📐Mathematics Capabilities

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# Senior Secondary Mathematics Examination
**Grade:** XII | **Time Allowed:** 2 Hours | **Maximum Marks:** 50

---

## Section A: Multiple-Choice Questions (1 Mark Each)

1. The value of $\lim_{x \to 0} \frac{\sin(5x)}{\tan(2x)}$ is: {{marks:1}}
   (A) $\frac{2}{5}$
   (B) $\frac{5}{2}$
   (C) $1$
   (D) $0$

2. If matrix $A = \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}$, then $\det(A^{-1})$ equals: {{marks:1}}
   (A) $5$
   (B) $\frac{1}{5}$
   (C) $-5$
   (D) $-\frac{1}{5}$

3. The solution set of the inequality $|2x - 3| \le 5$ on the real number line is: {{marks:1}}
   (A) $[-1, 4]$
   (B) $(-1, 4)$
   (C) $[1, 4]$
   (D) $(-\infty, -1] \cup [4, \infty)$

```diagram kind=number-line alt="Number line showing the closed interval from -1 to 4"
range: -3..6 step 1
interval: -1..4 closed
point: -1 label="x = -1" filled
point: 4 label="x = 4" filled
```

---

## Section B: Structured Algebra & Matrix Calculus

4. Consider the system of linear equations:
   $$\begin{cases} 2x + 3y - z = 7 \\ x - y + 2z = 3 \\ 3x + 2y + z = 10 \end{cases}$$
   **(a)** Express the system in matrix form $A \mathbf{x} = \mathbf{b}$. {{marks:2}}  
   {{blank:40}}  
   **(b)** Compute the determinant $\det(A)$ and determine if the system has a unique solution. {{marks:3}}  
   {{blank:60}}

5. Evaluate the definite integral:
   $$\int_{0}^{\frac{\pi}{2}} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}}\,dx$$
   Show each step of the symmetry property $I = \int_{a}^{b} f(a+b-x)\,dx$. {{marks:4}}

{{blank:80}}

---

## Section C: Coordinate Geometry & Function Analysis

6. **Right-Angled Triangle Analysis:**  
   In $\triangle ABC$, the right angle is situated at $B(4,0)$. The vertices are $A(0,0)$ and $C(4,3)$.

```diagram kind=geometry alt="Right-angled triangle ABC with base 4 cm and height 3 cm"
shape: triangle A=(0,0) B=(4,0) C=(4,3)
side: AB label="4 cm"
side: BC label="3 cm"
side: AC label="5 cm"
angle: B right
```

   **(a)** Verify the Pythagorean relation $AC^2 = AB^2 + BC^2$. {{marks:2}}  
   **(b)** Find $\cos(\angle BAC)$ and $\tan(\angle BAC)$. {{marks:2}}

{{blank:60}}

7. **Quadratic Function & Extremum:**  
   Analyze the curve defined by $f(x) = x^2 - 3$.

```diagram kind=plot alt="Graph of y = x^2 - 3, a parabola with vertex at (0, -3) and marked root point P"
x: -4..4 step 1 label="x"
y: -5..13 step 2 label="y"
fn: x^2 - 3
point: (2, 1) label="P(2, 1)"
```

   **(a)** Determine the coordinates of the local minimum vertex. {{marks:2}}  
   **(b)** Find the exact x-intercept roots where $f(x) = 0$. {{marks:2}}

{{blank:50}}

---

## Section D: Higher-Order Derivative Proof

8. If $y = e^{m \sin^{-1} x}$, prove rigorously that:
   $$(1 - x^2) \frac{d^2y}{dx^2} - x \frac{dy}{dx} - m^2 y = 0$$
   {{marks:5}}

{{blank:100}}

---

:::answer
### Official Marking Scheme & Complete Solutions

1. **Answer: (B)** $\frac{5}{2}$  
   *Working:* $\lim_{x \to 0} \frac{\sin(5x)/5x \cdot 5x}{\tan(2x)/2x \cdot 2x} = \frac{1 \cdot 5}{1 \cdot 2} = \frac{5}{2}$ (1 mark).

2. **Answer: (B)** $\frac{1}{5}$  
   *Working:* $\det(A) = (2)(4) - (3)(1) = 8 - 3 = 5$. Therefore, $\det(A^{-1}) = \frac{1}{\det(A)} = \frac{1}{5}$ (1 mark).

3. **Answer: (A)** $[-1, 4]$  
   *Working:* $-5 \le 2x - 3 \le 5 \implies -2 \le 2x \le 8 \implies -1 \le x \le 4$ (1 mark).

4. **Matrix System Solution:**  
   - (a) $\begin{pmatrix} 2 & 3 & -1 \\ 1 & -1 & 2 \\ 3 & 2 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 7 \\ 3 \\ 10 \end{pmatrix}$ (2 marks).  
   - (b) $\det(A) = 2(-1-4) - 3(1-6) - 1(2+3) = -10 + 15 - 5 = 0$. Since $\det(A) = 0$, system does not have a unique solution (3 marks).

5. **Definite Integral:**  
   - Applying $I = \int_{0}^{\pi/2} \frac{\sqrt{\cos x}}{\sqrt{\cos x} + \sqrt{\sin x}}\,dx$ (2 marks).  
   - Adding $2I = \int_{0}^{\pi/2} 1\,dx = \frac{\pi}{2} \implies I = \frac{\pi}{4}$ (2 marks).

6. **Geometry:**  
   - (a) $4^2 + 3^2 = 16 + 9 = 25 = 5^2$ (2 marks).  
   - (b) $\cos(A) = \frac{4}{5}$, $\tan(A) = \frac{3}{4}$ (2 marks).

7. **Parabola Analysis:**  
   - (a) Vertex at $(0, -3)$ (2 marks).  
   - (b) Roots at $x = \pm \sqrt{3}$ (2 marks).

8. **Derivative Proof:**  
   - $y_1 = \frac{m e^{m \sin^{-1} x}}{\sqrt{1-x^2}} \implies (1-x^2)(y_1)^2 = m^2 y^2$ (2 marks).  
   - Differentiating both sides with respect to $x$: $(1-x^2) 2 y_1 y_2 - 2x (y_1)^2 = 2 m^2 y y_1$ (2 marks).  
   - Dividing through by $2y_1$: $(1-x^2)y_2 - x y_1 - m^2 y = 0$ [Q.E.D.] (1 mark).
:::