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Q15 from Senior Mathematical Challenge 2018: A square is inscribed in a circle of radius 1. An isosceles triangle is inscribed in the square. What is the ratio of the area of this triangle to the area of the shaded region? (Requires Diagram))

Radius = 1, therefore diameter = 2Let x be the length of one side of the square.Using Pythagoras,x² + x² = 2²2x² = 4x = sqrt(2)Area of isosceles triangle = side...

Answered by Thomas H. • Maths tutor

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Solve e^(2x) = 5e^(x) - 6, giving your answers in exact form

Solve e^2x = 5e^x- 6e^2x - 5e^x+ 6 = 0(e^x)² - 5e^x<...

Answered by Thomas H. • Maths tutor

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A curve C is mapped by the equation ( 1+x)(4-x). The curve intersects the x-axis at x = –1 and x = 4. A region R is bounded by C and the x-axis. Use calculus to find the exact area of R.

First draw the curve. Figure out and write the integration problem. Integral⁴_-1 (1+x)(4-x) dx.Expand integral⁴_-1 4 + 3x - x² dx.= ⁴_...

Answered by Vishesh D. • Maths tutor

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Describe and explain the change in the shape of the graph y=x^2 and y=x^2 + 2.

The graph is translated by two in the positive y-axis direction. For example, for the equation y=x², taking the x-axis values of 0,1,2 and 3, y= 0²,1², 2

Answered by Matthew S. • Maths tutor

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(a) Find the set of values of k that satisfy the inequality k^2 - k - 12 < 0. (b) We have a triangle ABC, of lengths AC = 4 and BC = 2. Given that cos B < 1/4 , find the range of possible values for AB:

Solution: Let us consider the function f(k) = k²- k -12. Firstly, we want to find out the roots of this function. We calculate the discriminant: Δ = b² – 4ac = 49 hence we ...

Answered by Alexandru H. • Maths tutor

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