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Basic Skills, Differentiation, Quadratic Theory & Remainder Theorem 1. Express as a single fraction: (a) 2. 3 7 + a+2 a 3 5 + w+2 w (c) 5 8 + e−6 e (d) 4 7 − m m +1 (e) 2 7 + r r −9 (f) 6 9 − v v−3 (g) 2 3 + b +1 b + 2 (h) 5 2 + n−4 n+6 (i) 4 7 − s+5 s+8 (j) 2 5 − t −7 t +9 (k) 1 7 + x − 5 x −1 (l) 8 5 − y −9 y −6 (c) f ( x) = Find the derivative of each function: (a) 3. (b) 3 f ( x) = 4 x − x 2 (b) 6x 3 − x 2 + 5 f ( x) = x x( x − 5) x For each of the following functions: (i) Find the points where the curve intercepts both axes. (ii) Find the stationary points and justify their nature. (iii) Sketch the curve and label each important point. (a) y = x(x – 3)2 (b) y = 2x2 – x4 4. For what values of p does each equation have real roots? (a) 2x2 – 3x + p = 0 5. (b) x2 + (p – 3)x + p = 0 Find c such that the line y = x + c is a tangent to the curve y = x2 – 3x. What are the coordinates of the point of contact? Unit 2: Mathematics 2 Higher 6. Solve the following inequalities: (a) 4x2 – 12x + 5 ≥ 0 (b) 12 + x – x2 > 0 (c) 2x2 + 4x + 1 > 0 7. Use synthetic division to find the quotient and remainder in each of the following: (a) 2x2 + 3x + 4 divided by x – 1 (b) x3 + 4x2 – 3x – 11 divided by x + 4 (c) x4 – x2 + 7 divided by x + 1 (d) 2x3 + x2 + x + 10 divided by 2x + 3 8. Factorise x3 –12 x – 16 9. Find the remainder when 3x3 – 7x2 –15 is divided by x + 3 10. n = 2 is a solution of 2n3 – n2 – 5n – 2 = 0. Find the other two solutions. 11. If x – 2 is a factor of 3x3 + 2x2 + kx + 6, find k and then factorise this expression fully. 12. Show that x = –1 is the only real root of the equation x3 – x2 + 3x + 5 = 0 Unit 1: Mathematics 1 Higher