PABSON SECONDARY EDUCATION
PRE-BOARD EXAM – 2081
PABSON SECONDARY EDUCATION
PRE-BOARD EXAM – 2081
Subject: Optional Mathematics
Full Marks: 75
Time: 3 Hours
Instructions:Candidates are required to answer all the questions. सबै प्रश्नहरूको उत्तर दिनुहोस्।
Group ‘A’ [10 × 1 = 10]
1. Write the period of the function \( f(x) = \tan x \).
फन्कसन \( f(x) = \tan x \) को आवधि लेख्नुहोस्।
2. If the first term of an arithmetic sequence is ‘a’ and the common difference is ‘d’, write the formula for the nth term.
यदि पहिलो पद ‘a’ हो र समान अन्तर ‘d’ हो भने n औं पदको सूत्र लेख्नुहोस्।
3. Write the interval notation for the given number line:
दिइएको सङ्ख्या रेखाको लागि अन्तराल सङ्केत लेख्नुहोस्।
`←-------●-----●-----●-----○-----○-----→`
` -3 -2 -1 2 3`
4. Determine the determinant:
\[ \left| \begin{array}{cc} 4 & -3 \\ 0 & 1 \end{array} \right| \]
5. State the condition that the pair of lines represented by \( ax^2 + 2hxy + by^2 = 0 \) are perpendicular.
6. Which geometric figure is formed when a plane intersects a right circular cone parallel to the generator?
समकोण शङ्खुलाई जनरेटरसँग समानान्तर समतलले छेर्दा कस्तो आकृति बन्छ?
7. Express \( \cos 2M \) in terms of sine and cosine ratios.
8. Solve: \( \sin \alpha = \cos \alpha \), for \( 0^\circ \leq \alpha \leq 180^\circ \)
9. If \( \vec{a} \) and \( \vec{b} \) are parallel, what is \( \vec{a} \cdot \vec{b} \)?
10. If A’ is the image of A with respect to an inversion circle with center O and radius r, write the relation between OA, OA’ and r.
Group ‘B’ [8 × 2 = 16]
11. If \( f(x) = x^3 + kx^2 - 4x + 12 \) is divisible by \( x + 2 \), find the value of k.
12. Find the vertex of the parabola \( y = x^2 - 4x - 5 \) without graphing.
13. Find a 2×2 matrix A whose inverse is
\[ A^{-1} = \frac{1}{3} \begin{bmatrix} 2 & 4 \\ 3 & 1 \end{bmatrix} \]
14. Find the value of k if the lines \( 4x - 3y + 3 = 0 \) and \( 7x + 5y + 4 = 0 \) are perpendicular.
15. Prove that:
\[ \frac{1 - \sin \theta + \cos \theta}{1 + \sin \theta + \cos \theta} = \tan\left( \frac{\pi}{4} - \frac{\theta}{2} \right) \]
16. Solve: \( \sin 40^\circ = \cos(2\theta) \), for \( 0^\circ \leq \theta \leq 90^\circ \)
17. Find the position vector that divides the line segment joining
\( \vec{A} = \begin{bmatrix} 2 \\ 4 \end{bmatrix} \) and
\( \vec{B} = \begin{bmatrix} 8 \\ 2 \end{bmatrix} \) internally in the ratio 1:2.
18. Given:
\[ N = 20, \sum fm = 140, \text{Mean} = 16 \]
Find the mean deviation and its coefficient.
Group ‘C’ [11 × 3 = 33]
19. If \( f(x) = 3x + 5 \), \( g(x) = \frac{x - 1}{5} \), and \( g \circ f(x) = f \circ g(x) \), find the value of x.
20. Find the sum of the first 9 terms of a geometric series whose 3rd term is 20 and 6th term is 320.
21. Is the function
\[ f(x) = \begin{cases} \frac{2x + 5}{x - 3}, & x < 3 \\ x^2 - 5x + 6, & x > 3 \end{cases} \]
continuous at \( x = 3 \)? Justify your answer.
22. Solve by matrix method:
\( 7x + 4y = 13 \)
\( 3x + 5y = 9 \)
23. Find the equation of the straight line passing through the origin and perpendicular to \( 3x + 8y - 5 = 0 \)
24. Prove that:
\[ \sin^2 A + \sin^2 B + \sin^2 C = 2 + 2 \cos A \cos B \cos C \]
where \( A + B + C = \pi \)
25. Prove:
\[ \sin^2 \theta = \frac{1 - \cos 2\theta}{2}, \quad \cos^2 \theta = \frac{1 + \cos 2\theta}{2} \]
26. From the top of a 25m high building, the angle of elevation of the top of a tower is 15°, and the angle of depression of the bottom is 45°. Find the height of the tower.
27. Find the transformation matrix that maps the unit square using:
\[ \begin{bmatrix} 3 & 2 \\ 1 & 2 \end{bmatrix} \]
28. Find the average deviation from the data:
| Marks | 10–20 | 20–30 | 30–40 | 40–50 |
|---|---|---|---|---|
| Freq. | 2 | 3 | 5 | 1 |
29. Find the standard deviation and its coefficient from the following data:
| Class | 30–40 | 40–50 | 50–60 | 60–70 | 70–80 | 80–90 |
|---|---|---|---|---|---|---|
| Freq. | 6 | 4 | 3 | 2 | 5 | 3 |
Group ‘D’ [4 × 4 = 16]
30. Find the maximum value of the objective function \( E = 6x + 4y \) subject to:
\( x + y \leq 6, \quad x \geq 2, \quad x \geq 0, \quad y \geq 0 \)
31. Find the equation of the circle passing through the points (0, 2) and (2, 0), and its center lies on the line \( 2x - 3y + 1 = 0 \)
32. Prove vectorially that the median of an isosceles triangle is perpendicular to its base.
33. A triangle with vertices A(2,5), B(–2,3), and C(4,1) is rotated 90° clockwise about the origin and then reflected over the y-axis. Draw all the triangles on the same graph paper and write the single transformation matrix representing both transformations.
THE END
