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Formula used: 

Calculation: 

By using the BODMAS rule

\( \Rightarrow \frac{8}{3}\left[ {\frac{7}{2} + \left{ {\frac{4}{5} - \left( {\frac{7}{2} \times \frac{5}{{14}}} \right)} \right}} \right]\)

\( \Rightarrow \frac{8}{3}\left[ {\frac{7}{2} + \left{ {\frac{4}{5} - \frac{5}{4}} \right}} \right]\)

\( \Rightarrow \frac{8}{3}\left[ {\frac{7}{2} - \frac{9}{{20}}} \right]\)

\( \Rightarrow \frac{8}{3} \times \frac{{61}}{{20}}\)

\( \Rightarrow \frac{{122}}{{15}}\)

∴ The required value is \(\frac{{122}}{{15}}\).

Given:

Sum of square of two numbers = 74

Product of two number = 35

Formula:

(a + b)² = a² + b² + 2ab

Calculation:

(a + b)² = 74 + 2 × 35

⇒ (a + b)² = 74 + 70

⇒ (a + b)² = 144

⇒ a + b = √144

⇒ a + b = 12

∴ Sum of two numbers is 12.

Given:

Two trains each 200 m long are moving in opposite directions.

They cross each other in 10 seconds or 1/360 hours

The speed of one train is 90 km/h

Concept used:

Relative speed (while moving in opp. direction) = x + y

Calculations:

Total length = 200 + 200 = 400 m or 0.4 km

Let the speed of the other train be x  

When crossed each other (moving in opp. direction)

Distance traveled = 400 m

x + 90 = 0.4/(1/360)

⇒ x = (0.4 × 360) - 90

⇒ x = 144 - 90 = 54 km/h

∴ The speed of the other train is 54 km/h.

Shortcut Trick

Ratio of Amount to Principal for 2 years = 17745 : 10500

Simplify the ratio (divide by 105): 169 : 100

For a 2-year period, take the square root of the ratio: √169 : √100 = 13 : 10

Rate % = (Difference / Base value) × 100 = (13 − 10) / 10 × 100 = 3 / 10 × 100 = 30%

∴ The correct answer is 30%.

Alternate Method

Given:

Principal (P) = ₹10500

Amount (A) = ₹17745

Time (n) = 2 years

Formula Used:

A = P(1 + R/100)ⁿ

Calculations:

⇒ 17745 = 10500(1 + R/100)²

⇒ 17745 / 10500 = (1 + R/100)²

⇒ 1.69 = (1 + R/100)²

⇒ √1.69 = 1 + R/100

⇒ 1.3 = 1 + R/100

⇒ R/100 = 1.3 − 1 = 0.3

⇒ R = 0.3 × 100 = 30%

∴ The correct answer is 30%.

Additional Information

Compound Interest (CI) Basic Formula

Interest is calculated on the initial principal and also on the accumulated interest of previous periods. Formula: CI = P[(1 + R/100)ⁿ − 1].

Effective Rate for 2 Years

The effective interest rate for two years can be calculated using the successive percentage formula: R_eff = x + y + (xy/100).

Difference between CI and SI for 2 Years

The difference between compound interest and simple interest for a period of 2 years is given by the formula: Difference = P(R/100)².

Quarterly and Half-Yearly Compounding

If interest is compounded half-yearly, the rate becomes R/2 and time becomes 2n. For quarterly, the rate is R/4 and time is 4n.

Shortcut Trick

Ratio CSA/TSA = 1/3 ⇒ 2πrh / [2πr(h+r)] = 1/3 ⇒ h/(h+r) = 1/3 ⇒ r = 2h.

Volume = πr²h = π(2h)²h = 4πh³ = 4312.

4 × (22/7) × h³ = 4312 ⇒ h³ = 343 ⇒ h = 7 cm.

Radius r = 2 × 7 = 14 cm.

CSA = 2 × (22/7) × 14 × 7 = 616 cm².

∴ The correct answer is 616 cm².

Alternate Method

Given: Volume = 4312 cm³, Curved Surface Area (CSA) = 1/3 Total Surface Area (TSA).

Formula Used: Volume = πr²h; CSA = 2πrh; TSA = 2πr(h+r).

⇒ CSA = 1/3 TSA

⇒ 2πrh = 1/3 [2πr(h + r)]

⇒ 3h = h + r ⇒ r = 2h

⇒ Volume = πr²h = 4312

⇒ (22/7) × (2h)² × h = 4312

⇒ (22/7) × 4h³ = 4312

⇒ h³ = (4312 × 7) / 88

⇒ h³ = 343 ⇒ h = 7 cm

⇒ r = 2 × 7 = 14 cm

⇒ CSA = 2 × (22/7) × 14 × 7 = 616

∴ The correct answer is 616 cm².

Additional Information

Volume of Cylinder

It is the capacity of the cylinder, calculated as Base Area × Height, V = πr²h.

Surface Areas of Cylinder

Curved Surface Area (CSA) = 2πrh. Total Surface Area (TSA) includes the two circular bases, TSA = 2πrh + 2πr².

Special Case: CSA = 1/2 TSA

If CSA is half of TSA, then 2πrh = 2πr², which means the height of the cylinder is equal to its radius (h = r).

Shortcut Trick

For a toy where Volume of Cone = Volume of Hemisphere with the same radius (r):

(1/3) × π × r² × h = (2/3) × π × r³ ⇒ h = 2r.

The total height of the toy is the sum of the cone height (h) and hemisphere radius (r):

Total Height = h + r = 2r + r = 3r.

Given r = 7 cm, Total Height = 3 × 7 = 21 cm.

∴ The correct answer is 21 cm.

Alternate Method

⇒ Given: Radius (r) of both cone and hemisphere = 7 cm.

⇒ Formula Used: Volume of Cone = (1/3)πr²h | Volume of Hemisphere = (2/3)πr³.

⇒ Volume of conical part = Volume of hemispherical part

⇒ (1/3) × π × r² × h = (2/3) × π × r³

⇒ (1/3) × π × 7² × h = (2/3) × π × 7³

⇒ h = 2 × 7

⇒ h = 14 cm

⇒ Total height of the toy = Height of cone (h) + Radius of hemisphere (r)

⇒ Total height = 14 + 7 = 21 cm.

∴ The correct answer is 21 cm.

Additional Information

Volume of a Sphere

The volume of a complete sphere is (4/3) × π × r³. A hemisphere is exactly half of this volume.

Slant Height of a Cone

The slant height (l) is the distance from the apex to any point on the base circle: l = √(h² + r²).

Curved Surface Area (CSA)

For the toy, Total CSA = CSA of cone (πrl) + CSA of hemisphere (2πr²).

Shortcut Trick

Side of one cube = ³√216 = 6 cm.

When 5 cubes are joined, Length (L) = 5 × 6 = 30 cm, Breadth (B) = 6 cm, Height (H) = 6 cm.

Lateral Surface Area = 2 × (L + B) × H

⇒ 2 × (30 + 6) × 6 = 2 × 36 × 6 = 432 cm².

∴ The correct answer is 432 cm².

Alternate Method

Given:

Volume of one cube = 216 cm³

Number of cubes = 5

Formula Used:

Volume of cube = a³

Lateral Surface Area (LSA) of cuboid = 2h(l + b)

Calculations:

⇒ a³ = 216

⇒ a = √³(216) = 6 cm

⇒ New length (l) = 5 × a = 5 × 6 = 30 cm

⇒ Breadth (b) = 6 cm and Height (h) = 6 cm

⇒ LSA = 2 × 6 × (30 + 6)

⇒ LSA = 12 × 36

⇒ LSA = 432 cm²

∴ The correct answer is 432 cm².

Additional Information

Total Surface Area of Cuboid

It includes the top and bottom faces: TSA = 2(lb + bh + hl).

Lateral Surface Area vs Total Surface Area

Lateral area excludes top and bottom bases. For a cube, LSA = 4a² and TSA = 6a².

Diagonal of Cuboid

The longest rod that can be placed inside is given by √(l² + b² + h²).

Shortcut Trick

Average of 4 numbers = 27.

New average of 5 numbers = 28.

Increase in average = 28 − 27 = 1.

Added number = New Average + (Old count × Average increase) = 28 + (4 × 1) = 32.

∴ The correct answer is 32.

Alternate Method

Given: Average of 4 consecutive even numbers = 27. New average = 28.

Formula: Sum = Average × Number of items

⇒ Total sum of 4 consecutive even numbers = 27 × 4 = 108

⇒ Let the added number be x.

⇒ Total count of numbers after adding x = 4 + 1 = 5

⇒ New Sum ÷ 5 = 28

⇒ (108 + x) ÷ 5 = 28

⇒ 108 + x = 28 × 5

⇒ 108 + x = 140

⇒ x = 140 − 108

⇒ x = 32

∴ The correct answer is 32.

Additional Information

Average of Consecutive Numbers

For an arithmetic progression like consecutive even numbers, the average is exactly the middle value. For 4 numbers, it is the mean of the 2^nd and 3^rd terms.

Sum Property

The Sum of observations is always equal to Average × Number of observations. If a new member joins, the new sum = Old Sum + New Value.

Average Inclusion Formula

Value of new member = (New Count × New Average) − (Old Count × Old Average).

Shortcut Trick

The average of 4 consecutive odd numbers is the middle value between the 2^nd and 3^rd numbers.

Given Average = 34, so the numbers are (34−3), (34−1), (34+1), (34+3) → 31, 33, 35, 37.

Largest number = 37 and Second smallest number = 33.

Required Average = (37 + 33) ÷ 2 = 35.

Since 35 is not divisible by 2, it is an odd number.

∴ The correct answer is An odd number.

Alternate Method

Let the four consecutive odd numbers be x, (x + 2), (x + 4), and (x + 6).

Given: Average of these numbers = 34.

⇒ [x + (x + 2) + (x + 4) + (x + 6)] ÷ 4 = 34

⇒ (4x + 12) ÷ 4 = 34

⇒ 4x + 12 = 136

⇒ 4x = 124

⇒ x = 31

⇒ The numbers are 31, 33, 35, 37.

⇒ Largest number = 37

⇒ Second smallest number = 33

⇒ Required Average = (37 + 33) ÷ 2 = 70 ÷ 2 = 35

⇒ 35 is an odd number.

∴ The correct answer is An odd number.

Additional Information

Consecutive Numbers in A.P.

For any sequence in Arithmetic Progression, the average of the entire set is equal to the average of any two terms equidistant from the center.

Average of Odd/Even Numbers

The average of n consecutive odd or even numbers is always the mid-point. If n is even, the average is an integer between the two middle terms.

Sum of First n Odd Numbers

The sum of the first n consecutive odd numbers starting from 1 is always n², and their average is n.

Given:

Selling price of an article is 2,28,528.

Calculation:

Let the cost price of the article be Rs. x

According to question,

⇒ x × (115 / 100) × (90 / 100) = 2,28,528

⇒ x = 228528 × (100 / 90) × (100 / 115)

⇒ x =  2,20,800

The cost price of the article is 2,20,800

Shortcut Trick

We know, 15% → \(\frac{3}{20}\)

Let's assume that the CP is 20,

Then, MP is 15% more than CP so, MP = 23

A discount of 10% is given, so SP = \(23 \times \frac{90}{100} = 20.7\) 

According to the question,

⇒ 20.7 unit → 228528

⇒ 20 unit → \(\frac{228528}{20.7} \times 20 = 220800\)