The shaded region between the curve $y = e^{-x^2}$, the x-axis, and the lines $x=0$ and $x=N$, where $N > 0$, is rotated about the y-axis to form a solid of revolution. (i) Use the method of cylindrical shells to find the volume of this solid in terms of $N$. (ii) What is the limiting value of this volume as $N \rightarrow \infty$? (b) Suppose $\alpha, \beta, \gamma$ and $\delta$ are the four roots of the polynomial equation $x^4 + px^3 + qx^2 + rx + s = 0$. (i) Find the values of $\alpha + \beta + \gamma + \delta$ and $\alpha\beta + \alpha\gamma + \alpha\delta + \beta\gamma + \beta\delta + \gamma\delta$ in terms of $p, q, r$ and $s$. (ii) Show that $\alpha^2 + \beta^2 + \gamma^2 + \delta^2 = p^2 - 2q$. (iii) Apply the result in part (ii) to show that $x^4 - 3x^3 + 5x^2 + 7x - 8 = 0$ cannot have four real roots. (iv) By evaluating the polynomial at $x=0$ and $x=1$, deduce that the polynomial equation $x^4 - 3x^3 + 5x^2 + 7x - 8 = 0$ has exactly two real roots. (c) The point $P(x_1, y_1)$ lies on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, where $a > b > 0$. The equation of the normal to the ellipse at $P$ is $y = -\frac{b^2}{a^2}x + (\frac{b^2}{a^2})x_1 + y_1$. (i) The normal at $P$ passes through the point $B(0, -b)$. Show that $y_1 = \frac{b^3}{a^2 - b^2}$ or $y_1 = \pm b$. (ii) Show that if $y_1 = \frac{b^3}{a^2 - b^2}$, the eccentricity of the ellipse is at least $\frac{1}{\sqrt{2}}$.

SSCE HSC Mathematics Extension 2 Vector equation of a curve: The shaded region between the cu

The shaded region between the curve $y = e^{-x^2}$, the x-axis, and the lines $x=0$ and $x=N$, where $N > 0$, is rotated about the y-axis to form a solid of revolution - HSC - SSCE Mathematics Extension 2 - Question 4 - 2005 - Paper 1

Question 4

$The-shaded-region-between-the-curve-$y-=-e^{-x^2}$,-the-x-axis,-and-the-lines-$x=0$-and-$x=N$,-where-$N->-0$,-is-rotated-about-the-y-axis-to-form-a-solid-of-revolution-HSC-SSCE Mathematics Extension 2-Question 4-2005-Paper 1.png$

The shaded region between the curve $y = e^{-x^2}$, the x-axis, and the lines $x=0$ and $x=N$, where $N > 0$, is rotated about the y-axis to form a solid of revoluti... show full transcript

Worked Solution & Example Answer:The shaded region between the curve $y = e^{-x^2}$, the x-axis, and the lines $x=0$ and $x=N$, where $N > 0$, is rotated about the y-axis to form a solid of revolution - HSC - SSCE Mathematics Extension 2 - Question 4 - 2005 - Paper 1

Step 1

Use the method of cylindrical shells to find the volume of this solid in terms of $N$.

96%

114 rated

Answer

To find the volume of the solid formed by rotating the shaded area about the y-axis using cylindrical shells, we use the formula:

$V = 2\pi \int_{0}^{N} x \cdot f(x) \, dx$

where $f(x) = e^{-x^2}$ . Thus, we have:

$V = 2\pi \int_{0}^{N} x e^{-x^2} \, dx.$

Substituting $u = -x^2$ leads to:

$V = -\pi \int_{0}^{-N^2} e^{u} \, du = \pi (1 - e^{-N^2}).$

Step 2

What is the limiting value of this volume as $N \rightarrow \infty$?

99%

104 rated

Answer

As $N \rightarrow \infty$ , the term $e^{-N^2} \rightarrow 0$ . Thus, the limiting value of the volume $V$ is:

$\lim_{N \to \infty} \pi (1 - e^{-N^2}) = \pi.$

Step 3

Find the values of $\alpha + \beta + \gamma + \delta$ and $\alpha\beta + \alpha\gamma + \alpha\delta + \beta\gamma + \beta\delta + \gamma\delta$ in terms of $p, q, r$ and $s$.

96%

101 rated

Answer

According to Vieta's formulas:

The sum of the roots is:

$\alpha + \beta + \gamma + \delta = -p.$

The sum of the products of the roots taken two at a time is:

$\alpha\beta + \alpha\gamma + \alpha\delta + \beta\gamma + \beta\delta + \gamma\delta = q.$

Step 4

Show that $\alpha^2 + \beta^2 + \gamma^2 + \delta^2 = p^2 - 2q$.

98%

120 rated

Answer

We start with:

$\alpha^2 + \beta^2 + \gamma^2 + \delta^2 = (\\alpha + \beta + \gamma + \delta)^2 - 2(\alpha\beta + \alpha\gamma + \alpha\delta + \beta\gamma + \beta\delta + \gamma\delta).$

Substituting the known values:

$= (-p)^2 - 2q = p^2 - 2q.$

Step 5

Apply the result in part (ii) to show that $x^4 - 3x^3 + 5x^2 + 7x - 8 = 0$ cannot have four real roots.

97%

117 rated

Answer

Calculating:

Here, $p = -3$ , $q = 5$ , thus: $\alpha^2 + \beta^2 + \gamma^2 + \delta^2 = 9 - 10 = -1.$ Since the sum of squares of real numbers cannot be negative, the polynomial cannot have four real roots.

Step 6

By evaluating the polynomial at $x=0$ and $x=1$, deduce that the polynomial equation $x^4 - 3x^3 + 5x^2 + 7x - 8 = 0$ has exactly two real roots.

97%

121 rated

Answer

Evaluating at $x=0$ :

$P(0) = -8 < 0.$ At $x=1$ :

$P(1) = 1 - 3 + 5 + 7 - 8 = 2 > 0.$ Applying the Intermediate Value Theorem, there is at least one root between $0$ and $1$ . Since there are only two real roots, the polynomial crosses the x-axis exactly twice.

Step 7

Show that $y_1 = \frac{b^3}{a^2 - b^2}$ or $y_1 = \pm b$.

96%

114 rated

Answer

Substituting Point P in the normal equation gives:

$y_1 = -\frac{b^2}{a^2} \cdot 0 + (\frac{b^2}{a^2})x_1 + y_1 = \frac{b^3}{a^2 - b^2}.$ Thus, $y_1 = \frac{b^3}{a^2 - b^2}$ or for the vertical tangent case, $y_1 = \pm b$ .

Step 8

Show that if $y_1 = \frac{b^3}{a^2 - b^2}$, the eccentricity of the ellipse is at least $\frac{1}{\sqrt{2}}$.

99%

104 rated

Answer

The eccentricity $e$ is given by:

$e = \sqrt{1 - \frac{b^2}{a^2}}.$ Setting up the inequality:

$y_1 = \frac{b^3}{a^2 - b^2} \Rightarrow a^2 \geq 2b^2.$ This leads us to:

$e = \sqrt{1 - \frac{b^2}{2b^2}} = \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}}.$

The shaded region between the curve $y = e^{-x^2}$, the x-axis, and the lines $x=0$ and $x=N$, where $N > 0$, is rotated about the y-axis to form a solid of revolution - HSC - SSCE Mathematics Extension 2 - Question 4 - 2005 - Paper 1

Worked Solution & Example Answer:The shaded region between the curve $y = e^{-x^2}$, the x-axis, and the lines $x=0$ and $x=N$, where $N > 0$, is rotated about the y-axis to form a solid of revolution - HSC - SSCE Mathematics Extension 2 - Question 4 - 2005 - Paper 1

Use the method of cylindrical shells to find the volume of this solid in terms of $N$.

What is the limiting value of this volume as $N \rightarrow \infty$?

Find the values of $\alpha + \beta + \gamma + \delta$ and $\alpha\beta + \alpha\gamma + \alpha\delta + \beta\gamma + \beta\delta + \gamma\delta$ in terms of $p, q, r$ and $s$.

Show that $\alpha^2 + \beta^2 + \gamma^2 + \delta^2 = p^2 - 2q$.

Apply the result in part (ii) to show that $x^4 - 3x^3 + 5x^2 + 7x - 8 = 0$ cannot have four real roots.

By evaluating the polynomial at $x=0$ and $x=1$, deduce that the polynomial equation $x^4 - 3x^3 + 5x^2 + 7x - 8 = 0$ has exactly two real roots.

Show that $y_1 = \frac{b^3}{a^2 - b^2}$ or $y_1 = \pm b$.

Show that if $y_1 = \frac{b^3}{a^2 - b^2}$, the eccentricity of the ellipse is at least $\frac{1}{\sqrt{2}}$.

Join the SSCE students using SimpleStudy...