X - Class 10 Optional Mathematics SEE question paper 2081 (with solution)

SEE 2081 (2025)
Optional Mathematics (ऐच्छिक प्रथम गणित)

समय: ३ घण्टा  पूर्णाङ्क: ७५

दिइएका निर्देशनका आधारमा आफ्नै शैलीमा सिर्जनात्मक उत्तर दिनुहोस्।

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समूह 'क' (Group 'A') [10×1=10]

1. Write the definition of quadratic function.

2. Write the formula to find the sum of a geometric series having n terms, where common ratio 

$r$ is more than 1:

$$S_n=\frac{a(r^n-1)}{r-1}$$

3. Write the sentence for this limit:

$$\underset{x\to a^{-}}{\lim }f(x)$$

4. If $A^{-1}$ is the inverse of matrix $A$, then:

$$A\cdot A^{-1}=I$$

5. Write the angle formula between two lines having slopes $m_1$ and $m_2$:

$$\tan \theta =\mid \frac{m_1-m_2}{1+m_1{m}_2}\mid$$

6. In which condition does a plane intersect a cone to form a parabola?
When the plane is parallel to the slant/side of the cone.

7. Express $\cos 2A$ in terms of $\cos A$:

$$\cos 2A=2{\cos }^{2}A-1$$

8. Express $\sin C-\sin \mathrm{D\; in\; terms\; of\; product:}$

$$\sin C-\sin D=2\cos \left(\frac{C+D}{2}\right)\sin \left(\frac{C-D}{2}\right)$$

9. If vectors $\vec{a}$ and $\vec{b}$ are perpendicular, then:

$$\vec{a}\cdot \vec{b}=0$$

10. If point $P^{\mathrm{\prime }}$ is the inversion of point $P$ in a circle centered at $O$ with radius $r$, then:

$$OP\cdot O{P}^{\mathrm{\prime }}=r^2$$

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समूह 'ख' (Group 'B') [8×2=16]

11. If one factor of the polynomial
$f(x)=x^3-5x^2+(k+1)x+\mathrm{8 }$is$,\; find\; the\; value\; of$$k$.

12. Draw the graph of the inequality:
$$2x+y\le 3$$

13. Use Crammer's Rule to find the values of $D_x$ and $D_y$ from the system:
$$\{\begin{array}{l}2x-y=5\\ x-2y=1\end{array}$$

14. If two lines given by equations
$cx+dy+e=0$ and $fx+gy+h=0$ are parallel, prove that:
$$cg-df=0$$

15. Prove:

$$\frac{\sin \theta +\sin \frac{\theta }{2}}{\cos \theta +\cos \frac{\theta }{2}+1}=\tan \frac{\theta }{2}$$

16. Solve:
$$\sqrt{3}\tan A-1=0(0^{\circ }\le A\le {180}^{\circ })$$

17. If$\overrightarrow{\mathrm{ |a}}\mathrm{\mid }=4$,$\overrightarrow{\mathrm{ |b}}\mathrm{\mid }=6$, and
$\vec{a}\cdot \vec{b}=12$, find the angle between $\vec{a}$ and $\vec{b}$.

18. In a continuous series, the third quartile is two times the first quartile. If the sum of the first and third quartiles is 90, find the quartile deviation.

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समूह 'ग' (Group 'C') [11×3=33]

19. If$(x)=4x-17$,$(x)=\frac{2x+8}{5}$, and$(g(x))=f^{-1}(x)$, find the value of $x$.

21. Solve graphically:

$$x^2+x-2=0$$

21. If

$f(x)=\{\begin{array}{ll}2x+4& \text{for }x<3\\ 4x-2& \text{for }x\ge 3\end{array}$
Is the function continuous at $x=3$? Justify your answer.

22. Solve by matrix method:

$$x=\frac{2}{3}y,4x-3y=1$$

23. Find the equation of a circle whose center is (2, 3) and which passes through the center of the circle

$$x^2+y^2-10x+4y+13=0$$

24. Prove:

$$\cot (\frac{A}{2}+\frac{\pi }{4})-\tan (\frac{A}{2}-\frac{\pi }{4})=\frac{2\cos A}{1+\sin A}$$

25. If $A+B+C=\pi$, then prove:

$$\cos A+\cos B-\cos C=4\cos \frac{A}{2}\cos \frac{B}{2}\sin \frac{C}{2}-1$$

26. The angles of depression and elevation of the pinnacle of a temple 10 m high are ${60}^{\circ }$ and ${30}^{\circ }$, respectively. Find the height of the tower.

27. If the matrix

$$\left[\begin{array}{cc}a& 2\\ b& 2\end{array}\right]$$

transforms a unit square into the parallelogram with vertices (0, 4), (1, 3), (c, 2), (2, d), find the values of $a,b,c,d$.

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

Marks obtained	0–10	10–20	20–30	30–40	40–50
No. of students	3	8	5	6	4

29. Find the standard deviation from the following data:

Age (years)	0–4	4–8	8–12	12–16	16–20	20–24
No. of students	7	8	10	12	9	6

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समूह 'घ' (Group 'D') [4×4=16]

30. The sum of three numbers in an arithmetic series is 18.
If the geometric mean between the first and third numbers is $4\sqrt{2}$, find the numbers.

31. Separate the pair of lines from the equation:

$$x^2-2xy\csc \theta +y^2=0$$

Also, find the angle between them.

32. Prove using vector method that:
If $PQ$ is the diameter of a semicircle with center $O$, and $M$ is any point on the circumference, then:

$$\mathrm{\angle }PMQ={90}^{\circ }$$

33. A triangle has vertices P(4, 3), Q(-2, 0), and R(5, 2).
It is first translated by the vector

$$T=\left(\begin{array}{c}2\\ 2\end{array}\right)$$

and then rotated ${90}^{\circ }$ clockwise (negative direction) about the origin.
Find the coordinates of the new image and plot both triangles on the graph.

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