Rithvik's Blog GCSE

  In a mythical labyrinth, a treasure chest is locked behind a door with a log-based code. The code is the solution to the following logarithmic equation: log ⁡ 2 ( � ) + log ⁡ 2 ( � − 4 ) = 3 lo g 2 ​ ( x ) + lo g 2 ​ ( x − 4 ) = 3 . To unlock the chest and claim the treasure, you must find the value of � x that satisfies this equation. Remember, logarithmic properties might be helpful in solving this equation.

The function f(x) is defined as f(x) = 3x^2 - 2x + 5. Find the value of x when f(x) = 0.

The circle has equation x^2 + y^2 + 6x - 4y - 12 = 0 Find the centre and radius of the circle

Given that x and y are positive integers, solve the simultaneous equations: 3^x * 2^y = 72 3^(x-1) * 2^(y+1) = 48

A function f(x) is defined as f(x) = 2^x + k, where k is a constant. The function passes through the point (4, 40). Find the value of k. For this value of k, solve the equation f(x) = 3^x.

Solve the quadratic 3x^2 + 5x - 2 = 0

Given the function f(x) = 2x²-5x+3, for what values of x does f(x) equal zero.

A rectangle has a length that is 3 cm more than its width. If the area of the rectangle is 40 cm^2, find the dimensions of the rectangle. Let the width of the rectangle be x cm.

The diagram shows a trapezium ABCD with AB parallel to DC. E is a point on AD such that DE is perpendicular to DC. The length of AD is x cm and the length of DE is y cm. Given that: The area of triangle AED is x²/4 The area of the trapezium ABCD is x² + 16cm^2 Find an expression for the length of BC in terms of x

A box contains red, blue, and green marbles. There are twice as many red marbles as blue marbles, and there are 5 more green marbles than blue marbles. If the total number of marbles is 37, how many of each colour marble are there?

A train travels a distance of 180 miles in 3 hours. If the train travels at a constant speed, how long will it take the train to travel 270 miles?

A rectangle has a length that is 3 times its width. If the perimeter of the rectangle is 48 cm, find the dimensions of the rectangle.