Cambridge IGCSE And O Level Additional Mathematics 1st Edition Val Hanrahan, Jeanette Powell - Solutions

The diagram shows a parallelogram OABC with O̅A̅(vector) = a and O̅B̅(vector) = b. Write the following vectors in terms of a and b:a. A̅B̅(vector)b. B̅A̅(vector)c. C̅B̅(vector)d. B̅C̅(vector)e. O̅B̅(vector)f. B̅O̅(vector)g. A̅C̅(vector) h. C̅A̅(vector) B A a
A̅B̅(vector) = j, B̅C̅(vector) is the vector 2i + 2j and C̅D̅(vector) = i.a. Sketch the shape ABCD.b. Write AD as a column vector.c. Describe shape ABCD.
ABCD is a rhombus where A is the point (−1, −2).a. Write the vectors A̅D̅(vector) and D̅C̅(vector) as column vectors.b. Write the diagonals A̅C̅(vector) and B̅D̅(vector) as column vectors.c. Use the properties of a parallelogram to find the coordinates of the point of intersection of
A is the point (−3, −2), B is the point (5, 4) and C is the point (2, 8).a. Sketch the triangle ABC.b. Find the vectors representing the three sides AB, BC and CA in the form xi + yj.c. Find the lengths of each side of the triangle.d. What type of triangle is triangle ABC?
ABCD is a kite and AC and BD meet at the origin O. A is the point (−4, 0), B is (0, 4) and D is (0, −8).The diagonals of a kite are perpendicular and O is the midpoint of AC.a. Find each of the following in terms of i and j:i. O̅C̅(vector)ii. A̅B̅(vector)iii. B̅C̅(vector)iv.
A. (4, 4), B(24, 19) and C (48, 12) form the vertices of a triangle.a. Sketch the triangle.b. Write the vectors A̅B̅(vector), B̅C̅(vector) and A̅C̅(vector) as column vectors.c. Find the lengths of the sides of the triangle.d. What type of triangle is ABC?
Salman and Aloke are hiking on a flat level ground. Their starting point is taken as the origin and the unit vectors i and j are in the directions east and north. Salman walks with constant velocity 3i + 6j kilometres per hour. Aloke walks on a compass bearing of 300o at a steady speed of 6.5
Ama has her own small aeroplane. One afternoon, she flies for 1 hour with a velocity of 120i + 160j km h−1 where i and j are unit vectors in the directions east and north.Then she flies due north for 1 hour at the same speed. Finally, she returns to her starting point; flying in a straight line
Differentiate the following functions using the rulesy = kxn ⇒ dy/dx = nkxn-1and y = f(x) + g(x) ⇒ dy/dx = f'(x) + g'(x).a. y = x4b. y = 2x3c. y = 5d. y = 10x
For each curve in question.i. Find dy/dx and the value(s) of x for which dy/dx = 0ii. Classify the point(s) on the curve with these x-valuesiii. Find the corresponding y-value(s)iv. Sketch the curve.y = 1 + x − 2x2
Find dy/dx and d2y/dx2 for each of the following functions:a. y = x3 − 3x2 + 2x − 6b. y = 3x4 − 4x3c. y = x5 − 5x + 1
The sketch graph shows the curve of y = 5x− x2. The marked point, P, has coordinates (3, 6).Find:a. The gradient function dy/dxb. The gradient of the curve at Pc. The equation of the tangent at Pd. The equation of the normal at P. y. P (3, 6) y = 5x -x?
Differentiate each of the following functions:a. y = 3 sin x − 2 tan xb. y = 5 sin θ − 6c. y = 2 cos θ − 2 sin θd. y = 4 ln x e. y = ln 4x f y = 3exf. y = 2ex − ln 2x
Differentiate the following functions using the rulesy = kxn ⇒ dy/dx = nkxn-1and y = f(x) + g(x) ⇒ dy/dx = f'(x) + g'(x).a. y = x/12b. y = 5√xc. P = 7t3/2d. y = 1/5 x5/2
For each curve in question.i. Find dy/dx and the value(s) of x for which dy/dx = 0ii. Classify the point(s) on the curve with these x-valuesiii. Find the corresponding y-value(s)iv. Sketch the curve.y = 12x + 3x2 − 2x3
For each of the following curvesi. Find any stationary pointsii. Use the second derivative test to determine their nature.a. y = 2x2 − 3x + 4b. y = x3 − 2x2 + x + 6c. y = 4x4 − 2x2 + 1d. y = x5 − 5x
The sketch graph shows the curve of y = 3x2 − x3. The marked point, P, has coordinates (2, 4).a. Find:i. The gradient function dy/dxii. The gradient of the curve at Piii. The equation of the tangent at Piv. The equation of the normal at P.b. The graph touches the x-axis at the origin O and
Use the product rule to differentiate each of the following functions:a. y = x sin xb. y = x cos xc. y = x tan xd. y = ex sin x e. y = ex cos x f. y = ex tan x
Differentiate the following functions using the rulesy = kxn ⇒ dy/dx = nkxn-1and y = f(x) + g(x) ⇒ dy/dx = f'(x) + g'(x).
For each curve in question.i. Find dy/dx and the value(s) of x for which dy/dx = 0ii. Classify the point(s) on the curve with these x-valuesiii. Find the corresponding y-value(s)iv. Sketch the curve.y = x3 − 4x2 + 9
For y = 2x3 − 3x2 − 36x + 4a. Find dy/dx and the values of x when dy/dx = 0.b. Find the value of d2y/dx2 at each stationary point and hence determine its nature.c. Find the value of y at each stationary point.d. Sketch the curve.
The sketch graph shows the curve of y = x5 − x3.Find:a. The coordinates of the point P where the curve crosses the positive x-axisb. The equation of the tangent at Pc. The equation of the normal at P.The tangent at P meets the y-axis at Q and the normal meets the y-axis at R.d. Find the
Use the quotient rule to differentiate each of the following functions:a. y = sinx/xb. y = x/sin xc. y = cos x/x2d. y = x2/cos xe. y = x/tan xf. y = tan x /x
Differentiate the following functions using the rulesy = kxn ⇒ dy/dx = nkxn-1and y = f(x) + g(x) ⇒ dy/dx = f'(x) + g'(x).
A farmer has 160 m of fencing and wants to use it to form a rectangular enclosure next to a barn.Find the maximum area that can be enclosed and give its dimensions. fence barn wall
For each curve in question.i. Find dy/dx and the value(s) of x for which dy/dx = 0ii. Classify the point(s) on the curve with these x-valuesiii. Find the corresponding y-value(s)iv. Sketch the curve.y = x2(x− 1)2
a. Given that f(x) = x3 − 3x2 + 4x + 1, find f´(x).b. The point P is on the curve y = f(x) and its x-coordinate is 2.i. Calculate the y-coordinate of P.ii. Find the equation of the tangent at P.iii. Find the equation of the normal at P.c. Find the values of x for which the curve has a gradient
Use the chain rule to differentiate each of the following functions:a. y = (x + 3)4b. y = (2x + 3)4c. y = (x2 + 3)4d. y = √x + 3e. y = √2x + 3f. y = √x2 + 3
Differentiate the following functions using the rulesy = kxn ⇒ dy/dx = nkxn-1and y = f(x) + g(x) ⇒ dy/dx = f'(x) + g'(x).
For each curve in question.i. Find dy/dx and the value(s) of x for which dy/dx = 0ii. Classify the point(s) on the curve with these x-valuesiii. Find the corresponding y-value(s)iv. Sketch the curve.y = x4 − 8x2 + 4
A cylinder has a height of h metres and a radius of r metres where h + r = 3.a. Find an expression for the volume of the cylinder in terms of r.b. Find the maximum volume.
The sketch graph shows the curve of y = x3 − 9x2 + 23x − 15.The point P marked on the curve has its x-coordinate equal to 2.Find:a. The gradient function dy/dxb. The gradient of the curve at Pc. The equation of the tangent at Pd. The coordinates of another point on the curve, Q, at which the
Use an appropriate method to differentiate each of the following functions:a. y = sin x/1 + cos xb. y = 1 + cos x/sin xc. y = sin x(1 + cos x)d. y = cos x(1 + sin x)e. y = sin x(1 + cos x)2 f. y = cos x(1 + sin x)2
Find the gradient of the curve y = x2 − 9 at the points of intersection with the x- and y-axes. yA -9
For each curve in question.i. Find dy/dx and the value(s) of x for which dy/dx = 0ii. Classify the point(s) on the curve with these x-valuesiii. Find the corresponding y-value(s)iv. Sketch the curve.y = x3 − 48x
A rectangle has sides of length x cm and y cm.a. If the perimeter is 24 cm, find the lengths of the sides when the area is a maximum, confirming that it is a maximum.b. If the area is 36 cm2, find the lengths of the sides when the perimeter is a minimum, confirming that it is a minimum.
The point (2, −8) is on the curve y = x3 − px + q.a. Use this information to find a relationship between p and q.b. Find the gradient function dy/dx.The tangent to this curve at the point (2, −8) is parallel to the x-axis.c. Use this information to find the value of p.d. Find the coordinates
Use an appropriate method to differentiate each of the following functions:a. y = ex ln xb. y = ex/lnxc. y = ln x/ex
a. Copy the curve of y = 4 − x2 and draw the graph of y = x − 2 on the same axes.b. Find the coordinates of the points where the two graphs intersect.c. Find the gradient of the curve at the points of intersection. y 4 22 2
For each curve in question.i. Find dy/dx and the value(s) of x for which dy/dx = 0ii. Classify the point(s) on the curve with these x-valuesiii. Find the corresponding y-value(s)iv. Sketch the curve.y = x3 + 6x2 − 36x + 25
The sketch graph shows the curve of y = x2 − x − 1.a. Find the equation of the tangent at the point P(2, 1). The normal at a point Q on the curve is parallel to the tangent at P.b. State the gradient of the tangent at Q.c. Find the coordinates of the point Q. y ソ=x?-x-1 P (2, 1)
Use an appropriate method to differentiate each of the following functions:a. e−x sin xb. y = e-x/sin xc. y = sin x/e-x
A curve has the equation y = (x − 3)(7 − x).a. Find the gradient function dy/dx.b. Find the equation of the tangent at the point (6, 3).c. Find the equation of the normal at the point (6, 3).d. Which one of these lines passes through the origin?
A curve has the equation y = sin x − cos x where x is measured in radians.a. Show that the curve passes through the points (0, −1) and (π, 1).b. Find the equations of the tangents and normals at each of these points.
The graph of y = px + qx2 passes through the point (3, −15). Its gradient at that point is −14.a. Find the values of p and q.b. Calculate the maximum value of y and state the value of x at which it occurs.
A curve has the equation y = 1.5x3 − 3.5x2 + 2x.a. Show that the curve passes through the points (0, 0) and (1, 0).b. Find the equations of the tangents and normals at each of these points.c. Prove that the four lines in b form a rectangle.
A curve has the equation y = 2tan x − 1 where x is measured in radians.a. Show that the curve passes through the points (0, −1) and (π/4, 1).b. Find the equations of the tangents and normals at each of these points.
a. Find the stationary points of the function f(x) = x2(3x2 − 2x −3) and distinguish between them.b. Sketch the curve y = f(x).
A curve has the equation y = 2ln x − 1.a. Show that the curve passes through the point (e, 1)b. Find the equations of the tangent and normal at this point.
A curve has the equation y = ex − ln x.a. Sketch the curves y = ex and y = ln x on the same axes and explain why this implies that ex − ln x is always positive.b. Show that the curve y = ex − ln x passes through the point (1, e).c. Find the equations of the tangent and normal at this point.
Simplify the following:a.b.c. 1 + 3.
Find unit vectors parallel to each of the following:a. 3i – 4j b. 5i + 7jc. d.e. 5if. 5 12
For the function f(x) = 3x + 4, find:a. f(3)b. f(−2) c. f(0)d. f(1/2)
Sketch the graph of each function:a. y = x + 2b. y = |x + 2|c. y = |x + 2| + 3
Sketch the following graphs, indicating the points where they cross the x-axis:a. y = x(x – 2)(x + 2)b. y = |x(x – 2)(x + 2)|c. y = 3(2x – 1)(x + 1)(x + 3)d. y = |3(2x – 1)(x + 1)(x + 3)|
Solve the following quadratic equations. Leave your answers in the formx = p ± √q.a. x² + 4x − 9 = 0b. x² − 7x − 2 = 0c. 2x² + 6x − 9 = 0d. 3x² + 9x − 15 = 0
For each of the following functions:i. Use the method of completing the square to find the coordinates of the turning point of the graph.ii. State whether the turning point is a maximum or a minimum.iii. Sketch the graph.a. f(x) = x² + 6x + 15b. y = 8 + 2x − x²c. y = 2x² + 2x − 9d. f : x
Sketch the graph and find the corresponding range for each function and domain.a. y = x² − 7x + 10 for the domain 1 ≤ x ≤ 6b. f(x) = 2x² − x − 6 for the domain −2 ≤ x ≤ 2
a. y = x and y = |x|b. y = x − 1 and y = |x – 1|c. y = x − 2 and y = |x – 2|For this question, sketch each pair of graphs on the same axes.
Write each of the following inequalities in the form k |x - a| ≤ b:a. -3 ≤ x ≤ 15b. -4 ≤ x ≤ 16c. -5 ≤ x ≤ 17
Where possible, use the substitution x = u² to solve the following equations:a. x − 4 √x = −4b. x + 2 √x = 8c. x − 2 √x = 15d. x + 6 √x = −5
a. y = 2x and y = |2x|b. y = 2x – 1 and y = |2x − 1|c. y = 2x – 2 and y = |2x − 2|For this question, sketch each pair of graphs on the same axes.
Write each of the following expressions in the form a ≤ x ≤ b:a. |x - 1| ≤ 2b. |x – 2| ≤ 3c. |x – 3| ≤ 4
a. y = 2 – x and y = |2 – x|b. y = 3 – x and y = |3 – x|c. y = 4 – x and y = |4 – x|For this question, sketch each pair of graphs on the same axes.
Solve the following inequalities and illustrate each solution on a number line:a. |x - 1| < 4b. |x – 1| > 4c. |2x + 3| < 5d. |2x + 3| > 5
Solve the following equations graphically. You will need to use graph paper.a. x(x + 2)(x − 3) ≥ 1b. x(x + 2)(x − 3) ≤ −1c. (x + 2)(x − 1)(x − 3) > 2d. (x + 2)(x − 1)(x − 3) < −2
a. Draw the graph of y = |x + 1|.b. Use the graph to solve the equation |x + 1| = 5.c. Use algebra to verify your answer to part b.
a. Draw the graph of y = |x – 1|.b. Use the graph to solve the equation |x – 1| = 5.c. Use algebra to verify your answer to part b.
Write each of the following in its simplest form:a. √12b. √75c. √300d. 3√5 + 6√5e. √48 + √27f. 3√45 – 2√20
Given that f(x) = 3x + 2, g(x) = x2 and h(x) = 2x, find:a. fg(2)b. fg(x)c. gh(x)d. fgh(x)
Given that f(x) = 2x +1 and g(x) = 4 − x, find:a. fg(−4)b. gf(12)c. fg(x)d. gf(x)
For the function h:x → 3x2 + 1, find:a. h(2)b. h(−3)c. h(0)d. h(1/3)
Given that f(x) = x + 4, g(x) = 2x2 and h(x) = 1/2x + 1, find:a. f2(x) b. g2(x) c. h2(x)d. hgf(x)
For the function f: 2x + 6/3, find:a. f(3)b. f(−6)c. f(0)d. f(1/4)
For each function, find the inverse and sketch the graphs of y = f(x) and y = f−1(x) on the same axes. Use the same scale on both axes.a. f(x) = 3x − 1b. f(x) = x3, x > 0
For the function f(x) = √2x + 1:a. Draw a mapping diagram to show the outputs when the set of inputs is the odd numbers from 1 to 9 inclusive.b. Draw a mapping diagram to show the outputs when the set of inputs is the even numbers from 2 to 10 inclusive.c. Which number must be excluded as an
Solve the following equations:a. |x − 3| = 4b. |2x + 1| = 7c. |3x − 2| = 5d. |x + 2| = 2
Which value(s) must be excluded from the domain of these functions?a. f(x) = 1/xb. f(x) = √x −1c. f(x) = 3/2x − 3d. f(x) = √2 − x2
Sketch these graphs for 0° ≤ x ≤ 360°:a. y = cos xb. y = cos x + 1c. y = |cos x|d. y = |cos x| + 1
Find the inverse of each function:a. f(x) = 7x − 2b. g(x) = 3x + 4/2c. h(x) = (x – 1)2 for x ≥ 1d. f(x) = x2 + 4 for x ≥ 0
Graph 1 represents the line y = 2x − 1. Graph 2 is related to Graph 1 and Graph 3 is related to Graph 2.Write down the equations of Graph 2 and Graph 3. Graph 1 Graph 2 Graph 3 yA 3- /0.5 0.5 0.5 4.
a. Find the inverse of the function f(x) = 3x – 4.b. Sketch f(x), f–1(x) and the line y = x on the same axes. Use the same scale on both axes.
The graph shows part of a quadratic curve and its inverse.a. What is the equation of the curve?b. What is the equation of the inverse? yA / y =x (2, 5) * (1, 2)
a. Plot the graph of the function f(x) = 4 − x2 for values of x such that 0 ≤ x ≤ 3. Use the same scale on both axes.b. Find the values of f–1(−5), f–1(0), f–1(3) and f–1(4).c. Sketch y = f(x), y = f–1(x) and y = x on the same axes. Use the range −6 to +6 for both axes.
a. Sketch the graphs of these functions:i. y = 1 − 2xii. y = |1 − 2x|iii. y = −|1 − 2x|iv. y = 3 −|1 − 2x|b. Use a series of transformations to sketch the graph of y = |3x + 1| − 2.
For each part:a. Sketch both graphs on the same axes.b. Write down the coordinates of their points of intersection.i. y = |x| and y = 1 − |x|ii. y = 2|x| and y = 2 − |x|iii. y = 3|x| and y = 3 − |x|
Solve each equation by factorising:a. x² + x − 20 = 0 b. x² − 5x + 6 = 0c. x² − 3x − 28 = 0d. x² + 13x + 42 = 0
For each of the following equations, decide if there are two real and different roots, two equal roots or no real roots. Solve the equations with real roots.a. x² + 3x + 2 = 0 b. t² − 9 = 0 c. x² + 16 = 0d. 2x² − 5x = 0 e. p² + 3p − 18 = 0 f. x² + 10x + 25 = 0g.
Solve each equation by factorising:a. 2x² − 3x + 1 = 0 b. 9x² + 3x − 2 = 0c. 2x² − 5x − 7 = 0 d. 3x² + 17x + 10 = 0
Solve the following equations by:i. Completing the square ii. Using the quadratic formula.Give your answers correct to two decimal places.a. x² − 2x − 10 = 0b. x² + x = 0c. 2x² + 2x − 9 = 0d. 2x² + x − 8 = 0
Solve each equation by factorising:a. x² − 169 = 0b. 4x² − 121 = 0c. 100 − 64x² = 0d. 12x² − 27 = 0
Try to solve each of the following equations. Where there is a solution, give your answers correct to two decimal places.a. 4x² + 6x − 9 = 0 b. 9x² + 6x + 4 = 0c. (2x + 3)² = 7 d. x(2x − 1) = 9
For each of the following curves:i. Factorise the function.ii. Work out the coordinates of the turning point.iii. State whether the turning point is a maximum or minimum.iv. Sketch the graph, labelling the coordinates of the turning point and any points of intersection with the axes.a. y = x² + 7x
Use the discriminant to decide whether each of the following equations has two equal roots, two distinct roots or no real roots:a. 9x² − 12x + 4 = 0 b. 6x² − 13x + 6 = 0 c. 2x² + 7x + 9 = 0d. 2x² + 9x + 10 = 0 e. 3x² − 4x + 5 = 0 f. 4x² + 28x + 49 = 0
Write each quadratic expressions in the form (x + a)² + b:a. x² + 4x + 9 b. x² − 10x − 4c. x² + 5x − 7 d. x² − 9x − 2
For each pair of equations determine if the line intersects the curve, is a tangent to the curve or does not meet the curve. Give the coordinates of any points where the line and curve touch or intersect.a. y = x² + 12x; y = 9 + 6xb. y = 2x² + 3x − 4; y = 2x − 6c. y = 6x² − 12x + 6; y =
Write each quadratic expression in the form c(x + a)² + b.a. 2x² − 12x + 5 b. 3x² + 12x + 20c. 4x² − 8x + 5 d. 2x² + 9x + 6
Solve the following inequalities and illustrate each solution on a number line:a. x² − 6x + 5 > 0 b. a² + 3a − 4 ≤ 0 c. 4 − y² > 0d. x² − 4x + 4 > 0 e. 8 − 2a > a² f. 3y² + 2y − 1 > 0
Illustrate each of the following inequalities graphically by shading the unwanted region:a. y – 2x > 0b. y – 2x ≤ 0c. 2y – 3x > 0d. 2y – 3x ≤ 0
Identify the following cubic graphs: y A b y 12 11- 2 10- 1- 8 -2 6- 4 -3 2 4 3. -1

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Cambridge IGCSE And O Level Additional Mathematics 1st Edition Val Hanrahan, Jeanette Powell - Solutions

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