MATHEMATICS

1. It is given that f(x)
= (2x − 5)
³ + x, for x ∈ R . Show that f is an increasing function.

2. (i) In the expression (1− px
)⁶ , p is a non-zero constant. Find the first three terms when (1− px
)⁶ is expanded in ascending powers of x.

(ii) It is given that the coefficient of x² in the expansion of (1−x)
(1− px
)⁶6 is zero. Find the value  of p .

3.

In the diagram, OAB is a sector of a circle with centre O and radius 8 cm. Angle BOA is α radians. OAC is a semicircle with diameter OA. The area of the semicircle OAC is twice the area of the sector OAB.

(i) Find α in terms of Π.

(ii) Find the perimeter of the complete figure in terms of Π.

4 The third term of a geometric progression is −108 and the sixth term is 32. Find

(i) the common ratio,

(ii) the first term,

(iii) the sum to infinity.

5. (i) Show that (sin Θ / (sin Θ + cos Θ)) + (cos Θ (sin Θ − cos ȏ) Ξ (1 / (sin²Θ− cos²Θ)).

(ii) Hence solve the equation (sin Θ / (sin Θ + cos Θ)) + (cos Θ (sin Θ − cos ȏ)= 3, for 0^° ≤ Θ ≤ 360^°.

6. Relative to an origin O, the position vectors of three points, A, B and C, are given by

where p and q are constants.

(i) Show that (OA)^→ is perpendicular to (OC)^→ for all non-zero values of p and q.

(ii) Find the magnitude of (CA)^→ in terms of p and q.

(iii) For the case where p = 3 and q = 2, find the unit vector parallel to (BA)^→.

7. A curve has equation y = x² − 4x + 4 and a line has equation y = mx, where m is a constant.

(i) For the case where m = 1, the curve and the line intersect at the points A and B. Find the coordinates of the mid-point of AB.

(ii) Find the non-zero value of m for which the line is a tangent to the curve, and find the coordinates of the point where the tangent touches the curve.

8. (i) Express 2x² − 12x + 13 in the form a(x + b)² + c, where a, b and c are constants.

(ii) The function f is defined by f(x)
= 2x² − 12x + 13 for x ≥ k, where k is a constant. It is given that f is a one-one function. State the smallest possible value of k.

The value of k is now given to be 7.

(iii) Find the range of f.

(iv) Find an expression for f⁻¹ x
and state the domain of f⁻¹ .

9. A curve has equation y = fx
and is such that f′ x
= 3x ^1/2 + 3x^−1/2 − 10.

(i) By using the substitution u = x ^1/2, or otherwise, find the values of x for which the curve y = f(x)
has stationary points.

(ii) Find f′′(x)
and hence, or otherwise, determine the nature of each stationary point.

(iii) It is given that the curve y = fx
passes through the point (4,−7)
.

10.

The diagram shows part of the curve y = (x− 2
)⁴ and the point A1, 1
on the curve. The tangent at A cuts the x-axis at B and the normal at A cuts the y-axis at C.

(i) Find the coordinates of B and C.

(ii) Find the distance AC, giving your answer in the form (√a/b), where a and b are integers.

(iii) Find the area of the shaded region.