Complete Pure Mathematics 1 For Cambridge International AS & A Level 2nd Edition Jean Linsky, Brian Western, James Nicholson - Solutions

Find the equation of the tangent to the circle x² + y² − 4x + 2y + 3 = 0 at the point (3, −2).
The diagram shows a straight line AC with the mid-point B. BD is perpendicular to AC. Find the area of triangle ACD. A(-2,3) B C(6, 7) D(5, -1) X
Show that the line y = 2x does not intersect the circle (x − 5)² + (y − 2)² = 10.
The line 3x + y = 17 intersects the curve y = x² + 2x − 7 at the point P(3, 8) and the point Q.a) Find the coordinates of Q.b)Find the exact length of the line PQ.
A circle with equation x² + (y − k)² = 10 passes through the point (3, 5).Determine the two possible values of k.
A straight line L passes through the point (1, 0) and has gradient m.a) Write down the equation of the line. The line L is a tangent to the curve y = x² + 2x − 3.b) Find the value of m.The line L passes through the point (−2, k).c) Find the value of k.
The diagram shows triangle PQR with PQ perpendicular to QR. P is the point (6,5), Q is the point (2, 10), and R is the point (−8, y). Find the value of y. R(-8, y) yi 0 Q(2, 10) P(6,5) X
The perpendicular bisector of the line joining the points (3, 2) and (7,4) cuts the y-axis at the point (0, k). Find the value of k.
Find the equation of the line joining the point (1, 4) to the centre of the circle with equation x² + y² − 8x + 4y + 11 = 0.
Show that when k > 2, the line 2x + y = 1 does not intersect the curve with equation y = x² + k.
ABC is a triangle with vertices A(−2, −3), B(6, −3) and C(2, 3).Show that triangle ABC is an isosceles triangle.
A line has equation y = 2kx − 7 and a curve has equation y = x² + kx − 3, where k is a positive integer.a) Find the value of k for which the line is a tangent to the curve.b) For this value of k, find the coordinates of the point where the line touches the curve.
A(5, −5), B(3, −7), C(12, 3), and D(−3, −6) are four points.The lines AB and CD meet at P. Find the coordinates of P.
The diagram shows triangle ABC with AB perpendicular to BC. A is the point (−6, 6) and B is the point (−5, 2). The gradient of AC is −2/9. Find the coordinates of C. A(-6, 6) B(-5,2) YA X
The diagram shows the straight line AB that cuts the x-axis at P. PC is perpendicular to AB. The point A is (−3, −4), the point B is(6, 2) and the point C is (5, −3).Show that point P lies on the x-axisand find its coordinates. A(-3,-4) УА 0 P B(6,2) C(5,-3)
The points A, B and C have coordinates (4, 0), (p, 6) and (7, 1) respectively. The length of AB is twice the length of AC. Find the possible values of p.
P is the point (k, 3k) and Q is the point (5k, −5k). Find the equation of theperpendicular bisector of PQ.
Show that the line with equation y = 2x − 3 cannot be a tangent to the curve with equation y = x² − 3x + 5.
The straight line with equation 3x − 2y = 6 cuts the x-axis at P and the y-axis at Q.a) Find the coordinates of P and Q.b) Find the equation of the line that is perpendicular to PQ and passes through the point (4, 3).
PQR is a triangle with vertices P(4, −2), Q(−1,7) and R(k, 0). PR = QR.Find the value of k.
The line with equation x + y = 2 meets the line with equation 2y − x = 7 at the point P.Find the equation of the straight line thatpasses through P and is perpendicular to theline with equation y = 2x. Give your answerin the form ax + by + c = 0.
ABC is a triangle with angle ABC = 90°. The points A, B and C have coordinates (2, 5), (1, 3) and (5, 1) respectively. Find the area of triangle ABC.
P is the point with coordinates (−2, 5) and Q is the point with coordinates (−6, −3).a) Find the gradient of the line PQ.b) Find the equation of the line PQ. Give your answer in the form ax + by + c = 0.c) Find the equation of the line that isparallel to PQ and passes through thepoint R with
A line has equation y = 2kx − 9 and a curve has equation y = x² = kx, where k is a constant.a) Find the two values of k for which the line is a tangent to the curve.b) For each value of k, find the coordinates of the point wherethe line is a tangent to the curve, and find the equation of the
A(1, −2), B(−1, −3), C(−2, 14), and D(2, 2) are four points.The lines AB and CD meet at P. Find the coordinates of P.
ABCD is a square with vertices A(-1, −2), B(1, 2), C(5, 0) and D(3, −4).a) Show that the diagonals intersect at the point (2, −1).b) Find the area of ABCD.
ABC is an isosceles triangle with AB = BC. A has coordinates (3, 2), B has coordinates (2, −5) and C has coordinates (9, −4). Findthe equation of the line of symmetry of thetriangle.
Find the equation of the tangent to the circle x² + y² − 12x + 26 = 0at the point P(3, 1).Give your answer in the form ax + by + c = 0, where a, b and c are integers.
A(9, 0), B(5, 8), C(1, 1) and D(3, 2) are four points. The lines AB and CD meet at P. Find the equations of AB and CD and hence the coordinates of P.
The mid-point of the line joining the points P(b, −1) and Q(−2,7) is (−2a, a).Work out the values of a and b.
Find the equation of the straight line through the point P(7, 1) and parallel to the line with equation 2x − 5y + 3 = 0.
The line PQ is perpendicular to the line with equation 2x + y = 4. P has coordinates (5, 4). Find the coordinates of Q. 2х+у= 4 УА 0 P(5,4) X
A circle with centre C has equation x² + y² − 8x − 6y − 20 = 0.a) Find the coordinates of C and the radius of the circle.b) A(10, 0) lies on the circle. Find the equation of the tangent to the circle at A.
The gradient of the line joining the points A(1, k) and B(5, 1) is 1. Find the mid-point of the line AB.
The equation of a line is y = x − k, where k is a constant, and the equation of a curve is x² + 2y = k.a) When k = 1, the line intersects the curve at the points A and B. Find the coordinates of A and the coordinates of B.b) Find the value of k for which the line y = x − k is a tangent to the
A circle with equation x² + y² + 4x − 6y = 12 has centre C.The circle cuts the x-axis at the points A and B.Calculate the area of the triangle ABC.
Find the equation of the line through the point (1, 8) perpendicular to the line with equation 3x − 4y + 4 = 0.
ABC is a right-angled triangle with AB perpendicular to BC. The coordinates of the points A, B and C are (−5, 0), (x, −5) and (3,−2) respectively. Find the possible values of x.
M is the mid-point of the line joining the points A(−5, −2) and B(3, −8). Given that a point C has the coordinates (1, 4), finda) The gradient of the line MCb) The length of the line MC.
a) Show that the lines with equations 2y = x + 3 and 2x + y = 9 are perpendicular.b) Find their point of intersection.
The diagram shows a kite PQRS. The point P is (−3, 2), the point Q is (4, −1),point Ris (1, −8) and the point S is (−11, −7).The diagonal SQ bisects PR at M.Show that the ratio SM: MQ = 2:1 S(-11,-7) P(-3,2) M 0 R(1,-8) Q(4,-1)
A curve has equation y = 2x² + kx − 1 and a line has equation x + y + k = 0, where k is a constanta) State the value of k for which the line is a tangent to the curve.b) For this value of k find the coordinates of the point where the line touches the curve.
a) Show that (x − 5)(x − 2) + (y − 7)(y − 1) = 0 represents a circle. b) Find the centre and radius of this circle.
Find the equation of the line through the point (−1, 3) perpendicular to the line with equation 6x + 9y − 7 = 0.
The line that passes through the points (3, 0) and (−4, y) is perpendicularto the line with equation 7x + 4y = 5. Find the value of y.
Find if the lines with equations 9x − 6y = 8and 4y − 6x = 7 are parallel.
The straight line joining the points A (p, 3) and B(1, −1) has a length of 5.Work out the values of p.
Determine the shortest distance from the point A(1, 3) to the circle with equation x² + y² − 10x − 12y + 45 = 0.
PQ is a diameter of the circle, where P is (−8, 3) and Q is (2, −5).Find the equation of the circle.
Find the equation of the line through the point (−1, 3) parallel to theline with equation 2x − 5y = 10.
The line joining points A(x, 3) and B(2, −3) is perpendicular to BCwhere C is the point (10, 1).Find the value of x.
The straight line joining the points A(3, −5) and B(6, k) has a gradient of 4.Work out the value of k.
The line y = 5 - kx, where k is an integer, is a tangent to the curve y = 2k − x².a) Find the possible values of k. When k = 2, the line y = 5 − kx is a tangent to the curve y = 2k − x² at point A.b) Find the coordinates of A.
The circle with equation (x − 2)² + (y + a)² = 26 passes through the point (3, −1).Find the value of a where a < 0.
The line PQ has a gradient of 1/3. PQ is parallel to the line joining the 3 points (3, k) and (−6, 5).Find k.
The point P has coordinates (−5, 10). Thepoint Q has coordinates (1, 6). The point Rhas coordinates (3, 5). Show that the threepoints are not collinear.
The mid-point of the line joining the points A(p, −2) and B(6, −8) is (1, q).Work out the values of p and q.
The point (−4, 8) is the mid-point ofthe line joining the points (a + b, 3b) and(−2a, 2b − 3a).Find the values of a and b.
A circle with centre C has equation x² + y² − 6x + 4y + 8 = 0a) Express the equation in the form (x − a)² + (y − b)² = r².b) Find the coordinates of C and the radius of the circle.
The graph shows a sketch of f(x). On separate diagrams, sketch the graphs ofa) y = f(x − 2)b) y = −f(x)c) y = f(2x)d) y = −f(−x)e) y = 3 + f(1 − x).Mark the new position of the points O, A and B on each transformation, stating their coordinates. y, 0 A(2, -3) B(6, 0) X
The points P(−1,− 1) and Q(2, 1/2) lie on the graph of f(x) = 1/x. Sketch the graph of y = 2f(x − 1) + 2.Mark the images of the points P and Q,stating their coordinates, and write down theequations of any asymptotes.
Find the lengths of the straight lines joining each of the following sets of points. a) (−8, 4) and (−2,−4) b) (−1, 1) and (−5, −2)c) (6, -1) and (9,−4) d) (−7, −3) and (−6, −8) e) (4q, −4q) and (7q, −8q)f) (2p, −6p) and (−3p, 6p)
The graph of y = f(x) is shown. It cuts the axes at (0, 4) and (4, 0).a) Sketch y = − 3f(x) and give the coordinates where the graph cuts the axes.b) The graph of y = f(x) is transformed to the graph of y = f(− 1/2x). Describe fullythe two single transformations that havebeen combined to give
The curve shown has a maximum at A(− 1, 4) and a minimum at B(2, − 8). On separate diagrams sketcha) y = f(x) + 4b) y = 1/2f(x)Show clearly the coordinates of themaximum and minimum points and wherethe curve cuts the y-axis. УА A(-1, 4) +2 0 B(2,-8) ш X
Find the equation of the line AB with the given pairs of coordinates.a) A(1, 5) and B(3, 7) b) A(−1, −2) and B(−3, 4) c) A(5, −3) and B(−2, 4)d) A(-2, 4) and B(−6, 0)e) A(-8, −1) and B(2, −6) f) A(9,−3) and B(− 6, −2)
Find the gradient of a line that is (i) parallel, (ii) perpendicular to the lines with these equations.a) y = 7x − 3 b) x + y = 1c) 5y = 2 − 15xd) 4x − 3y = 6e) 3x = 4 + 3yf) y = −4x
The mid-point M of the line PQ has coordinates (−2, −1). P has coordinates(−3, 4). R has coordinates (5, −2). Find thelength of the line QR.
The graph of a function is reflected in the x-axis and then translated by the vector After the transformation its equation is y = 3 + 1/(x − 4)². Determine the equation of the originalfunction. 4 3
Find the gradients of the straight lines joining each of the following sets of points.a) (−3,−6) and (2, −1)b) (4, −6) and (1, 0)c) (5, −7) and (−3, 9)d) (−8, −3) and (2, −7)e) (−3k, −4k) and (−k, 6k) f) (4q, −4q) and (7q, 8q)
Find the value of k for which the line y = 2kx + 7 is a tangent to the curve y = 3 + kx².
Find the equation of each of these circles.a) Centre (9, 1), radius 4b) Centre (−5, 3), radius 7c) Centre (−4,−7), radius 5d) Centre (6, −2), radius 3
The diagram shows a sketch of the graph with equation f(x) = 6x² − 2x³. There is a minimum at the origin, a maximum at the point A(2, 8) and it cuts the x-axis at B(3, 0).a) On separate diagrams, sketch the graphs with equationsi) y = f(2x)ii) y = f(x + 1).On each sketch, show where the curve
The function f is defined by a) Find f−¹(x).b) Show that f-¹(x) = −2 has no solution. -fix-1, xe R, x = -2. fix→ x+2
Find the equation of the lines with the given gradient (m) passing through the given point P.a) m = 2 P(5, 7)b) m = −1 P(−1, −2)c) m = −3 P(−4,1) d) m = 4 P(3, −2)e) m =4/3 P(−6, −1)f) m = −1/5 P(−4,0)
The straight line joining the points P(9, −1) and Q(k, 5) has a gradient of −3. Work outthe value of k.
Find the mid-points of the straight lines joining each of the following sets of points.a) (1, 3) and (7, 13)b) (2, −1) and (− 4,− 5)c) (−6, −3) and (− 2, 7)d) (−8,5) and (3,−2) e) (−k, −9) and (−3k, 5)f) (3p, −4p) and (−p, 6p)
The diagram shows the graph of f(x).On separate diagrams, sketch the graphs ofa) y = f(4x)b) y = f(3x)c) y = − f(x + 1)d) y = 2f(− x).On each sketch, mark the new position ofthe points O, A, B and C, writing down theircoordinates. y A(2,4) B(4,0) C(8,4) X
The diagram shows the graph of y = f(x).On separate diagrams, sketch the graphs ofa) y = f(−x)b) y = f(x − 1)c) y = −f(2x)d) y = 3f(4x).On each sketch, write the equation of anyasymptotes and, where possible, write thecoordinates where the graph cuts the axes. У 2 О -2 X
The graph of y = f(x) is transformed to the graph of y = 5 − f(x).Describe fully the two single transformationsthat have been combined to give theresulting transformation.
a) Find an expression for f−¹(x), stating its domain.b)Find an expression for a in terms of x,a ≠ 0, when f(x) = f(2x + a). f(x)=√√2x+a, x = R, 2x ≥-a.
A function F is such that a) Find f−1(x) in the form ax² + bx + c, where a, b and c are constants.b) Find the domain of f−1. f(x) = 2+1, for x ≥ 1. 3
The functions f and g are defined for x ∈ R byf: x ↦ 2x − 1g: x ↦ x² + x.Express gf(x) in the form a(x + b)² + c, where a, b and care constants.
The functions f and g are defined for x ∈ R by f: x ↦ 4x − a g: x ↦ b + ax where a and b are constants. Given that f−¹(1) = 2 and g−¹(−1) = −3, find the values of a and b.
Functions f and g are defined by f: x ↦ x − 2, x ∈ R and g: x ↦ x² − x, x ∈ R. Find the set of values of x which satisfy fg(x) ≥ 0.
Given that f(x) = 3x − 1, g(x) = x² + 4 and fg(x) = gf(x), where x ∈ R, show that x² − x − 1 = 0.
The functions f and g are defined for x ∈ R by f: x ↦ 3x + 2 g: x ↦ x² + 1a) Find the values of x for which fg(x) = gf(x).b) Find an expression for (fg)−¹(x), stating its domain. c) Explain why (fg)−¹(x) is not a function.
Find if the following functions are self-inverse functions.a)b) g(x) =2x − 1, x ∈ Rc) f(x) = ₂x²R₁x #0 X
The functions f is defined bySolve ff(x) = −2. x-1 f: x→ *-¹, x € R, x ± 0. Xx
The functions f, g, and h are defined byFor each of these functions determine whether or not the inverse function exists. If it does, write down the inverse in its simplest form. If it does not, explain why. f: x + 3x + 5 g: x4-2x² h: x 4 + 2x² -1 ≤x≤5 x 20 * 2-1
The function f is defined by f: x ↦ 4x − x² for x ≥ 2.a) Express 4x − x in the form a − (x − b)², where a and b are positive constants.b) Express f−1(x) in terms of x.
The functions f and g are defined for x ∈ R byf: x ↦ a − x g: x ↦ x² + ax + b, where a and b are constants.Given that fg(−1) = −1 and gf(−1) = −1, find the values of a and b.
a) Find an expression for f−¹(x). b) Find the domain for which this inverse function is defined.c) Show that ff−¹(x) = x.f(x) = x² +1/2x² +1 , x ∈ R, x ≥ 2
a) The function f: x ↦ 2x² − 12x + 8 is defined for x ∈ R.i) Express f(x) in the form a(x + b)² + c, where a, b and care constants.ii) Find the range of f(x).b) The function g: x ↦ 2x² − 12x + 8 is defined for x ≥ D.i) Find the smallest value of D for which g has an inverse.ii) For
The functions fand g are defined for x ∈ R, by f: x ↦ x + 4, x ∈ R and g: x ↦ x² + 2, x ∈ R. Find the range of fg(x).
The function g: x ↦ x² + 10x + p is defined for the domain x ∈ R, where p is a constant.a) Express g(x) in the form (x + a)2 + b + p, where a and b are constants.b) State the range of g in terms of p.
Express each of the quadratic equations in the form y = a (x + b)² + c. Sketch the curve, stating the coordinates of the vertex and whether there is a maximum or minimum value of y.y = x² − 4x − 8
Solve each of these inequalities. Sketch the curve for each, showing the interval that satisfies each inequality.a) (x + 2)(x – 5) ≤ 0b) (x – 1)(x – 3) > 0c) (3x + 2)(x + 4) > 0d) (x – 7)(x + 10) < 0e) (6 – x)(3x) ≤ 0f) (5 + x)(1 – 2x) ≥ 0
Write each of these expressions in the form (x + p)² + q or q – (x + p)², where p and q are constants.a) x² + 2x – 5b) x² – 10x + 20 c) x² – 4x + 1d) 6 – 8x – x² e) 10 – 16x – x² f) x² + 20x – 45 g) x² + 12x + 16 h) 3 – 6x – x²
Solve each of these quadratic equations. Write your answers correct to 2 decimal places.a) 2x² – 3x – 4 = 0 b) 5x2 – 11x + 4 = 0c) 3x² + 12x + 5 = 0 d) x² + 5x – 2 = 0e) 4x²+2x – 5 = 0f) x²+ x – 4 = 0 g) 3x² – 6x – 2h) 9x = 6x² + 2i) 3 – 8x = 2x² j)
Express each of the quadratic equations in the form y = a (x + b)² + c.Sketch the curve, stating the coordinates of the vertex and whether there is a maximum orminimum value of y. y = x² + 2x

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Complete Pure Mathematics 1 For Cambridge International AS & A Level 2nd Edition Jean Linsky, Brian Western, James Nicholson - Solutions

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