Description

Chapter: 2 -Problem: 5 >> Find the range of values of k for which 3x2 ? 4x + k = 0 has no real solutions.
Answer Preview: To determine the range of values of k for which the quadratic equation 3x 2 …

, Chapter: 11 -Problem: 8 >> A boat has a position vector of (2i + j) km and a buoy has a position vector of (6i ? 4j) km, relative to a fixed origin O.a. Find the distance of the boat from the buoy.b. Find the bearing of the boat from the buoy.The boat travels with constant velocity (8i ? 10j) km/h.c. Verify that the boat is travelling directly towards the buoyd. Find the speed of the boat.e. Work out how long it will take t
Answer Preview: a Finding the distance of the boat from the buoy The distance between two points with position vectors P and Q can be found using the formula distance …

, Chapter: 3 -Problem: 1 >> In each case: i. Draw the graphs for each pair of equations on the same axes ii. Find the coordinates of the point of intersection. Transcribed Image Text: a y = 3x - 5 y = 3-x b y = 2x - 7 y = 8-3x c y = 3x + 2 3x + y + 1 = 0
Answer Preview: a y 3x 5 1 y 3 x 2 To graph line y 3x 5 put x 0 we get y 5 put y 0 we get x 5 3 Thus draw a line tha…

, Chapter: 11 -Problem: 2 >> OP(vector) = 4i ? 3j, OQ(vector) = 3i + 2j a. Find PQ(vector) b. Find, in surd form: Transcribed Image Text: i |OP| ii |00| iii |PQ|
Answer Preview: To find PQ vector we need to subtract the coordinates of point O from the coordinates of poi…

, Chapter: 2 -Problem: 4 >> The functions p and q are given by p(x) = x2 ? 3x and q(x) = 2x ? 6, x????. Find the two values of x for which p(x) = q(x).
Answer Preview: To find the values of x for which p x q x we set the two functions equal to each other …

, Chapter: 11 -Problem: 5 >> In the triangle PQR, PQ = 2a and QR = 2b. The midpoint of PR is M. Find, in terms of a and b: Transcribed Image Text: a PR b PM ? QM ??
Answer Preview: In the triangle PQR let PQ 2a and QR 2b PR Length of side PR The length …

, Chapter: 2 -Problem: 6 >> The function g(x) = x2 + 3px + (14p ? 3), where p is an integer, has two equal roots.a. Find the value of p.b. For this value of p, solve the equation x2 + 3px + (14p ? 3) = 0.
Answer Preview: To find the value of p in the equation g x x 2 3px 14p 3 where g x has two equal roots we can set th…

, Chapter: 10 -Problem: 36 >> a. Show that the equation 2cos2 x = 4 ? 5 sin x may be written as 2sin2 x ? 5sin x + 2 = 0.b. Hence solve, for 0 ? x < 360°, the equation 2cos2 x = 4 ? 5 sin x.
Answer Preview: a To show that the equation 2cos 2 x 4 5sin x can be written as 2sin 2 x 5sin x 2 0 we ll start by u…

, Chapter: 2 -Problem: 3 >> f(x) = x2 + 3x ? 5 and g(x) = 4x + k, where k is a constant.a. Given that f(3) = g(3), find the value of k. (3 marks)b. Find the values of x for which f(x) = g(x).
Answer Preview: a To find the value of k we ll set f 3 equal to g 3 and solve for k Given f …

, Chapter: 11 -Problem: 5 >> A particle P of mass m = 0.3 kg moves under the action of a single constant force F newtons. The acceleration of P is a = (5i + 7j) m s?2.a. Find the angle between the acceleration and i. Force, mass and acceleration are related by the formula F = ma.b. Find the magnitude of F.
Answer Preview: a To find the angle between the acceleration vector a 5i 7j m s and the vector i we can use the dot …

, Chapter: 3 -Problem: 8 >> Given that x ? 3, find the set of values for which Transcribed Image Text: 5 x-3 <2.
Answer Preview: To solve the inequality 5 x 3 2 we can follow these steps Start by multiplying both sides of the …

, Chapter: 11 -Problem: 3 >> OQ = 4i ? 3j, PQ(vector) = 5i + 6j a. Find OP(vector) b. Find, in surd form: Transcribed Image Text: i |OP| ii |OQ| iii |PQ|
Answer Preview: To find OP vector we need to subtract OQ vector fro…

, Chapter: 3 -Problem: 7 >> Find the set of values of x for whicha. 2(3x ? 1) < 4 – 3x.b. 2x2 – 5x – 3 < 0.c. Both 2(3x – 1) < 4 – 3x and 2x2 – 5x – 3 < 0.
Answer Preview: a 2 3x 1 4 3x First simplify both sides of the inequality 6x 2 4 3x Next bring like terms to one sid…

, Chapter: 12 -Problem: 1 >> Find the least value of the following functions:a. f(x) = x2 ? 12x + 8 b. f(x) = x2 ? 8x ? 1 c. f(x) = 5x2 + 2x
Answer Preview: To find the least value of a quadratic function we can determine the vertex of the parabola The x co…

, Chapter: 1 -Problem: 1 >> Simplify:a. x3 ÷ x?2 b. x5 ÷ x7 c. x3/2 × x5/2d. (x2)3/2 e. (x3)5/3 f. 3x0.5 × 4x?0.5g. 9x2/3 ÷ 3 x1/6 h. 5x7/5 ÷ x2/5 i. 3x4 × 2x?5j. ?x × 3?x k. (?x)3 × (?x)4l. (3?x)2/?x
Answer Preview: Let s simplify each expression one by one a x 3 x 2 When dividing with the same base we subtract the …

, Chapter: 11 -Problem: 6 >> Given that the point A has position vector 4i ? 5j and the point B has position vector 6i + 3j,a. Find the vector AB(vector).b. Find |AB(vector)| giving your answer as a simplified surd.
Answer Preview: To find the vector AB we subtract the position vector of point A from …

, Chapter: 3 -Problem: 6 >> Solve the simultaneous equations:x + 2y = 3x2 ? 2y + 4y2 = 18
Answer Preview: To solve the simultaneous equations we ll use a method called substitution L…

, Chapter: 11 -Problem: 10 >> The vector a = pi + qj, where p and q are positive constants, is such that |a| = 15. Given that a makes an angle of 55° with i, find the values of p and q.
Answer Preview: We are given that vector a pi qj has a magnitude of a 15 and makes an angle of 55 wi…

, Chapter: 3 -Problem: 3 >> Given the simultaneous equationsx ? 2y = 13xy ? y2 = 8a. Show that 5y2 + 3y ? 8 = 0.b. Hence find the pairs (x, y) for which the simultaneous equations are satisfied.
Answer Preview: To find the pairs x y that satisfy the simultaneous equations let s solve the equations step by step …

, Chapter: 11 -Problem: 6 >> In triangle ABC,Find BC. Transcribed Image Text: AB = 4i + 3j and AC = 5i + 2j.
Answer Preview: To find BC we can use the vector subtraction …

, Chapter: 3 -Problem: 5 >> a. Given that 3x = 9y ? 1, show that x = 2y ? 2.b. Solve the simultaneous equations:x = 2y ? 2x2 = y2 + 7
Answer Preview: To solve the given equations let s start with part a and show that 3x 9y 1 implies x 2y 2 a Given 3x …

, Chapter: 11 -Problem: 6 >> OPQ is a triangle. a. Show that OS(vector) = 2a + b. b. Point T is added to the diagram such that OT(vector) = ?b. Prove that points T, P and S lie on a straight line. Transcribed Image Text: 2PR RQ and 3OR = OS OP = a and 00 = b. 18 =
Answer Preview: To prove the given statements let s analyze the given information and apply vector operations and pr…

, Chapter: 3 -Problem: 9 >> The equation kx2 – 2kx + 3 = 0, where k is a constant, has no real roots. Prove that k satisfies the inequality 0 ? k < 3.
Answer Preview: To prove that k satisfies the inequality 0 k 3 given that the quadratic equation kx 2 2kx 3 0 has no …

, Chapter: 11 -Problem: 11 >> Given that |3i ? kj| = 3?5, find the value of k.
Answer Preview: To find the value of k we need to solve the equation 3i kj 35 Let s calculate the magnitude of t…

, Chapter: 4 -Problem: 12 >> Sketch on separate axes the graphs of:f(x) = x(x ? 2)2a. y = f(x)b. y = f(x + 3)Show on each sketch the coordinates of the points where each graph crosses or meets the axes.
Answer Preview: 0 5 y 2 1 1 2 0 5 1 0 1 5 Z O …

, Chapter: 3 -Problem: 14 >> Find the values of k for which kx2 + 8x + 5 = 0 has real roots.
Answer Preview: To find the values of k for which the quadratic equation kx 2 8x …

, Chapter: 12 -Problem: 2 >> f(x) = (x + 1)(x ? 4)2a. Sketch the graph of y = f(x).b. On a separate set of axes, sketch the graph of y = f´(x).c. Show that f´(x) = (x ? 4)(3x ? 2).d. Use the derivative to determine the exact coordinates of the points where the gradient function cuts the coordinate axes.
Answer Preview: a To sketch the graph of y f x x 1 x 4 2 we can start by finding the x intercepts y intercept and analyzing the behavior around the critical points X …

, Chapter: 7 -Problem: 1 >> Prove that n2 ? n is an even number for all values of n.
Answer Preview: To prove that n 2 n is an even number for all values of n we can demonstrate this by showin…

, Chapter: 2 -Problem: 6 >> Find the value k for which the equation 5x2 ? 2x + k = 0 has exactly one solution.
Answer Preview: To find the value of k for which the equation 5x 2 2x k 0 has exactly o…

, Chapter: 11 -Problem: 9 >> In triangle ABC, P is the midpoint of AB and Q is the midpoint of AC. a. Write in terms of a and b: b. Show that PQ is parallel to BC. Transcribed Image Text: AB a and AC = b. =
Answer Preview: a Writing the vectors in terms of a and b i BC The vector BC can be expressed as the displacement fr…

, Chapter: 7 -Problem: 14 >> Divide x4 ? 16 by (x + 2).
Answer Preview: To divide x 4 16 by x 2 we can use polynomial long divi…

, Chapter: 3 -Problem: 8 >> a. On a coordinate grid, shade the region that satisfies the inequalities y < x + 4, y + 5x + 3 > 0, y > ?1 and x < 2.b. Work out the coordinates of the vertices of the shaded region.c. Which of the vertices lie within the region identified by the inequalities?d. Work out the area of the shaded region.
Answer Preview: a The shaded region is the region to the right and below of the line a right and upward of line b up…

, Chapter: 11 -Problem: 7 >> Draw a sketch for each vector and work out the exact value of its magnitude and the angle it makes with the positive x-axis to one decimal place.a. 3i + 4j b. 2i ? j c. ?5i + 2j
Answer Preview: a The vector 3i is a vector of magnitude 3 units and parallel …

, Chapter: 7 -Problem: 12 >> The equation kx2 + 5kx + 3 = 0, where k is a constant, has no real roots. Prove that k satisfies the inequality 0 ? k < 12/25.
Answer Preview: To prove that the constant k satisfies the inequality 0 k 12 25 given that the quadratic equation kx …

, Chapter: 3 -Problem: 7 >> The curve and the line given by the equationswhere k is a non-zero constant, intersect at a single point. a. Find the value of k. b. Give the coordinates of the point of intersection of the line and the curve. Transcribed Image Text: kx² - xy + (k+1)x= 1 k 2x+y=1
Answer Preview: To find the value of k and the coordinates of the point of intersection of the line and the curve we …

, Chapter: 11 -Problem: 2 >> A small boat S, drifting in the sea, is modelled as a particle moving in a straight line at constant speed. When first sighted at 09:00, S is at a point with position vector (?2i ? 4j) km relative to a fixed origin O, where i and j are unit vectors due east and due north respectively. At 09:40, S is at the point with position vector (4i ? 6j) km.a. Calculate the bearing on which S is drifting.b. F
Answer Preview: To calculate the bearing on which the boat S is drifting we need to find the angle between the boat …

, Chapter: 4 -Problem: 12 >> a. Sketch the graph of y = x(x + 1)(x + 3)2.b. Find the possible values of b such that the point (2, 0) lies on the curve with equation y = (x + b)(x + b + 1)(x + b + 3)2.
Answer Preview: A We have y x x 1 x 3 2 The graph for the equation is shown below in red B We have the foll…

, Chapter: 3 -Problem: 5 >> a. By eliminating y from the equationsy = 2 ? 4x3x2 + xy + 11 = 0show that x2 ? 2x – 11 = 0.b. Hence, or otherwise, solve the simultaneous equationsy = 2 ? 4x3x2 + xy + 11 = 0giving your answers in the form a ± b?3, where a and b are integers.
Answer Preview: a To eliminate y from the equations we substitute the value of y from the first equation into the se…

, Chapter: 12 -Problem: 3 >> Find the y-coordinate and the value of the gradient at the point P with x-coordinate 1 on the curve with equation y = 3 + 2x ? x2.
Answer Preview: To find the y coordinate and the value of the gradient at the point P wit…

, Chapter: 7 -Problem: 11 >> a. Show that (x ? 2) is a factor of f(x) = x3 + x2 ? 5x ? 2.b. Hence, or otherwise, find the exact solutions of the equation f(x) = 0.
Answer Preview: To show that x 2 is a factor of f x x 3 x 2 5x 2 we need to prove that when x 2 f x becomes zero …

,

,

, Chapter: 7 -Problem: 4 >> a. Show that (x ? 3) is a factor of 2x3 ? 2x2 ? 17x + 15.b. Hence express 2x3 ? 2x2 ? 17x + 15 in the form (x ? 3)(Ax2 + Bx + C), where the values A, B and C are to be found.
Answer Preview: To show that x 3 is a factor of the polynomial 2x 3 2x 2 17x 15 we can use polynomial long …

, Chapter: 3 -Problem: 13 >> Find the set of values of x that satisfy Transcribed Image Text: 6/00 +1 < x2 6 0*x+x
Answer Preview: To find the s…

, Chapter: 12 -Problem: 4 >> f(x) = px3 ? 3px2 + x2 ? 4 When x = 2, f"(x) = ?1. Find the value of p.
Answer Preview: To find the value of p we need to differentiate the …

, Chapter: 6 -Problem: 24 >> The points A(?4, 0), B(4, 8) and C(6, 0) lie on the circumference of circle C.Find the equation of the circle.
Answer Preview: To find the equation of the circle passing through the points A 4 0 B 4 8 and C 6 0 we can use the general form of the equation for a circle x h 2 y k …

, Chapter: 3 -Problem: 1 >> Find the set of values of x for which:a. x2 ? 11x + 24 < 0 b. 12 ? x ? x2 > 0 c. x2 ? 3x ? 10 > 0d. x2 + 7x + 12 ? 0 e. 7 + 13x ? 2x2 > 0 f. 10 + x ? 2x2 < 0g. 4x2 ? 8x + 3 ? 0 h. ?2 + 7x ? 3x2 < 0 i. x2 ? 9 < 0j. 6x2 + 11x ? 10 > 0 k. x2 ? 5x > 0 l. 2x2 + 3x ? 0
Answer Preview: Let s solve each inequality one by one a x 2 11x 24 0 Factoring the quadratic expression x 3 x 8 0 The critical points are x 3 and x 8 Testing the int…

, Chapter: 11 -Problem: 3 >> Find the speed and the distance travelled by a particle moving in a straight line with:a. velocity (?3i + 4j) m s?1 for 15 seconds b. velocity (2i + 5j) m s?1 for 3 secondsc. velocity (5i ? 2j) km h?1 for 3 hours d. velocity (12i ? 5j) km h?1 for 30 minutes.
Answer Preview: To find the speed and distance travelled by a particle moving in a straight line we use the follow…

, Chapter: 8 -Problem: 5 >> Work out the 5th number on the 12th row from Pascal’s triangle.
Answer Preview: To find the fifth number on the twelfth row of Pascal's triang…

, Chapter: 7 -Problem: 1 >> Prove that when n is an integer and 1 ? n ? 6, then m = n + 2 is not divisible by 10.
Answer Preview: To prove that when n is an integer and 1 n 6 the expression m n 2 is not divis…

, Chapter: 3 -Problem: 7 >> Determine the number of points of intersection for these pairs of simultaneous equations. Transcribed Image Text: a y = 6x² + 3x - 7 y = 2x + 8 by=4x² 18x + 40 y = 10x -9 - cy=3x² - 2x +4 7x + y + 3 = 0
Answer Preview: To determine the number of points of intersection for each pair of simultaneous equations we need to solve the systems of equations and see how many s…

, Chapter: 11 -Problem: 7 >> OABC is a parallelogram. The point P divides OB in the ratio 5:3. Find, in terms of a and b: Transcribed Image Text: ?? a and OC = b.
Answer Preview: a To find OB we can use the fact that OABC is a parallel…

, Chapter: 12 -Problem: 25 >> The volume, V cm3, of a tin of radius r cm is given by the formula V = ?(40r ? r2 ? r3).Find the positive value of r for which dV/dr = 0, and find the value of V which corresponds to this value of r.
Answer Preview: To find the positive value of r for which dV/dr = 0, we need to take the derivative of V with respec…

, Chapter: 6 -Problem: 19 >> The circle C has equation x2 + 6x + y2 ? 2y = 7. The lines l1 and l2 are tangents to the circle. They intersect at the point R(0, 6). a. Find the equations of lines l1 and l2, giving your answers in the form y = mx + b. b. Find the points of intersection, P and Q, of the tangents and the circle. c. Find the area of quadrilateral APRQ.
Answer Preview: a To find the equations of lines l1 and l2 we can use the fact that they are tangent to the circle Tangent lines to a circle are perpendicular to the …

, Chapter: 3 -Problem: 4 >> 3x + ky = 8x ? 2ky = 5are simultaneous equations where k is a constant.a. Show that x = 3.b. Given that y = 1/2 determine the value of k.
Answer Preview: To solve the given simultaneous equations Equation 1 3x ky 8 1 Equation 2 x …

, Chapter: 11 -Problem: 8 >> The resultant of the vectors a = 4i ? 3j and b = 2 pi ? pj is parallel to the vector c = 2i ? 3j. Find:a. The value of p.b. The resultant of vectors a and b.
Answer Preview: a Finding the value of p Since the resultant is parallel to vector c the d…

, Chapter: 13 -Problem: 1 >> Find the following integrals: a. ?x3 dx b. ?x7 dx c. ?3x?4 dx d. ?5x2 dx
Answer Preview: Let's find the integrals: a. x^3 dx: Using the power rule of integration, we …

, Chapter: 7 -Problem: 9 >> a. Prove that for any positive numbers p and q: b. Show, by means of a counter-example, that this inequality does not hold when p and q are both negative. Transcribed Image Text: p+q>?4pq
Answer Preview: a To prove the inequality for any positive numbers p and q we can start by squaring both sides of th…

, Chapter: 3 -Problem: 3 >> a. On the same axes sketch the curve with equation x2 + y = 9 and the line with equation 2x + y = 6.b. Find the coordinates of the points of intersection.c. Verify your solutions by substitution.
Answer Preview: a The given curve is x 2 y 9 which is a parabola To draw a parabola we first need to find the vertex …

, Chapter: 12 -Problem: 3 >> Given that r = 12/t, find the value of dr/dt when t = 3.
Answer Preview: To find the value of dr dt when t 3 we need to differentiate the expressio…

, Chapter: 13 -Problem: 1 >> Find:a. ?(x + 1)(2x ? 5)dx b. ?(x1/3 + x?1/3) dx
Answer Preview: a. (x + 1)(2x 5)dx: To solve this integral, we can use the distributive property and then apply the …

, Chapter: 4 -Problem: 11 >> a. Factorise completely x3 ? 6x2 + 9x.b. Sketch the curve of y = x3 ? 6x2 + 9x showing clearly the coordinates of the points where the curve touches or crosses the axes.c. The point with coordinates (?4, 0) lies on the curve with equation y = (x ? k)3 ? 6(x ? k)2 + 9(x ? k) where k is a constant. Find the two possible values of k.
Answer Preview: a To factorize the expression x 3 6x 2 9x completely we can first factor out the common factor of x …

, Chapter: 3 -Problem: 12 >> Find the set of values of x for which: a. 6x ? 7 b. 2x2 ? 11x + 5 c. d. Both 6x ? 7 2 ? 11x + 5 Transcribed Image Text: 5 V 20 X
Answer Preview: Let s solve each part of the problem separately a 6x 7 To find the set of values of x we need to sol…

, Chapter: 12 -Problem: 1 >> Find the values of x for which f(x) is an increasing function, given that f(x) equals:a. 3x2 + 8x + 2 b. 4x ? 3x2 c. 5 ? 8x ? 2x2 d. 2x3 ? 15x2 + 36xe. 3 + 3x ? 3x2 + x3 f. 5x3 + 12x g. x4 + 2x2 h. x4 ? 8x3
Answer Preview: To determine the values of x for which each function is increasing we need to find the intervals where the first derivative of the function is positiv…

, Chapter: 12 -Problem: 20 >> f(x) = 3x4 ? 8x3 ? 6x2 + 24x + 20a. Find the coordinates of the stationary points of f(x), and determine the nature of each of them.b. Sketch the graph of y = f(x).
Answer Preview: To find the coordinates of the stationary points of f(x) = 3x^4 - 8x^3 - 6x^2 + 24x + 20, we need to …

, Chapter: 7 -Problem: 2 >> Prove that Transcribed Image Text: X 1 + ?2 ==x?2 - x.
Answer Preview: To prove that x 2 is equivalent to x2 x we need to show that both expressions simplify to …

, Chapter: 3 -Problem: 1 >> 2kx ? y = 44kx + 3y = ?2are two simultaneous equations, where k is a constant.a. Show that y = ?2.b. Find an expression for x in terms of the constant k.
Answer Preview: To solve the given simultaneous equations Equation 1 2kx y 4 1 Equation 2 4k…

, Chapter: 12 -Problem: 1 >> Find dy/dx when y equals:a. 2x2 ? 6x + 3 b. 1/2x2 + 12x c. 4x2 ? 6d. 8x2 + 7x + 12 e. 5 + 4x ? 5x2
Answer Preview: To find the derivative of a function we can apply the power rule and sum difference rule of differen…

, Chapter: 12 -Problem: 19 >> A curve has equation y = x3 ? 6x2 + 9x. Find the coordinates of its local maximum.
Answer Preview: To find the coordinates of the local maximum of the curve with the equation y = x^3 6x^2 + 9x, we ne…

, Chapter: 7 -Problem: 13 >> Divide x3 ? 1 by (x ? 1).
Answer Preview: To divide the polynomial x 3 1 by x 1 we can use polyno…

, Chapter: 1 -Problem: 3 >> Simplify these fractions: Transcribed Image Text: d 6x4 + 10x6 2x 8r3+5x 2x b 3x5-x7 X 7x7 +5x² 5x ? f 2x4-4x2 4x 9x5 - 5x³ 3.x
Answer Preview: When possible we ll divide the numerator by the denominator to simplify the sup…

, Chapter: 12 -Problem: 2 >> f(x) = x2 a. Show that b. Hence deduce that f'(x) = 2x. Transcribed Image Text: f'(x) = lim (2x + h). h?0
Answer Preview: To show that f x lim 2x h as h approaches 0 we ll start by find…

, Chapter: 8 -Problem: 4 >> If x is so small that terms of x3 and higher can be ignored, and (2 ? x)(3 + x)4 ? a + bx + cx2 find the values of the constants a, b and c.
Answer Preview: To find the values of the constants a, b, and c, we need to expand the expression (2 x)(3 + x)^4 and …

, Chapter: 7 -Problem: 7 >> Prove that for all real values of x(x + 6)2 ? 2x + 11
Answer Preview: To prove that the inequality x 6 2 2x 11 holds for all real values of x we will use algebraic ma…

, Chapter: 2 -Problem: 4 >> By completing the square, show that the solutions to the equation x2 + 2bx + c = 0 are given by the formula Transcribed Image Text: x=-b± ?b² - c.
Answer Preview: To complete the square and find the solutions to the equation x 2 2bx c 0 w…

, Chapter: 11 -Problem: 5 >> The position vectors of 3 vertices of a parallelogram are Find the possible position vectors of the fourth vertex. Transcribed Image Text: (2), (3) and (8).
Answer Preview: Let D have coordinates W Z A 4 2 B 3 5 C 8 6 D W Z The diagonals AD and BC of the parallelo…

, Chapter: 12 -Problem: 7 >> f(x) = ax2, where a is a constant. Prove, from first principles, that f'(x) = 2ax.
Answer Preview: To prove that f'(x) = 2ax for the function f(x) = ax^2 using first principles (al…

,

, Chapter: 2 -Problem: 4 >> A football stadium has 25 000 seats. The football club know from past experience that they will sell only 10 000 tickets if each ticket costs £30. They also expect to sell 1000 more tickets every time the price goes down by £1.a. The number of tickets sold t can be modelled by the linear equation t = M ? 1000p, where £p is the price of each ticket and M is a constant. Find the value of M. The tota
Answer Preview: a The number of tickets sold t can be modeled by the linear equation t M 1000p where p is the price …

, Chapter: 11 -Problem: 9 >> For each of the following vectors, findi. A unit vector in the same direction ii. The angle the vector makes with ia. a = 8i + 15j b. b = 24i ? 7j c. c = ?9i + 40j d. d = 3i ? 2j
Answer Preview: To find the unit vector in the same direction as each given vector and the angle each vector makes w…

, Chapter: 12 -Problem: 6 >> Find the gradients of the curve y = 2x2 at the points C and D where the curve meets the line y = x + 3.
Answer Preview: To find the gradients of the curve y = 2x^2 at the points where it meets the line y = x + 3, we need …

, Chapter: 7 -Problem: 1 >> Write each polynomial in the form (x ± p)(ax2 + bx + c) by dividing: a. x3 + 6x2 + 8x + 3 by (x + 1) b. x3 + 10x2 + 25x + 4 by (x + 4)c. x3 ? x2 + x + 14 by (x + 2) d. x3 + x2 ? 7x ? 15 by (x ? 3)e. x3 ? 8x2 + 13x + 10 by (x ? 5) f. x3 ? 5x2 ? 6x ? 56 by (x ? 7)
Answer Preview: a To divide x 3 6x 2 8x 3 by x 1 we perform long division as follows x 2 5x 3 x 1 x 3 6x 2 8x 3 x 3 …

, Chapter: 9 -Problem: 8 >> Show that cos P = ?1/4 Transcribed Image Text: 2 cm 4 cm P 3 cm R
Answer Preview: To solve this problem we can use the Law of Cosines which relates the lengths …

, Chapter: 3 -Problem: 4 >> a. By eliminating y from the equationsx + y = 2x2 + xy ? y2 = ?1show that x2 ? 6x + 3 = 0.b. Hence, or otherwise solve the simultaneous equationsx + y = 2x2 + xy ? y2 = ?1giving x and y in the form a ± b?6, where a and b are integers.
Answer Preview: To eliminate y from the given equations we can solve the first equation for y and substitute it into …

, Chapter: 12 -Problem: 2 >> Find the gradient of the curve with equation:a. y = 3x2 at the point (2, 12) b. y = x2 + 4x at the point (1, 5)c. y = 2x2 ? x ? 1 at the point (2, 5) d. y = 1/2x2 + 3/2x at the point (1, 2)e. y = 3 ? x2 at the point (1, 2) f. y = 4 ? 2x2 at the point (?1, 2)
Answer Preview: To find the gradient or slope of a curve at a given point we need to find the derivative of the func…

, Chapter: 12 -Problem: 4 >> Given that 2y2 ? x3 = 0 and y > 0, find dy/dx.
Answer Preview: To find dy dx we can differentiate the given equation wit…

, Chapter: 11 -Problem: 1 >> These vectors are drawn on a grid of unit squares. Express the vectors v1, v2, v3, v4, v5 and v6 in: (i) i, j notation  (ii) Column vector form Transcribed Image Text: 5, V3 V6
Answer Preview: (i) A vector is represented by a magnitude and direction. In (i, j) notation, the i component is …

, Chapter: 4 -Problem: 21 >> a. Solve the simultaneous equations: b. Hence, or otherwise, find the set of values of x for which: 2x2 ? 3x ? 16 > 5 ? 2x. Transcribed Image Text: y + 2x = 5 2x² 3x - y = 16.
Answer Preview: a To solve the simultaneous equations Equation 1 y 2x 5 Equation 2 2x 2 3x y 16 We can solve this sy…

, Chapter: 8 -Problem: 15 >> The coefficient of x2 in the binomial expansion of (2 + kx)8, where k is a positive constant, is 2800.a. Use algebra to calculate the value of k.b. Use your value of k to find the coefficient of x3 in the expansion.
Answer Preview: To find the coefficient of x 2 in the binomial expansion of 2 kx 8 we can use the …

, Chapter: 3 -Problem: 17 >> Find the set of values of x for which the curve with equation y = 2x2 + 3x ? 15 is below the line with equation y = 8 + 2x.
Answer Preview: To find the set of values of x for which the curve with equation y 2x 2 3x 15 is below the line with …

, Chapter: 11 -Problem: 6 >> Given that find: a. a + b + c  b. a ? 2b + c  c. 2a + 2b ? 3c Transcribed Image Text: = (2), b = (12) and c = ( a = -5) -3)
Answer Preview: To perform the given operations on matrices a b and c we need to add or subtr…

, Chapter: 7 -Problem: 19 >> f(x) = 12x3 + 5x2 + 2x – 1a. Show that (4x – 1) is a factor of f(x) and write f(x) in the form (4x – 1)(ax2 + bx + c).b. Hence, show that the equation 12x3 + 5x2 + 2x – 1 = 0 has exactly 1 real solution.
Answer Preview: a To show that 4x 1 is a factor of f x we need to prove that f x 4x 1 ax 2 bx c for some values …

, Chapter: 10 -Problem: 23 >> The coefficient of x in the binomial expansion of (2 ? 4x)q, where q is a positive integer, is ?32q. Find the value of q.
Answer Preview: To find the value of q, we can use the formula for the coefficient of x in the binomial expansion. I…

, Chapter: 7 -Problem: 3 >> Simplify (2x3 + 3x + 5)/(x + 1)
Answer Preview: To simplify the expression 2x 3 3x 5 x 1 we can use polynomial l…

, Chapter: 9 -Problem: 10 >> A windmill has four identical triangular sails made from wood. If each triangle has sides of length 12 m, 15 m and 20 m, work out the total area of wood needed.
Answer Preview: To find the total area of wood needed for the four identical triangular sails we first calcu…

, Chapter: 3 -Problem: 2 >> Solve the simultaneous equations:x + 2y = 3x2 ? 4y2 = ?33
Answer Preview: To solve the simultaneous equations x 2y 3 x 2 4y 2 33 We …

, Chapter: 10 -Problem: 37 >> Find all of the solutions in the interval 0 ? x < 360° of 2 tan2 x ? 4 = 5 tan x giving each solution, in degrees, to one decimal place.
Answer Preview: To find all the solutions in the interval 0 x 360 for the equation 2 tan 2 x 4 5 tan x we can use a …

, Chapter: 7 -Problem: 14 >> Prove that A(3, 1), B(1, 2) and C (2, 4) are the vertices of a right-angled triangle.
Answer Preview: To prove that triangle ABC with vertices A 3 1 B 1 2 and C 2 4 is a right angled …

, Chapter: 12 -Problem: 6 >> The function f is defined by f(x) = x + 9/x, x ? R, x ? 0.a. Find f'(x).b. Solve f'(x) = 0.
Answer Preview: To find the derivative of the function f(x) = x + 9/x, we'll use the rules of differentiation. Let's …

, Chapter: 7 -Problem: 5 >> Prove that Transcribed Image Text: 2 ? ( 7 ) ? ? ( ² + x) = xq + ? x - zx
Answer Preview: To prove the given equation we ll start by expanding the right hand side RHS of th…

, Chapter: 10 -Problem: 2 >> Without using your calculator, work out the values of:a. cos 270° b. sin 225° c. cos 180° d. tan 240° e. tan 135°
Answer Preview: Certainly Let s work through each of these trigonometric values step by step a cos 270 We know that …

, Chapter: 3 -Problem: 8 >> A person throws a ball in a sports hall. The height of the ball, h m, can be modelled in relation to the horizontal distance from the point it was thrown from by the quadratic equation: The hall has a sloping ceiling which can be modelled with equation Determine whether the model predicts that the ball will hit the ceiling.
Answer Preview: To determine whether the ball will hit the ceiling we need to compare the heights predicted by the q…

, Chapter: 12 -Problem: 1 >> The diagram shows the curve with equation y = x2 ? 2x. a. Copy and complete this table showing estimates for the gradient of the curve. b. Write a hypothesis about the gradient of the curve at the point where x = p. c. Test your hypothesis by estimating the gradient of the graph at the point (1.5, ?0.75). Transcribed Ima
Answer Preview: a To estimate the gradient of the curve y x 2 2x we can calculate the differen…

, Chapter: 13 -Problem: 12 >> a. Find ?(x1/2 ? 4)(x?1/2 ? 1)dx. b. Use your answer to part a to evaluate giving your answer as an exact fraction. Transcribed Image Text: (x-4)(x-²-1)dx
Answer Preview: a To find x 1 2 4 x 1 2 1 dx we ll expand the expression x 1 2 …

, Chapter: 13 -Problem: 1 >> Find an expression for y when dy/dx is the following: Transcribed Image Text: a x5 g -21-6 m -3x- b 10x4 h x- n -5 c -x-2 i 5x-12/ o 6x d -4x-³ j 6x P 2x-0.4 e x? k 36x¹1 f 4x² 1-14x-8
Answer Preview: a) To find an expression for y when dy/dx is given as x^5, we can integrate the given expression with respect to x. Integrating x^5 with respect to x gives us: x^5 dx = (1/6) x^6 + C where C is the co…

, Chapter: 4 -Problem: 13 >> Given that a. Sketch the graph of y = f(x) – 2 and state the equations of the asymptotes. b. Find the coordinates of the point where the curve y = f(x) – 2 cuts a coordinate axis. c. Sketch the graph of y = f(x + 3). d. State the equations of the asymptotes and the coordinates of the point where the curve cuts a coordinate axis.
Answer Preview: a To sketch the graph of y f x 2 where f x 1 x x 0 we can follow these steps 1 Plot the vertical asymptote The original function f x 1 x has a vertica…

, Chapter: 9 -Problem: 11 >> Triangle ABC is such that BC = 5?2 cm, ?ABC = 30° and ?BAC = ?, where sin ? = ?5/8 Work out the length of AC, giving your answer in the form a ?b , where a and b are integers.
Answer Preview: To find the length of AC we can use the Law of Sines which states th…

, Chapter: 2 -Problem: 8 >> The function f is defined as f(x) = 22x ? 20(2x) + 64, x????.a. Write f(x) in the form (2x ? a)(2x ? b), where a and b are real constants.b. Hence find the two roots of f(x).
Answer Preview: a To write f x in the form 2x a 2x b we need to expand the expression and fin…

, Chapter: 12 -Problem: 1 >> Find d?/dt where ? = t2 ? 3t.
Answer Preview: To find d dt we need to take the derivative of with respect to t …

, Chapter: 7 -Problem: 23 >> Prove that the distance between opposite edges of a regular hexagon of side length ?3 is a rational value.
Answer Preview: To prove that the distance between opposite edges of a regular hexagon of side length 3 is a rationa…

, Chapter: 12 -Problem: 7 >> Given that find dy/dx. Transcribed Image Text: y = 3?x - 4 x > 0,
Answer Preview: To find the derivative of the function y = 3(x) - 4/((x)), we can use the rule…

, Chapter: 4 -Problem: 10 >> a. Sketch the graph of y = x2(x – 3)(x + 2), marking clearly the points of intersection with the axes.b. Hence sketch y = (x + 2)2(x – 1)(x + 4).
Answer Preview: a b 20 …

, Chapter: 10 -Problem: 8 >> a. Sketch the graphs of y = 2 sin x and y = cos x on the same set of axes (0 ? x ? 360°).b. Write down how many solutions there are in the given range for the equation 2 sin x = cos x.c. Solve the equation 2 sin x = cos x algebraically, giving your answers in exact form.
Answer Preview: a To sketch the graphs of y 2 sin x and y cos x on the same set of axes we ll first plot the key poi…

, Chapter: 3 -Problem: 3 >> Find the set of values of x for which the curve with equation y = f(x) is below the line with equation y = g(x). Transcribed Image Text: a f(x) = 3x² - 2x - 1 g(x) = x + 5 2 d f(x) = , x #0 g(x) = 1 b f(x) = 2x² - 4x + 1 g(x) = 3x - 2 e f(x) = 3 4 .x2 g(x) = -1 X X #0 e f(x) = 5x2x² - 4 g(x) = -2x
Answer Preview: a f x 3x 2x 1 g x x 5 For part of f x to be below g x g x f x 0 x 5 3x 2x 1 0 …

, Chapter: 11 -Problem: 4 >> Find the angle that each of these vectors makes with the positive x-axis.a. 3i + 4j b. 6i ? 8j c. 5i + 12j d. 2i + 4j
Answer Preview: To find the angle that each of the given vectors makes with the positive …

, Chapter: 13 -Problem: 9 >> The graph shows part of the curve C with equation y = x2(2 ? x). The region R, shown shaded, is bounded by C and the x-axis. Use calculus to find the exact area of R. Transcribed Image Text: R y = x²(2-x) C 2 X
Answer Preview: To find the exact area of the shaded region R bounded by the curve C and the x axis we ne…

, Chapter: 12 -Problem: 22 >> The diagram shows the part of the curve with equation y = 5 ? 1/2 x2 for which y > 0. The point P(x, y) lies on the curve and O is the origin. a. Show that OP2 = 1/4 x4 ? 4x2 + 25. Taking f(x) = 1/4 x4 ? 4x2 + 25: b. Find the values of x for which f'(x) = 0. c. Hence, or otherwise, find the minimum distance from O to the curve, showing that your answer is a minimum.
Answer Preview: a. To show that OP^2 = (1/4)x^4 - 4x^2 + 25, we need to find the distance squared between the point P(x, y) on the curve and the origin O(0, 0). The d…

, Chapter: 7 -Problem: 2 >> Prove that every odd integer between 2 and 26 is either prime or the product of two primes.
Answer Preview: To prove that every odd integer between 2 and 26 is either prime or the product of two primes we can …

, Chapter: 10 -Problem: 6 >> Given that ? is an acute angle, express in terms of cos ? or tan ?:a. cos (180° ? ?) b. cos (180° + ?) c. cos (??) d. cos (?(180° ? ?))e. cos (? ? 360°) f. cos (? ? 540°) g. tan (??) h. tan (180° ? ?)i. tan (180° + ?) j. tan (?180° + ?) k. tan (540° ? ?) l. tan (? ? 360°)
Answer Preview: To express the given expressions in terms of cos or tan we can use the trigonome…

, Chapter: 2 -Problem: 5 >> Solve the equation Transcribed Image Text: 5x+3 =v3x+7.
Answer Preview: To solve the equation 5x 3 3x 7 we can follow these step…

, Chapter: 12 -Problem: 4 >> a. Given that the function f(x) = x2 + px is increasing on the interval [?1, 1], find one possible value for p.b. State with justification whether this is the only possible value for p.
Answer Preview: a To determine a possible value for p we need to examine the behavior of the funct…

, Chapter: 13 -Problem: 6 >> Evaluate giving your answer in the form a?+?b?3 , where a and b are integers. Transcribed Image Text: 12 2 X dx,
Answer Preview: To evaluate the integral 4 12 2 x dx we can rewrite it as 4 12 2x 1 2 dx U…

, Chapter: 12 -Problem: 6 >> Find the maximum value and hence the range of values for the function f(x) = 27 ? 2x4.
Answer Preview: To find the maximum value of the function f(x) = 27 - 2x^4, we need to determine the critical points …

, Chapter: 4 -Problem: 10 >> The curve C1 has equation y = ?a/x2 where a is a positive constant. The curve C2 has the equation y = x2 (3x + b) where b is a positive constant.a. Sketch C1 and C2 on the same set of axes, showing clearly the coordinates of any point where the curves touch or cross the axes.b. Using your sketch state, giving reasons, the number of solutions to the equation x4(3x + b) + a = 0.
Answer Preview: a x y a x 2 7 0 071428 6 0 09722 5 0 14 4 0 21875 2 0 875 1 5 1 55 1 3 5 0 5 14 0 0 5 …

, Chapter: 10 -Problem: 3 >> Solve the following equations in the interval given:a. 3 sin 3? = 2 cos 3?, 0 ? ? ? 180°b. 4 sin (? + 45°) = 5 cos (? + 45°), 0 ? ? ? 450°c. 2 sin 2x – 7 cos 2x = 0, 0 ? x ? 180°d. ?3 sin (x – 60°) + cos(x – 60°) = 0, –180° ? x ? 180°
Answer Preview: a To solve 3 sin 3 2 cos 3 in the interval 0 180 we can rearrange the equation using the trigonometr…

, Chapter: 3 -Problem: 10 >> Find the set of values of x for which (x ? 1)(x ? 4) < 2(x ? 4).
Answer Preview: To find the set of values of x for which x 1 x 4 2 x 4 we can start by expanding an…

, Chapter: 11 -Problem: 5 >> In triangle ABC the position vectors of the vertices A, B and C are Find: a. |AB(vector)  b. |AC(vector)|  c. |BC(vector)| d. The size of ?BAC, ?ABC and ?ACB to the nearest degree. Transcribed Image Text: (3), (3) and (7).
Answer Preview: To find the required values for triangle ABC let s use the given position vectors of the vertices a …

, Chapter: 12 -Problem: 21 >> The diagram shows part of the curve with equation y = f(x), where: The curve cuts the x-axis at the points A and C. The point B is the maximum point of the curve. a. Find f'(x). b. Use your answer to part a to calculate the coordinates of B. Transcribed Image Text: f(x) = 200- 250 X - x², x>0
Answer Preview: To find the derivative of the function f(x), we can use the power rule and the quotient rule. Let's …

, Chapter: 7 -Problem: 11 >> Given that (x + 1) and (x ? 2) are factors of cx3 + dx2 – 9x – 10, find the values of c and d.
Answer Preview: To find the values of c and d we can use the factor theorem …

, Chapter: 9 -Problem: 16 >> A farmer has a field in the shape of a quadrilateral as shown. The angle between fences AB and AD is 74°. Find the angle between fences BC and CD. Transcribed Image Text: 75m A D 120m 135 m B 60m
Answer Preview: To find the angle between fences BC and CD we can us…

, Chapter: 3 -Problem: 2 >> a. Use graph paper to draw accurately the graphs of 2y = 2x + 11 and y = 2x2 ? 3x – 5 on the same axes.b. Use your graph to find the coordinates of the points of intersection.c. Verify your solutions by substitution.
Answer Preview: a The graphs of the two functions plotted on the same …

, Chapter: 4 -Problem: 13 >> Given that f(x) = x2 ? 6x + 18, x ? 0,a. Express f(x) in the form (x ? a)2 + b, where a and b are integers. The curve C with equation y = f(x), x ? 0, meets the y-axis at P and has a minimum point at Q.b. Sketch the graph of C, showing the coordinates of P and Q.  The line y = 41 meets C at the point R.c. Find the x-coordinate of R, giving your answer in the form p + q?2,where p and q are integers
Answer Preview: a f x x 6x 18 x 0 f x x 2 3 x 9 9 x 0 x 2 3 x 3 9 x 0 x 3 9 x 0 Therefore expressing f x in the form …

,

, Chapter: 3 -Problem: 6 >> The sketch shows the graphs of the straight lines with equations: y = x + 1, y = 7 – x and x = 1. a. Work out the coordinates of the points of intersection of the functions. b. Write down the set of inequalities that represent the shaded region shown in the sketch. Transcribed Image Text: ??????? 5
Answer Preview: a The three points of intersections of the functions are A B and C as seen in the figure below …

, Chapter: 7 -Problem: 8 >> f(x) = 2x2 + px + q. Given that f(?3) = 0, and f(4) = 21:a. Find the value of p and qb. Factorise f(x).
Answer Preview: To find the values of p and q in the quadratic function f x 2x 2 px q we will use the given inf…

, Chapter: 9 -Problem: 8 >> Town B is 6 km, on a bearing of 020°, from town A. Town C is located on a bearing of 055° from town A and on a bearing of 120° from town B. Work out the distance of town C from:a. Town A b. Town B
Answer Preview: To determine the distances between the towns we can use trigonometry and the given bearings a Distance from Town A to Town C To find the distance from Town A to Town C we need to break it down into tw…

, Chapter: 2 -Problem: 4 >> Solve the following equations, giving your answers correct to 3 significant figures:a. k2 + 11k ? 1 = 0 b. 2t2 ? 5t + 1 = 0 c. 10 ? x ? x2 = 7 d. (3x ? 1)2 = 3 ? x2
Answer Preview: Let s solve the given equations a k 2 11k 1 0 To solve this quadratic equation we can use the quadra…

, Chapter: 6 -Problem: 15 >> The line segment AB is a chord of a circle centre (2, ?1), where A and B are (3, 7) and (?5, 3) respectively. AC is a diameter of the circle. Find the area of ?ABC.
Answer Preview: To find the area of triangle ABC we need to first find the length of the base AB and the height the …

, Chapter: 9 -Problem: 10 >> In ?ABC, AB = (2 ? x) cm, BC = (x + 1) cm and ?ABC = 120°.a. Show that AC2 = x2 ? x + 7.b. Find the value of x for which AC has a minimum value.
Answer Preview: To solve this problem we ll use the Law of Cosines and the concept of minimizing a quadratic e…

, Chapter: 3 -Problem: 1 >> The diagram shows a sketch of L1 and L2. L1 has equation 2y + 3x = 6. L2 has the equation x ? y = 5. a. Find the coordinates of P, the point of intersection. b. Hence write down the solution to the inequality 2y + 3x > x – y. Transcribed Image Text: 0 L??:x=y=5 X L?: 2y + 3x = 6
Answer Preview: To find the coordinates of the point of intersection P between L1 and L2 we need to solve the system …

, Chapter: 12 -Problem: 7 >> f(x) = x2 ? 2x ? 8a. Sketch the graph of y = f(x).b. On the same set of axes, sketch the graph of y = f'(x).c. Explain why the x-coordinate of the turning point of y = f(x) is the same as the x-coordinate of the point where the graph of y = f'(x) crosses the x-axis.
Answer Preview: a. To sketch the graph of y = f(x) = x^2 - 2x - 8, we can start by finding the x-intercepts, y-inter…

, Chapter: 7 -Problem: 6 >> Divide:a. x3 + x2 ? 36 by (x ? 3) b. 2x3 + 9x2 + 25 by (x + 5)c. ?3x3 + 11x2 ? 20 by (x ? 2)
Answer Preview: To divide polynomials we can use the long division method Let s go thr…

, Chapter: 10 -Problem: 8 >> Given that tan ? = ? ?3 and that ? is reflex, find the exact value of: a. sin ? b. cos ?
Answer Preview: Given that tan 3 and that is a reflex angle we can determine the exact values of sin and cos using t…

, Chapter: 2 -Problem: 6 >> The function f is defined as f(x) = x2 ? 2x + 2, x????.a. Write f(x) in the form (x + p)2 + q, where p and q are constants to be found.b. Hence, or otherwise, explain why f(x) > 0 for all values of x, and find the minimum value of f(x).
Answer Preview: a To write f x in the form x p 2 q we need to complete the square Le…

, Chapter: 12 -Problem: 3 >> Find the gradient of the curve with equation y = 3?x at the point where:a. x = 4 b. x = 9c. x = 1/4 d. x = 9/16
Answer Preview: To find the gradient of the curve with equation y 3x we need t…

, Chapter: 12 -Problem: 7 >> f(x) = (2 ? x)9a. Find the first 3 terms, in ascending powers of x, of the binomial expansion of f(x), giving each term in its simplest form.b. If x is small, so that x2 and higher powers can be ignored, show that f'(x) ? 9216x ? 2304.
Answer Preview: To find the first three terms of the binomial …

, Chapter: 7 -Problem: 10 >> Use completing the square to prove that ?n2 ? 2n ? 3 is negative for all values of n.
Answer Preview: To prove that the expression n 2 2n 3 is negative for all values of n we can use c…

, Chapter: 10 -Problem: 6 >> a. Given that 2(sin x + 2 cos x) = sin x + 5 cos x, find the exact value of tan x.b. Given that sin x cos y + 3 cos x sin y = 2 sin x sin y ? 4 cos x cos y, express tan y in terms of tan x.
Answer Preview: a To find the exact value of tan x given the equation 2 sin x 2 cos x sin x 5 cos x we can simplify …

, Chapter: 3 -Problem: 1 >> On a coordinate grid, shade the region that satisfies the inequalities:y > x – 2, y < 4x and y ? 5 – x.
Answer Preview: We have to draw these inequalities on the s…

, Chapter: 12 -Problem: 2 >> Find the greatest value of the following functions:a. f(x) = 10 ? 5x2 b. f(x) = 3 + 2x ? x2c. f(x) = (6 + x)(1 ? x)
Answer Preview: To find the greatest value of each function we need to determine the vertex point of …

, Chapter: 12 -Problem: 4 >> Find the coordinates of the point on the curve with equation y = x2 + 5x ? 4 where the gradient is 3.
Answer Preview: Let's find the derivative first: y = x^2 + 5x - 4 Taki…

, Chapter: 7 -Problem: 7 >> Find the value of k if (x ? 2) is a factor of x3 ? 3x2 + kx ? 10.
Answer Preview: To find the value of k we need to apply the factor …

, Chapter: 10 -Problem: 1 >> Draw diagrams to show the following angles. Mark in the acute angle that OP makes with the x-axis.a. ?80° b. 100° c. 200° d. 165° e. ?145°f. 225° g. 280° h. 330° i. ?160° j. ?280°
Answer Preview: To draw the diagrams representing the given angles and mark the acute angle …

, Chapter: 2 -Problem: 13 >> Find all of the roots of the function r(x) = x8 ? 17x4 + 16.
Answer Preview: To find all the roots of the function r x x 8 17x 4 16 we can fact…

, Chapter: 12 -Problem: 1 >> For the function f(x) = x2, use the definition of the derivative to show that: a. f'(2) = 4 b. f'(?3) = ?6 c. f'(0) = 0 d. f'(50) = 100
Answer Preview: To show the values of the derivatives using the definition of the derivative for the function …

, Chapter: 13 -Problem: 1 >> Find the equation of the curve with the given derivative of y with respect to x that passes through the given point: Transcribed Image Text: D ? e dy dx dy d.x dy dx = 3x² + 2x; = ?x + = x²; = (x + 2)²; point (2, 10) point (4, 11) point (1,7) b d f dy dx = 4x³ + dy 3 dx ?x = +33- dy dx ?x = X; x² +
Answer Preview: (a) dy/dx = 3x + 2x Integrating both sides w r t x (dy/dx) dx = dy = 3x + 2x + c c is the constant o…

, Chapter: 7 -Problem: 4 >> Divide:a. 3x4 + 8x3 ? 11x2 + 2x + 8 by (3x + 2) b. 4x4 ? 3x3 + 11x2 ? x ? 1 by (4x + 1)c. 4x4 ? 6x3 + 10x2 ? 11x ? 6 by (2x ? 3) d. 6x5 + 13x4 ? 4x3 ? 9x2 + 21x + 18 by (2x + 3)e. 6x5 ? 8x4 + 11x3 + 9x2 ? 25x + 7 by (3x ? 1) f. 8x5 ? 26x4 + 11x3 + 22x2 ? 40x + 25 by (2x ? 5)g. 25x4 + 75x3 + 6x2 ? 28x ? 6 by (5x + 3) h. 21x5 + 29x4 ? 10x3 + 42x ? 12 by (7x ? 2)
Answer Preview: To divide the given polynomials we can use polynomial long division I ll go through each division st…

, Chapter: 9 -Problem: 15 >> In ?ABC, AB = x cm, BC = 5 cm, AC = (10 ? x) cm. a. Show that  b. Given that cos ?ABC = ?1/7 , work out the value of x. Transcribed Image Text: LABC= 4x - 15 2x
Answer Preview: a To show that angle ABC is equal to 4x 15 2x we can use the Law of Cosines According to the Law of …

, Chapter: 2 -Problem: 7 >> a. Find the discriminant of h(x) in terms of k.b. Hence or otherwise, prove that h(x) has two distinct real roots for all values of k.h(x) = 2x2 + (k + 4)x + k, where k is a real constant.
Answer Preview: a The discriminant of a quadratic equation in the form of ax 2 bx c i…

, Chapter: 11 -Problem: 14 >> A particle P is accelerating at a constant speed. When t = 0, P has velocity u = (3i + 4j) m s?1 and at time t = 2 s, P has velocity v = (15i ? 3j) m s?1.The acceleration vector of the particle is given by the formula: a =  (v ? u)/t.Find the magnitude of the acceleration of P.
Answer Preview: To find the magnitude of the acceleration of particle …

, Chapter: 13 -Problem: 23 >> The graph shows a sketch of part of the curve C with equation y = (x ? 4)(2x + 3). The curve C crosses the x-axis at the points A and B. a. Write down the x-coordinates of A and B. The finite region R, shown shaded, is bounded by C and the x-axis. b. Use integration to find the area of R. Transcribed Image Text:
Answer Preview: a. To find the x-coordinates of the points where the curve C crosses the x-axis, we need to set y eq…

, Chapter: 7 -Problem: 1 >> Use the factor theorem to show that:a. (x ? 1) is a factor of 4x3 ? 3x2 ? 1 b. (x + 3) is a factor of 5x4 ? 45x2 ? 6x ? 18c. (x ? 4) is a factor of ?3x3 + 13x2 ? 6x + 8.
Answer Preview: To use the factor theorem to show that a given polynomial expression is divisible by a specific bino…

, Chapter: 9 -Problem: 5 >> In ?ABC, AB = 10 cm, BC = a?3 cm, AC = 5?13 cm and ?ABC = 150°. Calculate:a. The value of ab. The exact area of ?ABC.
Answer Preview: To find the value of a and the exact area of ABC we can use the Law of Cosines and the formula for t…

, Chapter: 12 -Problem: 4 >> Find the equations of the normals to the curve y = x + x3 at the points (0, 0) and (1, 2), and find the coordinates of the point where these normals meet.
Answer Preview: To find the equations of the normals to the curve y x x 3 at the points 0 0 and 1 2 we first need to find the slopes of the tangent lines at these poi…

, Chapter: 12 -Problem: 6 >> A rectangular garden is fenced on three sides, and the house forms the fourth side of the rectangle.a. Given that the total length of the fence is 80 m, show that the area, A, of the garden is given by the formula A = y(80 ? 2y), where y is the distance from the house to the end of the garden.b. Given that the area is a maximum for this length of fence, find the dimensions of the enclosed garden,
Answer Preview: a. To find the formula for the area of the garden, we can start by visualizing the rectangular garde…

, Chapter: 7 -Problem: 10 >> Show that when 3x4 ? 8x3 + 10x2 ? 3x ? 25 is divided by (x + 1) the remainder is ?1.
Answer Preview: To show that the remainder is 1 when the polynomial 3x 4 8x 3 10x …

, Chapter: 12 -Problem: 2 >> Find the gradient of the curve with equation y = f(x) at the point A where:a. f(x) = x3 ? 3x + 2 and A is at (?1, 4) b. f(x) = 3x2 + 2x?1 and A is at (2, 13)
Answer Preview: To find the gradient of a curve at a given point we need to find the derivative of the fu…

, Chapter: 10 -Problem: 22 >> If x is so small that terms of x3 and higher can be ignored, (2 ? x)(1 + 2x)5 ? a + bx + cx2. Find the values of the constants a, b and c.
Answer Preview: To find the values of the constants a, b, and c in the expression (2 - x)(1 + 2x)^5 a + bx + cx^2, w…

, Chapter: 7 -Problem: 5 >> Divide: a. x3 + x + 10 by (x + 2) b. 2x3 ? 17x + 3 by (x + 3)c. ?3x3 + 50x ? 8 by (x ? 4)
Answer Preview: a To divide x 3 x 10 by x 2 we can use polynomial long divi…

, Chapter: 12 -Problem: 3 >> Find the coordinates of the points where the gradient is zero on the curves with the given equations. Establish whether these points are local maximum points, local minimum points or points of inflection in each case. Transcribed Image Text: a y = 4x² + 6x d y = x(x² - 4x - 3) g y=x-3?x by=9+x-x² 1
Answer Preview: Let s find the coordinates where the gradient is zero for each given equation and determine the nature of those points a y 4x 6x To find where the gradient is zero we need to find the derivative and s…

, Chapter: 10 -Problem: 31 >> The graph shows the curve y = sin (x + 45°), ?360° ? x ? 360°. a. Write down the coordinates of each point where the curve crosses the x-axis. b. Write down the coordinates of the point where the curve crosses the y-axis. Transcribed Image Text: YA ÄÄ y = sin(x + 45°) X
Answer Preview: a. To find the points where the curve crosses the x-axis, we need to look for the values of x where …

, Chapter: 6 -Problem: 25 >> The points A(?7, 7), B(1, 9), C(3, 1) and D(?7, 1) lie on a circle.a. Find the equation of the perpendicular bisector of:i. AB ii. CDb. Find the equation of the circle.
Answer Preview: a i To find the equation of the perpendicular bisector of AB we need to find the midpoint of AB and the slope of AB Midpoint of AB x1 x2 2 y1 y2 2 7 1 …

, Chapter: 11 -Problem: 5 >> Find the angle that each of these vectors makes with j.a. 3i ? 5j b. 4i + 7j c. ?3i + 5j d. ?4i ? j
Answer Preview: The dot product of two vectors a and b is given by a b a b cos where a b is the dot product a and b …

, Chapter: 10 -Problem: 15 >> a. Show that (2x ? 1) is a factor of 2x3 ? 7x2 ? 17x + 10.b. Factorise 2x3 ? 7x2 ? 17x + 10 completely.c. Hence, or otherwise, sketch the graph of y = 2x3 ? 7x2 ? 17x + 10, labelling any intersections with the coordinate axes clearly.
Answer Preview: a. To show that (2x 1) is a factor of 2x 3 7x 2 17x + 10, we are using synthetic division. 2 | 2 -7 …

, Chapter: 4 -Problem: 9 >> a. On the same axes sketch the curves with equations y = (x – 2)(x + 2)2 and y = –x2 – 8.b. Find the coordinates of the points of intersection.
Answer Preview: a On the same axes sketch the graphs of y x 2 x 2 2 and y x 2 8 Dra…

, Chapter: 10 -Problem: 6 >> Find, without using your calculator, the values of:a. sin ? and cos ?, given that tan ? = 5/12 and ? is acute.b. sin ? and cos ?, given that cos ? = ?3/5 and ? is obtuse.c. cos ? and tan ?, given that sin ? = ?7/25 and 270° < ? < 360°.
Answer Preview: To find the values of trigonometric functions without using a calculator we can use the definitions …

, Chapter: 12 -Problem: 3 >> Find the coordinates of the point where the tangent to the curve y = x2 + 1 at the point (2, 5) meets the normal to the same curve at the point (1, 2).
Answer Preview: To show that the function f x 4 x 2x 2 3 is decreasing for all x R we need to demonstrate that the derivative of the function is always negative First …

, Chapter: