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2001_Paper I

# 2001_Paper I - Section A Pure Mathematics 1 The points A B...

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Section A: Pure Mathematics 1 The points A , B and C lie on the sides of a square of side 1 cm and no two points lie on the same side. Show that the length of at least one side of the triangle ABC must be less than or equal to ( 6 - 2) cm. 2 Solve the inequalities (i) 1 + 2 x - x 2 > 2 /x ( x = 0) , (ii) (3 x + 10) > 2 + ( x + 4) ( x - 10 / 3) . 3 Sketch, without calculating the stationary points, the graph of the function f ( x ) given by f ( x ) = ( x - p )( x - q )( x - r ) , where p < q < r . By considering the quadratic equation f ( x ) = 0, or otherwise, show that ( p + q + r ) 2 > 3( qr + rp + pq ) . By considering ( x 2 + gx + h )( x - k ), or otherwise, show that g 2 > 4 h is a sufficient condition but not a necessary condition for the inequality ( g - k ) 2 > 3( h - gk ) to hold. 4 Show that tan 3 θ = 3 tan θ - tan 3 θ 1 - 3 tan 2 θ . Given that θ = cos - 1 (2 / 5) and 0 < θ < π/ 2, show that tan 3 θ = 11 / 2 . Hence, or otherwise, find all solutions of the equations (i) tan(3 cos - 1 x ) = 11 / 2 , (ii) cos( 1 3 tan - 1 y ) = 2 / 5 . 2

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5 Show that (for t > 0) (i) 1 0 1 (1 + tx ) 2 d x = 1 (1 + t ) , (ii) 1 0 - 2 x (1 + tx ) 3 d x = - 1 (1 + t ) 2 . Noting that the right hand side of (ii) is the derivative of the right hand side of (i), conjecture the value of 1 0 6 x 2 (1 + x ) 4 d x . (You need not verify your conjecture.) 6 A spherical loaf of bread is cut into parallel slices of equal thickness. Show that, after any number of the slices have been eaten, the area of crust remaining is proportional to the number of slices remaining. A European ruling decrees that a parallel-sliced spherical loaf can only be referred to as ‘crusty’ if the ratio of volume V (in cubic metres) of bread remaining to area A
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