328

Given:

Height of the lighthouse = 100m

Calculation:

In triangle ADC, AD/DC = tan 45°

⇒ AD/DC = 1   [tan 45° = 1]

⇒ AD = DC = 100m

In triangle ABD, AD/BD = tan 30°

⇒ 100/BD = 1/√3     [tan 30° = 1/√3]

⇒ BD = 100 × √3 = 173m [ √3 = 1.73]

⇒ BC = BD + DC

⇒ 173 + 100 = 273 m

∴ The distance between two ships is 273 m

Given:

In the given figure, O is the point of intersection of two chords AB and CD such that OB = OD

Concept Used:

According to the chord intersection theorem, the product of two intersecting inside-the-circle chords' segments is equal.

Solution:

Given that, O is where the chords AB and CD converge and  OB = OD

The type of triangles OAC and ODB must be determined.

According to the chord intersection theorem, the product of two intersecting inside-the-circle chords' segments is equal.
As a result, the segment product is: OA x OB = OC x OD

∴  OA = OC and OB = OD.

In ΔOAC, OA and OC lengths are equal.

∴ The triangle is isosceles.

In ΔOBD, OB and OD lengths are equal.

∴ The triangle is isosceles.

Due to the resemblance characteristic, identical triangles have congruent corresponding angles and proportionate corresponding sides.

OA = OB = OC = OD (radius)

∴ The corresponding sides and angles are equal.

The triangles OAC and OBD are hence comparable.

Hence, the triangles OAC and OBD are similar and isosceles.

Given:

2/3 of A = 75% of B = 0.6 of C, then A ∶ B ∶ C

Calculations:

So, B : C = 4 : 5

Now, A : B = (9 : 8) × 1

And, B : C = (4 : 5) × 2 = 8 : 10

Thus, A : B : C = 9 : 8 : 10

∴ The answer is 9 : 8 : 10.

Given:

Ratio of milk and water is 2 : 3

60 liters of mixture is taken out

Then ratio of milk and water is 1 : 2

Calculation:

Let the milk and water in the total mixture be 2x and 3x.

⇒ Milk in the total mixture = 2x/5x

⇒ Milk in the total mixture = 2/5

⇒ Water in the total mixture = 3x/5x

⇒ Water in the total mixture = 3/5

In 60 liters of mixture 

⇒ Milk = 2/5 × 60

⇒ Milk = 24 liters

⇒ Water = 3/5 × 60

⇒ Water = 36 liters 

When 60 liters of mixture is taken out, 

Replaced with 60 liters of water.

Then the ratio of Milk and Water is 1 : 2

⇒ (2x – 24) : (3x – 36 + 60) = 1 : 2

⇒ (2x – 24)/(3x + 24) = 1 : 2

⇒ 2(2x – 24) = 1(3x + 24)

⇒ 4x – 48 = 3x + 24

⇒ 4x – 3x = 24 + 48

⇒ x = 72

⇒ Total capacity of container = 2x + 3x

⇒ Total capacity of container = 5x

⇒ Total capacity of container = 5 × 72

⇒ Total capacity of container =  360 liters

∴ Total capacity of a container is 360 liters.