L.7 PRE-LEAVING CERTIFICATE EXAMINATION, 00 MATHEMATICS HIGHER LEVEL PAPER (300 marks) TIME : ½ HOURS Attempt SIX QUESTIONS (50 marks each). WARNING: Marks will be lost if all necessary work is not clearly shown. Answers should include the appropriate units of measurement, where relevant. 00 L.7 /8 Page of 5
. (a) Simplify fully 4 + + 3. 3 (b) (i) Let g (x) = ax + bx + c, where a, b, c R. Given that k is a real number such that g (k) = 0, prove that x k is a factor of g (x). x t is a factor of x 3 + px + p. 3 t Show that p =. ( t + ) (c) Show that z 4z + 5 is a factor of z 3 + (i 4)z + (5 4i)z + 5i. Hence, find the three roots of z 3 + (i 4)z + (5 4i)z + 5i = 0.. (a) Solve the simultaneous equations x + y + z = 6 x y + 3z = 4 5x + y = 8. (b) (i) Let f (x) = 3tx 6x + (t ), where t R. Find the values of t for which f (x) = 0 has equal roots. Hence, for each value of t, find the roots of f (x) = 0. Find the range of values of x for which x 3 x + 4 >, where x R and x 4. (c) α and β are the roots of the equation x + 3ax + a = 0. (i) Find the value of α 3 + β 3. Find the quadratic equation whose roots are β α and α β. 00 L.7 /8 Page of 5
3. (a) i Let z =, where i =. i Express z in the form a + bi and plot it on an Argand diagram. k k simplifies to a constant. (b) (i) Show that ( k ) 4 6 3 Let A = and B =. 9 Find the matrix C, such that CA = B. (c) (i) w is a complex number such that w w = + 4i where i =. Express w in the form p + qi, where p, q R. Use De Moivre s theorem to find the three roots of z 3 8 = 0. Give your answer in the form x + yi, with x and y fully evaluated. Hence, show that the sum of the three roots is zero. n n n + 4. (a) Show that + = for all natural numbers n 3. 3 3 (b) A sequence is defined by u n = (an + b)3 n, where a, b R. (i) Show that u n + 6u n + + 9u n = 0, for all n 0. Given that u 3 = 54 and u 4 = 405, find the value of a and the value b. (c) Let g(x) = + x + 5x +, where x <. 4x Show that g(x) =. ( x) 00 L.7 3/8 Page 3 of 5
5. (a) The sum to infinity of a geometric series is 0. The common ratio of the series is. Find the first term of the series. (b) (i) Solve log (3x + ) log (x ) =, x >, x R. In the expansion of (x + a) 8 35, the middle term is x 4. 8 Find the values of a, a R. (c) (i) Use induction to prove that n! < for all n 4, n N. n Hence, deduce that n = 4 n! <. 8 6. (a) Differentiate 4 3x with respect to x. (b) Let y = xe x. (i) dy d y Find and. Find the value of k R for which d y dy + 4 + ky = 0. (c) Let f (x) = 5 3x x 3, x R. (i) Find the co-ordinates of the local maximum and local minimum points of the curve y = f (x). Show that the only real root of f (x) = 0 lies between and and hence draw a sketch of the curve y = f (x). (iii) Take x = as the first approximation of the real root of the equation f (x) = 0. Find, using the Newton-Raphson method, x, the second approximation. 00 L.7 4/8 Page 4 of 5
7. (a) Prove from first principles, the addition rule d du dv (u + v) = + where u = u(x) and v = v(x). (b) Given that y = ln x + for x >, x 4 dy find and express your answer in the form b a a x, where a, b N. (c) The parametric equations of a curve are x = cos t + t sin t y = sin t t cos t, where t 4 π. (i) dy Find in terms of t. Hence, show that x y = π is a tangent to the curve. 8. (a) Find (i) 3 x e3x. (b) Evaluate (i) 6 x x. 4x + 0 6 x. 4x + 0 (c) The diagram shows the curve y = x(x ) and the tangent to the curve at its local maximum point. (i) Find the area enclosed by the curve and the x-axis. Hence, find the area of the shaded region. 00 L.7 5/8 Page 5 of 5
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