Pure Mathematics Year 1 (AS) Unit Test 1: Algebra and Functions

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1 Pure Mathematics Year (AS) Unit Test : Algebra and Functions Simplify 6 4, giving your answer in the form p 8 q, where p and q are positive rational numbers. f( x) x ( k 8) x (8k ) a Find the discriminant of f(x) in terms of k giving your answer as a simplified quadratic. b If the equation f(x) = 0 has two equal roots, find the possible values of k. c Show that when k = 8, f(x) > 0 for all values of x. A stone is thrown from the top of a cliff. The height h, in metres, of the stone above the ground level after t seconds is modelled by the function h( t) 5.5t 4.9t a Give a physical interpretation of the meaning of the constant term 5 in the model. b Write h(t) in the form A B(t C), where A, B and C are constants to be found. c Using your answer to part b, or otherwise, find, with justification i the time taken after the stone is thrown for it to reach ground level ii the maximum height of the stone above the ground and the time after which this maximum height is reached. 4 p( x) x, q( x) x 0x 0 a Solve the equation q(x) = 0. Write your answer in the form a b, where a and b are integers to be found. b Sketch the graphs of y = p(x) and y = q(x) on the same set of axes. Label all points where the curves intersect the coordinate axes. c Use an algebraic method to find the coordinates of any point of intersection of the graphs y = p(x) and y = q(x). d Write down, using set notation, the set of values of x for which p(x) < q(x). 5 4 g( x) 5, x R x 6 Sketch the graph y = g(x). Label any asymptotes and any points of intersection with the coordinate axes. Pearson Education Ltd 07. Copying permitted for purchasing institution only. This material is not copyright free.

2 Pure Mathematics Year (AS) Unit Test : Algebra and Functions f x x x 6 The figure below shows a sketch of part of the curve with equation y = f(x). Figure a On a separate set of axes, sketch the curve with equation y = f(x) showing the location and coordinates of the images of points A, B, C and D. b On a separate set of axes, sketch the curve with equation y = f( x) showing the location and coordinates of the images of points A, B, C and D. 7 a On a coordinate grid (x and y axes running from 6 to 6), shade the region comprising all points whose coordinates satisfy the inequalities y x + 5, y + x 6 and y b Work out the area of the shaded region. Pearson Education Ltd 07. Copying permitted for purchasing institution only. This material is not copyright free.

3 Pure Mathematics Year (AS) Unit Test : Coordinate geometry in the (x, y) plane The points A and B have coordinates (k 4, ) and (, k + ) respectively, where k is a constant. Given that the gradient of AB is : a show that k = b find an equation of the line through A and B c find an equation of the perpendicular bisector of A and B, leaving your answer in the form ax by c 0 where a, b and c are integers. a Find an equation of the straight line passing through the points with coordinates (4, 7) and ( 6, ), giving your answer in the form ax by c 0,where a, b and c are integers. The line crosses the x-axis at point A and the y-axis at point B and O is the origin. b Find the area of triangle AOB. The line with equation mx y 0 touches the circle with equation x 6x y 8y 4. Find the two possible values of m, giving your answers in exact form. 4 The equations of two circles are x 0x y y and x 6x y qy 9. a Find the centre and radius of each circle, giving your answers in terms of q where necessary. (6 marks) b Given that the distance between the centres of the circles is possible values of q. 80, find the two Pearson Education Ltd 07. Copying permitted for purchasing institution only. This material is not copyright free.

4 Pure Mathematics Year (AS) Unit Test : Coordinate geometry in the (x, y) plane 5 A is the centre of circle C, with equation x 8x y 0y 0 P, Q and R are points on the circle and the lines l, l and l are tangents to the circle at these points respectively. Line l intersects line l at B and line l at D. Figure a Find the centre and radius of C. b Given that the x-coordinate of Q is 0 and that the gradient of AQ is positive, find the y-coordinate of Q, explaining your solution. c Find the equation of l, giving your answer in the form y = mx + b. d Given that APBQ is a square, find the equation of l in the form y = mx + b. l intercepts the y-axis at E. e Find the area of triangle EPA. Pearson Education Ltd 07. Copying permitted for purchasing institution only. This material is not copyright free.

5 Pure Mathematics Year (AS) Unit Test : Further Algebra f ( x) x x x 6 Use the factor theorem and division to factorise f(x) completely. (6 marks) a Expand x 8 in ascending powers of x, up to and including the term in simplifying each coefficient in the expansion. x, b Showing your working clearly, use your expansion to find, to 5 significant figures, 8 an approximation for.0 a Find the first four terms, in ascending powers of x, of the binomial expansion of px 9 Given that the coefficient of the x term in the expansion is 84 b i Find the value of p. ii Find the numerical values for the coefficients of the x and x terms. 4 a Fully expand 5 p q A fair four-sided die, numbered,, and 4, is rolled 5 times. Let p represent the probability that the number 4 is rolled on a given roll and let q represent the probability that the number 4 is not rolled on a given roll. b Using the first three terms of the binomial expansion from part a, or otherwise, find the probability that the number 4 is rolled at least times. 5 f ( x) x x px q where p and q are constants Given that f(5) = 0 and f( ) = 8 a find the values of p and q. b factorise f(x) completely. 6 Prove that, for all values of x, x 6x 8 x 7 a Prove that if x x x then x > 0 b Show, by means of a counter example, that the inequality x x x is not true for all vaues of x. Pearson Education Ltd 07. Copying permitted for purchasing institution only. This material is not copyright free.

Pure Mathematics Year (AS) Unit Test 4: Trigonometry The diagram shows ABC with AC 8x, BC 4x, ABC 0 o and ACB 5 o. Figure a Show that the exact value of x is 9 6.

6 Pure Mathematics Year (AS) Unit Test 4: Trigonometry The diagram shows ABC with AC 8x, BC 4x, ABC 0 o and ACB 5 o. Figure a Show that the exact value of x is b Find the area of ABC giving your answer to decimal places. The diagram shows the position of three boats, P, Q and R. Boat Q is 7 km from boat P on a bearing of 7. Boat R is 5 km from boat P on a bearing of 04. Figure a Find the distance between boats Q and R to decimal place. b Find the figure bearing of boat R from boat Q. Find all the solutions, in the interval 0 x 60, to the equation 8 7 cos x = 6 sin x, giving solutions to decimal place where appropriate. (6 marks) Pearson Education Ltd 07. Copying permitted for purchasing institution only. This material is not copyright free.

7 Pure Mathematics Year (AS) Unit Test 4: Trigonometry 4 a Calculate the value of tan ( 0 ). b On the same set of axes sketch the graphs of y = sin (x 60 ) and y = tan x, in the interval 80 x 80, showing the coordinates of points of intersection with the coordinate axes in exact form. c Explain how you can use the graph to identify solutions to the equations y = sin (x 60 ) + tan x = 0 in the interval 80 x 80. d Write down the number of solutions of the equation y = sin (x 60 ) + tan x = 0 in the interval 80 x Find, to decimal place, the values of θ in the interval 0 θ 80 for which 4 sin 0 o 4cos 0 o. (6 marks) 6 A teacher asks one of her students to solve the equation cos x 0 for 0 x 80. The attempt is shown below. cos x = cos x = x = cos o x =50 o x =75 o o o w or x =60 75 =95 so reject as out of range. a Identify the mistake made by the student. b Write down the correct solutions to the equation. 7 A buoy is a device which floats on the surface of the sea and moves up and down as waves pass. For a certain buoy, its height, above its position in still water, y in metres, is modelled by a sine function of the form sin 80 o y t, where t is the time in seconds. a Sketch a graph showing the height of the buoy above its still water level for 0 t 0 showing the coordinates of points of intersection with the t-axis. b Write down the number of times the buoy is 0.4 m above its still water position during the first 0 seconds. c Give one reason why this model might not be realistic. Pearson Education Ltd 07. Copying permitted for purchasing institution only. This material is not copyright free.

Circles, Mixed Exercise 6

Circles, Mixed Exercise 6 a QR is the diameter of the circle so the centre, C, is the midpoint of QR ( 5) 0 Midpoint = +, + = (, 6) C(, 6) b Radius = of diameter = of QR = of ( x x ) + ( y y ) = of ( 5