Question: FGEIs C/G WAH REGION
1st TERM EXAMINATION 2025 – 26
Subject: Mathematics
Class: 10th
Marks: 60
Time: 2 hrs 30 Min
Section – B (Subjective)
Attempt any nine parts from the following. All parts carry equal marks. (9 x 4 = 36)
i. Solve by completing square: $x^2 + 7 = 0$
"OR"
Solve the following exponential equation $2^{2x} – 2 \times 2^x + 1 = 0$
ii. If roots of $x^2 + kx + 9 = 0$ are equal, find $k$?
"OR"
Simplify: $(-2i)^4$
iii. Find multiplicative inverse of $10 – 12i$
"OR"
If $|P| = 9$ and $P = \begin{bmatrix}3 & 3 \\ 1 & 4k\end{bmatrix}$ then find $k$.
iv. Solve: $y^3 = 125$
"OR"
Find the unit vector of $\vec{u} = [-5,12]$
v. Find the value of $P$, if $\vec{a} = 3i + 6j$ is parallel to $\vec{v} = i + Pj$
"OR"
For any complex number $Z$ show that $(Z – \overline{Z})^2$ is real number.
vi. Find matrix $Z$ from equation, $4 \begin{bmatrix}5 \\ 10\end{bmatrix} – 5Z = \sqrt{5} \begin{bmatrix}\sqrt{45} \\ \sqrt{5}\end{bmatrix}$
"OR"
Find quadratic equation with roots $3 + 2i$ and $3 – 2i$
vii. Find $Z \overline{Z}$ if $Z = -\frac{1}{2} + i$
"OR"
Find $A^{-1}$, if $A = \begin{bmatrix}5 & 4 \\ 2 & 6\end{bmatrix}$
viii. If $\vec{a} = 2i – 4j$ and $\vec{b} = 2i + xj$ find the value of $x$ if $|\vec{a} + 2\vec{b}| = 6$
"OR"
Solve $3x^2 + 12x + 8 = 0$ by completing square method.
ix. If $A = \begin{bmatrix}1 & 0 \\ 2 & 3\end{bmatrix}$ and $B = \begin{bmatrix}2 & 1 \\ 1 & -3\end{bmatrix}$ then verify that $(AB)^T = B^T A^T$
"OR"
Show that $A (0,2)$, $B(\sqrt{3}, 1)$ and $C(0, -2)$ are the vertices of right angle triangle.
Section – C (Marks – 24)
Note: Attempt all questions:
Q.3 Solve the following system of linear equation $Z – 4\omega = 3i$, $2Z + 3\omega = 11 – 5i$
"OR"
If Asad bought 3 Notebook and 2 book paying Rs. 600 and Ali bought 4 notebooks and 1 book paying Rs. 550. Find cast of 1 notebook and 1 book.
Q.4 Show that the roots of $2x^2 + (mx – 1)^2 = 3$ are equal if $3m^2 + 4 = 0$
"OR"
Prove that the line segments joining the midpoints of two sides of triangle is half of third side.
Q.5 Use matrix Inversion method to solve, if possible;
$-x = -2y + 1.5$
$5x + 4y – 3 = 0$
"OR"
The sum of square of two consecutive positive odd number is 34. Find Numbers
graph/shape: a single page of a mathematics exam paper with no graph or plotted shape visible, position_hint=center
1. **Problem:** Solve by completing the square: $x^2 + 7 = 0$
2. **Formula:** To complete the square for $x^2 + bx + c = 0$, rewrite as $(x + d)^2 = e$ where $d$ and $e$ are constants.
3. **Step 1:** Write the equation:
$$x^2 + 7 = 0$$
4. **Step 2:** Move constant to the right side:
$$x^2 = -7$$
5. **Step 3:** Take square root on both sides:
$$x = \pm \sqrt{-7}$$
6. **Step 4:** Simplify the square root of negative number using imaginary unit $i$:
$$x = \pm \sqrt{7}i$$
7. **Answer:** The solutions are
$$x = \pm i\sqrt{7}$$