In this blog, we will discuss the model question paper for Optional Mathematics Grade X along with solutions.

The question paper is prepared by the PABSON Examination Committee, Syangja, Gandaki Province. It was issued on Chaitra month of 2077 BS, Nepal for SEE preparation of students.

Question Paper: Optional Mathematics Class 10


SEE Pre-board / SENT UP Examination 2077 / 2021
PABSON Examination Committee, Syangja, Gandaki Province
Class - 10 (X)
Subject: Optional Mathematics   Subject Code: 241
Full Marks: 100                        Pass Marks: 40
Time: 3 hours
Candidates are required to write their answers according to the instructions given.

Group A

1 a) Define constant function.

1 b) If 'a' and 'b' are two positive numbers, write the relation between their AM and GM.

2 a) Which quadrants do the inequality y $\geq$ 0 represent in a graph?

2 b) Define the singular matrix.

3 a) Under what condition the pair of straight lines $\text{ax^2 + 2hxy + by^2 =0}$ will be perpendicular?

3 b) State the diameter form of equation of circle.

4 a) Write sin 3A in the form of sin A.

4 b) What is the scalar product between $\vec{a}$ and $\vec{b}$ if the angle between them is $\theta$?

5 a) Where is the inverse of point P if the point P is outside the circle of inversion?

5 b) What transformation does $\left ( \displaylines{ -1 & 0 \\ 0 & 1} \right ) $ denote?

Group B

6 a) If $f(x) = 4x + 5$ and $fog(x) = 8x +13$ then find $g(x)?$

6 b) If $g(x) = -10x + 6$ find the value of $g^{-1}(2)$.

6 c) If $2x^3 + 3x^2 -px  + 4$ is divided by $(x+3)$, the remainder is 10. Find the value of p using the remainder theorem.

7 a) If $P = \left ( \displaylines{3 & -4 \\ 2 & 1} \right )$ and $Q = \left ( \displaylines{1 & -2 \\ 3 & -1} \right )$, find the determinant of 3P - 2Q.

7 b) If $D = \left | \displaylines{ 1 & -2 \\3 & 7} \right |$, $Dx = \left | \displaylines{ -7 & -2 \\5 & 7} \right |$ and $Dy= \left | \displaylines{ 1 & -7 \\3 & 5} \right |$.

8 a) If the lines $l_1x +m_1y + n_1 = 0$ and $l_2x + m_2y+n_2 =  0$ are perpendicular to each other, prove that $l_1l_2 + m_1m_2 = 0$.

8 b) Find the acute angles between the lines represented by the equation $2x^2 -7xy +3y^2 = 0$.

9 a) If $cos 330^o$ = $\dfrac{\sqrt{3}}{2}$, prove that $sin165^o$ = $\dfrac{\sqrt{3 }- 1}{2 \sqrt{2}}$.

9 b) Prove that: $cos^2 \left ( \dfrac{\pi}{4} - B  \right )$ - $sin^2 \left ( \dfrac{\pi}{4} - B  \right )$ $= sin 2B$.

9 c) Solve $4 - 3 sec^2 \theta = 0$.

10 a) If $\vec{p} + \vec{q} + \vec{r} = 0$, $| \vec{p} | = 6, |\vec{q} | = 7, and | \vec{r} | = \sqrt{127}$, find the angled between $\vec{p} and \vec{q}$.

10 b) In the given diagram prove that: $\vec{AB} + \vec{BC} + \vec{CD} + \vec{DE} + \vec{EA} = 0$.

10 c) The third quartile and inter-quartile range of a data are 58 and 28 respectively. Find the coefficient of quartile deviation.


Group C

11. Solve $6x^3 - 5x^2 -3x + 2 = 0$.

12. Maximize $p = 10x + 12y$, under the following constraints
$x + 2y \leq 12, 3x + 2y \leq 24, x \geq 0, y \geq 0$.

13. If $f(x) = 2x - 3$, then
  1. Find f(1.99), f(1.999), f(1.9999).
  2. Find $\displaylines{lim \\ x \to 2+} f(x) = f(2)$.
14. Solve by matrix method:
$4x + 3y = 5$
$y - 3x = 7$

15. Find the equation of the straight lines passing through the point (5,0) and making an angle 45$^o$ with the line 4x - 5y + 9 = 0.

16. Prove that: $cosec 2A + cosec 4A = cot A - cot4A$

17. If $A + B + C = \pi ^c$, Prove that: $sin \frac{A}{2} + sin \frac{B}{2} + sin \frac{C}{2} = 1 + 4 sin\frac{B+C}{4} sin \frac{A+C}{4} sin \frac{A+B}{4}$.

18. The angles of elevation of the top of a tower as observed from the points at the distance of 144 meter and 121 meter from the foot of the tower are found to be complementary. Find the height of the tower.

19. $\triangle$ ABC having the vertices A(3,6), B(5,-3) and C(-4,2) is transformed by a 2x2 matrix so that the co-ordinate of the vertices of its image are A'(-3,-6), B(-5,3) and C(4,-2). Find the 2x2 matrix.

20. Find the mean deviation from the median of the following data:

21. Find the standard deviation from the data given below:

Group D

22. In a geometric series having positive value of a common ratio, the sum of first four terms is 40 and the sum of first two terms is 4. Find the sum of first eight terms of the series.

23. The equations of two diameters of a circle passing through the points (5,1) are $x-y=3$ and $2x + y = 21$ respectively. Find the equation of circle. Also, find the equation of diameter.

24. OABC is parallelogram, the point P and Q divide OC and BC such that CP:PO = CQ:QB = 1:3. If $\vec{OA} = \vec{a} \ \text{and} \ \vec{OC} = \vec{c}$. Find the vector which represents $\vec{PQ}$ and prove that $\vec{PQ}$ and $\vec{OB}$ are parallel.

25. Sketch the graph of $\triangle ABC$ having vertices A(1,0), B(0,2) and C(0,1) in a graph. Enlarge the $\triangle ABC$ by E[0,2] first and then reflect the image $\triangle A'B'C'$ in the y-axis. Show all triangles on same graph.



Solution: Optional Mathematics Class 10

1 a) Ans: A function that always returns the same value for any input value is said to be a constant function. It is represented by $y = f(x) = c$.

1 b) Ans: If $'a' \ and \ 'b'$ are two numbers, then the relation between their AM and GM is
 $AM \geq GM$

2 a) Ans: The inequality $y \geq 0$ represents 1st and 2nd quadrant.

2 b) Ans: A square matrix whose determinant is equal to zero is called singular matrix.

3 a) Ans: The pair of straight line $ax^2 + 2hxy + by^2 = 0$ will be perpendicular if $a + b = 0$.

3 b) Ans: The diameter form of equation of circle is $(x-x_1)(x-x_2) + (y-y_1)(y-y_2) = 0$.

4 a) Ans: $sin 3A = 3sinA - 4 sin^3 A$

4 b) Ans: Scalar product $\vec{a}.\vec{b} = |a||b| cos \theta$.

5 a) Ans: Since the point lies outside the circle of inversion, its inverse point lies within the circle of inversion.

5 b) Ans: Matrix $\left ( \displaylines{ -1 & 0 \\ 0 & 1} \right ) $ denotes reflection in y-axis.

6 a) Solution:

Given,

$f(x) = 4x + 5$

$fog(x) = 8x + 13$

We know,

$fog(x) = f[g(x)]$

$or, 8x + 13 = 4[g(x)] + 5$

$or, 8x + 13 -5 = 4[g(x) +5 -5$

$or,, 8x + 8 = 4[g(x)]$

$or, 4(2x + 2) = 4[g(x)]$

$\therefore g(x) = 2x + 2


6 b) Solution:

Given,

$g(x) = -10x + 6$

Let $y = g(x) = -10x + 6$

$or, y = -10x + 6$

Interchanging values of x and y,

$or, x = -10y + 6$

$or, 10 y = 6 - x$

$or, y = \dfrac{6-x}{10}$

$\therefore g^{-1}(x) = \dfrac{6--x}{10}$


6 c) Solution:

Let g(x) = $2x^3 + 3x^2 -px  + 4$

Comparing (x-a) with (x+3), we get a = -3

Given, remainder = 10

According to remainder theorem,

g(a) = remainder

$or, 2(3)^3 + 3(3)^2 - p(3) + 4 = 10$

$or, 2 * 27 + 27 - 3p = 6$

$or, 81 - 6 = 3p$

$or, 75 = 3p$

$\therefore p = 25$


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9 a) Solution:

We know,

$cos 2\theta = 1 - 2sin^2 \theta$

$\implies sin \theta = \dfrac{1- cos 2\theta}{2}$

Here,

$\theta = 165°$

$\2 \theta = 330°$