Questions and Answers of Complete Pure Mathematics

For each of the following improper integrals, find the value of the integral or explain briefly why it does not have a value.a)b) Da 27 dx
a) Find the first 3 terms in the expansion of (3x − 2/x)6 in descending powers of x.b) Hence find the coefficient of x² in the expansion of 1 + * |
Find the maximum value of W.W = pq and 2p + 5q = 100.
The diagram shows parts of the curves y = x² and y = x4.Find the shaded area. 3 2 1 0 y=x² y=x² 2 X
A solid cylinder with radius x cm has a volume of 1000 cm³.a) Show that the total surface area of thecylinder is given by A = 2πx² +2000/x.b) Find the minimum total surface area.Give your answer
The first 3 terms of a geometric progression are x, 10 − x and 2x + 1 where x > 0.Find the value of the common ratio, r.
The curve C, with equation y = f(x) passes through the point (−2, −1) and f´(x) = x(3 − x).Find the equation of C in the form y = f(x).
Explain briefly whyis an important integral 100 S 0 3x ²dx
The sum to n terms of an arithmetic progression is n². Find the nth term.
Find f(x).f´(x) = 1/(5x − 3)4 and f(1) = −90.
Find an expression for y in terms of x if dy/dx = (3x − 5)(x − 1) and y = 0 when x = 5.
Find the area bounded by the curve y = x³, the x-axis and the lines x = −1 and x = 2.
A curve is such that d2y/dx2 = −8x. The curve has a maximum point when x = 1, and the point (2, −1) lies on the curve. Find the equation of the curve.
A circle with centre C(4, − 2) passes through the point P(7, 2).a) Find the equation of the circle in the form (x − a)² + (y − b)² = k.b) Find the equation of the tangent to the circle at P.
Find y in terms of x. d²y/dx2 = (1/4 x + 1)7. When x = 4, dy/dx = 6 and when x = 4, y = 0.
Show that only one of the following improper integrals has a finite value and find that value.a)b) 0 3 X 4 -dx
Find the area bounded by the two curves. YA 130- -10 -5 25- 20- 154 10- 5 0 -5- -10 -15- -20 5 y=x² - 4x 10 X y=16-x²
Find the area bounded by the curve y = x² − 3x + 8 and the line y = x + 5. YA 12 10 8 -1 LO 6 4 2- 0 y=x²-3x + 8 1 2 3 y=x+ 5 4 X
Find a)b)c)d) 3 2 f (x + ¹)² dx X 1
a) Prove the identity cos4 θ − sin4 θ = 2 cos²θ − 1.b) Solve the equation cos4 θ − sin4 θ = 0 for 0° ≤ θ ≤ 360°.
For each of the following improper integrals, find the value of the integral or explain briefly why it does not have a value.a)b)c) 100 Jз 0 3x ² dx 2
a) Find, in terms of a and b, the value of the integralb) Show that only one of the following improper integrals has a finite value and find that value.i)ii) b 3 2x² dx.
The diagram shows the line QS where Q has coordinates (−2, −8) and S has coordinates (3, −6).a) Find the exact length of QS. P has coordinates (−6, 0) and M is the mid-point of PQ. MR is
Find 1 (3x+2) 5 dx.
Find the area bounded by the curve x = (y + 3)(y − 1) and the y-axis.
Show that only one of the following improper integrals has a finite value and find that value.a)b) 16 0 X -dx
Find the area between the curve with equation y = x(x − 1)and the line y = 3x.
The function f is defined by f: x ↦ 7 − 5 cos 2x for 0 ≤ x ≤ π. The function g is defined by g: x ↦ x + π/2 for 0 ≤ x ≤ π.a) State the range of f.b) Sketch the graph of y = f(x).c)
The diagram shows part of the curve with equation y = 1/x2 + x. At point P on the curve, x = 1 and at point Q on the curve, x = 2. Find the shaded area. YA 5 4. 3- 2 1 0 FO P Pr I 1 I y= + X 1 2 3 4 X
Find f(x). f´(x) = 8x³ − 4 + 3x−1/2 and f(4) = 3,
Find the values of k for which the line y = 7x − 4 is a tangent to the curvey = 4x² − kx + k.
Find the volume obtained when region bounded by the curve y = x(3 − x)and the x-axis is rotated through 360° about the x-axis.
In the diagram, ADC is a semicircle. ABC is the arc of a circle, centre O and radius r. Angle COA = π/3 radians.Find the area of the shaded region, leaving your answer interms of √3, π and r.
Find the area bounded by the curve y = 2√x, the x-axis and the lines x = 1 and x = 9.
The diagram shows the curve with equation y = 25 + 2x − x².The straight line with equation y = x + 5 cuts the curve in two places.Find the exact area of the shaded region. YA 25 20 15 10- -10
a) Show that the curve y = 2x² + 3 and the curve y = 10x − x² meet at thepoints where x = 3 and x = 1/3. b) Find the area between these two curves.
Find in terms of p and q the value of the integrals. p q 3 x 4 dx.
For each of the following integrals explain briefly why it is an improper integral.a)b)c)d) Find whether each of the integrals has a finite value and, where possible, find its value. 16 S = 0 -dx
Given that d2y/dx2 = −3x + 2 and that when x = −1, dy/dx = 5 and y = 0, find y in terms of x.
The diagram shows part of the curve with equation y = 16/√x + √x. Find the area bounded by the curve, the lines x = 1, x = 4and the x-axis. y 20 15- 10 5 0 1 2 3 16 4 주 5 5 6 X
The coefficient of x in the expansion of (3 − x)6 − (kx − 3)5 is 375. Find the possible values of k.
a) Draw the graph of x + y = 5 and find the area bounded by the line x + y = 5, the x-axis and the y-axis.b) Check this area using integration.
The diagram shows a trapezium. All measurements are in centimetres.a) Show that the area of the trapezium is (6 + x − x²) cm².b) The area of the trapezium can be written in the form a(x + b)² +
a) Differentiate with respect to xb) Find 5x4+6VX -6√x - 3-x² 2.x²
The curve C passes through the point (3, 10) and its gradient at any point is given by dy/dx = 6x² − 4x + 3. a) Find the equation of the curve C.b) Show that the point (2, −21) lies on the
The diagram shows a plan for a rectangular piece of land ABCD. AD = 40 metres. Vegetables are going to be planted in the triangular vegetable patch ACP. BP = CD= x metres.a) Show that the area, A m²
Find the volume obtained when the shaded region is rotated through 360° about the x-axis. Give your answer to the nearest whole number. X=-3 6- 42 2- 0 -6-4-22 -6- -8- y = x(5 + x) 2
a) Solve the equation 4 sin²x + 7 cos x = 7 for 0° ≤ x ≤ 360°.b) Hence solve the equation 4 sin²(θ + 20)° +7 cos (θ + 20)° = 7 for 0° ≤ θ ≤ 360°.
The diagram shows part of the curve with equation y = (x − 5)(x + 2)(x − 3).a) Write down the coordinates of A and B.b) Show that the equation may be written as y = x³ − 6x² − x + 30.c)
A curve is such that d2y/dx2 = 6x. The curve has a maximum point when x = −1, and the point (3, −2) lies on the curve. Find the equation of the curve.
The diagram shows part of the curve with equation y = x³ − 12x² + 36x. The x-axis is a tangent to the curve at P.a) Find the coordinates of P. The curve has a maximum turning point at Q.b) Find
The curve y² = x and the line y = x meet at the points A and B.a) Find the coordinates of the points A and B.b) Sketch the curve y² = x and the line y = x on one set of axes.c) Find the volume
The diagram shows a sector OAB of a circle with centre O and radius r cm. The angle AOB is θ radians and the perimeter of the sector is 30 cm. a) Show that θ = 30/r − 2. b) Find the area of the
The diagram shows the lines y = 2 and y = 5 and part of the curve y = 2x² + 1. The shaded region is rotated through 360° about the y-axis. Find the exact value of the volume of revolution obtained.
The diagram shows the graph of y = x2/25, x ≥ 0, which passes through the points P and Q. The x-coordinate of P is 5 and the x-coordinate of Q is 10.a) Find the y-coordinates of P and Q.b) Find the
The diagram shows the curve y = x(x − 1).Find the total shaded area. y 2 1. 0 -1- y = x(x - 1) 1 2 3 X
The diagram shows part of the curve y² = 32x and part of the curve y = x³. The shaded region is rotated through 360° about the x-axis. Find the exact value of the volume of revolution obtained.
The gradient of a curve is given by dy/dx = ax + b. Given that the curve passes through (0, 0), (1, 1) and (−2, 16), find the equation of the curve.
The diagram shows the curve y = x(4 − x)and the line y = 3. Find the shaded area. y) 5 4. 3 2 1 لي 0 y = x4 - x) 12 y لي = 3 3 4 *
The diagram shows a sketch of the curve y = 3x − x² and the line y = −2x. Find thearea of the shaded region. YA 4 2 -2 -1 0 -2 -4 -6 -8- -10-1 1 2 y=3x-x² 3 4 y=-2x 5 X
Function F and g are defined bya) Express in terms of xi) f−¹(x)ii) g−¹(x).b) Sketch in a single diagram the graphs of y = g(x) and y = g−¹(x), makingclear the relationship between the
Find the volume generated when the shaded area is rotated through 360° about the x-axis. -1 УА 3 2- 1. N -2 -3- y=x2(x-2) 1 2 3 х
The diagram shows the line y = 4 and part of the curve with equation a) Show that the equation y = √x + 4 can be written in the form x = y² − 4.b) Find the area of the shaded region.c) Find the
The diagram shows the line y = 1 and part of the curve with equation y² = 4 − x.a) Find the volume obtained when the shaded region is rotated through 360° about the x-axis.b) Find the volume
Find the value of and explain the significance of the answer. -4 x³ dx
The diagram shows the region enclosed by the curve with equation y = 6/x, the y-axis and the lines y = 2 and y = 3. Find, in terms of π, the volumeobtained when this region is rotatedthrough 360°
Find the area enclosed by the curvethe line x = 1/2 and the line x = 4. y = 2x³ +3 x²
The area between curve y² = x(4 − x)² and the x-axis is rotated through 360° about the x-axis. Find the exact value of the volume of the solid of revolution formed by this rotation.
Find the area bounded by the curve y = 3x² − 2x + 1, the x-axis and thelines x = 1 and x = 2.
Find the area bounded by the curve y = 2 + x − x² and the line y = x + 1.
a) Sketch the curve y = x(x² − 1) showing clearly where the curve crosses the x-axis.b) Find the area between the curve and the x-axis.c) Find the volume obtained when thearea bounded by the curve
The region bounded by the curve 4y = x², the line x = 4 and the line y = 1, is rotated through 360° about the x-axis. Find the volume of the solid formed.
The area bounded by the curve y = 15kx − 15x² and the x-axis is rotated through 360° about the x-axis, where k is an integer. The volume of the solid formed is 240π. Find the value of k.
Evaluatea)b)c)d)e)f) dx X px² + ₂x
a) Find, in terms of a and b, the value of the integralb)Show that only one of the following improper integrals has a finitevalue and find that value.i)ii) a 6 dx. in
A curve passes through the point (7, 10) and its gradient function is 6/x3 +2. Find the equation of the curve.
The function f(x) is defined for x ≤ = 4/3 byi) Find f´(x) and f´´(x).The first, second and third terms of ageometric progression are f(1), f´(1) andkf´´(1). ii) Find the value of the
Finda)b)c)d)e)f)g)h)i)j)k)l) fox 6x dx
Find these integrals. a)b)c)d)e)f)g)h)i)j)k)l) S(2x- (2x - 1)ºdx
Finda)b)c)d)e)f) (6x² - 2x) dx
Find f(x) given f´(x) in each case.a) 9/8x1/2b) -2xc) √x/3d) 12x5e) 3 − x − 2x5f)g)h)i) (x − 4)(x + 7) j) −5/6k) x² + x−²l) 1/1 +5) - 1.2²
The diagram shows the curve y = 9 − x² and the line y = 5.Find the shaded area. YA 10. -4 -3 -2 -1 00 8 6 4 2-4 0 -1 1 y = 5 y=9-x² 2 3 X
Describe the geometric transformation which mapsonto the graph ofa)b) y=√√√x² +16
Finda)b)Hence, find whether the following integrals exist and, if they do, find their value.c)d) xp x 2
Find these integrals.a)b)c)d)e)f) 12x² dx
Find the equation of the curve, given dy/dx and a point on the curve.a)b)c)d) dy dx =9x² - 2x + 1 point = (-3,5)
Find the area of each of the shaded regions.a)b)c)d)e)f) y 12 10 8 6- 4 2 0 -11 -2 -1 y=3x² 12 X
Finda)b)c)d)e)f) fo 0 6x dx
The curve y = f(x) has a maximum at A(−2, 20), a minimum at B(1, −7),and passes through the origin. The graph of y = f(x) is shown below.On separate diagrams, sketcha) y = f(x) + 4b) y =
Find the equation of each curve, given dy/dx and a point on the curve.a)b)c)d)e)f) dy d.x 3x² - 6x + 2 point = (-2,-10)
The area of a rectangle is (5− √2) cm². The base of the rectangle is (3 − √2) cm and the height is x cm. Find the height of the rectangle. Give your answer in the form a +b√c. (3-√2)cm x
Find the area bounded by the curve y = √x, the y-axis and the lines y = 1 and y = 3.
Find y given dy/dx in each case.a) 2xb) x7c) 3x²d) −15e) x − x³f) 10x + 8x7g) 5 − 1/2x h) x(x4 − 6)i) √xj) (2x − 3)²k)l) 2x³ +7x X
Find the shaded areas.a)b) Y 3 ليا 2 1 0 1 2 y=x y=x (3-X) 3 X
Given that f´(x) = 2x² − x + 3√x andf(4) = 0, find f(x).
A curve is such thatand the point (1, − 6) lies on the curve. Find the equation of the curve. dy dx || 5 - 10vx3 X
Show that only one of the following improper integrals has a finite value and find that value.a)b) 1 0 4 志 dx
A curve is such that dy/dx = (7 − x)4 and the point (5, −3) lies on the curve. Find the equation of the curve.
Find the points of intersection of the graphs 2x − y = 4 and 4x² + y² = 10.
Find the area of the region bounded by the curve with equation y = x² and the curve with equation y = 2x − x².

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