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Calculation:

The surface area of sphere = 4πr²

Let r1 and r2 be the initial and final radius.

Initial surface area = 4πr12

Increase in new radius by 50%

∴ r2 = r1 + 50% of r1 = (3/2)r1

New surface area = 4π × (9/4)r12 = 9πr12

∴ Percentage increase in surface area = [(9πr12 - 4πr12)/4πr12] × 100

∴ Percentage increase = (5/4) × 100 = 125%

We know equation of line: y = mx + c

y = −x + 1    

y = − x + 2  

Both the slopes are same = -1

We know that 2 lines never intersect each other if they have same slope

∴ No intersection point exists.

Arithmetic mean of two +ve number a and b = (a + b)/2

The geometric mean of two positive numbers a and b = √ab

(a + b)/2 = 5

⇒ a + b = 10       ---- (i)

and √ab = 3

⇒ ab = 9       ---- (ii)

from (i) , b = 10 - a

⇒ a(10 - a) = 9

⇒ a² -10a + 9 = 0

⇒ (a - 9)(a + 1) = 0 or a = 9, (neglecting -ve, since +ve number is asked)

∴ b = 9/9 = 1

∴ a = 9 and b = 1

Let the two volume be a₁ and a₂

Volume of cube = a³

Total surface area of cube = 6a²

a₁³/a₂³ = 27/64

⇒ a₁/a₂ = 3/4

Then, 6a₁²/6a₂² = (6 × 9)/(6 × 16)

Ratio of their total surface area = 9: 16

x⁴ + 1/x⁴ = 14

⇒ x⁴+ 1/x⁴ + 2 – 2 = 14

⇒ {(x²)² + (2x² × 1/x²) + (1/x²)²} = 14 + 2

⇒ (x² + 1/x²)² = 16

⇒ (x² + 1/x²)² = 4²

⇒ x² + 1/x² = 4

⇒ x² + 1/x² + 2 – 2 = 4

⇒ x² + (2x² × 1/x²) + 1/x² = 4 + 2

⇒ (x + 1/x)² = 6

⇒ x + 1/x = √6

⇒ x³ + 1/x³ = (x + 1/x)³ - 3x × (1/x) (x + 1/x)

⇒ (√6)3 – 3(√6) = 6√6 - 3√6

⇒ 3√6

Given:

∠BOC = 110°

Concept:

The single point in which the three bisectors of the interior angles of a triangle intersect and which is the centre of the inscribed circle

Here from triangle:

When bisectors of ∠B and ∠C intersect at O, then

∠BOC = 90 + ∠A/2

Calculation:

∠BOC = 90 + ∠A/2

⇒ 110° = 90° + ∠A/2

⇒ ∠A/2 = 20°

⇒ ∠A = 40°

∴ The value of ∠BAC is 40°

Calculation:

Let the number be x

75% of x + 75 = x

⇒ 3/4 of x = x - 75

⇒ x/4 = 75 or x = 300

∴ The number is 300

2⁵⁰ = (2⁵)¹⁰ = 32¹⁰

3⁴⁰ = (3⁴)¹⁰ = 81¹⁰

4³⁰ = (4³)¹⁰ = 64¹⁰

5²⁰ = (5²)¹⁰ = 25¹⁰

∴ 3⁴⁰ is the greatest number.

∠ACD = 40° 

We know ∠ACB = 90° (∵ Angle subtended by half circle at the circumference of a circle is equal to 90 degrees)

∠BCD =∠ACB + ∠ACD = 40 + 90 = 130°

We know that in a cyclic quadrilateral,

Sum of opposite angles of cyclic quadrilateral = 180° 

∠BAD + ∠BCD = 180° 

∴ ∠BAD = 180 - 130 = 50°

Given:

The given range is: 0 ≤ x ≤ 5 and 5 ≤ y ≤ 10

Concept used:

The maximum value of (x/y) will be when we will take maximum value of x and minimum value of y

Calculation:

Maximum value of x = 5 and minimum value of y = 5

∴ The maximum value of (x/y) is 5/5 = 1