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mathematics
cambridge international as & a level mathematics probability & statistics

Questions and Answers of Cambridge International AS & A Level Mathematics Probability & Statistics

The line y = k − 3x is a tangent to the curve x2 + 2xy − 20 = 0.a. Find the possible values of k.b. For each of these values of k, find the coordinates of the point of contact of the tangent with
Find the set of values of k for which the line y = kx − 3 intersects the curve y = x2 − 9x at two distinct points.
Solve the equationGive your answers correct to 3 significant figures. 5 2 + x-3x+1 = 1.
The diagram shows a right-angled triangle with sides 2x cm, (2x + 1) cm and 29 cm.a. Show that 2x2 + x − 210 = 0.b. Find the lengths of the sides of the triangle. 2x 29 2x+1
Express each of the following in the form p − q(x + r)2.a. 7 − 8x − 2x2 b. 3 − 12x − 2x2 c. 13 + 4x − 2x2 d. 2 + 5x − 3x2
Find the values of k for which the following equations have two equal roots.a. x2 + kx + 4 = 0 b. 4x2 + 4(k − 2)x + k = 0c. (k + 2)x2 + 4k = (4k + 2)x d. x2 − 2x + 1 = 2k(k − 2)
The graph shows y a(22x) b(2x) c.The graph crosses the axes at the points (2, 0), (4, 0) and (0, 90).Find the value of a, the value of b and the value of c. 90 O A/ 2 14 x
Solve the quadratic equation ax2 − bx + c = 0, giving your answers in terms of a, b and c.How do the solutions of this equation relate to the solutions of the equation ax2 + bx + c = 0?
Find the minimum value of x2 − 7x + 8 and the corresponding value of x.
The sum of the circumferences of two circles is 36-cm and the sum of the areas is 170πcm2. Find the radius of each circle.
The area of the trapezium is 35.75cm2.Find the value of x. x-1 X x+3
Find the values of m for which the line y = mx + 6 is a tangent to the curve y = x2 − 4x + 7. For each of these values of m, find the coordinates of the point where the line touches the curve.
Find the set of values of the constant k for which the line y = 2x + k meets the curve y = 1 + 2kx − x2 at two distinct points.
Find the set of values of x for which:a. x2 − 3x ≥ 10 and (x − 5)2 , 4b. x2 + 4x − 21 ≤ 0 and x2 − 9x + 8 > 0c. x2 + x − 2 > 0 and x2 − 2x − 3 ≥ 0
Express each of the following in the form (ax + b)2 + c.a. 9x2 − 6x − 3 b. 4x2 + 20x + 30 c. 25x2 + 40x − 4 d. 9x2 − 42x + 61
Find the values of k for which the following equations have two distinct roots.a. x2 + 8x + 3 = k b. 2x2 − 5x = 4 − kc. kx2 − 4x + 2 = 0 d. kx2 + 2(k − 1)x + k = 0e. 2x2 = 2(x −
a. Write 1 + x − 2x2 in the form p − 2(x − q)2.b. Sketch the graph of y = 1 + x −2x2.
A cuboid has sides of length 5 cm, x cm and y cm. Given that x + y = 20.5 and the volume of the cuboid is 360 cm3, find the value of x and the value of y.
Solve a minimum point. a.b.c.d.e.f. x-1 >3 W
Find the set of values of k for which the line y = 2x − 1 intersects the curve y = x2 + kx + 3 at two distinct points.
The diagram shows a solid formed by joining a hemisphere, of radius r cm, to a cylinder, of radius r cm and height h cm. The total height of the solid is 18cm and the surface area is 205π cm2. Find
a. Find the coordinates of the vertex of the parabola y = 4x2 − 12x + 7.b. Find the values of the constant k for which the line y = kx + 3 is a tangent to the curve y = 4x2 − 12x + 7.
Find the range of values of x for which 2x2 − 3x − 40 > 1.
Solve by completing the square.a. x2 + 8x − 9 = 0 b. x2 + 4x − 12 = 0 c. x2 − 2x − 35 = 0d. x2 − 9x + 14 = 0 e. x2 + 3x − 18 = 0 f. x2 + 9x − 10 = 0
Find the values of k for which the following equations have no real roots.a. kx2 − 4x + 8 = 0 b. 3x2 + 5x + k + 1 = 0c. 2x2 + 8x − 5 = kx2 d. 2x2 + k = 3(x − 2)e. kx2 + 2kx = 4x −
Prove that the graph of y 4x2 + 2x + 5 = does not intersect the x-axis.
Find the equations of parabolas A, B and C. 4 B y 12 10 8 4- 2 0 -2 -4 -6 8 4 6 C A x
SolveLeave your answers in surd form. 5 x + 2 + 3 x-4 = 2.
Find the set of values of k for which the line x + 2y = k intersects the curve xy = 6 at two distinct points.
A curve has equation y = 5 − 2x + x2 and a line has equation y = 2x + k, where k is a constant.a. Show that the x-coordinates of the points of intersection of the curve and the line are given by
Solve by completing the square. Leave your answers in surd form.a. x2 + 4x − 7 = 0 b. x2 − 10x + 2 = 0 c. x2 + 8x − 1 = 0d. 2x2 − 4x − 5 = 0 e. 2x2 + 6x + 3 = 0 f.
The equation kx2 + px + 5 = 0 has repeated real roots. Find k in terms of p.
The diagram shows a right-angled triangle with sides xm, (2x + 5)m and 10m.Find the value of x. Leave your answer in surd form. X 10 2x+5
The line y = 2 − x cuts the curve 5x2 − y2 = 20 at the points A and B.a. Find the coordinates of the points A and B.b. Find the length of the line AB.
Find the set of values of k for which the line y = k − x cuts the curve y = 5 − 3x − x2 at two distinct points.
A curve has equation y = x2 − 5x + 7 and a line has equation y = 2x − 3.a. Show that the curve lies above the x-axis.b. Find the coordinates of the points of intersection of the line and the
Find the range of values of k for which the equation kx2 − 5x + 2 = 0 has real roots.
The diagram shows eight parabolas. The equations of two of the parabolas are y = x2 − 6x + 13 and y = −x2 − 6x − 5.a. Identify these two parabolas and find the equation of each of the other
Find the set of values of m for which the line y = mx + 5 does not meet the curve y = x2 − x + 6.
A curve has equation y = 10x − x2.a. Express 10x − x2 in the form a − (x + b)2.b. Write down the coordinates of the vertex of the curve.c. Find the set of values of x for which y ≤ 9.
Prove that the roots of the equation 2kx2 + 5x − k = 0 are real and distinct for all real values of k.
A parabola passes through the points (0, −24), (−2, 0) and (4, 0). Find the equation of the parabola.
Find the set of values of k for which the line y = 2x − 10 does not meet the curve y = x2 − 6x + k
The path of a projectile is given by the equationwhere x and y are measured in metres.a. Find the range of this projectile.b. Find the maximum height reached by this projectile. y
A line has equation y = kx + 6 and a curve has equation y = x2 + 3x + 2k, where k is a constant.i. For the case where k = 2, the line and the curve intersect at points A and B. Find the distance AB
Find the real solutions of the equation (3x2 + 5x − 7)4 = 1.
A parabola passes through the points (−2, −3), (2, 9) and (6, 5). Find the equation of the parabola.
The line 7y − x = 25 cuts the curve x2 + y2 = 25 at the points A and B. Find the equation of the perpendicular bisector of the line AB.
Find the value of k for which the line y = kx + 6 is a tangent to the curve x2 + y2 − 10x + 8y = 84.
The line y = mx + c is a tangent to the curve ax2 + by2 = c, where a, b, c and m are constants.Prove that m² = abc-a b
A curve has equation y = x2 − 4x + 4 and a line has the equation y = mx, where m is a constant.i. For the case where m = 1, the curve and the line intersect at the points A and B. Find the
Prove that x2 + kx + 2 = 0 has real roots if k ≥ 2 √2. For which other values of k does the equation have real roots?
Prove that any quadratic that has its vertex at ( p, q) has an equation of the form y = ax2 − 2apx + ap2 + q for some non-zero real number a.
The straight line y = x + 1 intersects the curve x2 − y = 5 at the points A and B.Given that A lies below the x-axis and the point P lies on AB such that AP : PB = 4 : 1, find the coordinates of P.
The line y = mx + c is a tangent to the curve y = x2 − 4x + 4. Prove that m2 + 8m + 4c = 0.
i. Express 2x2 − 4x + 1 in the form a(x + b)2 + c and hence state the coordinates of the minimum point, A, on the curve y = 2x2 − 4x + 1. The line x − y + 4 = 0 intersects the curve y = 2x2 −
Find an expression for f−1(x) for each of the following functions.a. f(x) = 5x − 8 for x ∈ ℝ b. f(x) = x2 + 3 x x + for x ∈ ℝ, x ≥ 0c. f(x) = (x − 5)2 + 3 for x ∈ ℝ, x ≥
The line x − 2y = 1 intersects the curve x + y2 = 9 at two points, A and B.Find the equation of the perpendicular bisector of the line AB.
Find the equation of each graph after the given transformationa.b.c.d.e.f.g.h. y = 2x² (8) after translation by
Functions f and g are defined for x ∈ ℝ by:Express gf(x) in the form a − b(x − c)2, where a, b and c are constants. t* - XS
The diagram shows the graph of y = g(x).Sketch the graphs of each of the following functions.a. y = −g(x) b. y = g(−x) 4 -2 3. 1- 의 -2- -3 4 y = g(x) 123 4x
a. Split 10 into two parts so that the difference between the squares of the parts is 60.b. Split N into two parts so that the difference between the squares of the parts is D.
Find: f(x) x2 + 6 for x ∈ ℝ g(x) = √x + 3 − 2 for x ∈ ℝ, x ≥ −3a. fg(6) b. gf(4) c. ff(−3)
The diagram shows the graph of y = f(x). Sketch the graphs of each of the following functions.a. y = 3f(x)b. y = f(2x) -6 4. 2- -20 -2- 4. y = f(x)
The diagram shows the graph of y = g(x) . Sketch the graph of each of the following.a. y = g(x + 2) + 3 b. y = 2g(x) + 1c. y = 2 − g(x) d. y = 2g(−x) + 1e. y = −2g(x) − 1 f. y
Which of these graphs represent functions? If the graph represents a function, state whether it is a one-one function or a many-one function.a. y = 2x − 3 for x ∈ ℝ b. y = x2 − 3 for x
a. Represent on a graph the function:b. Explain why this relation is not a function. X→ [9-x² for xe R, -3 ≤ x ≤ 2 2x+1 for xeR, 2≤x≤4
The diagram shows the graph of y = f(x). Write down, in terms of f(x), the equation of the graph of each of the following diagrams.a. b. c. 4-3-2 y 2 0 _y=f(x) 2 3 x
Find the translation that transforms the graph.a. y = x2 + 5x − 2 to the graph y = x2 + 5x + 2b. y = x3 + 2x2 + 1 to the graph y = x3 + 2x2 − 4c. y = x2 − 3x to the graph y = (x + 1)2 − 3(x +
a. State the domain and range of f−1.b. Find an expression for f−1(x).f : x ↦ x2 + 4x for x ∈ ℝ, x > −2
The diagram shows a sketch of the curve with equation y = f(x).a. Sketch the graph ofb. Describe fully a sequence of two transformations that maps the graph of y = f(x) onto the graph of y = f(3 −
a. Find an expression for f−1(x).b. State the domain and range of f−1.c. Sketch, on the same diagram, the graphs of y = f(x) and y = f−1(x), making clear the relationship.f : x ↦ 2x − 1 for
The diagram shows the graph ofa. State the range of f .b. Find an expression for f−1(x).c. State the domain and range of f−1.d. On a copy of the diagram, sketch the graph of y = f−1(x), making
Find the equation of each graph after the given transformation.a. y = 5x2 after reflection in the x-axis.b. y = 2x4 after reflection in the y-axis.c. y = 2x2 − 3x + 1 after reflection in the
a. Represent on a graph the relation:b. Explain why this relation is not a function. y = [x²+1 2x-3 for 0≤x≤2 for 2 ≤x≤4
a. Find an expression for f−1(x).b. Find the domain of f−1. f: xH 5 2x+1 for x = R, x 2
Find the equation of each graph after the given transformationa. y = 3x2 after a stretch parallel to the y-axis with stretch factor 2.b. y = x3 − 1 after a stretch parallel to the y-axis with
Describe the single transformation that maps the graph:a. y = x2 + 2x − 5 onto the graph y = 4x2 + 4x − 5b. y = x2 − 3x + 2 onto the graph y = 3x2 − 9x + 6c. y = 2x + 1 onto the graph y
The diagram shows the graph of y = f(x). Sketch the graphs of each of the following functions.a. y = f(x) − 4b. y = f (x − 2)c. y = f (x + 1) − 5 YA 4- 3- 2. 4-32-18| -2. 3. •y = f(x) 1 2 3 4
Given that f(5) = 3 and f(3) = −3:f(x) = ax + b for x ∈ ℝa. Find the value of a and the value of bb. Solve the equation ff(x) = 4.
State the domain and range for the functions represented by these two graphs.a.b. (-1,4) y (1,8) y=7+2x-x² (5,-8) x
Given that y = x2, find the image of the curve y = x2 after each of the following combinations of transformations.a. A stretch in the y-direction with factor 3 followed by a translation by the
A curve has equation y = x2 + 6x + 8.a. Sketch the curve, showing the coordinates of any axes crossing points.b. The curve is translated by the vectorthen stretched vertically with stretch factor
For each of the following functions, find an expression for f −1(x) and, hence, decide if the graph of y = f(x) is symmetrical about the line y = x.a.b.c.d. f(x): = x + 5 2x - 1 for x ER, x 7 1
a. Find gf(x).b. Solve the equation gf(x) = 2.f : x ↦ 2x + 3 for x ∈ ℝ 12 1-x g: xH. for x = R, x # 1
Find the equation of each graph after the given transformation.a. y = 5x2 after reflection in the x-axis.b. y = 2x4 after reflection in the y-axis.c. y = 2x2 − 3x + 1 after reflection in the
a. On the same diagram, sketch the graphs of y = 2x and y = 2x + 2.b.Find the value of a.c.Find the value of b. y = 2x can be transformed to y = 2x + 2 by a translation of (9) a
Find the range for each of these functions.a. f(x) = x + 4 for x > 8 b. f(x) = 2x − 7 for −3 ≤ x ≤ 2c. f(x) = 7 − 2x for −1≤ x ≤ 4d. f : x ↦ 2x2 for 1 ≤ x ≤ 4e. f(x) =
Find the equation of the image of the curve y = x2 after each of the following combinations of transformations and, in each case, sketch the graph of the resulting curve.a. A stretch in the
a. Find an expression for f−1(x).b. Find the domain of f−1.f : x ↦ (x + 1)3 − 4 for x ∈ ℝ, x ≥ 0
A cubic graph has equation y = ( x + 3)( x − 2 )( x − 5 ).Write, in a similar form, the equation of the graph after a translation of (3)
a.where a and b are constants. Prove that this function is self-inverse.b.where a, b, c and d are constants. Find the condition for this function to be self-inverse. f(x) = x + a bx - 1 for x = R, x
The function f : x ↦ x2 − 2 is defined for the domain x ≥ 0.a. Find f−1(x) and state the domain of f−1.b. On the same diagram, sketch the graphs of f and f−1.
a. Find gh(x).b. Solve the equation gh(x) = 14.g(x) = x2 − 2 for x ∈ ℝ h(x) = 2x + 5 for x ∈ ℝ
Given that f(x) = x2 + 1, find the image of y = f(x) after each of the following combinations of transformations.a. Translation followed by a stretch parallel to the y-axis with stretch factor
The graph of y = x2 − 4x + 1 is translated by the vectorFind, in the form y = ax2 + bx + c, the equation of the resulting graph. (¹) 2
a. Explain why g has an inverse.b. Find an expression for g−1(x).g : x ↦ 2x2 − 8x + 10 for x ∈ ℝ, x ≥ 3
Solve the equation fg(x) = 5.f(x) x2 + 1 for x ∈ ℝ 3 x-2 نیا for x ER, x 2
Solve the equation hg(x) = 11. g(x)= = 2 x+1 for x = R, x = −1 h(x) = (x + 2)² - 5 for x = R
The function f : ↦ −x2 + 6x − 5 is defined for x ≥ m, where m is a constant.i. Express -x2 +6x −5 in the form a(x + b)2 + c, where a, b and c are constants.ii State the smallest

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