2003_Paper II

# 2003_Paper II - STEP II 2003 Section A 1 2 Pure Mathematics...

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STEP II, 2003 2 Section A: Pure Mathematics 1 Consider the equations ax - y - z = 3 , 2 ax - y - 3 z = 7 , 3 ax - y - 5 z = b , where a and b are given constants. (i) In the case a = 0 , show that the equations have a solution if and only if b = 11 . (ii) In the case a 6 = 0 and b = 11 show that the equations have a solution with z = λ for any given number λ . (iii) In the case a = 2 and b = 11 ﬁnd the solution for which x 2 + y 2 + z 2 is least. (iv) Find a value for a for which there is a solution such that x > 10 6 and y 2 + z 2 < 1 . 2 Write down a value of θ in the interval π/ 4 < θ < π/ 2 that satisﬁes the equation 4 cos θ + 2 3 sin θ = 5 . Hence, or otherwise, show that π = 3 arccos(5 / 28) + 3 arctan( 3 / 2) . Show that π = 4 arcsin(7 2 / 10) - 4 arctan(3 / 4) . 3 Prove that the cube root of any irrational number is an irrational number. Let u n = 5 1 / (3 n ) . Given that 3 5 is an irrational number, prove by induction that u n is an irrational number for every positive integer n . Hence, or otherwise, give an example of an inﬁnite sequence of irrational numbers which converges to a given integer m . [An irrational number is a number that cannot be expressed as the ratio of two integers.]

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STEP II, 2003 3 4 The line y = d , where d > 0 , intersects the circle x 2 + y 2 = R 2 at G and H . Show that the area of the minor segment GH is equal to R 2 arccos ± d R ² - d p R 2 - d 2 . ( * ) In the following cases, the given line intersects the given circle. Determine how, in each case, the expression ( * ) should be modiﬁed to give the area of the minor segment. (i) Line: y = c ; circle: ( x - a ) 2 + ( y - b ) 2 = R 2 . (ii) Line: y = mx + c ; circle: x 2 + y 2 = R 2 . (iii) Line: y = mx + c ; circle: ( x - a ) 2 + ( y - b ) 2 = R 2 .
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2003_Paper II - STEP II 2003 Section A 1 2 Pure Mathematics...

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