TOPICS, QUESTIONS AND TACKLING TECHNIQUES FOR 9709 PAPER 1

Syllabus/ topics Sample Questions Expected Answers1. Quadratic functions

(a) Maximum and minimum points by completing square method and state its range f(x).

(b) Solving quadratic inequalities

(c) Finding quadratic functions, domain, x, range f(x) and converting the function into a one-one function and hence find the inverse of the function and the domain x, range f-1(x).

1. Express 3 – 4x – 2x2 in the form of a – b(x + c)2, and use the result to find (a) the range of the functions, which is defined

for all real values of x.(b) the the maximum value that f(x) can take and

the corresponding value of x.

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

3. Restrict the domain of the function to x ≥ k, f(x) = x2 – 2x, so that an inverse function exists.(a) Find the least possible value of k.(b) Find an expression for f-1.

4. The function f(x) = x2 – 4x +3 which is defined for all real values of x, and x ≤ k.(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).

Syllabus/ topics Sample Questions Expected Answers2. Quadratic equations

(a) Recognising and turning an equation into quadratic equation.

(b) Solving equations by using factorization or formula.

(c) Discriminant b2 – 4ac for equal roots, real and distinct roots, unreal roots

(d) Application of b2 – 4ac to intersection of a curve and a line, tangent, intersects at two points and no intersections.

5. Solve the equation (a) (b) y6 – 7y3 = 8.

6. (a) Find the values of k for -3 + kx – 2x2 = 0 has a repeated root.

(b) Find the range of k for kx2 – 2x – 7 = 0 has two real roots.

7. Show that the line y = 3x – 3 and the curve y = (3x + 1)(x + 2) do not meet.

8. Find the value of k for which the line x + 2y = 3 and the curve 2x2 + ky2 = 4 has two points of intersections.

Syllabus/ topics Sample Questions Expected Answers3. Functions, composite, inverse,

domain and range(a) finding composite functions:

ff(x) , fg(x), gf(x)

(b) finding the inverse of the functions f-1(x) and its domain x, and range f - 1(x)

(c) sketch the graph of a function and its inverse and state the relationship between them. (reflection of one another in the line y = x)

9. Given that f(x) = x2 and g(x) = 3x – 2, for all values of x, find a, and b such that(a) fg(a) = 100(b) gg(4) = b.

10. Find the inverse of the function,

where .

11. Sketch the graph of g(x) = 3x – 2 and g-1(x) for x ≥ 0.

Syllabus/ topics Sample Questions Expected Answers4. Surds

(a) Adding, subtracting, multiplying surds

(b) Rationalise the denominator(c) Pythagoras theorem and

trigonometric ratio to get sides of a triangle

12. In the triangle PQR, Q is a right angle, PQ = cm and QR = cm.

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

5. Indices(a) Zero, negative, fractional

indices

(b) Multiplying, dividing indices (law of indices)

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

14. Simplify (a) (b)

Syllabus/ topics Sample Questions Expected Answers6. Coordinate Geometry

(a) Finding distance between two points, midpoints of two points

(b) perpendicular bisector of line 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.

(e) finding y-intercept or x-intercept of a line (x = 0, y = 0)

(f) finding interception points of two lines or a line and a curve simultaneously

15. Points A and B are (3,2) and (4, -5) respectively. Find the coordinates of the mid-point of AB and the gradient of AB. Hence find the equation of the perpendicular bisector of AB.

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.

20. The curve y = 1 + crosses the x-axis at A

and y-axis at B. (a) Calculate the coordinates of A and B.(b) Find the equation of the line AB.(c) Calculate the coordinates of the point of

intersection of the line AB and the line with equation 3y = 4x.

Syllabus/ topics Sample Questions Expected Answers7. Sequences

(a) AP: Term and sum of nth term(b) GP: term , sum of nth term,

sum to infinity

(c) Applications of AP and GP to daily life situation

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

of the first eight terms.

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.

23. Melisa is given an interest-free loan to buy a car. She repays

the loan in unequal monthly instalments; these start at $30

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.

Syllabus/ topics Sample Questions Expected Answers8. Circular measure

(a) conversion between radians and degrees

(b) length of arc and area of the sector, area of triangle, area of a segment

26. The diagram shows two intersecting circles of radius 6cm and 4cm with centres 7cm apart. Find the perimeter and area of the shaded region common to both circles.

Syllabus/ topics Sample Questions Expected Answers9. Binomial expansion

(a) Expand (a + bx)n, where n >0

(b) Finding the binomial coefficients of xr, for (a + bx)n, (c +dx) (a + bx)n

(c) Finding the rth power of a decimal number by binomial expansion

(d) Expanding (a + bx)n involving a or b in surds

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

28. Find the coefficient of x6y6 in the expansion of (2x + y)12.

29. Find the coefficient of x2 in the expansion of

.

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

31. Simplify (1 – x)8 + (1 + x)8. Substitute a suitable value of x to find the exact value of 0.998 + 1.018.

32. Expand where

a and b are integers.

Syllabus/ topics Sample Questions Expected Answers10. Trigonometry

(a) Finding the exact value of the trigonometric ratios, either in form of fraction or surds.

(b) Draw or sketch the graph of the sine, cosine and tangent functions and determine its maximum and miniumum

(c) Solving trigonometric equations for angles in all quadrants.

33. Given that cos θ = - , find all the possibles value

of sin θ and tan θ.

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

35. Prove that and hence solve

(d) Proving identitiesthe equation for θ ≤ θ ≤ 180°.

Syllabus/ topics Sample Questions Expected Answers11. Vectors

(a) Addition and subtraction of vectors

(b) Unit vector, position vectors

(c) Parallel and perpendicular vectors (Dot product and scalar common factor)

(d) Angles between two vectors

36. Points A and B have coordinates (2, 7) and (-3, -3) respectively. Use a vector method to find (a) C, where the point C is such that AC = 3AB.(b) a unit vector which is in the same direction as

AB.

37. Show that the vectors are

perpendicular.

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.

Syllabus/ topics Sample Questions Expected Answers12. Differentiation

(a) Composite/chain rule(b) Product rule(c) Quotient rule(d) Increasing or decreasing

functions using differentiation

(e) Stationary points(f) Second derivatives for nature

of stationary points

(g) Equation of tangent and normal line to the curve when given a point and the equation of the curve, or the gradient and the equation of the curve,

39. Differentiate the following functions:

(a) (1 + )2 (b) (c)

40. For each of the following functions f(x), find f’(x) and

any intervals in which f(x) is (i) decreasing and (ii)

increasing.

(a) 2x3 – 18x + 5 (b) x4 – 4x3

41. Find the coordinates of the stationary point on the

graph of the function and find whether the

point is maxima or minima.

42. Find the equation of the tangent to the curve y =

(x2 – 5)6 at the point (2, 1).

43. The equation of a curve is y = 2x2 – 5x + 14. The

normal to the curve at the point (1, 11) meets the

curve again at the point P. Find the coordinates of P.

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.

Syllabus/ topics Sample Questions Expected Answers13. Integration

(a) Finite and infinite integration

(b) Area under the curve by integration

(c) Volume under the curve by integration

45. Evaluate the following:

(a) (b)

46. The diagram shows the curve y = (x – 2)2 + 1 with minimum point at P. The point Q on the curve is such that the gradient of PQ is 2.

(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.