*X100/301* X100/301 MATHEMATICS HIGHER. Units 1, 2 and 3 Paper 1 (Non-calculator) Read Carefully

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1 X00/0 NATINAL QUALIFICATINS 007 TUESDAY, 5 MAY 9.00 AM 0.0 AM MATHEMATICS HIGHER Units, and Paper (Non-calculator) Read Carefull Calculators ma NT be used in this paper. Full credit will be given onl where the solution contains appropriate working. Answers obtained b readings from scale drawings will not receive an credit. LI X00/0 6/770 *X00/0*

2 FRMULAE LIST Circle: The equation + + g + f + c = 0 represents a circle centre ( g, f) and radius The equation ( a) + ( b) = r represents a circle centre (a, b) and radius r. g + f c. Scalar Product: a.b = a b cos θ, where θ is the angle between a and b or a.b = a b + a b + a b where a = a a a b and b =. b b Trigonometric formulae: sin (A ± B) = sin A cos B ± cos A sin B ± cos (A ± B) = cos A cos B sin A sin B sin A = sin A cos A cos A = cos A sin A = cos A = sin A Table of standard derivatives: f ( ) f ( ) sin a cos a acosa asina Table of standard integrals: f ( ) fd ( ) sin a a cosa + C cosa a sin a + C [X00/0] Page two

3 ALL questions should be attempted. Marks. Find the equation of the line through the point (, ) which is parallel to the line with equation + = 0.. Relative to a suitable coordinate sstem A and B are the points (,, ) and (,, ) respectivel. C A, B and C are collinear points and C is positioned such that BC = AB. Find the coordinates of C. A B. Functions f and g, defined on suitable domains, are given b f() = + and g() =. Find: (a) g(f()); (b) g(g()).. Find the range of values of k such that the equation k = 0 has no real roots. 5. The large circle has equation = 0. Three congruent circles with centres A, B and C are drawn inside the large circle with the centres ling on a line parallel to the -ais. This pattern is continued, as shown in the diagram. A B C D Find the equation of the circle with centre D. 5 [Turn over [X00/0] Page three

4 Marks 6. Solve the equation sin = 6cos for A sequence is defined b the recurrence relation u n+ = un 6, u = (a) Calculate the values of u, u and u. Four terms of this sequence, u, u, u and u are plotted as shown in the graph. As n, the points on the graph approach the line u n = k, where k is the limit of this sequence. u n k u n = k (b) (i) Give a reason wh this sequence has a limit. n (ii) Find the eact value of k. 8. The diagram shows a sketch of the graph of = (a) Show that the graph cuts the -ais at (, 0). (b) Hence or otherwise find the coordinates of A. A B (c) Find the shaded area A function f is defined b the formula f() =. (a) Find the eact values where the graph of = f() meets the - and -aes. (b) Find the coordinates of the stationar points of the function and determine their nature. (c) Sketch the graph of = f(). 7 [X00/0] Page four

5 Marks d 0. Given that = +, find. d. (a) Epress f() = cos + sin in the form kcos ( a), where k > 0 and π 0 < a <. (b) Hence or otherwise sketch the graph of = f() in the interval 0 π. [END F QUESTIN PAPER] [X00/0] Page five

6 X00/0 NATINAL QUALIFICATINS 007 TUESDAY, 5 MAY 0.0 AM.00 NN MATHEMATICS HIGHER Units, and Paper Read Carefull Calculators ma be used in this paper. Full credit will be given onl where the solution contains appropriate working. Answers obtained b readings from scale drawings will not receive an credit. LI X00/0 6/770 *X00/0*

7 FRMULAE LIST Circle: The equation + + g + f + c = 0 represents a circle centre ( g, f) and radius The equation ( a) + ( b) = r represents a circle centre (a, b) and radius r. g + f c. Scalar Product: a.b = a b cos θ, where θ is the angle between a and b or a.b = a b + a b + a b where a = a a a b and b =. b b Trigonometric formulae: sin (A ± B) = sin A cos B ± cos A sin B ± cos (A ± B) = cos A cos B sin A sin B sin A = sin A cos A cos A = cos A sin A = cos A = sin A Table of standard derivatives: f ( ) f ( ) sin a cos a acosa asin a Table of standard integrals: f ( ) fd ( ) sin a a cosa + C cosa a sin a + C [X00/0] Page two

8 ALL questions should be attempted. Marks. ABCDEFG is a cube with side units, as shown in the diagram. B has coordinates (,, 0). P is the centre of face CGD and Q is the centre of face CBFG. z D P G Q E F C B (,, 0) A (a) Write down the coordinates of G. (b) Find p and q, the position vectors of points P and Q. (c) Find the size of angle PQ. 5. The diagram shows two right-angled triangles with angles c and d marked as shown. (a) Find the eact value of sin (c + d). c d (b) (i) Find the eact value of sin c. (ii) Show that cos d has the same eact value.. Show that the line with equation = 6 is a tangent to the circle with equation = 0 and find the coordinates of the point of contact of the tangent and the circle. 6. The diagram shows part of the graph of a function whose equation is of the form = asin (b ) + c. (a) Write down the values of a, b and c. (b) Determine the eact value of the -coordinate of P, the point where the graph intersects the -ais as shown in the diagram. P 60 0 [Turn over [X00/0] Page three

9 5. A circle centre C is situated so that it touches the parabola with equation = 8+ at P and Q. (a) The gradient of the tangent to the parabola at Q is. Find the coordinates of Q. P C Q Marks 5 (b) Find the coordinates of P. (c) Find the coordinates of C, the centre of the circle. 6. A householder has a garden in the shape of a right-angled isosceles triangle. It is intended to put down a section of rectangular wooden decking at the side of the house, as shown in the diagram. S 0 m Decking Side Wall 0 m m T (a) (i) Find the eact value of ST. (ii) Given that the breadth of the decking is metres, show that the area of the decking, A square metres, is given b ( ) A= 0. (b) Find the dimensions of the decking which maimises its area Find the value of 0 sin(+ ) d. 8. The curve with equation = log ( )., where >, cuts the -ais at the point (a, 0). Find the value of a. [X00/0] Page four

10 9. The diagram shows the graph of = a, a >. n separate diagrams, sketch the graphs of: Marks = a (a) = a ; (b) = a. (, a) 0. The diagram shows the graphs of a cubic function = f() and its derived function = f (). Both graphs pass through the point (0, 6). The graph of = f () also passes through the points (, 0) and (, 0). 6 = f() = f () (a) Given that f () is of the form k( a)( b): (i) write down the values of a and b; (ii) find the value of k. (b) Find the equation of the graph of the cubic function = f().. Two variables and satisf the equation =. (a) Find the value of a if (a, 6) lies on the graph with equation =. (b) If ( ) also lies on the graph, find b., b (c) A graph is drawn of log 0 against. Show that its equation will be of the form log 0 = P + Q and state the gradient of this line. [END F QUESTIN PAPER] [X00/0] Page five