Questions and Answers of Complete Pure Mathematics

For each, find the area bounded by the curve and the x-axis.a) y = 3x² − 3b) y = x(x + 2)(x − 1)c) y = x³ − 5x² − 6xd) y = 1 − 4x²
A curve passes through the point (1, 5) and its gradient function is (3x − 4)5.Find the equation of the curve.
A circle has equation x² + y² − 6x + 4y − 3 = 0.Find the centre and radius of the circle.
A sector of a circle, radius r cm, has an area of 100 cm².Find the minimum value for the perimeter. X rcm
Solve 6x² − 7x − 5 > 0.
The diagram shows a cuboid with a square base.Find the maximum volume of the cuboid if the sum of the height and the length of the base is 12 cm. x cm hcm xcm
The equation of a curve is y = 5x² + 3/x,i) Obtain an expression for dy/dxii) A point is moving along the curve insuch a way that the x-coordinate isincreasing at a constant rate of 0.04 unitsper
The equation of a curve is y = 4x³ + px² + qx where p and q are positive constants.i) Show that the curve has only one stationary point when p = 23√q. ii) In the case where p = 1/2 and q =
The diagram shows a semicircle on top of a rectangle. The perimeter of the shape is 20 cm. Find the maximum area of the rectangle. rcm hcm
The equation of a curve is y = x³ − 5x² + 7x − 14.i) Find the coordinates of the stationary points on the curve.ii) Find the equation of the tangent to the curve at the point where the curve
A spherical balloon is being blown up so that its volume increases at a constant rate of 1.5 cm³ s−¹.Find the rate of increase of the radius of theballoon when the volume is 56 cm³.
The volume of a cube is increasing at the rate of 10 cm³ s−¹.Find the rate of increase of the surface areaof the cube when the side of the cube is 8 cm.
The radius of a sphere is increasing at a rate of 2 m s−¹.Find the exact rate of increase of the volumewhen the radius is equal to 4 m.
The curve with equation y = (2x + 1)(x² − k) has a stationary point where x = 1.a) Find k.b)Find the coordinates of the stationarypoints and determine their nature.
An open tank made of metal is in the shape of a cuboid with a square base. The volume of the tank is 13.5 m³. Find the dimensions of the tank that uses as little metal as possible.
Paint is poured onto a table, forming a circle which increases at a rate of 2.5 cm² s−¹. Find the rate the radius is increasing when the area of the circle is 20π cm².
The curve y = ax² + bx + c has a minimum when x = 1/2 and passes through the points (2, 0) and (1, −3).Find the values of a, b and c.
Triangle ABC is a right-angle triangle with angle ACB = 90°. AC + BC = 6 cm. Show that the maximum area of the triangle is 4.5 cm².
A cuboid has a square base. The height of the cuboid is twice the length of the side of the base. The surface area of the cuboid is increasing at a rate of 10 cm² s−¹.Find the rate of increase of
A tank in the shape of a right circular cylinder with no top has a surface area of 3πm².What height and base radius will maximise the volume of the cylinder?
a) Show that there is one value of x for which f(x) and g(x) both have the same stationary value.b) On the same axes sketch the graphs of f(x) and g(x).f(x) = 3x² + 2x + 5 g(x) = x³ − 4x²
Ifshow that dy X d.x (5 - 3.x²)º
Find the gradient of the tangent to the curveat the point (−1, 25) y = (4x-1)² .x²
Given thatfind the value of d²y/dx² when x = −1. 2.x² 4 xp 1-x9 p
Find the coordinates of the points on the curve y = x4 + 2x³ where the gradient is parallel to the x-axis.
Find the coordinates of the point on the curvewhere the gradient is zero. y = 2x −5+√x X
The gradient of the tangent to the curve y = 2x³ + ax² − x + 3 at the point x = 1 is 3.Find the value of a.
Find the y-coordinate and the gradient of y = (x − 3)² when x = −2.
An open box with a square base has a total surface area of 300 cm². Find the greatest possible volume of the box. xcm hcm xcm
Find the equations of the tangents to the curve y = x³ − 6x² + 12x + 2 which are parallel to the line y = 5 + 3x.
Find the coordinates of the points on the curve y = 6 + 9x − 3x² − x³ wherethe gradient is 9.
Fig. 1 shows a square sheet of metal of side 40 cm. A square x cm by x cm is cut from each corner. The sides are then bent upwards to form an open box as shown in Fig. 2. Find the value of x that
a) Find the equations of the tangent and the normal to the curve y = 4√x at the point P(9, 12). The tangent and normal to the curve at P cut the x-axis at Q and R.b) Find the area of triangle PQR.
Find the gradient of the tangent to the curve y = (√x + 3)(3√x − 5) at the pointwhere x = 1.
Given that f(x) = 3x4 + 5x³ − 6x², find the value ofa) f´(−2)b) f´´(1/2)c) 24/f(−1)
Given that y = ax² + 3x − a² has a gradient of5 when x = −2, find the value of y when x = −3.
For each of the following functions, find the coordinates of the stationary points and determine their nature.a) f(x) = 2x³ − 3x²b) f(x) = 2x³ − 9x² + 12x + 4c)d) 1 f(x)=x+,x=0
A farmer has a rectangular piece of land for pigs. One of the sides of the rectangle is a wall. The other three sides have fencing. The fencing is 80 m in length. Find the maximum possible area of
A spherical balloon is being blown up so that its radius increases at a rate of 0.4 cm s−¹.Find the rate of increase of the surface area of the balloon when theradius is 20 cm.
Find the coordinates of the stationary points of the curveand determine their nature. y= y = x² 3 4 -X
Find the values of x for which y = 3x² − 2x + 7is an increasingfunction.
For each of the following functions find the range of values of x for which f(x) is increasing.a) f(x) = x² − 6xb) f(x) = 8 + 3x − 2x²c) f(x) = x³ − 48x + 2d) f(x) = 2x³ − 9x² + 12x −
For each of the following functions, find the coordinates of the stationary points and determine their nature.a) y = x² − 2x + 5b) y = 3 + x − x²c) y = 2x² + 4x − 3d) y = x³ −5x² + 3x +
The radius of a circular ink blob is increasing at a rate of 5 cm s−¹.Find the exact rate of increase of the circumference of the circle.
The diagram shows a sporting track made up of a rectangle with semicircles at each end.The rectangle has dimensions p meters by 2r meters where r is the radius of each semicircle. The perimeter of
Find the range of values of x for which y = 3x² − 2x³ isdecreasing.
Find the range of values of x for which y is decreasing.a) y = x² − x + 2b) y = 9 − 2x − x²c) y = 3x² + 6x + 5 d) y = x³ − 5x² + 3x − 4e) y = 2x³ − 54x − 1f) y = x³ −
For each of the following functions, find the coordinates of the stationary points and determine their nature. Sketch the curve.a) y = x² − 8x + 12b) y = x² − x³ c) y = 4x³ − x4d) y =
Find the coordinates of the stationary points of the curveand determine their nature. y 1+54x³ x²
The side of a cube is increasing at 0.2 cm s−¹.Find the rate of increase of the volume when the length of the side is 4 cm.
Find the range of values of x for which f(x) = x³ + x² − x + 5 is decreasing.
Find the range of values of x for which y is increasing.a) y = 5x² + 5xb) y = 3 + 16x − 4x²c) y = x³ − 3x − 4d) y = 6x² − 6x − 7e) y = 4 − 12x + 9x² − 2x³ f) y = 3x³ −
Given that x + y = 3, find the least possible value of x² + 14y.
A spherical balloon is inflated so that its volume increases at a rate of 50 cm³ s−¹. Find the rate of increase of the radius of the balloon when the radius is 12 cm.
Find the value of p.f(x) = x² + px + 3. f(x) has a turning point when x = −3.
Find the range of values of x for which f(x) is decreasing.a) f(x) = 9x² − 2x³b) f(x) = 2 − 5x − 10x²c) f(x) = x³ + 6x² + 12x − 1d) f(x) = 5x³ − 15x − 3e) f(x) = 4x² − 12x³f)
Find the coordinates of the stationary points of the curve y = (3x − 2)³− 9x and determine their nature.
The diagram shows a shape made from a rectangle and a semicircle. y + 1/4 x (π + 2) = 25a) Show that the area A square metres of the shape is given by A = 100x − 2x² − 1/2πx2b) Find the
The side of a cube is decreasing at a rate of 0.4 cm s−¹.Find the rate of decrease of the surface area when the length of the side is 3 cm.
Find the turning point of the curve y = (x3 + 2)/x and determine its nature.
Find the maximum possible value of x2y if x + 2y = 8.
A cone has a height of 7 cm. The radius of the base of the cone is increasing at a rate of 8 cm s−¹.Find the rate of change of the volume of the cone when the base radius is 5 cm.
a) Find the coordinates of the turning points on the curve y = 2x³ + 3x² − 12x + 6 and determine their nature.b) Find the range of values of x for which y is increasing.c) Sketch the graph of y.
Find the coordinates of the stationary points of the curve y = x4 − 2x³ + x² − 2 and determine their nature. Sketch the curve.
A cylinder with an open top has radius r cm and a volume of 512πcm³.a) Write down the surface area of the cylinder in terms of r.b)Find the minimum surface area. Leave your answer in terms of π.c)
The volume of a cube is increasing at the rate of 12 cm³ s−¹.Find the rate of increase of the surface area of the cube when the side ofthe cube is 7 cm.
Find the coordinates of the stationary points of the curve y = (x + 1)(2x - 1)² and determine their nature. Sketch the curve.
Show that the curve y = 3x³ − 5x² + 3x + 4 has no stationary points.
A sector of a circle, radius r has a perimeter of 20 cm. The angle of the sector is θ radiansand the area is A cm².Find the maximum possible area of the sector.
The surface area of a cube is increasing at 0.3 m² s−¹.Find the rate of increase of the volume of the cube when the lengthof the side is 5 m.
Given that y = ax4 − 3x² and d2y/dx²= 42 when x = 2, determine the value of a.
The normals at the points (0, 0) and (1, 2) to the curve y = x + x³ meet at point P. Find the coordinates of P.
a) Find the equation of the tangent to the curve y = x(2 − x) at the origin.b) Find a point on the curve where thetangent is perpendicular to the tangentat the origin.
The normal to the curve y = √x + 3√x at the point where x = 1 meetsthe axes at (p, 0) and (0, q). Find p and q.
Given that f(x) = 2x4 − 3x³ − x², find the value ofa) f´(3)b) f´´(−2)c) 1/f(1).
Find the values of x where the tangents to the curve y = x³ − x² − 42x − 7are parallel to the line y = −2x.
Find the points on the curve y = x³ + 3x² − 4where the tangent is parallel to the liney = 9x − 2.
The curve y = (5 − x)(2 + x) crosses the x-axis at points A and B.The tangents at the points A and B meet at point C.Find the coordinates of C.
Given thatfind f"(x). f'(x) = = 5 (5 - 2x)*'
The curve cuts the y-axis at thepoint P. Find the equation of the tangent tothe curve at the point P. y=√9-4x
Find a)b)c)d)e)f)g)h) dx (3x¹2x³ + 5x² - x - 1)
Find the coordinates of the points on the curve y = 2x³ − 15x + 7 where the gradient is 9.
Differentiate each expression with respect to x.a)b)c)d)e)f)g) (3x2 − 2)4h)i)j) 1 9x + 8
Find f´(x) in each case.a)b)c) f(x) = x (x − 2)²d)e)f) f(x)=(√x − 4)²g)h) 6 =√x+ f(x) = +4√x
a) Find the equation of the tangent to the curve y = 5 − 2x − x² at the point (1, 2).b) This tangent meets the x-axis at P and the y-axis at Q. Find the area of triangle OPQ.
Given that y = 4x² − x³, show that dydy - 3y = 8 + 2x - 15x² + 3x². d.x² dx
By considering the gradient of the chord joining (x, f(x)) and ((x + h), f(x + h)) for the following functions f(x), deduce what f´(x) will be.f(x) = xn
Given that y = 6x³ − 2x², show that d²y/dx² − 4 dy/dx + 20 = 4(4 + 13x − 18x²).
Find the derivative in each case.a) y = 4x²b) y = −2/3xc) y = 5/7 x21d) y = 16√xe)f) y = −4/√xg) y = −5xh) y = 25x1/5i) y = (5x9 × 2x)j)k) y = −x2 × −3x3l) y = −x y 9 X
Find the gradient of the tangent to the curve y = (x + 2)(x − 3) at each of the points where the curve crosses the x-axis.
Find f´(x) whena)b)c)d) f(x) = 2√√x² + √√x X
a) Find the coordinates of the points on the curve y = x² − x + 2 at which the gradients are 3 and −3.b) Find the equations of the tangents at these points.c) Show that these tangents intersect
Find the equations of the tangents to the curve y = x² − x − 12 at each ofthe points where the curve crosses the x-axis.
By considering the gradient of the chord joining (x, f(x)) and ((x + h), f(x + h)) for the following functions f(x), deduce what f´(x) will be.f(x) = 1/x
Find the derived function in each case.a) f(x) = x7 + x³ b) f(x) = 6x8 − 4xc) f(x) = 3x² + 5x − 9d) f(x) = (3x + 7)(2x − 9)e) f(x) = x−4(x² + x6) f) f(x) = 5x²(3x² − 7x −
Find d²y/dx² when dy/dx = 1 − 7x2.
Given thatfind the value of f´´(2). f(x) 4 V3x-2
Find the gradient of the tangent to the curve y = (2x + 3)² at the point where x = 0.
Find d2y/dx² in each case. a) y = 2x² + 4x³b) y = 3x9 − 7x c) y = (2x + 1)(x − 8)d) y = x−³(x + 5x7)e) y = 6x³(2x² − x + 1)f) )y = (9 − x)5g)h)i) y = 4 x³ + x² +² X
The grid shows the graph of y = x³ − 3x + 1.a) Estimate the gradient of the tangent to the curve at thepoint where x = −1.5.b) Write down the coordinates of the points on the curvewhere the

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