Revision 9709 Paper 1

TOPICS, QUESTIONS AND TACKLING TECHNIQUES FOR 9709 PAPER 1

1.

Syllabus/ topics Quadratic functions

Sample Questions 1. Express 3 – 4x – 2x2 in the form of a – b(x + c)2, and use the result to find (a) Maximum and minimum (a) the range of the functions, which is defined points by completing square for all real values of x. method and state its range (b) the the maximum value that f(x) can take and f(x). the corresponding value of x.

(b) Solving quadratic inequalities

2. Solve the inequality (a) 2x2 – 3x + 1 < 0 (b) x (x + 1) > 12.

(c) Finding quadratic functions, 3. Restrict the domain of the function to x ≥ k, domain, x, range f(x) and f(x) = x2 – 2x, so that an inverse function exists. converting the function into a (a) Find the least possible value of k. one-one function and hence (b) Find an expression for f-1. find the inverse of the 4. The function f(x) = x2 – 4x +3 which is defined function and the domain x, for all real values of x, and x ≤ k. -1 range f (x). (a) Find the greatest possible value of k. (b) Determine the range of f. (c) Find the inverse function f-1 and state its domain and range. (d) Sketch the graphs of y = f(x) and f-1(x).

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2.

Syllabus/ topics Quadratic equations

Sample Questions

12 6 3 (b) y – 7y = 8. (a) Recognising and turning an 5. Solve the equation (a) t  4  t equation into quadratic equation. (b) Solving equations by using factorization or formula.

(c) Discriminant b2 – 4ac for 6. (a) Find the values of k for -3 + kx – 2x2 = 0 has a equal roots, real and distinct repeated root. roots, unreal roots (b) Find the range of k for kx2 – 2x – 7 = 0 has two real roots.

(d) Application of b2 – 4ac to 7. Show that the line y = 3x – 3 and the curve intersection of a curve and a y = (3x + 1)(x + 2) do not meet. line, tangent, intersects at two 8. Find the value of k for which the line x + 2y = 3 points and no intersections. and the curve 2x2 + ky2 = 4 has two points of intersections.

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3.

Syllabus/ topics Sample Questions Functions, composite, inverse, 9. Given that f(x) = x2 and g(x) = 3x – 2, for all domain and range values of x, find a, and b such that (a) finding composite functions: (a) fg(a) = 100 ff(x) , fg(x), gf(x) (b) gg(4) = b.

x2 (b) finding the inverse of the 10. Find the inverse of the function, f ( x)  -1 functions f (x) and its domain x2 -1 x, and range f (x) where x  , and x  2 .

(c) sketch the graph of a function 11. Sketch the graph of g(x) = 3x – 2 and g-1(x) for and its inverse and state the x ≥ 0. relationship between them. (reflection of one another in the line y = x)

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Syllabus/ topics 4.

5.

Surds (a) Adding, subtracting, multiplying surds (b) Rationalise the denominator (c) Pythagoras theorem and trigonometric ratio to get sides of a triangle

Indices (a) Zero, negative, indices

Sample Questions 12. In the triangle PQR, Q is a right angle, PQ = (6  2 2 ) cm and QR = (6  2 2 ) cm.

(a) Find the area of the triangle. (b) Show that the length of PR is 2 22 cm.

fractional 13. Solve the equation 42x × 8x-1 = 32.

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1

(5b) 1 14. Simplify (a) (b) (2 x 6 y 8 ) 4  (8 x 2 ) 4 1 (b) Multiplying, dividing indices (8b 6 ) 3 (law of indices)

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6.

Syllabus/ topics Sample Questions Coordinate Geometry (a) Finding distance between two 15. Points A and B are (3,2) and (4, -5) respectively. points, midpoints of two Find the coordinates of the mid-point of AB and points the gradient of AB. Hence find the equation of the (b) perpendicular bisector of line perpendicular bisector of AB. joining two points

(c) Finding equation of a line using two points, a point and its gradient (d) Finding equation of a perpendicular to line 1 using a point and the equation of the line 1.

16. Find the equation of a line through the point (-2, 3) with gradient -1. 17. Find the equation of the line joining the points (3, 4) and (-1, 2). 18. Find the equation of the line through (4, -3) parallel to y + 2x = 7. 19. Find the equation of the line through (1, 7) parallel to the x-axis.

1 (e) finding y-intercept or 20. The curve y = 1 + crosses the x-axis at A x-intercept of a line (x = 0, y = 2 x and y-axis at B. 0) (a) Calculate the coordinates of A and B. (f) finding interception points of (b) Find the equation of the line AB. two lines or a line and a curve (c) Calculate the coordinates of the point of simultaneously intersection of the line AB and the line with equation 3y = 4x.

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7.

Syllabus/ topics Sample Questions Sequences (a) AP: Term and sum of nth term 21. The sum of the first two terms of an arithmetic progression is 18 and the sum of the first four terms is 52. Find the sum (b) GP: term , sum of nth term, of the first eight terms. sum to infinity 22. In a sequence, 1.0, 1.1, 1.2,…, 99.9, 100.0, each number after the first is 0.1 greater than the preceding number. Find (a) how many numbers there are in the sequence, (b) the sum of all the numbers in the sequence.

(c) Applications of AP and GP to 23. Melisa is given an interest-free loan to buy a car. She repays the loan in unequal monthly instalments; these start at $30 daily life situation in the first month and increase by $2 each month after that. She makes 24 payments. (a) Find the amount of her final payment. (b) Find the amount of her loan. 24. The population of pythagora is decreasing steadily at a rate of 4% each year. The population in 1998 was 21000. Estimate (a) the population in 2002 (b) the population in 1990.

25. Sera’s grandparents put $1000 into a savings bank account for her every year since her age of 10. The account pays interest at 6% per year. (a) How much money is in the account at her age of 18. (b) Find the estimate age in year, that the amount in the account exceeding $20,000.

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8.

Syllabus/ topics Sample Questions Circular measure 26. The diagram shows two intersecting circles of (a) conversion between radians radius 6cm and 4cm with centres 7cm apart. Find and degrees the perimeter and area of the shaded region (b) length of arc and area of the common to both circles. sector, area of triangle, area of a segment

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9.

Syllabus/ topics Binomial expansion (a) Expand (a + bx)n, where n >0

Sample Questions 27. Expand (3x +2)2(2x+3)3, in ascending power of x up to and including the term in x3.

(b) Finding the binomial 28. Find the coefficient of x6y6 in the expansion of coefficients of xr, for (a + (2x + y)12. bx)n, (c +dx) (a + bx)n 29. Find the coefficient of x2 in the expansion of 3

 4 4 x   . x 

30. Given that the expansion of (1 + ax)n begins 1 + 36x + 576x2, find the values of a and n.

(c) Finding the rth power of a 31. Simplify (1 – x)8 + (1 + x)8. Substitute a suitable decimal number by binomial value of x to find the exact value of 0.998 + 1.018. expansion

(d) Expanding (a + bx)n involving 4 32. Expand 2 2  3 in the form of a  b 6 , where a or b in surds a and b are integers.





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Syllabus/ topics Sample Questions 10. Trigonometry 1 (a) Finding the exact value of the 33. Given that cos θ = - , find all the possibles value trigonometric ratios, either in 3 of sin θ and tan θ. form of fraction or surds.

(b) Draw or sketch the graph of 34. Sketch the following functions for all x, where the sine, cosine and tangent 0°≤ x ≤360° and state its maximum and minimum functions and determine its values of each functions and state the values of x at which they occur. maximum and miniumum (a) y = 2 + sinx (b) y = 5 + 8cos2x (c) y = 7 – 4cosx

1 cos  (c) Solving trigonometric 35. Prove that  sin   and hence solve equations for angles in all sin  tan  1 quadrants. the equation  sin   2 for θ ≤ θ ≤ 180°. (d) Proving identities sin 

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Syllabus/ topics Sample Questions 11. Vectors (a) Addition and subtraction of 36. Points A and B have coordinates (2, 7) and vectors (-3, -3) respectively. Use a vector method to find (b) Unit vector, position vectors (a) C, where the point C is such that AC = 3AB. (b) a unit vector which is in the same direction as AB.

(c) Parallel and perpendicular 37. Show that the vectors (Dot product and scalar common factor) perpendicular.

(d) Angles between two vectors

vectors

 3   2   and    2  3 

are

38. OABCDEFG, shown in the figure, is a cuboid. The position vectors of A, C and D are 4i, 2j and 3k respectively.Calculate, (a) |AG| (b) The angle between AG and OB.

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Syllabus/ topics Sample Questions 39. Differentiate the following functions: 12. Differentiation (a) Composite/chain rule 2  1  1 r3 2 (b) Product rule  (c) (a) (1 + x ) (b) x1  r2 x  (c) Quotient rule (d) Increasing or decreasing 40. For each of the following functions f(x), find f’(x) and any intervals in which f(x) is (i) decreasing and functions using differentiation (ii) increasing. (a) 2x3 – 18x + 5

(b) x4 – 4x3

41. Find the coordinates of the stationary point on the (e) Stationary points 1 3 (f) Second derivatives for nature graph of the function y   2 and find whether of stationary points x x the point is maxima or minima.

(g) Equation of tangent and 42. Find the equation of the tangent to the curve y = (x2 – 5)6 at the point (2, 1). normal line to the curve when given a point and the equation 43. The equation of a curve is y = 2x2 – 5x + 14. The normal to the curve at the point (1, 11) meets the of the curve, or the gradient curve again at the point P. Find the coordinates of P. and the equation of the curve, or the equation of the curve and a parallel/perpendicular line to the curve. (h) Rate of change

44. The length of the side of a square is increasing at a constant rate of 1.2m/s. At the moment when the length of the side is 10cm, find (a) the rate of increase of the perimeter, (b) the rate of increase of the area.

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Syllabus/ topics 13. Integration (a) Finite and infinite integration

Sample Questions 45. Evaluate the following: (a)

 1 2   x 3  x  dx

(b)



4

0

2 x  1 dx

(b) Area under the curve by 46. The diagram shows the curve y = (x – 2)2 + 1 with integration minimum point at P. The point Q on the curve is (c) Volume under the curve by such that the gradient of PQ is 2. integration (a) Find the area of the shaded region between PQ and the curve.

(b) Find the volume generated when the region bounded by PQ and the x-axis between P and Q is rotated through 360° about the x-axis.

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