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Concept:

The only number which is even and prime both is 2

∴ m = 2

Calculation:

According to question:

2 × (2 + 2) × (2 + 4) = 48

Now, 48/3 = 16, 48/6 = 8 and 48/2 = 24

We can see that 48 is divisible by 3, 6 and 2.

∴ From the options given 48 is not divisible by 5

Concept used:

am ÷ an = a(m - n)

(am)n = amn

Calculation:

(9/7)³ × (7/9)^{2(2x - 6)} = (7/9)⁹

⇒ (7/9)^{4x - 12} = (7/9)⁹ × (7/9)³

⇒ (7/9)^{4x - 12} = (7/9)¹²

⇒ 4x -12 = 12 (∵ Same base can be equated)

⇒ 4x = 24 or x = 6

∴ The value of x is 6.

Given:

There are 10 triangles, out of which 5 are black and 5 are of grey colour.

Formula used:

Area of triangle = √s(s − a)(s − b)(s − c),

s = (a + b + c)/2​

Calculation:

Area of of each cloth = 5 × area of 1 triangle

s = (a + b + c)/2​ = (50 + 20 + 50​)/2

∴ s = 60 cm.

∴ Area Δ = 60(60 − 50)(60 − 20)(60 − 50) = 60 × 10 × 40 × 10

6 × 10 × 10 × 4 × 10 × 10 = 2 × 10 × 106 = 200√6 cm²

Area of each cloth required = 5 × 200√6 cm² = 1000√6 cm²

Let a be the first root of the quadratic equation,

Then the other root = 1/a

The quadratic equation ax² + bx + c = 0 

∴ Product of root = c/a

⇒ c/a = a × 1/a or c = a

Given:

Time = 10 years

Formula used:

SI = (PRT/100)

Amount = P + SI

Where, SI → Simple interest, P → Principal, R → Rate, T → Time

Calculation:

Let the principal be P

Sum of money is doubled = 2P

∴ Interest = amount – principal = 2P – P = P

Time = 10 years

SI = (PRT/100)

⇒ P = (P × R × 10)/100

⇒ R = 10

∴ The rate of interest is 10%

Given:

Number of B2 birds are more than B1 birds  by a multiple of 9 and 7,

B1, B2, and B3 are three kinds of birds on a tree in the ratio 3 ∶ 7 ∶ 5 

Calculation:

Let the ratio be 3x: 7x: 5x

According to the question:

7x - 3x = Multiple of 9 and 7

⇒ 4x = Multiple of 9 and 7

Number of birds on the tree will be = 3x + 7x + 5x = 15x or multiple of 15

According to option,  630 and 945 is divisible by 15

Let 630 be the number of birds, then x = 630/15 = 42

Now it must satisfy, 4x =  Multiple of 9 and 7

4x = 4 × 42 = 168 is not multiple of 9

Let 945 be the number of birds, then x = 945/15 = 63

4x = 4 × 63 = 252 is divisible by both 7 and 9.

The minimum number of birds on the tree are 945

Given:

x = 1/(2 - √3)

Formula used:

(a + b)² = a² + b² + 2ab

Calculation:

x = 1/(2 - √3)

Rationalizing

x = {1/(2 - √3)} × {(2 + √3)/(2 + √3)}

⇒ x = (2 + √3)/(4 - 3)

⇒ x = 2 +√3

⇒ (x - 2) = √3

Squaring both sides

⇒ x² – 4x + 4 = 3

⇒ x² – 4x + 1 = 0 ---(i)

⇒ x² = 4x -1

Multiplying x in both side

x³ = 4x² - x

According to question:

x³ -2x² – 7x + 5 = 0

putting value of x³

4x² – x – 2x² – 7x + 5 = 0

⇒ 2x² – 8x + 5

⇒ 2(x² – 4) +5 ---(ii)

Equation (i) can be written as

x² – 4x = -1 ---(iii)

value of equation (iii) can be put in equation (ii)

⇒ 2(-1) + 5 = -2 + 5 = 3

∴ the required value = 3

10³ A + A can be written as 1000A + A

⇒ A(1000 + 1)

⇒ 1001 × A

⇒ 7 × 11 × 13 × A

Here 7, 11, and 13 are prime numbers so to make it a perfect square we have to multiply with 7, 11, and 13

For this to be a perfect square:

7 × 11 × 13 × A = (7 × 11 × 13) × (7 × 11 × 13)

1001 × A = 1001 × 1001 

∴ A = 1001

∴ Minimum number of digits in A = 4

Given:

Sum of the present ages of father and son = 68 years

The ratio of their ages 8 years ago = 12: 1

Calculation:

Let the present age of father and son be x and y

According to question:

(x - 8)/(y - 8) = 12/1

⇒ x - 8 = 12 y - 96

⇒ x - 12y = - 88

And x + y = 68

From above equations:

⇒ -13y = - 156 or y = 12 years

∴ x = 56 years

4 years hence means 4 years after present age.

The ratio of their ages after 4 years hence = 60: 16 = 15: 4

Given: 

abx² = (a² + b² + 2ab) (x – 1) 

Formula used:

(a + b)² = (a2 + b2 + 2ab)

Concept used:

The general equation of the quadratic equation in terms of roots

x² – (α + β)x + αβ = 0 

Here, α  and β be the roots of the quadratic equation:

Calculation:

abx² = (a + b)²x - (a + b)²

⇒ x² = {(a + b)²/ab}x - {(a + b)²/ab}

⇒ x2 – {(a + b)2/ab}x + {(a + b)2/ab} = 0

Now, comparing the above equation with the general quadratic equation we'll get

α + β = {(a + b)2/ab} and αβ = {(a + b)2/ab}

Sum of the reciprocals of the root = 1/α + 1/β = (α + β)/αβ

⇒ (α + β)/αβ = {(a + b)²/ab}/{(a + b)²/ab} = 1