First Group
(1) If \( X = \begin{pmatrix} 3 & 4 \\ 5 & 2 \end{pmatrix} \), \( Y = \begin{pmatrix} 1 & 1 \\ 0 & -1 \end{pmatrix} \), find the matrix \( X + Y + 2I \).
Answer: \( \begin{pmatrix} 6 & 5 \\ 5 & 3 \end{pmatrix} \)(2) Find the values of \( a, b \) satisfying the equation \( 2 \begin{pmatrix} a & 3 \\ 6 & -1 \end{pmatrix} = \begin{pmatrix} 8 & 6 \\ 12 & b \end{pmatrix} \).
Answer: \( a = 4, b = -2 \)(3) Find the general solution of the equation \( 2 \sin \theta - 1 = 0 \).
Answer: \( \theta = \frac{\pi}{6} + 2n\pi \) or \( \theta = \frac{5\pi}{6} + 2n\pi \)(4) If \( \vec{A} = (0, 5) \), \( \vec{B} = (1, 3) \), find \( \vec{A} - 2\vec{B} \).
Answer: \( (-2, -1) \)(5) Express using fundamental unit vectors a displacement of 40 cm in the direction Eastern South.
Answer: \( 20\sqrt{2} \vec{i} - 20\sqrt{2} \vec{j} \) cmSecond Group
(1) If \( X = \begin{pmatrix} 1 & -2 \\ 0 & 5 \end{pmatrix} \), \( Y = \begin{pmatrix} 1 & 0 \\ -1 & -1 \end{pmatrix} \), find the matrix \( X - Y + I \).
Answer: \( \begin{pmatrix} 1 & -2 \\ 1 & 7 \end{pmatrix} \)(2) Find the values of \( a, b \) satisfying \( 3 \begin{pmatrix} a & 3 \\ 6 & 2 \end{pmatrix} = \begin{pmatrix} 6 & 9 \\ 18 & b \end{pmatrix} \).
Answer: \( a = 2, b = 6 \)(3) Find the general solution of the equation \( \sqrt{2} \cos \theta - 1 = 0 \).
Answer: \( \theta = \pm \frac{\pi}{4} + 2n\pi \)(4) If \( \vec{A} = (1, 3) \), \( \vec{B} = (2, 0) \), find \( 2\vec{A} + \vec{B} \).
Answer: \( (4, 6) \)(5) Express using fundamental unit vectors a displacement of 20 cm in the direction Western North.
Answer: \( -10\sqrt{2} \vec{i} + 10\sqrt{2} \vec{j} \) cmThird Group
(1) If \( X = \begin{pmatrix} 0 & 0 \\ 1 & 3 \end{pmatrix} \), \( Y = \begin{pmatrix} 3 & -2 \\ 2 & 4 \end{pmatrix} \), find the matrix \( X - Y + I \).
Answer: \( \begin{pmatrix} -2 & 2 \\ -1 & 0 \end{pmatrix} \)(2) Find \( a, b \) satisfying \( \begin{pmatrix} a & 6 \\ 6 & -9 \end{pmatrix} = 3 \begin{pmatrix} 1 & 2 \\ 2 & b \end{pmatrix} \).
Answer: \( a = 3, b = -3 \)(3) Find the general solution of the equation \( \tan \theta - 1 = 0 \).
Answer: \( \theta = \frac{\pi}{4} + n\pi \)(4) If \( \vec{A} = (1, 1) \), \( \vec{B} = (-2, 2) \), find \( 3\vec{A} - 2\vec{B} \).
Answer: \( (7, -1) \)(5) Express using fundamental unit vectors a displacement of 10 cm in the direction Western South.
Answer: \( -5\sqrt{2} \vec{i} - 5\sqrt{2} \vec{j} \) cmMastery & Detailed Explanations
01 First Group: Detailed Analysis
Q1: Matrix Addition \( X + Y + 2I \)
To solve this, we follow these steps:
1. Identify the identity matrix \( I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \).
2. Calculate \( 2I = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} \).
3. Perform element-wise addition:
\( \begin{pmatrix} 3 & 4 \\ 5 & 2 \end{pmatrix} + \begin{pmatrix} 1 & 1 \\ 0 & -1 \end{pmatrix} + \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} \).
4. Resulting matrix: \( \begin{pmatrix} 3+1+2 & 4+1+0 \\ 5+0+0 & 2-1+2 \end{pmatrix} = \mathbf{\begin{pmatrix} 6 & 5 \\ 5 & 3 \end{pmatrix}} \).
Q2: Solving for \( a, b \) in Matrix Equality
Distribute the scalar 2 on the left side: \( \begin{pmatrix} 2a & 6 \\ 12 & -2 \end{pmatrix} = \begin{pmatrix} 8 & 6 \\ 12 & b \end{pmatrix} \).
By comparing corresponding elements:
\( 2a = 8 \Rightarrow \mathbf{a = 4} \).
\( b = -2 \Rightarrow \mathbf{b = -2} \).
Q3: General Solution of \( 2\sin\theta - 1 = 0 \)
Isolate sine: \( \sin\theta = \frac{1}{2} \).
The reference angle is \( \frac{\pi}{6} \) (30°). Since sine is positive, solutions are in Q1 and Q2.
Q1: \( \theta = \frac{\pi}{6} + 2n\pi \).
Q2: \( \theta = (\pi - \frac{\pi}{6}) + 2n\pi = \frac{5\pi}{6} + 2n\pi \).
Q5: Displacement in Eastern South direction
Eastern South implies \( 45^\circ \) south of east, which is \( 315^\circ \).
\( \vec{v} = 40(\cos 315^\circ \vec{i} + \sin 315^\circ \vec{j}) \).
\( \vec{v} = 40(\frac{\sqrt{2}}{2} \vec{i} - \frac{\sqrt{2}}{2} \vec{j}) = \mathbf{20\sqrt{2} \vec{i} - 20\sqrt{2} \vec{j}} \).
02 Second Group: Key Concepts
Q1: Matrix Subtraction \( X - Y + I \)
1. Subtract element-wise: \( X-Y = \begin{pmatrix} 1-1 & -2-0 \\ 0-(-1) & 5-(-1) \end{pmatrix} = \begin{pmatrix} 0 & -2 \\ 1 & 6 \end{pmatrix} \).
2. Add the identity matrix \( I \): \( \begin{pmatrix} 0+1 & -2+0 \\ 1+0 & 6+1 \end{pmatrix} = \mathbf{\begin{pmatrix} 1 & -2 \\ 1 & 7 \end{pmatrix}} \).
Q3: General Solution of \( \sqrt{2}\cos\theta - 1 = 0 \)
Isolate cosine: \( \cos\theta = \frac{1}{\sqrt{2}} \).
The reference angle is \( \frac{\pi}{4} \) (45°).
General solution for cosine: \( \mathbf{\theta = \pm \frac{\pi}{4} + 2n\pi} \).
Q5: Displacement in Western North direction
Western North implies \( 45^\circ \) north of west, which is \( 135^\circ \).
\( \vec{v} = 20(\cos 135^\circ \vec{i} + \sin 135^\circ \vec{j}) \).
\( \vec{v} = 20(-\frac{\sqrt{2}}{2} \vec{i} + \frac{\sqrt{2}}{2} \vec{j}) = \mathbf{-10\sqrt{2} \vec{i} + 10\sqrt{2} \vec{j}} \).
03 Third Group: Advanced Steps
Q2: Solving Matrix Equation with 3 as Scalar
Distribute 3 on the right: \( \begin{pmatrix} a & 6 \\ 6 & -9 \end{pmatrix} = \begin{pmatrix} 3 & 6 \\ 6 & 3b \end{pmatrix} \).
Compare: \( a = 3 \) and \( 3b = -9 \Rightarrow b = -3 \).
Final values: \( \mathbf{a=3, b=-3} \).
Q3: General Solution of \( \tan\theta - 1 = 0 \)
Isolate tangent: \( \tan\theta = 1 \).
The reference angle is \( \frac{\pi}{4} \). Since tangent has a period of \( \pi \):
General Solution: \( \mathbf{\theta = \frac{\pi}{4} + n\pi} \).
Q5: Displacement in Western South direction
Western South implies \( 45^\circ \) south of west, which is \( 225^\circ \).
\( \vec{v} = 10(\cos 225^\circ \vec{i} + \sin 225^\circ \vec{j}) \).
\( \vec{v} = 10(-\frac{\sqrt{2}}{2} \vec{i} - \frac{\sqrt{2}}{2} \vec{j}) = \mathbf{-5\sqrt{2} \vec{i} - 5\sqrt{2} \vec{j}} \).