Figure 4 shows a sketch of part of the curve $\displaystyle C$ with equation $\displaystyle y = f(x)$ where

The curve $\displaystyle C$ has a maximum at the point $\displaystyle A$ with coordinates $\displaystyle \left(-\frac{1}{3}, \frac{100}{27}\right)$ and a minimum at the point $\displaystyle B$ with coordinates $\displaystyle (2, -9)$

Given that $\displaystyle a$, $\displaystyle b$ and $\displaystyle c$ are integers

(a) show that $\displaystyle a = -5$, $\displaystyle b = -4$ and $\displaystyle c = 3$

(b) (i) Show that $\displaystyle (x+1)$ is a factor of $\displaystyle f(x)$

(ii) Hence, or otherwise, use algebra to factorise $\displaystyle f(x)$ completely.

The curve $\displaystyle C$ crosses the $\displaystyle x$-axis at the points $\displaystyle M$, $\displaystyle N$ and $\displaystyle P$

The finite regions shown shaded in Figure 4 are bounded by the curve $\displaystyle C$ and parts of the $\displaystyle x$-axis from $\displaystyle M$ to $\displaystyle N$ and from $\displaystyle N$ to $\displaystyle P$

(c) Use algebraic integration to determine the total area of the shaded regions.

Give your answer as an exact fraction.

Figure 5 shows part of the curve $\displaystyle C$ with equation $\displaystyle y = \sqrt{4x - 8}$ and the line $\displaystyle l$ with equation $\displaystyle x = b$ where $\displaystyle b > 0$

The finite region bounded by the curve $\displaystyle C$, the $\displaystyle x$-axis and the line $\displaystyle l$, shown shaded in Figure 5, is rotated through $\displaystyle 360^\circ$ about the $\displaystyle x$-axis.

Given that the volume of the solid formed is $\displaystyle 50\pi$ units3

find the value of $\displaystyle b$

Given that

• $\displaystyle (2x+1)$ is a factor of $\displaystyle f(x)$

• when $\displaystyle f'(x)$ is divided by $\displaystyle (x-1)$ the remainder is $\displaystyle -27$

(a) show that $\displaystyle A = 3$ and find the value of $\displaystyle B$

(b) Hence, using algebra, solve the equation $\displaystyle f(x) = 0$

Figure 2 shows a sketch of part of the curve $\displaystyle C$ with equation $\displaystyle y = f(x)$

(c) Use calculus to find the exact area of the finite region bounded by $\displaystyle C$ and the $\displaystyle x$-axis, shown shaded in Figure 2

Figure 3 shows part of the curve $\displaystyle C$ with equation $\displaystyle y = 3\sqrt{4-x}$ and part of the line $\displaystyle l$ with equation $\displaystyle px + qy + r = 0$ where $\displaystyle p$ and $\displaystyle r$ are integers and $\displaystyle q$ is prime.

The curve $\displaystyle C$ cuts the $\displaystyle y$-axis at the point $\displaystyle A$

The line $\displaystyle l$ is the tangent to $\displaystyle C$ at $\displaystyle A$

(a) Find the value of $\displaystyle p$, the value of $\displaystyle q$ and the value of $\displaystyle r$

The finite region $\displaystyle R$, shown shaded in Figure 3, is rotated through $\displaystyle 360^\circ$ about the $\displaystyle x$-axis.

(b) Use algebraic integration to find the volume of the solid formed.
Give your answer in terms of $\displaystyle \pi$

Figure 4 shows part of a curve $\displaystyle C$ with equation $\displaystyle y = 3\mathrm{e}^{\frac{x}{3}}$.
The point $\displaystyle A$ with coordinates $\displaystyle (2, p)$ lies on $\displaystyle C$

(a) Write down the exact value of $\displaystyle p$

The straight line $\displaystyle n$, shown on Figure 4, is the normal to $\displaystyle C$ at $\displaystyle A$ and crosses the $\displaystyle x$-axis at point $\displaystyle B$.

The region $\displaystyle R$, shown shaded in Figure 4, is bounded by $\displaystyle C$, $\displaystyle n$, the $\displaystyle x$-axis and the $\displaystyle y$-axis.

(b) Use algebraic integration to find the area of $\displaystyle R$

Give your answer in the form $\displaystyle \frac{W}{V} \mathrm{e}^2 + W \mathrm{e}^{\frac{2}{3}} - W$

where $\displaystyle W$ and $\displaystyle V$ are integers to be found.

Figure 3 shows part of the curve $\displaystyle C$ with equation $\displaystyle y^2 = x - 1$ and part of the line $\displaystyle l$ with equation $\displaystyle 2y + x - 4 = 0$

The region $\displaystyle R$, bounded by the $\displaystyle x$-axis, the curve $\displaystyle C$ and the line $\displaystyle l$, is rotated through $\displaystyle 360^\circ$ about the $\displaystyle x$-axis.

Using algebraic integration, find the exact value of the volume of the solid generated.

(a) Show that $\displaystyle f(\theta) = \frac{3}{2} \mathrm{sin} 2\theta + 2 \mathrm{cos} 2\theta$

Figure 4 shows part of the curve $\displaystyle S$ with equation $\displaystyle y = f(\theta) + 2$

Given that $\displaystyle S$ intersects with the $\displaystyle \theta$-axis at the point with coordinates $\displaystyle (a, 0)$

(b) using $\displaystyle \mathrm{sin}^2 \theta + \mathrm{cos}^2 \theta = 1$, or otherwise, show that $\displaystyle a = \frac{\pi}{2}$

(c) Using algebraic integration, find the exact area bounded by $\displaystyle S$, the positive $\displaystyle \theta$-axis and the positive $\displaystyle y$-axis shown shaded in Figure 4

Figure 3 shows part of the curve $\displaystyle C$ with equation $\displaystyle y = \frac{1}{4x}$, $\displaystyle x > 0$ and part of the curve $\displaystyle S$ with equation $\displaystyle y = 2x^2$, $\displaystyle x \geq 0$

The curve $\displaystyle C$ and the curve $\displaystyle S$ intersect at the point $\displaystyle A$

(a) Find the coordinates of point $\displaystyle A$

The finite region $\displaystyle R$, shown shaded in Figure 3, bounded by the curve $\displaystyle C$, the curve $\displaystyle S$ and the straight line $\displaystyle y = 4$ is rotated through $\displaystyle 360^\circ$ about the $\displaystyle y$-axis.

(b) Find, using algebraic integration, the exact volume of the solid formed.

(a) Using a formula given on page 2, show that

(b) Hence show that

where $\displaystyle a$, $\displaystyle b$ and $\displaystyle c$ are integers to be found.

Figure 3 shows part of the curve $\displaystyle C_1$ with equation $\displaystyle y = 2 \mathrm{cos}^2 \theta - 1$ and part of the curve $\displaystyle C_2$ with equation $\displaystyle y = -\mathrm{cos} \theta$

Point $\displaystyle B$ is the intersection of $\displaystyle C_1$ and $\displaystyle C_2$ as shown in Figure 3

Point $\displaystyle A \left( \frac{3\pi}{4}, 0 \right)$ is the intersection of $\displaystyle C_1$ with the $\displaystyle \theta$-axis as shown in Figure 3

Point $\displaystyle E \left( \frac{\pi}{2}, 0 \right)$ is the intersection of $\displaystyle C_2$ with the $\displaystyle \theta$-axis as shown in Figure 3

The finite region $\displaystyle R$, shown shaded in Figure 3, is bounded by the $\displaystyle \theta$-axis, $\displaystyle C_1$ and $\displaystyle C_2$

(c) Use calculus to find, in its simplest form, the exact area of $\displaystyle R$

Figure 3 shows a sketch of part of the curve $\displaystyle S$ with equation $\displaystyle y = -2\mathrm{e}^{3x} + 4$ and the line $\displaystyle L$

The curve $\displaystyle S$ has intersections with the line $\displaystyle L$ at the points $\displaystyle A$ and $\displaystyle B$ with $\displaystyle x$ coordinates $\displaystyle x = -1$ and $\displaystyle x = 0$ respectively.

The finite region bounded by $\displaystyle S$ and $\displaystyle L$ is shown shaded in Figure 3

Use calculus to find the exact area of this region.

Give your answer in the form $\displaystyle \frac{a + be^{-c}}{c}$ where $\displaystyle a$, $\displaystyle b$ and $\displaystyle c$ are integers to be found.

Given that $\displaystyle f(x)$ can be written in the form $\displaystyle A(x + B)^2 + C$ where $\displaystyle A$, $\displaystyle B$ and $\displaystyle C$ are constants,

(a) find the value of $\displaystyle A$, the value of $\displaystyle B$ and the value of $\displaystyle C$

(b) Hence, or otherwise, find

(i) the value of $\displaystyle x$ for which $\displaystyle f(x)$ has its greatest value

(ii) the greatest value of $\displaystyle f(x)$

The curve $\displaystyle C$ has equation $\displaystyle y = f(x)$

The curve $\displaystyle S$ with equation $\displaystyle y = x^2 - x + 13$ intersects curve $\displaystyle C$ at two points.

(c) Find the $\displaystyle x$ coordinate of each of these two points.

(d) Use algebraic integration to find the exact area of the finite region bounded by the curve $\displaystyle C$ and the curve $\displaystyle S$.

Figure 2 shows the graph of part of the curve $\displaystyle C$ with equation $\displaystyle y = \sqrt{2x + 6}$. The finite region enclosed by the curve $\displaystyle C$ and the straight line with equation $\displaystyle 3y - x = 3$ is rotated through $\displaystyle 360^\circ$ about the $\displaystyle x$-axis.

Use algebraic integration to find the exact volume of the solid generated. Give your answer in terms of $\displaystyle \pi$

(a) Expand $\displaystyle (1-8x^2)^{\frac{1}{2}}$ in ascending powers of $\displaystyle x$, up to and including the term in $\displaystyle x^6$ giving each coefficient as an integer.

Given that the fourth and fifth terms, in ascending powers of $\displaystyle x$, in the series expansion of $\displaystyle g(x)$ are $\displaystyle 20x^3$ and $\displaystyle 48x^4$ respectively,

(b) find the value of $\displaystyle a$ and the value of $\displaystyle b$

Using the first five terms, in ascending powers of $\displaystyle x$, in the series expansion of $\displaystyle g(x)$

(c) obtain an estimate, to 4 significant figures, of $\displaystyle \int_0^{0.2} g(x) \, \mathrm{d}x$

$\displaystyle f(x) = 2x^3 + ax^2 - 14x + b$ where $\displaystyle a$ and $\displaystyle b$ are constants.

When $\displaystyle f(x)$ is divided by $\displaystyle (x - 4)$ the remainder is 39

Given that $\displaystyle (x - 1)$ is a factor of $\displaystyle f(x)$

(a) show that $\displaystyle a = -3$ and find the value of $\displaystyle b$

(b) Hence factorise $\displaystyle f(x)$ completely.

Figure 3 shows part of the curve $\displaystyle C$ with equation $\displaystyle y = f(x)$

Given that $\displaystyle C$ crosses the $\displaystyle x$-axis at the points with coordinates $\displaystyle (p, 0)$, $\displaystyle (q, 0)$ and $\displaystyle (r, 0)$

(c) write down the value of $\displaystyle p$, the value of $\displaystyle q$ and the value of $\displaystyle r$

The region shown shaded in Figure 3 is bounded by the curve and the $\displaystyle x$-axis.

(d) Use algebraic integration to find the exact area of the shaded region.

(a) Expand $\displaystyle (1 + 2x)^{-\frac{1}{3}}$ in ascending powers of $\displaystyle x$ up to and including the term in $\displaystyle x^3$ expressing each coefficient as a fraction in its lowest terms.

(b) Find the range of values of $\displaystyle x$ for which your expansion is valid.

(c) Obtain a series expansion of $\displaystyle f(x)$ in ascending powers of $\displaystyle x$ up to and including the term in $\displaystyle x^3$
Give your coefficients in terms of $\displaystyle k$ where appropriate.

The coefficient of $\displaystyle x^3$ in the series expansion of $\displaystyle f(x)$ is $\displaystyle -\frac{8}{3}$

(d) Find the exact value of $\displaystyle k$

(e) Hence, using algebraic integration, estimate the value of

Give your answer to 4 decimal places.

The curve $\displaystyle S$ with equation $\displaystyle y = \frac{x^2}{4} + 2$ where $\displaystyle x \geq 0$ and the line $\displaystyle l$ with equation $\displaystyle 2y - x - 4 = 0$ where $\displaystyle x \geq 0$ intersect at the points $\displaystyle A$ and $\displaystyle B$, as shown in Figure 2.

(a) (i) Show that the coordinates of point $\displaystyle A$ are $\displaystyle (0, 2)$

(ii) Find the coordinates of the point $\displaystyle B$

The finite region bounded by $\displaystyle S$ and $\displaystyle l$, shown shaded in Figure 2, is rotated through $\displaystyle 2\pi$ radians about the $\displaystyle y$-axis.

(b) Use algebraic integration to find the volume of the solid generated.

Give your answer in terms of $\displaystyle \pi$

(a) Expand $\displaystyle \left(1 + \frac{x}{3}\right)^{-3}$ in ascending powers of $\displaystyle x$ up to and including the term in $\displaystyle x^3$

Where appropriate express each coefficient as an exact fraction in its lowest terms.

(b) Write down the range of values of $\displaystyle x$ for which your expression is valid.

(c) Express $\displaystyle (3 + x)^{-3}$ in the form $\displaystyle P(1 + Qx)^{-3}$ where $\displaystyle P$ and $\displaystyle Q$ are rational numbers whose values should be stated.

(d) Obtain a series expansion for $\displaystyle f(x)$ in ascending powers of $\displaystyle x$ up to and including the term in $\displaystyle x^2$

(e) Hence, using algebraic integration, obtain an estimate of $\displaystyle \int_0^{0.2} f(x) \, \mathrm{d}x$

Give your answer to 5 significant figures.

Figure 1 shows part of the curve $\displaystyle S$ with equation $\displaystyle y = px^2 + qx + r$ where $\displaystyle p$, $\displaystyle q$ and $\displaystyle r$ are constants.

The points $\displaystyle A$, $\displaystyle B$ and $\displaystyle P$ with coordinates $\displaystyle (-2, 0)$, $\displaystyle (6, 0)$ and $\displaystyle (4, -6)$ respectively lie on $\displaystyle S$

(a) Show that an equation of $\displaystyle S$ is $\displaystyle y = \frac{x^2}{2} - 2x - 6$

The line $\displaystyle l$ is the normal to $\displaystyle S$ at the point $\displaystyle P$

(b) Show that an equation of $\displaystyle l$ is $\displaystyle 2y + x + 8 = 0$

The finite region shown shaded in Figure 1 is bounded by $\displaystyle S$ and $\displaystyle l$

(c) Use algebraic integration to find the exact area of the shaded region.

Given that $\displaystyle f(x)$ can be written in the form

(a) find the value of $\displaystyle a$, the value of $\displaystyle b$ and the value of $\displaystyle c$.

Figure 3 shows a sketch of part of the curve $\displaystyle C$ with equation

The curve $\displaystyle C$ crosses the $\displaystyle x$-axis at the point $\displaystyle M$, the origin and the point $\displaystyle B$. The point $\displaystyle N$ lies on the $\displaystyle x$-axis between $\displaystyle M$ and $\displaystyle O$. The point $\displaystyle A$ lies on $\displaystyle C$ such that $\displaystyle AN$ is parallel to the $\displaystyle y$-axis.

The area of the shaded region bounded by the curve and $\displaystyle OB$ is numerically equal to the area of the shaded region bounded by the curve, $\displaystyle ON$ and $\displaystyle NA$.

Given that the coordinates of $\displaystyle N$ are $\displaystyle (n,0)$,

(b) use algebraic integration to show that $\displaystyle n$ satisfies the equation

(c) Hence find the exact coordinates of $\displaystyle A$.

(a) Using a formula from page 2, show that

The finite region $\displaystyle R$ is bounded by the curve with equation

the $\displaystyle x$-axis, the $\displaystyle y$-axis and the line with equation

The region $\displaystyle R$ is rotated through $\displaystyle 360^\circ$ about the $\displaystyle x$-axis.

(b) Use calculus to find the volume of the solid generated. Give your answer to the nearest integer.

(a) Using a formula from page 2, show that

Figure 2 shows a sketch of part of the curves with equations

The points $\displaystyle A$, $\displaystyle B$ and $\displaystyle C$ shown in Figure 2 are three points that are common to both curves.

(b) Find the coordinates of each of these points.

$\displaystyle R_1$ and $\displaystyle R_2$, shown shaded in Figure 2, are two regions enclosed by the two curves.

(c) Use calculus to find, in its simplest form, the ratio

The region $\displaystyle R$, shown shaded in Figure 4, is bounded by the curve with equation

the curve with equation

the straight line with equation $\displaystyle x=a$, the $\displaystyle x$-axis and the $\displaystyle y$-axis.

When the region $\displaystyle R$ is rotated through $\displaystyle 360^\circ$ about the $\displaystyle x$-axis, the volume of the solid generated is

where $\displaystyle k$ is a constant.

Using algebraic integration, find a possible value of $\displaystyle a$ and the exact corresponding value of $\displaystyle k$.

The region $\displaystyle R$, shown shaded in Figure 2, is bounded by the $\displaystyle x$-axis, the curve $\displaystyle S$ with equation

and the curve $\displaystyle C$ with equation

As shown in Figure 2, $\displaystyle C$ crosses the $\displaystyle x$-axis at the point $\displaystyle A$.

(a) Write down the $\displaystyle x$ coordinate of $\displaystyle A$.

As shown in Figure 2, $\displaystyle C$ and $\displaystyle S$ intersect at the point $\displaystyle B$.

(b) Find the $\displaystyle x$ coordinate of $\displaystyle B$.

(c) Using calculus, find the area of the shaded region $\displaystyle R$. Give your answer in the form

where $\displaystyle a$ and $\displaystyle b$ are integers.

The region enclosed by the curve

the $\displaystyle x$-axis, the $\displaystyle y$-axis and the line $\displaystyle x=3$ is rotated through $\displaystyle 360^\circ$ about the $\displaystyle x$-axis.

Use algebraic integration to find, in terms of $\displaystyle \pi$ and $\displaystyle \mathrm{e}$, the volume of the solid generated.

(a) Show that $\displaystyle f(1)=0$.

(b) Hence, using an algebraic method, solve $\displaystyle f(x)=0$.

The region $\displaystyle R$, shown shaded in Figure 4, is bounded by the curve $\displaystyle C_1$ with equation

the curve $\displaystyle C_2$ with equation

and the line with equation $\displaystyle x=a$.

The curves $\displaystyle C_1$ and $\displaystyle C_2$ intersect at the point $\displaystyle B$, with $\displaystyle x$ coordinate $\displaystyle p$, where $\displaystyle p<a$.

(c) Find the value of $\displaystyle p$.

The region $\displaystyle R$ is rotated through $\displaystyle 360^\circ$ about the $\displaystyle x$-axis to generate a solid with volume

(d) Use algebraic integration to find the value of $\displaystyle a$.

The curve $\displaystyle C$ has equation

The point $\displaystyle P$ lies on $\displaystyle C$ and has $\displaystyle x$ coordinate $\displaystyle 4$.

The line $\displaystyle k$ is the tangent to $\displaystyle C$ at $\displaystyle P$.

(a) Find an equation for $\displaystyle k$, giving your answer in the form

where $\displaystyle a$, $\displaystyle b$ and $\displaystyle c$ are integers.

The line $\displaystyle l$ is the normal to $\displaystyle C$ at $\displaystyle P$.

(b) Find an equation for $\displaystyle l$, giving your answer in the form

where $\displaystyle d$, $\displaystyle e$ and $\displaystyle f$ are integers.

(c) Find the area of the triangle bounded by the line $\displaystyle k$, the line $\displaystyle l$ and the $\displaystyle x$-axis.

The finite region bounded by $\displaystyle C$, the line $\displaystyle l$, the $\displaystyle x$-axis and the $\displaystyle y$-axis is rotated through $\displaystyle 360^\circ$ about the $\displaystyle x$-axis.

(d) Use algebraic integration to find, to the nearest whole number, the volume of the solid generated.

Given that $\displaystyle f(x)$ can be written in the form $\displaystyle D(x+E)^2+F$, where $\displaystyle D$, $\displaystyle E$ and $\displaystyle F$ are integers,

(a) find the value of $\displaystyle D$, the value of $\displaystyle E$ and the value of $\displaystyle F$.

(b) Find

(i) the maximum value of $\displaystyle f(x)$,

(ii) the value of $\displaystyle x$ for which the maximum occurs.

The curve $\displaystyle C$ has equation $\displaystyle y=f(x)$.

The curve $\displaystyle S$ has equation

The curve $\displaystyle S$ intersects the curve $\displaystyle C$ at two points.

(c) Find the coordinates of each of these two points.

The finite region $\displaystyle R$ is bounded by the curve $\displaystyle C$ and the curve $\displaystyle S$.

(d) Use algebraic integration to find the area of $\displaystyle R$.

The equation Figure line shows crosses part at of two the curve points. with equation $\displaystyle y = (2x + 3) 1 2$ and the line $\displaystyle l$ with

(a) Find the coordinates of each of these points.

The about finite the x-axis. region bounded by $\displaystyle C$ and $\displaystyle {\mathit{l}}_{\mathrm{,}}$ shown shaded in Figure $\displaystyle {\underline{{3}}},$ is rotated through $\displaystyle 360^{\circ}$

(b) Use algebraic integration to find, in terms of $\displaystyle \pi,$ the volume of the solid generated.

The finite region $\displaystyle R$ enclosed by the $\displaystyle y$-axis, the straight line with equation

and the curve with equation

is defined for points with coordinates $\displaystyle (x,y)$ with $\displaystyle x\geqslant0$.

The region $\displaystyle R$ is rotated through $\displaystyle 360^\circ$ about the $\displaystyle y$-axis.

Use algebraic integration to find the volume of the solid generated.

Give your answer in terms of $\displaystyle \pi$.