Given:

a + b = 7 and ab = 12

Formula Used:

(a + b)² = a² + 2ab + b²

Calculation:

(a + b)² = 7²

⇒ a² + b² + 2ab = 49

⇒ a² + b² = 49 – 2 × 12

⇒ a² + b² = 25

Given:

Diameter of bigger plate = 3.5 cm

Diameter of smaller plate = 2.5 cm

Formula used:

Surface area of plate = π × radius² = π × diameter² / 4

Calculation:

⇒ π {(3.5)² - (2.5)²} / 4

⇒ π {6} / 4

⇒ 1.5 π

Shortcut Trick

Let the average of the 58 innings = a

Average after 59th inning,

⇒ 59(a + 1.5) = 58a + 117

⇒ 59a + 88.5 = 58a + 117

⇒ a = 28.5

Now,

⇒ Average after 59th inning = a + 1.5 = 28.5 + 1.5 = 30

∴ Average after 59th inning is 30.

Alternate Method

Given:

A batsman score of his 59th inning is 117 runs.

Concept used:

Average score of a batsman = (Total run made by batsman) ÷ (Total innings played by batsman)

Calculation:

Let the average of the 58 innings = a

⇒ Total runs made by batsman after 58th innings = 58a

If he made a runs in 59th inning his average remain same, but his average increases 1.5 runs, so

⇒ new average after 59th innings = a + 1.5

⇒ Total runs made by batsman after 59th innings = 59(a + 1.5)

According to question,

⇒ Total runs after 59th innings = Total runs after 58th innings + score of the 59th inning

⇒ 59(a + 1.5) = 58a + 117

⇒ 59a + 88.5 = 58a + 117

⇒ a = 117 - 88.5 = 28.5

⇒ a = 28.5

Now,

⇒ Average after 59th inning = a + 1.5 = 28.5 + 1.5 = 30

∴ Average after 59th inning is 30.

Important Points

Average score of a batsman = (Total run made by batsman) ÷ (Total innings played by batsman)

A cricketer's bowling average = (the numbers of runs they have conceded) ÷ (the number of wickets they have taken)

Calculation:

The minimum production of medicine by any company = 1.125 mn

This production is done by Company A in Medicine P.

The answer is Medicine P, Company A.

Given:

The angle subtended by a chord on the major arc of a circle is 50 degrees.

Concept used:

The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.

Calculation:

Let O, R, and PQ be the centre, a point on the major arc, and the aforementioned chord respectively.

So, the major arc for the PQ chord is PRQ.

Hence, ∠PRQ = 50°

Here, ∠POQ is the angle subtended by the same chord, PQ on the centre of the circle.

According to the concept,

∠POQ = 2 × ∠PRQ

⇒ ∠POQ = 2 × 50°

⇒ ∠POQ = 100°

∴ The angle subtended by the same chord on the centre of the circle is 100°.

∴ Option 1 is the correct answer.

Given:

Cash price = Rs. 400

Cash down payment = Rs. 100

Concept used:

Simple Interest = PRT/100

Calculation:

Balance to be paid in installments = Rs. 400 - Rs. 100 = Rs. 300

Let rate of interest be r% p.a.

At the end of 3 months, Rs. 300 will be amount to

⇒ Rs. 315 + 0.2625r   --------- (2)

From (1) and (2) we get

⇒ 300 + 0.75r = 315 + 0.2625r

⇒ 0.4875r = 15

⇒ r = 30.76%

∴ Rate of interest charged under instalment plan = 30.76%

Given:

radius = 14 cm

Concept used:

Perimeter(Semi-circle) = πr + 2r

Calculation:

⇒ 44 + 28 cm

⇒ 72 cm

∴ The correct answer is 72 cm.

Given data:

Initial  624 litres

Initial ratio of milk to water: 5 ∶ 3

Desired ratio of milk to water: 5 ∶ 4

Concept:

To keep the amount of milk constant, calculate the amount of water that needs to be added to achieve the desired ratio.

Calculation:

⇒ Volume of milk = 5 / (5 + 3) x 624 litres = 390 litres

⇒ Volume of water in new mixture = 4 / 5 x 390 litres = 312 litres

⇒ Volume of water to be added = (3 / (5 + 3) x 624 litres)  - 312 = 78 litres

Therefore, 78 litres of water needs to be added to achieve the desired ratio.

Given:

The number of Kannada books in bad condition = 92

⇒ (23%) × 2% = 92

⇒ 23% = 4600

⇒ 1% = 200

Number of English books in the library = 20%

⇒ 1% = 200

⇒ 20 × 1% = 200 × 20

⇒ 20% = 4000

Here, the number of English books in the library is 4000.

Hence, the correct answer is "4000".

Concept use:

Mean = Sum of the observation/ Total No of Observation.

Calculations:

The original sum of the numbers was 10 × 40 = 400 (since mean is the sum of the observations divided by the number of observations).

In the recheck, it was found that the number 45 was mistaken for 15, suggesting a difference of 45 - 15 = 30.

So, we add the difference to the original sum: 400 + 30 = 430.

Now, the correct mean is the adjusted sum divided by the number of observations: 430 / 10 = 43.

So, the correct mean is: 43