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mathematics
cambridge international as & a level mathematics probability & statistics

Cambridge International AS And A Level Mathematics Pure Mathematics 1 Coursebook 1st Edition Sue Pemberton, Julian Gilbey - Solutions

The diagram shows the curve and the line y = 1/2 x + 2 meeting at the points (−4, 0) and (0, 2). Find the area of the shaded region. y = √√√x +4
The curvemeets the y-axis at the point Q.The tangent at the point P(3, 3) to this curve meets the y-axis at the point R.a. Find the equation of the tangent to the curve at P.b. Find the exact value of the area of the shaded region PQR. y = √2x+3
The diagram shows part of the curve y = 5√x − x.The curve meets the x-axis at O and P.a. Find the coordinates of P.b. Find the volume obtained when the shaded region is rotatedthrough 360° about the x-axis. 0 y = 5√√x-x P X
a. Sketch the graph of y = (x − 2)2.b. Find the volume of the solid formed when the enclosed region bounded by the curve, the x-axis and the y-axis is rotated through 360° about the x-axis.
A function y = f(x) has gradient function f´(x) = 8 − 2x. The maximum value of the function is 20. Find f(x) and sketch the graph of y = f(x).
The diagram shows the curve y = 10 + 9x − x2. Points P(6, 28) and Q(10, 0) lie on the curve. The tangent at P intersects the x-axis at R.a. Find the equation of the tangent to the curve at P.b. Find the area of the shaded region. y = 10 + 9x-x² P(6, 28) R X
A curve is such that Given that the curve passes through the point (2, −7) find:a. The equation of the curveb. The set of values of x for which the gradient of the curve is positive. dy d.x 3x2 + x – 10-
The diagram shows part of the curvethat intercepts the y-axis at the point P. The shaded region is bounded by the curve, the y-axis and the line y = 1.a. Find the coordinates of P.b. Find the volume obtained when the shaded region is rotated through 360° about the y-axis. X = 9 -1
a. Show thatb. Use your result from part a to evaluate the area of the shaded region. dx x² +5 x √x² +5
A curve is such thatThe gradient of the curve at the point (0, 4) is 10.a. Express y in terms of x.b. Show that the gradient of the curve is never less than 4. d²y 12x + 12. dx²
The diagram shows the curve y = 4x − x3.The point P has coordinates (2, 0) and the point Q has coordinates (−4, 48).a. Find the equation of the tangent to the curve at P.b. Find the area of the shaded region. y = 4x-x³ 2 P *
The diagram shows part of the curveThe line y = 7 intersects the curve at the points P and Q.a. Find the coordinates of P and Q.b. Find the volume obtained when the shaded region is rotated through 360° about the x-axis. 2 y = 3x + = X
a. Sketch the curve y = (x − 3)2 + 2.b. The region enclosed by the curve, the x-axis, the y-axis, the line x = 3 is rotated through 360° about the x-axis. Find the volume obtained, giving your answer in terms of π.
A curve is such thatGiven that the curve has a minimum point at (−2, −6), find the equation of the curve. d²y dx² = -6x - 4.
The equation of a curve isi. Find an expression for dy/dx and determine, with a reason, whether the curve has any stationary points.ii. Find the volume obtained when the region bounded by the curve, the coordinate axes and the line x = 1 is rotated through 360° about the x-axis.iii. Find the set
A line has equation y = 2x + c and a curve has equation y = 8 − 2x − x2.i. For the case where the line is a tangent to the curve, find the value of the constant c.ii. For the case where c = 11, find the x-coordinates of the points of intersection of the line and the curve.Find also, by
A function f is defined asa. Find an expression, in terms of x, for f´(x) and explain how your answer shows that f is a decreasing function.b. Find an expression, in terms of x, for f−1(x) and find the domain of f−1.c. On a diagram, sketch the graph of y = f(x) and the graph of y = f−1(x) ,
Find the shaded area enclosed by the curve y = 2√x, the line x + y = 8 and the x-axis. 0 y = 2√5 x+y=8
The diagram shows the line y = 1 and part of the curvei. Show that the equationcan be written in the formii. Find Hence find the area of the shaded region.iii The shaded region is rotated through 360° about the y-axis. Find the exact value of the volume ofrevolution obtained. y= = 2 √√x+1
The diagram shows part of the curveThe point P(4, 3) lies on the curve.a. Find the volume obtained when the shaded region is rotated through 360° about the y-axis.b. Find the volume obtained when the shaded region is rotated through 360° about the x-axis. y = √25-x².
A curve is such thatThe curve passes through the pointi. Find the equation of the curve.ii. Find d2y/dx2.iii. Find the coordinates of the stationary point and determine its nature. dyx2-x = dx
The diagram shows the curvemeeting the x-axis at A and the y-axis at B.The y-coordinate of the point C on the curve is 3.i. Find the coordinates of B and C.ii. Find the equation of the normal to the curve at C.iii. Find the volume obtained when the shaded region is rotated through 360° about the
The diagram shows the curve y2 = 2x − 1 and the straight line 3y = 2x − 1.The curve and straight line intersect at x = 1/2 and x = a, where a is a constant.i. Show that a = 5.ii. Find, showing all necessary working, the area of the shaded region.
The tangent to the curve y = 8x − x2 at the point (2, 12) cuts the x-axis at the point P.a. Find the coordinates of P.b. Find the area of the shaded region. P YA (2, 12) y = 8x-x² X
The diagram shows part of the curveThe shaded area is rotated through 360° about the x-axis between x = 0 and x = p. Show that as p → ∞, the volume approaches the value 2π. y = 2 2x+1
The diagram shows part of the curveand the tangent to the curve at P(9, 4).a. Find the equation of the tangent to the curve at P.b. Find the area of the shaded region. Give your answer correct to 3 significant figures. y = 5-√10-x
A curve y = f(x) has a stationary point at P(2, −13) and is such that f´(x) = 2x2 + 3x − k, where k is a constant.a. Find the x-coordinate of the other stationary point, Q, on the curve y = f(x).b. Determine the nature of each of the stationary points P and Q.
A curve has equation y = f(x) and is such thati. By using the substitutionor otherwise, find the values of x for which the curve y = f(x) has stationary points.ii. Find f´´(x) and hence, or otherwise, determine the nature of each stationary point.iii. It is given that the curve y = f(x) passes
The function f is defined for x > 0 and is such thatThe curve y = f(x) passes through the point P(2, 6).i. Find the equation of the normal to the curve at P.ii. Find the equation of the curve.iii. Find the x-coordinate of the stationary point and state with a reason whether this point is a
A curve y = f(x) has a stationary point at (1, −1) and is such thatFind f´(x) and f(x). f"(x) = 2 + - 4 ایان
A curve is such that dy/dx = k + x where k is a constant.a. Given that the tangents to the curve at the points where x = 5 and x = 7 are perpendicular, find the value of k.b. Given also that the curve passes through the point (10, −8), find the equation of the curve.
The diagram shows part of the curveThe curve intersects the y-axis at A(0, 4) . The normal to the curve at A intersects the line x = 4 at the point B.i. Find the coordinates of B.ii. Show, with all necessary working, that the areas of the regions P and Q are equal. y = 8 √3x +4
The point P(3, 5) lies on the curvei. Find the x-coordinate of the point where the normal to the curve at P intersects the x-axis.ii. Find the x-coordinate of each of the stationary points on the curve and determine the nature of each stationary point, justifying your answers. 1 9 x-1 x-5
A curve is such thatGiven that the curve has a minimum point at (3, −49), find the coordinates of the maximum point. d²y d.x² 2 2x+8.
The diagram shows the curve y = (3 − 2x)3 and the tangent to the curve at the pointi. Find the equation of this tangent, giving your answer in the form y = mx + c.ii. Find the area of the shaded region. (1,8).
The figure shows part of the curve y = f(x). The points P(2, 4) and Q(7, 12) lie on the curve. Given that find the value of 7 2 y dx = 42,
The figure shows part of the curve y = g(x). The points A(2, 8) and B(6, 1) lie on the curve. Given thatfind the value of 2 ydx = 16,
The diagram shows parts of the curvesintersecting at points P(0, 1) and Q(2, 3). The angle between the tangents to the curves at Q is α.i. Find α, giving your answer in degrees correct to 3 significant figures.ii. Find by integration the area of the shaded region. y = (4x + 1)² and y = x² +1
The diagram shows part of the curveand a point P(6, 5) lying on the curve. The line PQ intersects the x-axis at Q(8, 0).i. Show that PQ is a normal to the curve.ii. Find, showing all necessary working, the exact volume of revolution obtained when the shaded region is rotated through 360° about the
The diagram shows part of the curve y = (x − 2)4 and the point A(1, 1) on the curve.The tangent at A cuts the x-axis at B and the normal at A cuts the y-axis at C.i. Find the coordinates of B and C.ii. Find the distance AC, giving your answer in the form √a/b, where a and b are integers.iii.
A curve is such thatand the point P(1, 6) is a point on the curve.a. Find the equation of the curve.b. Find the coordinates of the stationary point on the curve and determine its nature. dy d.x 3√x-6
Find Cambridge International AS 2 J (5-3) 5x - X d.x.
Find the equation of the curve, given dy/dx and a point P on the curve.a.b.c.d. dy dx = (2x – 1)3, P = 2.
a. Differentiate (2x2 − 1)5 with respect to x.b. Hence, find fx(2x² - 1)4 dx.
Find the volume obtained when the shaded region is rotated through 360° about the x-axis.a.b.c.d. y=x² + ²/ 2 X
Evaluate: Consider the area boundeda.b.c.d.e.f. √₁² ( 3x² − 2 + 1/72 ) ₁² - dx
A curve is such thati. A point P moves along the curve in such a way that the x-coordinate is increasing at a constant rate of 0.3 units per second. Find the rate of change of the y-coordinate as P crosses the y-axis. The curve intersects the y-axis where y = 4/3.ii. Find the equation of the curve.
Find f(x) in terms of x for each of the following.a. f´(x) = 5x4 − 2x3 + 2 b. f´(x) = 3x5 + x2 − 2xc.d. f(x) = 3 + + X +6x
A curve is such that dy/dx = 3 − 2x and (1, 11) is a point on the curve.a. Find the equation of the curve.b. A line with gradient 1/5 is a normal to the curve at the point (4, 5). Find the equation of this normal.
A curve is such thatwhere k is a constant. Given that the curve passes through the points (6, 2.5) and (−3, 1), find the equation of the curve. dy dx k x
The diagram shows the curve y = (x − 3)2 and the line y = 2x − 3 that intersect at points A and B. Find the area of the shaded region. A B y = (x-3)² X y=2x-3
The diagram shows part of the curveShow that as p → ∞, the shaded area tends to the value 2. y = 20 (2x + 5)²
The diagram shows the curve y = x(x − 2)(x − 4) that crosses the x-axis at the points O(0, 0), A(2, 0) and B(4, 0). Show by integration that the area of the shaded region R1 is the same as the area of the shaded region R2. R₁ A R₂ B x
A curve is such thatwhere k is a constant. The gradient of the normal to the curve at the point (4, 2) is 1/12. Find the equation of the curve. dy d.x k(x - 5)³,
a. Given thatand state the value of k.b. Hence, find y 1 x²-5 show that dy kx dx (x²-5)²²
A curve is such thatand the point (9, 2) lies on the curve.i. Find the equation of the curve.ii. Find the x-coordinate of the stationary point on the curve and determine the nature of the stationary point. dy dx = 3x2 - 6
Find the volume obtained when the shaded region is rotated through 360° about the y-axis.a.b. 11 2 y=x² + 2 X
Find y in terms of x for each of the following.a.b.c.d.e.f. dy dx = x(x+5)
A curve is such that and the point (3, 5.5) lies on the curve. Find the equation of the curve. || 치어 5.x
A curve is such thatwhere k is a constant. Given that the curve passes through the points (1, −3) and (3, 11), find the equation of the curve dy = kx²-12x+5, d.x
The diagram shows the curve y = −x2 + 11x − 18 and the line 2x + y = 12 . Find the area of the shaded region. y=-x² + 11x-18 B X 2x + y = 12
Show that none of the following improper integrals exists.a.b.c.d.e.f. 00 S 4 되어 6 dx
Evaluate: Consider the area bounded bya.b.c.d.e.f. -1 (2x + 3)³ dx
A curve is such thatGiven that the curve passes through the point P(2, 1), find:a. The equation of the normal to the curve at Pb. The equation of the curve. dy d.x 5 √2.x - 3
a. Differentiatewith respect to x.b. Hence, find 3x (4-3x²)²
The diagram shows part of the curve y = a/x , where a > 0. The volume obtained when the shaded region is rotated through 360° about the x-axis is 18π. Find the value of a. y 2 y = x
A curve has equation y = f(x) . It is given thatand that f(2) = 3. Find f(x). f'x) 3 Jx+2 8 X
Find each of the following.a.b.c.d.e.f. [12x³ dx
A curve is such thatwhere k is a constant. Given that the curve passes through the pointP(1, 6) and that the gradient of the curve at P is 9, find the equation of the curve. dy = kx² dx 6 X
Sketch the curve and find the total area bounded by the curve and the x-axis for each of these functions.a. y = x(x − 3)(x + 1) b. y = x ( x2 − 9)c. y = x(2x − 1)(x + 2) d. y = (x − 1)(x + 1)(x − 4)
A curve is such thatlies on the curve.i. Find the equation of the curve.ii. Find the set of values of x for which the gradient of the curve is less than 1/3. dy dx 3 (1+2x)² and the point (1,2
The diagram shows part of the curveFind the volume obtained when the shaded region is rotated through 360° about the x-axis. y = √√x³+4x²+3x+2
a. Given thatb. Hence, evaluate y = 2 x² +5 find dy dx
Sketch the curve and find the enclosed area for each of the following.a. y = x4 − 6x2 + 9, the x-axis and the lines x = 0 and x = 1.b. the x-axis and the lines x = 1 and x = 2c.the x-axis and the lines x = 2 and x = 5d. y = 3√x, the x-axis and the lines x = 1 and x = 4e. the x-axis
The diagram shows parts of the curves y = (2x − 1)2 and y2 = 1 − 2x , intersecting at points A and B.i. State the coordinates of A.ii. Find, showing all necessary working, the area of the shaded region. B y=(2x-1)² A ² = 1-2x X
The diagram shows part of the curveThe shaded region is bounded by the curve, the y-axis, and the lines y = 1 and y = 3. Find the volume, in terms of π, when this shaded region is rotated through 360° about the y-axis. x = 6 2 +1.
Find each of the following.a.b.c.d.e.f.g.h.i. f(x+1) (x + 1)(x + 4)dx
Given that the curve passes through the point (1, 3), find:a. The equation of the curveb. The equation of the tangent to the curve when x = 4. dy. dx = SXVX+2.
A curve is such thata. Show that the curve has a stationary point when x = 1 and determine its nature.b. Given that the curve passes through the point (0, 13) , find the equation of the curve. dy dx 12 √3.x + 1 - 4.x - 2.
a. Given thatb. Hence, evaluate y = (x - 2), find dy dx
Sketch the following curves and lines and find the area enclosed between their graphs.a. y = x2 − 3 and y = 6b. y = −x2 + 12x − 20 and y = 2x + 1c. y = x2 − 4x + 4 and 2x + y = 12
The diagram shows part of the line 3x + 8y = 24. Rotating the shaded region through 360° about the x-axis would give a cone of base radius 3 and perpendicular height 8.Find the volume of the cone using:a. Integrationb. The formula for the volume of a cone. УА 3 0 3x +8y = 24 8 X
A curve is such thatwhere k is a constant. The point P(3, 2) lies on the curve and the normal to the curve at P is x + 4y = 11. Find the equation of the curve. dy dx 4 √2x + k
A curve has equation y = f(x) and it is given thatThe point A is the only point on the curve at which the gradient is −1.i. Find the x-coordinate of A.ii. Given that the curve also passes through the point (4, 10), find the y-coordinate of A, giving your answer as a fraction. f'(x)= 3x2 - 2x 2.
a. Differentiate (√x + 3)8 with respect to x.b. Hence, find √(√x + 3)² dx.
Sketch the curve and find the enclosed area for each of the following.a. y = x3 , the y-axis and the lines y = 8 and y = 27b. x = y2 + 1 , the y-axis and the lines y = −1 and y = 2
Sketch the curves y = x2 and y = x(2 − x) and find the area enclosed between the two curves.
a. Given thatb. Hence, evaluate y = (√x + 1) ³ 10 find dy d.x
A curve is such that dy/dx = kx + 3, where k is a constant. The gradient of the normal to the curve at the point (1, −2) is − 1/7. Find the equation of the curve.
A function is defined for x ∈ Rand is such that f´(x) = 6x − 6. The range of the function is given by f(x) ≥ 5.a. State the value of x for which f(x) has a stationary value.b. Find an expression for f(x) in terms of x.
The diagram shows the curveThe shaded region is bounded by the curve, the y-axis and the line y = 3. Find the area of the shaded region. y = √2x +1.
Show that each of the following improper integrals has a finite value and, in each case, find this value.a.b.c.d.e.f.g.h.i. ܙ 1 2 -2 X -dx
Find the area of the region bounded by the curve y = 5 + 6x − x2, the line x = 4 and the line y = 5. 4 y = 5+ 6x-x² x
Find the area of each shaded region.a.b.c.d. y=x²-8x² + 16x 4 X
Find the equation of the curve, given dy/dx and a point P on the curve.a.b.c.d.e.f. dy d.x 3x² +1, P = (1, 4)
Find y in terms of x for each of the following.a. b. c. d.e.f. dy = 15x² dx
A curve is such thatGiven that the curve passes through the point (−3, −2), find the equation of the curve. dy - 2x²-3. dx ||
Evaluate:a.b.c.d.e.f. S₁²³ 3x² dx
a. Differentiate (x2 + 2)4 with respect to x.b. Hence, find [x(x² + 2)³ dx.
Find:quation of the curve, given dy/dxa.b.c.d.e.f.g.h.i. f (2x-7)³ dx

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