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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

i. Prove that 2 cosec 2θ tanθ Ξ sec2 θ.ii. Hencea. Solve the equation 2 cosec 2θ tanθ = 5 for 0 < x < π.b. Find the exact value of TU 2 cosec 4x tan 2x dx.
Express each of the following improper fractions as the sum of a polynomial and a proper fraction.a.b.c.d.e.f. 8x 2x-5
Express the following proper fractions as partial fractions.a.b.c.d.e.f. 6x-2 (x-2)(x + 3)
Expand the following, in ascending powers of x, up to and including the term in x3. State the range of values of x for which each expansion is valid.a. (1 + x)−2b. (1 + 3x)−1c. (1 −
Expand (1 − 2x)−4 in ascending powers of x, up to and including the term in x3, simplifying the coefficients.
a. Express 7x - 1/(1 - x)(1 + 2x) in partial fractions.b. Hence obtain the expansion of 7x - 1/(1 - x)(1 + 2x) in ascending powers of x, up to and including the term in x3.
a. Express 5x2 + x/(1 - x)2(1 - 3x) in partial fractions.b. Hence obtain the expansion of 5x2 + x/(1 - x)2(1 - 3x) in ascending powers of x, up to and including the term in x3.
The equation x3 + 5x − 7 = 0 has a root α between 1 and 1.2.The iterative formulacan be used to find the value of α.a. Using a starting value of x1 = 1.1, find and write down, correct to 4
a. On the same axes, sketch the graphs of y = √1+x and y = x2.b. Using your answer to part a, explain why the equation x2 − √1 + x = 0 has two roots.c. Show, by calculation, that the smaller of
The terms of the sequence generated by the iterative formula with initial value x1 = 1.5, converge to α.a. Use this formula to find α correct to 2 decimal places. Give the result of each
The diagram shows part of the graph of y = sin x. The points (π/6, 1/2) and (π/3, √3/2) lie on the curve.a. Find the exact value ofb. Hence show that y NI- N/5 B|6 B|M v = sinx X
The equation of a curve is 2x + y ln x = 4y . Find the equation of the tangent to the curve at the point with coordinates (1,1/2).
The parametric equations of a curve are x = 3(1 + sin2 t), y = 2 cos3 t. Find dy/dx in terms of t, simplifying your answer as far as possible.
Find the gradient of the curve 5exy2 + 2exy = 88 at the point (0.4)
The equation of a curve is 3x2 + 4xy + y2 = 24. Find the equation of the normal to the curve at the point (1, 3), giving your answer in the form ax + by + c = 0 where a, b and c are integers.
The equation x = cos x + sin x has a root α that lies between 1 and 1.4a. Show that α is also a root of the equationUsing the iterative formulafind the value of α, giving your answer correct to 2
a. Show graphically that the equation 2x = x + 4 has exactly two roots.b. Show, by calculation, that the larger of the two roots is between 2.7 and 2.8.
i. Prove thatii. Hencea. Find the exact value ofb. Evaluate tan 0 + cot 0 = 2 sin 20
The equationhas a root, α, between x = 1 and x = 1.4.a. Show that α also satisfies the equationb. Using an iterative formula based on the equation from part a, with a suitable starting value, find
The functions f and g are defined, for 0 ≤ x ≤ π, by f(x) = ex−2 and g(x) = 5 − cos x.The diagram shows the graph of y = f(x) and the graph of y = g(x).The gradients of the curves are equal
a. By sketching a suitable pair of graphs, show that the equation x3 + 4x = 7x + 4 has only one root for 0 < x < 5.b. Verify by calculation that this root lies between x = 2 and x = 3.
The equation of a curve is y = e−2x tan x, for 0 ≤ x < 1/2π.i. Obtain an expression for dy/dx and show that it can be written in the form e−2x (a + b tan x)2, where a and b are
The terms of a sequence, defined by the iterative formula xn+1 = In(xn2 + 4), converge to the value α. The first term of the sequence is 2.a. Find the value of α correct to 2 decimal places.
In the diagram, A, B and C are points on the circumference of the circle with centre O and radius r. The shaded region, ABC, is a sector of the circle with centre C. Angle OCA is equal to θ
A curve has parametric equationsThe point P on the curve has parameter p. It is given .that the gradient of the curve at P is −0.4.a. Show thatb. Use an iterative process based on the equation in
It is given that the positive constant a is such thati. Show thatii. Use the iterative formulato find α correct to 3 decimal places. Give the result of each iteration to 5 decimal places. -a (4e²x
The equation cosec x = x2 has a root, α, between 1 and 2.The equation can be rearranged either asa. Write down two possible iterative formulae, one based on each given rearrangement. Use the
The parametric equations of a curve arei. Show that dy/dx = sin t.ii. Hence show that the equation of the tangent to the curve at the point with parameter t is y = x sin t − tan t. X= 1 [, y
a. By sketching a suitable pair of graphs, show that the equation 7 - x6 = |x2 - 1|has exactly two real roots, α and β, where α is a positive constant.b. Show that α satisfies the
a. Show, by calculation, that (x + 2)e5x = 1 has a root between x = 0 and x = −0.2.b. Show, by sketching the graph of y = e5x and one other suitable graph, that this is the only root of this
The equation x4 − 1 − x = 0 has a root, α, between x = 1 and = 2.a. Show that α also satisfies the equationb. Write down an iterative formula based on the equation in part a.c. Use your
The equation of a curve is y = 6sin x − 2 cos 2x. Find the equation of the tangent to the curve at the point (1/6 π, 2) Give the answer in the form y = mx + c, where the values of m and c are
The equation of a curve is y3 + 4xy = 16.i. Show that dy/dx = -4y/3y2 + 4xii. Show that the curve has no stationary points.iii. Find the coordinates of the point on the curve where the tangent
Find the x-coordinate of the point on the curve y = (x + 2) √1 − 2x where the gradient is zero.
Find the gradient of the curve y = 8/5 + 2e2x at the point where x = 0.
The equation of a curve is 2x2 + 3xy + y2 = 3.i. Find the equation of the tangent to the curve at the point (2, −1), giving your answer in the form ax + by + c = 0, where a, b and c are
Find the gradient of the curve y = e2x − 5 ln(2x + 1) at the point where x = 0.
Find the equation of the tangent of the curve y = x - 4/2x + 1 at the point where the curve crosses the y-axis.
A curve has equation 2x2 + 3y2 − 2x + 4y = 4. Find the equation of the tangent to the curve at the point(1, -2).
Find the x-coordinate of the points on the curve y = (3 − x)3 (x + 1)2 where the gradient is zero.
Find the gradient of each of the following curves at the point for which x = 0.i. y = 3sin x + tan 2xii. y = 6/1 + e2x
Find the coordinates of the points on the curve y = 1 - 2x/x - 5 at which the gradient is 1.
Find the gradient of the curve 2x3 − 4xy + y3 = 16 at the point where the curve crosses the x-axis.
a. Sketch the graph of the function y = ln(2x − 3).b. Find the gradient of the curve y = ln(2x − 3) at the point where x = 5.
Find the gradient of the tangent to the curve y = (x + 2)(x - 1)3 at the point where the curve meets the y-axis.
The mass, m grams, of a radioactive substance remaining t years after a given time, is given by the formula m = 300e−0.00012t. Find the rate at which the mass is decreasing when t = 2000.
The parametric equations of a curve are x = e3t , y = t2et + 3.i. Show that y = dy/dx = t(t + 2)/3e2tii. Show that the tangent to the curve at the point (1, 3) is parallel to the x-axis.iii. Find the
Find the coordinates of the points on the curve y = (x - 1)2/2x + 5 where the tangent is parallel to the x-axis.
Find the gradient of the curve x2 + 3xy − 5y + y3 = 22 at the point (1, 3).
Differentiate with respect to x.a. x ln x b. 2x3 ln x c. x ln(2x + 1)d. 3x ln 2/x e. x ln(ln x)f. In5x/xg. 2/In xh. In(3x - 2)/xi. In(2x + 1)/4x - 1
a. Sketch the graph of the function y = 1 − e2−x.b. Find the equation of the normal to the curve y = 1 − e2−x at the point where y = 0.
Find the value of dy/dx when x = 4 in each of the following cases:i. y = x ln(x − 3)ii. y = x - 1/x +1
Find the gradient of the curve y = x2 √x + 4 at the point (−3, 9).
Find the gradient of the curve y = x - 5/x + 4 at the point (2, - 1/2).
Find dy/dx for each of these functions.a. x3 + 2xy + y3 = 10b. x2y + y2 = 5xc. 2x2 + 5xy + y2 = 8d. x ln y = 2x + 5e. 2exy + e2xy3 = 10f. ln(xy) = 4 − y2g. xy3 = 2 ln yh. ln x − 2 ln y + 5xy = 3
The answers to question 1 parts a and b are the same. Why is this the case? How many different ways can you justify this?
Differentiate with respect to x.a. 2 + sin x b. 2 sin x + 3 cos x c. 2 cos x − tan xd. 3sin2xe 4tan5x f. 2 cos 3x − sin2xg. tan( 3x + 2 ) h. sin (2x + π/3)i. 2 cos (3x
Differentiate each expression with respect to x.a. y5b. x3 + 2y2 c. 5x2 + ln yd. 2 + sin y e. 6x2y3 f. y2 + xyg. x3 − 7xy + y3 h. x sin y + y cos x i. x3 ln yj. x cos
Differentiate with respect to x.a. ln 3xb. ln 7xc. ln(2x + 1)d. 5 + ln(x2 + 1)e. ln(2x − 1)5f. ln √x − 3g. In(x + 3)2h. 3x + In (2/x)i. 5x + In(2/1 - 2x)j. In(In x)k. In(2 - √x)2l. In
Differentiate with respect to x:a.b.c. √√x 5x - 1
Find the exact value of the gradient of the curve y = 2 sin3x − 4 cos x at the point (π/3, -2).
The equation of a curve is 4x2y + 8 ln x + 2 ln y = 4. Find the equation of the normal to the curve at the point (1,1).
A curve has equation y = x2 ln5x.Find the value of dy/dx and d2y/dx2 at the point where x = 2.
Find the exact coordinates of the stationary point on the curve y = xex.
a. Sketch the curve y = (x − 1)2 (5 − 2x) + 3.b. The curve y = (x − 1)2 (5 − 2x) + 3 has stationary points at A and B. The straight line through A and B cuts the axes at P and Q. Find the
Given that y = sin2 x for 0 < x < π, find the exact values of the x-coordinates of the points on the curve where the gradient is √3/2.
The equation of a curve is 5x2 + 2xy + 2y2 = 45.a. Given that there are two points on the curve where the tangent is parallel to the x-axis, show by differentiation that, at these points, y =
Find the x-coordinate of the point on the curve y = x + 1/√x - 1where the gradient is 0.
The equation of a curve is y = x2 ln x. Find the exact coordinates of the stationary point on this curve and determine whether it is a maximum or a minimum point.
The equation of a curve is y = 6sin x − 2 cos 2x.Find the equation of the tangent to the curve at the point (1/6 π, 2Give the answer in the form y = mx + c, where the values of m and c are correct
The curve y = 2e2x + e−x cuts the y-axis at the point P. Find the equation of the tangent to the curve at the point P and state the coordinates of the point where this tangent cuts the x-axis.
Prove that the gradient of the curve y = 5/2 - tanx is always poitive.
The equation of a curve is y2 − 4xy − x2 = 20.a. Find the coordinates of the two points on the curve where x = 4.b. Show that at one of these points the tangent to the curve is parallel to the
The equation of a curve is y = In x/x. Find the exact coordinates of the stationary point on this curve and determine whether it is a maximum or a minimum point.
Find the exact coordinates of the stationary point on the curve y = (x − 4)ex and determine its nature.
a. By writing sec x as 1/cosx, find d/dx(secx).b. By writing cosec x as 1/sin x, find d/dx(cosecx).c. By writing cot x as cosx/sinx , find d/dx (cotx).
The equation of a curve is y3 − 12xy + 16 = 0.a. Show that the curve has no stationary points.b. Find the coordinates of the point on the curve where the tangent is parallel to the y-axis.
The line 2x − 2y = 5 intersects the curve 2x2y − x2 − 26y − 35 = 0 at three pointsa. Find the x-coordinates of the points of intersection.b. Find the gradient of the curve at each of the
Find the exact coordinates of the stationary point on the curve y = e2x/x2 and determine its nature.
Prove that the normal to the curve y = sin x at the point P(π/2, π/2) intersects the x-axis at the point (π, 0).
Use the laws of logarithms to help differentiate these expressions with respect to x.a. √5x - 1b. In(1/3x + 2) c. In [x(X + 1)5]d. In (2x + 3/x - 1)e. In (1 - 3x/x2)f. In [x(x - 2)/x+4]g. In
The equation of a curve is ln(xy) − y3 = 1.i. Show thatii. Find the coordinates of the point where the tangent to the curve is parallel to the y-axis, giving each coordinate correct to 3
The equation of a curve is y = 5sin3x − 2 cos x . Find the equation of the tangent to the curve at the point (π/3, -1). Give the answer in the form y = mx + c, where the values of m and c are
The equation of a curve is x2 − 4x + 6y + 2y2 = 12. Find the coordinates of the two points on the curve at which the gradient is 4/3.
Find dy/dx in terms of x, for each of the following.a. ey = 2x2 − 1 b. ey = 3x3 + 2xc. ey = (x + 1)(x − 5)
Find the exact value of the x-coordinates of the points on the curve y = x2e-2x at which d2y/dx2 = 0.
A curve has equation y = 3cos 2x + 4sin2x + 1 for 0 x < π. Find the x-coordinates of the stationary points of the curve, giving your answer correct to 3 significant figures.
Find the coordinates of the stationary point on the curve y = e2x+1/x.
A curve has equation y = sin 2x/e2x for 0 < x < π/2. Find the exact value for the x-coordinate of the stationary point of this curve.
The equation of a curve is y = xx. Find the exact value of the x-coordinate of the stationary point on this curve.
By writing 2 as eIn 2, prove that d/dx(2x) = 2x In2.
The curve with equation 6e2x + key + e2y = c, where k and c are constants, passes through the point P with coordinates (ln3, ln 2).i. Show that 58 + 2k = c.ii. Given also that the gradient of
A curve has equation y = sin 2x/e2x for 0 < x < π/2. Find the exact value for the x-coordinate of the stationary point of this curve.
Find the stationary points on the curve x2 − xy + y2 = 48. By finding d2y/dx2 determine the nature of each of these stationary points.
A curve has equation y = e3x/sin3x for 0 < x < π/2. Find the exact value for the x-coordinate of the stationary point of this curve and determine the nature of this point.
Find:a.b.c.d.e.f.g.h.i. e2x dx
Find the value:a. b.c.d.e.f. Л 6 0 1 cos² x cos²x-. dx
Use the trapezium rule with 2 intervals to estimate the value of each of these definite integrals. Give your answers correct to 2 decimal places.a.b.c.d.e.f. 4 S✓ 2 √x² - 2 dx
a. Use the trapezium rule with 6 intervals to estimate the value ofgiving your answer correct to 2 decimal places.b. Use a sketch of the graph of whether the trapezium rule gives an
The diagram shows part of the curve y = x2e−x. Use the trapezium rule with 4 intervals to estimate the value ofgiving your answer correct to 2 decimal places. XP x-əzX S₁²

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