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A circle with centre C has equation x^2+8x+y^2-12y=12. The points P and Q lie on the circle. The origin is the midpoint of the chord PQ. Show that PQ has length nsqrt(3) , where n is an integer.

First complete the square for both x and y. Move all constants to the right hand side. The square root of this is the radius of the circle. The two constants in the completed square bracket show the x and y ...
Answered by • Maths tutor
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Integrate(1+x)/((1-x^2)(2x+1)) with respect to x.

To simplify the fraction first notice (1-x 2 ) = (1-x)(1+x) so the common factor of (1+x) in the numerator and denominator can be cancelled. (1+x)/((1-x^2)(2x+1)) = 1/((1-x)(2x+1)), then we need to split thi...
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Answered by Ravinder H. • Maths tutor
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What is 7 to the power of 8? (

Rearrange to (7 4 ) 2 = ((7 2 ) 2 ) 2 7 2 = 49(7 2 ) 2 = 49 2 = (50-1) 2 = 50 2 - 2x50 + 1 = 24017 8 = (7 4 ) 2 =((7 2 ) 2 ) 2 = 2401 2 = (2000 + 400 + 1) 2 = 5,764,801
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Answered by Michael H. • Maths tutor
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The numbers a, b, c and d satisfy the following equations: a + b + 3c + 4d = k; 5a = 3b = 2c = d. What is the smallest value for k for which a, b, c and d are all positive integers

5a = 3b = 2c = d. d must be a multiple of 5, 3 and 2, therefore the smallest possible value for d is 30. This sets a = 6, b = 10 and c = 152) a + b + 3c + 4d = 6 + 10 + 3x15 + 4x30 = 181 k = 181
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Answered by Michael H. • Maths tutor
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How can I remember when a turning point of a function is a maximum or a minimum?

The key is to look at the first and second derivatives of that function. Remember that a turning point always has the first derivative equal to zero. Then, the sign of the second derivative indicates if that...
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Answered by Titus D. • Maths tutor
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