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Questions and Answers of
Financial Accounting Information For Decisions
A sector of a circle of radius r cm has an angle θ radians, where θ < π. The perimeter of the sector is 30 cm.a. Show that the area, A cm2, of the sector is given by A = 15r – r2.b. Given
Find the gradient of the curve y = 7x – 2/2x + 3 at the point where the curve crosses the y-axis.
The normal to the curve y = x3 – 2x + 1 at the point (2, 5) intersects the y-axis at the point P.Find the coordinates of P.
A curve has equation y = (x + 1) (2x – 3)4.Find, in terms of p, the approximate change in y as x increases from 2 to 2 + p, where p is small.
Variables x and y are connected by the equation y = 2x/x2 + 3.Given that x increases at a rate of 2 units per second, find the rate of change of y when x = 1.
A curve has equation y = 2x3 – 15x2 + 24s + 6. Copy and complete the table to show whether dy/dx and d2y/dx2 are positive (+), negative (-) or zero (0) for given values of x. 1 3 4 dy dx d'y
Find the gradient of the curve y = x-4/x at the point where the curve crosses the x-axis.
The curve y = x3 + ax + b has a stationary point at (1, 3).a. Find the value of a and the value of b.b. Determine the nature of the stationary point (1, 3).c. Find the coordinates of the other
A piece of wire, of length 60 cm, is bent to form a sector of a circle with radius r cm and sector angle θ radians. The total area enclosed by the shape is A cm2.a. Express θ in terms of r.b. Show
Find the gradient of the curve y = 8/(x – 2)2 at the point where the curve crosses the y-axis.
Find the x-coordinates of the points on the curve y = (2x – 3)3 (x + 2)4 where the gradient is zero.
a. Given that y = (1/4 x – 5)8, find dy/dx.b. Hence find the approximate change in y as x increases from 12 to 12 + p, where p is small.
Differentiate with respect to x:a. √x/2x + 1b. x/√1- 2xc. x2/√x2+2d. 5√x/3+x
Find the equation of the tangent and the normal to the curve y = x – 1/√x + 4 at the point where the curve intersects the y-axis.
A curve has equation y = (x - 2) √2x + 1.Find, in terms of p, the approximate change in y as x increases from 4 to 4 + p, where p is small.
Variables x and y are connected by the equation y = 2x – 5/x - 1.Given that x increases at a rate of 0.02 units per second, find the rate of change of y when y = 1.
A curve has equation y = 2x3 + 3x2 – 36x + 5. Find the range of values of x for which both dy/dx and d2y/dx2 are both positive.
Find the gradient of the curve y = x3 – 2x2 + 5x – 3 at the point where the curve crosses the y-axis.
The curve y = x2 + a/x + b has a stationary point at (1, - 1).a. Find the value of a and the value of b.b. Determine the nature of the stationary point (1, - 1).
The diagram shows a window made from a rectangle with base 2r m and height h m and a semicircle of radius r m. The perimeter of the window is 6 m and the surface area is A m2.a. Express h in terms of
Find the gradient of the curve y = x + 4/x – 5 at the points where the curve crosses the x-axis.
Find the x-coordinate of the point on the curve y = (x + 3) √4 – x where the gradient is zero.
Find the equation of the normal to the curve y = x2 + 8/x – 2 at the point on the curve where x = 4.
Find the gradient of the curve y = x-2/√x + 5 at the point (-4, -6).
The tangents to the curve y = x2 – 5x + 4, at the points (1, 0) and (3, - 2), meet at the point Q.Find the coordinates of Q.
The periodic time, T seconds, for a pendulum of length Lcm is T = 2π √L/10.Find the approximate increase in T as L increases from 40 to 41.
Variables x and y are connected by the equation 1/y = 1/8 – 2/x.Given that x increases at a rate of 0.01 units per second, find the rate of change of y when x = 8.
Given that y = x2 – 2x + 5, show that 4 d2y/dx2 + (x – 1) dy/dx = 2y.
The curve y = 2x2 + 7x – 4 and the line y = 5 meet at the point P and Q. Find the gradient of the curve at the point P and at the point Q.
The curve y = ax + b/x2 has a stationary point at (1, - 12).a. Find the value of a and the value of b.b. Determine the nature of the stationary point (- 1, - 12).
ABCD is rectangle with base length 2p units, and area A units2. The points A and B lie on the x-axis and the points C and D lie on the curve y = 4 – x2.a. Express BC in terms of p.b. Show that A =
Find the coordinates of the point on the curve y = √(x2 – 6x + 13) where the gradient is 0.
a. Find the equation of the tangent to the curve y = x3 + 2x2 – 3x + 4 at the point where the curve crosses the y-axis.b. Find the coordinates of the point where the tangent meets the curve again.
Find the coordinates of the point on the curve y = 2(x – 5)/√x+1 where the gradient is 5/4.
The tangent to the curve y = 3x2 – 10x – 8 at the point P is parallel to the line y = 2x – 5.Find the equation of the tangent at P.
The volume of the solid cuboid is 360 cm3 and the surface area is A cm2.a. Express y in terms of x.b. Show that A = 4x2 + 1080/x.c. Find, in terms of p, the approximate change in A as x increases
A square has sides of length x cm and area A cm2.The area is increasing at a constant rate of 0.2 cm2s-1.Find the rate of increase of x when A = 16.
Given that y = 8√x, show that 4x2 d2y/dx2 + 4x dy/dx = y.
The curve y = ax2 + bx has gradient 8 when x = 2 and has gradient – 10 when x = - 1.Find the value of a and the value of b.
The curve y = 2x3 – 3x2 + ax + b has a stationary point at the point (3, - 77).a. Find the value of a and the value of b.b. Find the coordinates of the second stationary point on the curve.c.
A solid cylinder has radius r cm and height h cm. The volume of this cylinder is 250π cm3 and the surface area is A cm2.a. Express h in terms of r.b. Show that A = 2πr2 + 500π/r.c. Find dA/dr and
The curve y = a/√bx + 1 passes through the point (1, 4) and has gradient – 3/2 at this point.Find the value of a and the value of b.
The normal to the curve y = x3 + 6x2 – 34x + 44 at the point P(2, 8) cuts the x-axis at A and the y-axis at B. Show that the mid-point of the line AB lies on the line 4y = x + 9.
The line 5x – 5y = 2 intersects the curve x2y – 5x + y + 2 = 0 at three points.a. Find the coordinates of the points of intersection.b. Find the gradient of the curve at each of the points of
A curve has equation y = x3 – x + 6.a. Find the equation of the tangent to this curve at the point P(-1, 6). The tangent at the point Q is parallel to the tangent at P.b. Find the coordinates of
A cube has sides of length x cm and area V cm3.The volume is increasing at a rate of 2 cm3s-1.Find the rate of increase of x when V = 512.
The gradient of the curve y = ax + b/x at the point (-1, -3) is – 7.Find the value of a and the value of b.
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 surface area of the solid is 288π cm2 and the volume is V cm3.a. Express h
Given that f(x) = x2 – 648/√x, find the value of x for which f”(x) = 0.
A curve has equation y = 4 + (x – 1)4.The normal at the point P(1, 4) and the normal at the point Q(2, 5) intersects at the point R.Find the coordinates of R.
A sphere has radius r cm and volume V cm3.The radius is increasing at a rate of 1/π cm s-1.Find the rate of increase of the volume when V = 972π.
Find the coordinates of the points on this curve y = x3/3 – 5x2/2 + 6x – 1 where the gradient is 2.
A piece of wire, of length 50 cm, is cut into two pieces. One piece is bent to make a square of side x cm and the other is bent to make circle of radius r cm. The total area enclosed by the two
A rectangular sheet of metal measures 60 cm by 45 cm. A scoop is made by cutting out squares, of side x cm, from two corners of the sheet and folding the remainder as shown.a. Show that volume, V cm3
A curve has equation y = (2 - √x)4.The normal at the point P(1, 1) and the normal at the point Q(9, 1) intersect at the point R.a. Find the coordinates of R.b. Find the area of triangle PQR.
A solid metal cuboid has dimensions x cm by x cm by 5x cm.The cuboid is heated and the volume increases at a rate of 0.5 cm3s-1.Find the rate of increase of x when x = 4.
The curve y = 1/3 x3 – 2x2 – 8x + 5 and the line y = x + 5 meet at the points A, B and C.a. Find the coordinates of the points A, B and C.b. Find the gradient of the curve at the points A, B and
The diagram shows a solid cylinder of radius r cm and height 2h cm cut from a solid sphere of radius 5 cm. The volume of the cylinder is V cm3.a. Express r in terms of h.b. Show that V = 2πh(25 –
The diagram shows an empty container in the form of an open triangular prism. The triangular faces are equilateral with a side of x cm and the length of each rectangular face is y cm. The container
A curve has equation y = √x(x – 2)3. The tangent at the point P(3, √3) and the normal at the point Q(9, 1) intersect at the point R.a. Show that the equation of the tangent at the point P(3,
A cone has base radius r cm and a fixed height 18 cm.The radius of the base is increasing at a rate of 0.1 cm s-1.Find the rate of change of the volume when r = 10.
Y = 4x3 + 3x2 – 6x – 1a. Find dy/dx.b. Find the range of values of x for which dy/dx ≥ 0.
The diagram shows a hollow cone with base radius 12 cm and height 24 cm. A solid cylinder stands on the base of the cone and the upper edge touches the inside of the cone. The cylinder has base
The equation of a curve is y = x2/x + 2.The tangent to the curve at the point where x = - 3 meets the y-axis at M.The normal to the curve at the point where x = - 3 meets the x-axis at N.Find the
Water is poured into the conical container at a rate of 5 cm3 s-1.After t seconds, the volume of water in the container, V cm3 is given by V = 1/12 πh3, where h cm is the height of the water in the
Y = x3 + x2 – 16x – 16a. Find dy/dx.b. Find the range of values of x for which dy/dx ≤ 0.
The diagram shows a right circular cone of base radius r cm and height h cm cut from a solid sphere 10 cm. The volume of the cone is V cm3.a. Express r in terms of h.b. Show that V = 1/3πh2 (20 –
The equation of a curve is y = x-3/x+2.The curve intersects the x-axis at the point P.The normal to the curve at P meets the y-axis at the Point Q.Find the area of the triangle POQ, where O is the
Water is poured into the hemispherical bowl at a rate of 4π cm3 s-1.After t seconds, the volume of water in the bow, V cm3, is given by V = 8πh2 – 1/3 πh3, where h cm is the height of the water
A curve has equation y = x5 – 5x3 + 25x2 + 145x + 10. Show that the gradient of the curve is never negative.
Variables x and y are connected by the equation y = 2x3 – 3x.Find the approximate change in y as x increases from 2 to 2.01.
Variables x and y are connected by the equation y = x2 – 5x.Given that x increases at a rate of 0.05 units per second, find the rate of change of y when x = 4.
Find d2y/dx2 for each of the following functions.a. y = 5x2 – 7x + 3b. y = 2x3 + 3x2 – 1c. y = 4 – 3/x2d. y = (4x + 1)5e. y = √2x + 1f. y = 4/√x + 3
Differentiate with respect to x.a. x4b. x9c. x-3d. x-6e. 1/xf. 1/x5g. √xh. √x5i. x1/5j. x1/3k. 3√x2l. 1/√xm. xn. x3/2o. 3√x5p. x2 × x4q. x2 × xr. x4/x2s. x/√xt. x√x/x3
Find the coordinates of the stationary points on each of the following curves and determine the nature of each of the stationary points.a. y = x2 – 12x + 8b. y = (5 + x) (1 – x)c. y = x3 – 12x
The sum of two numbers x and y is 8.a. Express y in terms of x.b. i. Given that P = xy, write down an expression for P in terms of x.ii. Find the maximum value of P.c. i. Given that S = x2 + y2,
Differentiate with respect to x.a. (x + 2)9b. (3x – 1)7c. (1 – 5x)6d. (1/2x – 7)4e. (2x + 1)6/3f. 2(x – 4)6g. 6(5 – x)5h. ½(2x + 5)8i. (x2 + 2)4j. (1 – 2x2)7k. (x2 – 3x)5l. (x2 + 2/x)4
Use the product rule to differentiate each of the following with respect to x.a. x(x + 4)b. 2x(3x + 5)c. x(x + 2)3d. x2(x – 1)3e. x√x – 5f. (x + 2) √xg. x2√x + 3h. √x(3 – x2)3i. (2x +
Find the coordinates of the stationary point on the curve y = x2 + 16/x.
Use the quotient rule to differentiate each of the following with respect to x:a. 1 + 2x/5x – 2b. 3x + 2/x + 4c. x – 1/3x + 4d. 5x – 2/3 – 8xe. x2/5x – 2f. x/x2 – 1g. 5/3x – 1h. x +
Find the equation of the tangent to the curve at the given value of x.a. y = x4 – 3 at x = 1b. y = x2 + 3x + 2 at x = -2c. y = 2x3 + 5x2 – 1 at x = 1d. y = 5 + 2/x at x = -2e. y = (x -3) (2x –
Variables x and y are connected by the equation y = 5x2 – 8/x3.Find the approximate change in y as x increases from 1 to 1.02.
Variables x and y are connected by the equation y = x + √x – 5.Given that x increases at a rate of 0.01 units per second, find the rate of change of y when x = 9.
Find d2y/dx2 for each of the following functions.a. y = x(x – 4)3b. y = 4x – 1/x2c. y = x +1/x-3d. y = x+2/x2 -1e. y = x2/x – 5f. y = 2x+5/3x -1
Differentiate with respect to x.a. 2x3 – 5x + 4b. 8x5 – 3x2 – 2c. 7 – 2x3 + 4xd. 3x2 + 2/x – 1/x2e. 2x – 1/x – 1/√xf. x + 5/√xg. x2 – 3/xh. 5x2 - √x/xi. x2 – x – 1/√xj.
Find the coordinates of the stationary points on each of the following curves and determine the nature of each of the stationary points.a. y = √x + f/√xb. y = x2 – 2/xc. y = 4/x + √xd. y =
The diagram shows a rectangular garden with a fence on three of its sides and a wall on its fourth side. The total length of the fence is 100 m and the area enclosed is Am2.a. Show that A = 1/2x(100
Differentiate with respect to x.a. 1/(x + 4)b. 3/(2x – 1)c. 5/(2 – 3x)d. 16/(2x2 – 5)e. 4/(x2 – 2x)f. 1/(x – 1)5g. 2/(5x + 1)3h. 1/2(3x – 2)4
Find the gradient of the curve y = x2 √x + 2 at the point (2, 8).
Given that y = x2/2 + x2, show that dy/dx = kx/(2 + x)2, where k is a constant to be found.
Find the gradient of the curve y = x + 3/x – 1 at the point (2, 5).
Find the equation of the normal to the curve at the given value of x.a. y = x2 + 5x at x = - 1b. y = 3x2 – 4x + 1 at x = 2c. y = 5x4 – 7x2 + 2x at x = - 1d. y = 4 – 2/x2 at x = - 2e. y = 2x(x
Variables x and y are connected by the equation y = x2y = 400.Find, in terms of p, the approximate change in y as x increases from 10 to 10 + p, where p is small.
Variables x and y are connected by the equation y = (x – 3) (x + 5)3.Given that x increases at a rate of 0.02 units per second, find the rate of change of y when x = - 4.
Given that f(x) = x3 – 7x2 + 2x + 1, finda. f(1)b. f’(1)c. f”(1).
Find the value of dy/dx at the given point on the curve.a. y = 3x2 – 4 at the point (1, - 1)b. y = 4 – 2x2 at the point (- 1, 2)c. y = 2 + 8/x at the point (-2, 2)d. y = 5x3 – 2x2
The equation of a curve is y = 2x + 5/x+1.Find dy/dx and hence explain why the curve has no turning points.
The volume of the solid cuboid is 576 cm3 and the surface area is A cm2.a. Express y in terms of x.b. Show that A = 4x2 + 1728/x.c. Find the maximum value of A and state the dimensions of the cuboid
Differentiate with respect to x.a.b.c.d.e.f.g.h. x+2
Find the gradient of the curve y = (x – 1)3 (x + 3)2 at the point where x = 2.
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