From the given data,

⇒ 2² – 3³ + 4³ – 6² = 4 - 27 + 64 - 36 = 5

According to the given information,

Let’s assume the number to be ‘x’

9/4 × 7/2 × x = 126

x = 16

7/2^th of that number is

⇒ 7x/2 = (7 × 16)/2  = 56

Let the three numbers be ‘x’, ‘y’ and ‘z’.

Given, x + y = 55; y + z = 75 and z + 3x = 100

Let the above three equations be (1), (2) and (3) respectively.

From equation (1), y = 55 – x and From (3), z = 100 – 3x

Substituting in (2) ⇒ 55 – x + 100 – 3x = 75

⇒ 155 – 4x = 75 ⇒ 155 – 75 = 4x ⇒ 80 = 4x ⇒ x = 80/4 = 20

⇒ z = 100 – 3 × 20 = 100 – 60 = 40

Given, thrice the square of number decreased by 4 times the number = 50 + number

Let the natural number be ‘n’.

⇒ 3n² – 4n = 50 + n

⇒ 3n² – 5n – 50 = 0

⇒ 3n² – 15n + 10n – 50 = 0                (Splitting middle term to factorise)

⇒ 3n(n – 5) + 10(n – 5) = 0

⇒ (3n + 10)(n – 5) = 0

⇒ (3n + 10) = 0 or (n – 5) = 0

∴ n = -10/3 or n = 5

∵ n is natural ⇒ n = 5

∴ n = 5

The formula for annual compound interest is:

Where:

A = the future value of the investment/loan, including interest

P = the principal investment amount (the initial deposit or loan amount)

r = the annual interest rate (decimal)

n = the number of times that interest is compounded per year

t = the number of years the money is invested or borrowed for.

In the given question, Interest = Rs 246

n = 1 [As the amount is compounded annually]

t = 2 years

r = 5% compounded annually

A = P + Interest (I)

Now from the formula,

Now the formula for simple interest:

Interest = Principal × Rate percent × Time ÷ 100

Here, principal = Rs. 2400

Rate percent = 6

Time = 3 years

Putting the values in the formula, we get

Simple interest for 3 years:

 = Rs. 2400 × 6 × 3 ÷ 100

= Rs. 432

Let price of an article be x

Price of an article with 42% cut = x – 42% of x ⇒ 0.58x

% increase = [(x – 0.58x)/0.58x] × 100 = 72.41%

Ram takes 10 days to finish the work.

Fraction of work done by Ram in day = 1/10

Raj takes 5 days to finish the work.

Fraction of work done by Raj in day = 1/5

Let Ram worked for X days to finish the work and Raj worked for 2 days

Given, vertices of a quadrilateral are (- 1, 1), (0, - 3), (5, 2) and (4, 6)

Distance between two points = √((x1 – x2)² + (y1 – y2)²)

Slope of a line = (y1 – y2)/(x1 – x2)

First of all we will find whether the adjacent sides are equal.

Let the points be A, B, C and D in order.

AB = √((- 1 – 0)² + (1 + 3)²) = √17

BC = √((0 – 5)² + (- 3 - 2)²) = 5√2

CD = √((5 – 4)² + (2 - 6)²) = √17

DA = √((4 + 1)² + (6 - 1)²) = 5√2

AB is not equal to BC, thus it can’t be a square or rhombus.

Slope of line AB = (- 3 – 1)/(0 + 1) = - 4

Slope of line BC = (- 2 + 3)/(5 – 0) = 1/5

AB and BC are not perpendicular as Slope of AB × slope of BC is not equal to - 1

AB and CD are parallel.

Thus, ABCD is a parallelogram