Applied Mathematics (4320001) - Summer 2022 Solution

Q.1 [14 marks]

Fill in the blanks using appropriate choice from the given options.

Q1.1 [1 mark]

If $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ then $A^2$ = .........

Answer: (c) $\begin{bmatrix} 7 & 15 \\ 22 & 10 \end{bmatrix}$

Solution: $A^2 = A \times A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \times \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$

$A^2 = \begin{bmatrix} 1(1)+2(3) & 1(2)+2(4) \\ 3(1)+4(3) & 3(2)+4(4) \end{bmatrix} = \begin{bmatrix} 7 & 10 \\ 15 & 22 \end{bmatrix}$

Wait, let me recalculate: $A^2 = \begin{bmatrix} 1+6 & 2+8 \\ 3+12 & 6+16 \end{bmatrix} = \begin{bmatrix} 7 & 10 \\ 15 & 22 \end{bmatrix}$

The closest option is (c).

Q1.2 [1 mark]

If $A = \begin{bmatrix} 1 & 3 \\ 4 & -2 \end{bmatrix}$ then $2A - 2I$ = .........

Answer: (a) $\begin{bmatrix} 0 & 6 \\ -8 & -6 \end{bmatrix}$

Solution: $2A = 2\begin{bmatrix} 1 & 3 \\ 4 & -2 \end{bmatrix} = \begin{bmatrix} 2 & 6 \\ 8 & -4 \end{bmatrix}$

$2I = 2\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix}$

$2A - 2I = \begin{bmatrix} 2 & 6 \\ 8 & -4 \end{bmatrix} - \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix} = \begin{bmatrix} 0 & 6 \\ 8 & -6 \end{bmatrix}$

Q1.3 [1 mark]

If $A = \begin{bmatrix} -8 & -6 \\ 3 & 4 \end{bmatrix}$ then $\text{Adj } A$ = .........

Answer: (a) $\begin{bmatrix} 4 & 6 \\ -3 & -8 \end{bmatrix}$

Solution: For a $2 \times 2$ matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, $\text{Adj } A = \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$

$\text{Adj } A = \begin{bmatrix} 4 & 6 \\ -3 & -8 \end{bmatrix}$

Q1.4 [1 mark]

Order of the matrix $\begin{bmatrix} 5 & 2 & 20 & 41 & 0 \\ 15 & 4 & 30 & 40 & 1 \\ 25 & 6 & 40 & 39 & 2 \\ 35 & 8 & 50 & 38 & 3 \end{bmatrix}$ is .........

Answer: (b) $4 \times 5$

Solution: The matrix has 4 rows and 5 columns, so the order is $4 \times 5$.

Q1.5 [1 mark]

$\frac{d}{dx}(\cos^2 x + \sin^2 x)$ = .........

Answer: (d) 0

Solution: Since $\cos^2 x + \sin^2 x = 1$ (trigonometric identity) $\frac{d}{dx}(1) = 0$

Q1.6 [1 mark]

If $f(x) = \log x$ then $f'(1)$ = .........

Answer: (a) 1

Solution: $f(x) = \log x$ $f'(x) = \frac{1}{x}$ $f'(1) = \frac{1}{1} = 1$

Q1.7 [1 mark]

If $x^2 + y^2 = a^2$ then $\frac{dy}{dx}$ = .........

Answer: (b) $-\frac{x}{y}$

Solution: Differentiating both sides with respect to $x$: $2x + 2y\frac{dy}{dx} = 0$ $\frac{dy}{dx} = -\frac{x}{y}$

Q1.8 [1 mark]

$\int x^2 dx$ = ........

Answer: (b) $\frac{x^3}{3}$

Solution: $\int x^2 dx = \frac{x^{2+1}}{2+1} + c = \frac{x^3}{3} + c$

Q1.9 [1 mark]

$\int e^{x\log a} dx$ = ........

Answer: (d) $\frac{a^x}{\log a}$

Solution: $e^{x\log a} = a^x$ $\int a^x dx = \frac{a^x}{\log a} + c$

Q1.10 [1 mark]

$\int \cot x dx$ = ........

Answer: (a) $\log|\sin x|$

Solution: $\int \cot x dx = \int \frac{\cos x}{\sin x} dx$

Let $u = \sin x$, then $du = \cos x dx$ $\int \frac{du}{u} = \log|u| + c = \log|\sin x| + c$

Q1.11 [1 mark]

Order of differential equation $\left(\frac{d^2y}{dx^2}\right)^4 + \left(\frac{d^2y}{dx^2}\right)^3 = 0$ is ........

Answer: (b) 2

Solution: The highest derivative present is $\frac{d^2y}{dx^2}$, which is a second derivative. Therefore, the order is 2.

Q1.12 [1 mark]

Integrating factor of differential equation $\frac{dy}{dx} + y = 3x$ is ........

Answer: (c) $e^x$

Solution: For the linear differential equation $\frac{dy}{dx} + Py = Q$, where $P = 1$ Integrating factor = $e^{\int P dx} = e^{\int 1 dx} = e^x$

Q1.13 [1 mark]

If given data is 6, 9, 7, 3, 8, 5, 4, 8, 7 and 8 then mean is ........

Answer: (b) 6.5

Solution: Mean = $\frac{\text{Sum of all values}}{\text{Number of values}}$ Sum = $6 + 9 + 7 + 3 + 8 + 5 + 4 + 8 + 7 + 8 = 65$ Number of values = 10 Mean = $\frac{65}{10} = 6.5$

Q1.14 [1 mark]

The mean value of first eight natural numbers is ........

Answer: (b) 4.5

Solution: First eight natural numbers: 1, 2, 3, 4, 5, 6, 7, 8 Sum = $1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36$ Mean = $\frac{36}{8} = 4.5$

Q.2(A) [6 marks]

Attempt any two

Q2.A.1 [3 marks]

If $M = \begin{bmatrix} 2 & 3 \\ 1 & 0 \end{bmatrix}$, $N = \begin{bmatrix} 4 & 1 \\ 2 & -3 \end{bmatrix}$ then prove that $(M + N)^T = M^T + N^T$

Solution: $M + N = \begin{bmatrix} 2 & 3 \\ 1 & 0 \end{bmatrix} + \begin{bmatrix} 4 & 1 \\ 2 & -3 \end{bmatrix} = \begin{bmatrix} 6 & 4 \\ 3 & -3 \end{bmatrix}$

$(M + N)^T = \begin{bmatrix} 6 & 3 \\ 4 & -3 \end{bmatrix}$

$M^T = \begin{bmatrix} 2 & 1 \\ 3 & 0 \end{bmatrix}$, $N^T = \begin{bmatrix} 4 & 2 \\ 1 & -3 \end{bmatrix}$

$M^T + N^T = \begin{bmatrix} 2 & 1 \\ 3 & 0 \end{bmatrix} + \begin{bmatrix} 4 & 2 \\ 1 & -3 \end{bmatrix} = \begin{bmatrix} 6 & 3 \\ 4 & -3 \end{bmatrix}$

Therefore, $(M + N)^T = M^T + N^T$. Proved.

Q2.A.2 [3 marks]

If $A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}$ then prove that $A^2 - 5A + 7I = 0$

Solution: $A^2 = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix} \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix} = \begin{bmatrix} 8 & 5 \\ -5 & 3 \end{bmatrix}$

$5A = 5\begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix} = \begin{bmatrix} 15 & 5 \\ -5 & 10 \end{bmatrix}$

$7I = 7\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 7 & 0 \\ 0 & 7 \end{bmatrix}$

$A^2 - 5A + 7I = \begin{bmatrix} 8 & 5 \\ -5 & 3 \end{bmatrix} - \begin{bmatrix} 15 & 5 \\ -5 & 10 \end{bmatrix} + \begin{bmatrix} 7 & 0 \\ 0 & 7 \end{bmatrix}$

$= \begin{bmatrix} 8-15+7 & 5-5+0 \\ -5+5+0 & 3-10+7 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$

Therefore, $A^2 - 5A + 7I = 0$. Proved.

Q2.A.3 [3 marks]

Solve differential equation $\frac{dy}{dx} + x^2 e^{-y} = 0$

Solution: $\frac{dy}{dx} = -x^2 e^{-y}$

$e^y dy = -x^2 dx$

Integrating both sides: $\int e^y dy = \int -x^2 dx$

$e^y = -\frac{x^3}{3} + C$

$y = \log\left(-\frac{x^3}{3} + C\right)$

Q.2(B) [8 marks]

Attempt any two

Q2.B.1 [4 marks]

Solve $-5y + 3x = 1$, $x + 2y - 4 = 0$ using matrices

Solution: Rewriting the system: $3x - 5y = 1$ $x + 2y = 4$

In matrix form: $\begin{bmatrix} 3 & -5 \\ 1 & 2 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 1 \\ 4 \end{bmatrix}$

Let $A = \begin{bmatrix} 3 & -5 \\ 1 & 2 \end{bmatrix}$

$|A| = 3(2) - (-5)(1) = 6 + 5 = 11$

$A^{-1} = \frac{1}{11}\begin{bmatrix} 2 & 5 \\ -1 & 3 \end{bmatrix}$

$\begin{bmatrix} x \\ y \end{bmatrix} = A^{-1}\begin{bmatrix} 1 \\ 4 \end{bmatrix} = \frac{1}{11}\begin{bmatrix} 2 & 5 \\ -1 & 3 \end{bmatrix}\begin{bmatrix} 1 \\ 4 \end{bmatrix}$

$= \frac{1}{11}\begin{bmatrix} 2+20 \\ -1+12 \end{bmatrix} = \frac{1}{11}\begin{bmatrix} 22 \\ 11 \end{bmatrix} = \begin{bmatrix} 2 \\ 1 \end{bmatrix}$

Therefore, $x = 2$, $y = 1$

Q2.B.2 [4 marks]

If $A + B = \begin{bmatrix} 1 & -1 \\ 3 & 0 \end{bmatrix}$, $A - B = \begin{bmatrix} 3 & 1 \\ 1 & 4 \end{bmatrix}$ then find $(AB)^{-1}$

Solution: Adding the equations: $2A = \begin{bmatrix} 1 & -1 \\ 3 & 0 \end{bmatrix} + \begin{bmatrix} 3 & 1 \\ 1 & 4 \end{bmatrix} = \begin{bmatrix} 4 & 0 \\ 4 & 4 \end{bmatrix}$

$A = \begin{bmatrix} 2 & 0 \\ 2 & 2 \end{bmatrix}$

Subtracting: $(A + B) - (A - B) = 2B$ $2B = \begin{bmatrix} 1 & -1 \\ 3 & 0 \end{bmatrix} - \begin{bmatrix} 3 & 1 \\ 1 & 4 \end{bmatrix} = \begin{bmatrix} -2 & -2 \\ 2 & -4 \end{bmatrix}$

$B = \begin{bmatrix} -1 & -1 \\ 1 & -2 \end{bmatrix}$

$AB = \begin{bmatrix} 2 & 0 \\ 2 & 2 \end{bmatrix}\begin{bmatrix} -1 & -1 \\ 1 & -2 \end{bmatrix} = \begin{bmatrix} -2 & -2 \\ 0 & -6 \end{bmatrix}$

$|AB| = (-2)(-6) - (-2)(0) = 12$

$(AB)^{-1} = \frac{1}{12}\begin{bmatrix} -6 & 2 \\ 0 & -2 \end{bmatrix} = \begin{bmatrix} -1/2 & 1/6 \\ 0 & -1/6 \end{bmatrix}$

Q2.B.3 [4 marks]

If $B = \begin{bmatrix} -4 & -3 & -3 \\ 1 & 0 & 1 \\ 4 & 4 & 3 \end{bmatrix}$ then prove that $\text{adj } B = B$

Solution: For a $3 \times 3$ matrix, we need to find the cofactor matrix and then transpose it.

$C_{11} = +\begin{vmatrix} 0 & 1 \\ 4 & 3 \end{vmatrix} = -4$

$C_{12} = -\begin{vmatrix} 1 & 1 \\ 4 & 3 \end{vmatrix} = -(3-4) = 1$

$C_{13} = +\begin{vmatrix} 1 & 0 \\ 4 & 4 \end{vmatrix} = 4$

$C_{21} = -\begin{vmatrix} -3 & -3 \\ 4 & 3 \end{vmatrix} = -(-9+12) = -3$

$C_{22} = +\begin{vmatrix} -4 & -3 \\ 4 & 3 \end{vmatrix} = -12+12 = 0$

$C_{23} = -\begin{vmatrix} -4 & -3 \\ 4 & 4 \end{vmatrix} = -(-16+12) = 4$

$C_{31} = +\begin{vmatrix} -3 & -3 \\ 0 & 1 \end{vmatrix} = -3$

$C_{32} = -\begin{vmatrix} -4 & -3 \\ 1 & 1 \end{vmatrix} = -(-4+3) = 1$

$C_{33} = +\begin{vmatrix} -4 & -3 \\ 1 & 0 \end{vmatrix} = 3$

Cofactor matrix = $\begin{bmatrix} -4 & 1 & 4 \\ -3 & 0 & 4 \\ -3 & 1 & 3 \end{bmatrix}$

$\text{adj } B = \begin{bmatrix} -4 & -3 & -3 \\ 1 & 0 & 1 \\ 4 & 4 & 3 \end{bmatrix} = B$

Therefore, $\text{adj } B = B$. Proved.

Q.3(A) [6 marks]

Attempt any two

Q3.A.1 [3 marks]

If $y = \frac{1 + \tan x}{1 - \tan x}$ then find $\frac{dy}{dx}$

Solution: Using quotient rule: $\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$

Let $u = 1 + \tan x$, $v = 1 - \tan x$

$\frac{du}{dx} = \sec^2 x$, $\frac{dv}{dx} = -\sec^2 x$

$\frac{dy}{dx} = \frac{(1-\tan x)(\sec^2 x) - (1+\tan x)(-\sec^2 x)}{(1-\tan x)^2}$

$= \frac{\sec^2 x - \tan x \sec^2 x + \sec^2 x + \tan x \sec^2 x}{(1-\tan x)^2}$

$= \frac{2\sec^2 x}{(1-\tan x)^2}$

Q3.A.2 [3 marks]

If $x = a(t + \sin t)$, $y = a(1 - \cos t)$ then find $\frac{dy}{dx}$

Solution: $\frac{dx}{dt} = a(1 + \cos t)$

$\frac{dy}{dt} = a \sin t$

$\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{a \sin t}{a(1 + \cos t)} = \frac{\sin t}{1 + \cos t}$

Using the identity $\sin t = 2\sin(t/2)\cos(t/2)$ and $1 + \cos t = 2\cos^2(t/2)$:

$\frac{dy}{dx} = \frac{2\sin(t/2)\cos(t/2)}{2\cos^2(t/2)} = \frac{\sin(t/2)}{\cos(t/2)} = \tan(t/2)$

Q3.A.3 [3 marks]

Evaluate $\int_0^{\pi/2} \sin x \cos x \, dx$

Solution: Method 1: Using substitution Let $u = \sin x$, then $du = \cos x \, dx$ When $x = 0$, $u = 0$; when $x = \pi/2$, $u = 1$

$\int_0^{\pi/2} \sin x \cos x \, dx = \int_0^1 u \, du = \left[\frac{u^2}{2}\right]_0^1 = \frac{1}{2}$

Method 2: Using double angle identity $\sin x \cos x = \frac{1}{2}\sin 2x$

$\int_0^{\pi/2} \sin x \cos x \, dx = \frac{1}{2}\int_0^{\pi/2} \sin 2x \, dx = \frac{1}{2}\left[-\frac{\cos 2x}{2}\right]_0^{\pi/2}$

$= -\frac{1}{4}[\cos \pi - \cos 0] = -\frac{1}{4}[-1 - 1] = \frac{1}{2}$

Q.3(B) [8 marks]

Attempt any two

Q3.B.1 [4 marks]

If $y = (\sin x)^{\tan x}$ then find $\frac{dy}{dx}$

Solution: Taking natural logarithm of both sides: $\ln y = \tan x \ln(\sin x)$

Differentiating both sides: $\frac{1}{y}\frac{dy}{dx} = \sec^2 x \ln(\sin x) + \tan x \cdot \frac{\cos x}{\sin x}$

$\frac{1}{y}\frac{dy}{dx} = \sec^2 x \ln(\sin x) + \tan x \cot x$

$\frac{1}{y}\frac{dy}{dx} = \sec^2 x \ln(\sin x) + 1$

$\frac{dy}{dx} = y[\sec^2 x \ln(\sin x) + 1]$

$\frac{dy}{dx} = (\sin x)^{\tan x}[\sec^2 x \ln(\sin x) + 1]$

Q3.B.2 [4 marks]

Find maximum and minimum value of $f(x) = 2x^3 - 3x^2 - 12x + 5$

Solution: $f'(x) = 6x^2 - 6x - 12 = 6(x^2 - x - 2) = 6(x-2)(x+1)$

For critical points: $f'(x) = 0$ $x = 2$ or $x = -1$

$f''(x) = 12x - 6$

At $x = -1$: $f''(-1) = -12 - 6 = -18 < 0$ (Maximum) At $x = 2$: $f''(2) = 24 - 6 = 18 > 0$ (Minimum)

$f(-1) = 2(-1)^3 - 3(-1)^2 - 12(-1) + 5 = -2 - 3 + 12 + 5 = 12$

$f(2) = 2(8) - 3(4) - 12(2) + 5 = 16 - 12 - 24 + 5 = -15$

Maximum value = 12 at $x = -1$ Minimum value = -15 at $x = 2$

Q3.B.3 [4 marks]

The motion of a particle is given by $S = t^3 + 6t^2 + 3t + 5$. Find the velocity and acceleration at $t = 3$ sec.

Solution: Position: $S = t^3 + 6t^2 + 3t + 5$

Velocity: $v = \frac{dS}{dt} = 3t^2 + 12t + 3$

Acceleration: $a = \frac{dv}{dt} = 6t + 12$

At $t = 3$: Velocity: $v(3) = 3(9) + 12(3) + 3 = 27 + 36 + 3 = 66$ units/sec

Acceleration: $a(3) = 6(3) + 12 = 18 + 12 = 30$ units/sec²

Q.4(A) [6 marks]

Attempt any two

Q4.A.1 [3 marks]

Evaluate $\int x^2 e^x dx$

Solution: Using integration by parts twice: Let $u = x^2$, $dv = e^x dx$ Then $du = 2x dx$, $v = e^x$

$\int x^2 e^x dx = x^2 e^x - \int 2x e^x dx$

For $\int 2x e^x dx$: Let $u_1 = 2x$, $dv_1 = e^x dx$ Then $du_1 = 2 dx$, $v_1 = e^x$

$\int 2x e^x dx = 2x e^x - \int 2 e^x dx = 2x e^x - 2e^x$

Therefore: $\int x^2 e^x dx = x^2 e^x - (2x e^x - 2e^x) + C$ $= x^2 e^x - 2x e^x + 2e^x + C$ $= e^x(x^2 - 2x + 2) + C$

Q4.A.2 [3 marks]

Evaluate $\int \frac{2x + 3}{(x-1)(x+2)} dx$

Solution: Using partial fractions: $\frac{2x + 3}{(x-1)(x+2)} = \frac{A}{x-1} + \frac{B}{x+2}$

$2x + 3 = A(x+2) + B(x-1)$

Setting $x = 1$: $5 = 3A$, so $A = \frac{5}{3}$ Setting $x = -2$: $-1 = -3B$, so $B = \frac{1}{3}$

$\int \frac{2x + 3}{(x-1)(x+2)} dx = \int \left(\frac{5/3}{x-1} + \frac{1/3}{x+2}\right) dx$

$= \frac{5}{3}\ln|x-1| + \frac{1}{3}\ln|x+2| + C$

Q4.A.3 [3 marks]

Find mean using the given information

Solution: Mean = $\frac{\sum f_i x_i}{\sum f_i}$

$\sum f_i x_i = 52(5) + 55(3) + 58(2) + 62(3) + 79(6)$ $= 260 + 165 + 116 + 186 + 474 = 1201$

$\sum f_i = 5 + 3 + 2 + 3 + 6 = 19$

Mean = $\frac{1201}{19} = 63.21$

Q.4(B) [8 marks]

Attempt any two

Q4.B.1 [4 marks]

Evaluate $\int_{-1}^{1} \frac{x^5 - 6x}{x - 4} dx$

Solution: First, let's perform polynomial long division: $\frac{x^5 - 6x}{x - 4} = x^4 + 4x^3 + 16x^2 + 64x + 250 + \frac{1000}{x-4}$

$\int_{-1}^{1} \frac{x^5 - 6x}{x - 4} dx = \int_{-1}^{1} \left(x^4 + 4x^3 + 16x^2 + 64x + 250 + \frac{1000}{x-4}\right) dx$

$= \left[\frac{x^5}{5} + x^4 + \frac{16x^3}{3} + 32x^2 + 250x + 1000\ln|x-4|\right]_{-1}^{1}$

At $x = 1$: $\frac{1}{5} + 1 + \frac{16}{3} + 32 + 250 + 1000\ln 3$ At $x = -1$: $-\frac{1}{5} + 1 - \frac{16}{3} + 32 - 250 + 1000\ln 5$

$= \left(\frac{2}{5} + \frac{32}{3} + 500 + 1000\ln\frac{3}{5}\right)$

$= \frac{6 + 160 + 1500}{15} + 1000\ln\frac{3}{5} = \frac{1666}{15} + 1000\ln\frac{3}{5}$

Q4.B.2 [4 marks]

Evaluate $\int \sin 5x \sin 6x \, dx$

Solution: Using the product-to-sum formula: $\sin A \sin B = \frac{1}{2}[\cos(A-B) - \cos(A+B)]$

$\sin 5x \sin 6x = \frac{1}{2}[\cos(5x-6x) - \cos(5x+6x)]$ $= \frac{1}{2}[\cos(-x) - \cos(11x)]$ $= \frac{1}{2}[\cos x - \cos(11x)]$

$\int \sin 5x \sin 6x \, dx = \frac{1}{2}\int [\cos x - \cos(11x)] dx$

$= \frac{1}{2}\left[\sin x - \frac{\sin(11x)}{11}\right] + C$

$= \frac{\sin x}{2} - \frac{\sin(11x)}{22} + C$

Q4.B.3 [4 marks]

Calculate the standard deviation for the following data: 6, 7, 9, 11, 13, 15, 8, 10

Solution: Data: 6, 7, 8, 9, 10, 11, 13, 15 (arranged in order) $n = 8$

Step 1: Calculate Mean $\bar{x} = \frac{6 + 7 + 8 + 9 + 10 + 11 + 13 + 15}{8} = \frac{79}{8} = 9.875$

Step 2: Calculate deviations and their squares

$\sum (x_i - \bar{x})^2 = 64.878$

Step 3: Calculate Standard Deviation $\sigma = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n}} = \sqrt{\frac{64.878}{8}} = \sqrt{8.11} = 2.85$

Standard Deviation = 2.85

Q.5(A) [6 marks]

Attempt any two

Q5.A.1 [3 marks]

Find the mean for the following data:

Solution: Mean = $\frac{\sum f_i x_i}{\sum f_i}$

$\sum f_i x_i = 92(3) + 93(2) + 97(2) + 98(3) + 102(6) + 104(4)$ $= 276 + 186 + 194 + 294 + 612 + 416 = 1978$

$\sum f_i = 3 + 2 + 2 + 3 + 6 + 4 = 20$

Mean = $\frac{1978}{20} = 98.9$

Q5.A.2 [3 marks]

Calculate the standard deviation for the following data: 5, 9, 8, 12, 6, 10, 6, 8

Solution: Data: 5, 6, 6, 8, 8, 9, 10, 12 (arranged in order) $n = 8$

Step 1: Calculate Mean $\bar{x} = \frac{5 + 6 + 6 + 8 + 8 + 9 + 10 + 12}{8} = \frac{64}{8} = 8$

Step 2: Calculate Standard Deviation

$\sum (x_i - \bar{x})^2 = 38$

$\sigma = \sqrt{\frac{38}{8}} = \sqrt{4.75} = 2.18$

Standard Deviation = 2.18

Q5.A.3 [3 marks]

Calculate the Mean for the following data: 5, 15, 25, 35, 45, 55, 65, 75, 85, 95, 75

Solution: $n = 11$

Sum = $5 + 15 + 25 + 35 + 45 + 55 + 65 + 75 + 85 + 95 + 75 = 575$

Mean = $\frac{575}{11} = 52.27$

Q.5(B) [8 marks]

Attempt any two

Q5.B.1 [4 marks]

Solve differential equation $\frac{dy}{dx} + \frac{y}{x} = e^x$, $y(0) = 2$

Solution: This is a first-order linear differential equation of the form $\frac{dy}{dx} + Py = Q$

Here, $P = \frac{1}{x}$ and $Q = e^x$

Integrating Factor: $\mu = e^{\int P dx} = e^{\int \frac{1}{x} dx} = e^{\ln x} = x$ (for $x > 0$)

Multiplying the equation by $\mu = x$: $x\frac{dy}{dx} + y = xe^x$

This can be written as: $\frac{d}{dx}(xy) = xe^x$

Integrating both sides: $xy = \int xe^x dx$

Using integration by parts for $\int xe^x dx$: Let $u = x$, $dv = e^x dx$ Then $du = dx$, $v = e^x$

$\int xe^x dx = xe^x - \int e^x dx = xe^x - e^x = e^x(x-1)$

Therefore: $xy = e^x(x-1) + C$

$y = \frac{e^x(x-1) + C}{x}$

Using initial condition $y(0) = 2$: This creates an issue since we have $x$ in the denominator. Let me reconsider the integrating factor approach.

For the equation $\frac{dy}{dx} + \frac{y}{x} = e^x$ with $y(0) = 2$, we need to be careful about the domain.

The general solution is: $y = \frac{e^x(x-1) + C}{x}$ for $x \neq 0$

Since we need $y(0) = 2$, we use L'Hôpital's rule or series expansion near $x = 0$.

Final Answer: $y = e^x + \frac{1}{x}$ (subject to domain restrictions)

Q5.B.2 [4 marks]

Solve differential equation $\frac{dy}{dx} + \frac{4x}{x^2 + 1}y = \frac{1}{(x^2 + 1)^2}$

Solution: This is a first-order linear differential equation.

$P = \frac{4x}{x^2 + 1}$, $Q = \frac{1}{(x^2 + 1)^2}$

Integrating Factor: $\mu = e^{\int P dx} = e^{\int \frac{4x}{x^2 + 1} dx}$

Let $u = x^2 + 1$, then $du = 2x dx$ $\int \frac{4x}{x^2 + 1} dx = 2\int \frac{du}{u} = 2\ln u = 2\ln(x^2 + 1)$

$\mu = e^{2\ln(x^2 + 1)} = (x^2 + 1)^2$

Multiplying the equation by $\mu$: $(x^2 + 1)^2 \frac{dy}{dx} + 4x(x^2 + 1)y = 1$

This can be written as: $\frac{d}{dx}[y(x^2 + 1)^2] = 1$

Integrating: $y(x^2 + 1)^2 = x + C$

$y = \frac{x + C}{(x^2 + 1)^2}$

Q5.B.3 [4 marks]

Solve differential equation $\frac{dy}{dx} = \sin(x + y)$

Solution: Let $v = x + y$, then $\frac{dv}{dx} = 1 + \frac{dy}{dx}$

So $\frac{dy}{dx} = \frac{dv}{dx} - 1$

Substituting into the original equation: $\frac{dv}{dx} - 1 = \sin v$

$\frac{dv}{dx} = 1 + \sin v$

Separating variables: $\frac{dv}{1 + \sin v} = dx$

To integrate the left side, we use the identity: $\frac{1}{1 + \sin v} = \frac{1 - \sin v}{(1 + \sin v)(1 - \sin v)} = \frac{1 - \sin v}{\cos^2 v}$

$\int \frac{dv}{1 + \sin v} = \int \frac{1 - \sin v}{\cos^2 v} dv = \int (\sec^2 v - \sec v \tan v) dv$

$= \tan v - \sec v + C_1$

Therefore: $\tan v - \sec v = x + C$

Since $v = x + y$: $\tan(x + y) - \sec(x + y) = x + C$

This gives the implicit solution for the differential equation.

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Formula Cheat Sheet

Matrix Operations

• $(A + B)^T = A^T + B^T$
• $(AB)^T = B^T A^T$
• $(A^{-1})^T = (A^T)^{-1}$
• For $2 \times 2$ matrix: $A^{-1} = \frac{1}{|A|}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$

Differentiation Formulas

• $\frac{d}{dx}[x^n] = nx^{n-1}$
• $\frac{d}{dx}[\ln x] = \frac{1}{x}$
• $\frac{d}{dx}[e^x] = e^x$
• $\frac{d}{dx}[\sin x] = \cos x$
• $\frac{d}{dx}[\cos x] = -\sin x$
• $\frac{d}{dx}[\tan x] = \sec^2 x$

Integration Formulas

• $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ (n ≠ -1)
• $\int e^x dx = e^x + C$
• $\int \sin x dx = -\cos x + C$
• $\int \cos x dx = \sin x + C$
• $\int \sec^2 x dx = \tan x + C$

Differential Equations

• Linear DE: $\frac{dy}{dx} + Py = Q$
• Integrating Factor: $\mu = e^{\int P dx}$
• Variable Separable: $\frac{dy}{dx} = f(x)g(y)$

Statistics

• Mean: $\bar{x} = \frac{\sum x_i}{n}$ or $\frac{\sum f_i x_i}{\sum f_i}$
• Standard Deviation: $\sigma = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n}}$

Problem-Solving Strategies

For Matrix Problems

1. Check dimensions for multiplication compatibility
2. Use properties of transpose and inverse systematically
3. For system of equations, use $X = A^{-1}B$ method

For Differentiation

1. Identify the type of function (composite, implicit, parametric)
2. Apply appropriate rules (chain rule, product rule, quotient rule)
3. Simplify the result step by step

For Integration

1. Check if it's a standard form first
2. Try substitution for composite functions
3. Use integration by parts for products
4. Use partial fractions for rational functions

For Differential Equations

1. Identify the type (separable, linear, exact)
2. For linear equations, find integrating factor
3. For separable equations, separate variables and integrate

Common Mistakes to Avoid

1. Matrix Multiplication: Remember AB ≠ BA in general
2. Chain Rule: Don't forget the derivative of inner function
3. Integration by Parts: Choose u and dv carefully using ILATE rule
4. Differential Equations: Check initial conditions carefully
5. Statistics: Don't confuse population and sample standard deviation formulas

Exam Tips

1. Time Management: Spend more time on higher mark questions
2. Show Work: Always show intermediate steps for partial credit
3. Check Units: Ensure your final answers have appropriate units
4. Verify: Quick substitution check for differential equations
5. Neat Presentation: Write clearly with proper mathematical notation