X - Class 10 Optional Mathematics question paper 2081

Optional Mathematics

Question paper

Pre-Board Examination 2081
General

Class 10

National Examination Board (NEB)

Subject: Optional Math

Class: 10
Time: 3 hrs
F.M.: 75

All questions are compulsory

Group "A" [10 × 1 = 10]

1. Write the period of function \( y = \cos x \).

2. Define inverse function.

3. Write \( \lim_{x \to c} f(x) \) in a sentence.

4. If \(\begin{vmatrix} 2 & -5 \\ -2 & -10 \end{vmatrix} = 0\), find the value of \(a\).

5. If the intersection plane is parallel to the axis of cone, what conic section is formed?

6. In which condition the pair of lines \( ax^2 + 2hxy + by^2 = 0 \) coincide?

7. Express \(\sin 2A\) in terms of \(\tan A\).

8. Solve: \(4 \sin^2 \theta - 3 = 0\), (\(0^\circ \leq \theta \leq 90^\circ\)).

9. If \(\vec{p} = (a_1, b_1)\) and \(\vec{q} = (a_2, b_2)\), write the scalar product \(\vec{p} \cdot \vec{q}\).

10. Where is the inverse of point \(A\) if it lies inside the inversion circle?

Group "B" [8 × 2 = 16]

11. If \( y^3 + 3y^2 - ky + 4 \) divided by \( y + 2 \) leaves remainder "k", find "k".

12. If \( f(x) = 5x + b, f^{-1}(3) = 6 \), find the value of "b".

13. Matrices \( P = \begin{bmatrix} 5 & 3 \\ 7 & 4 \end{bmatrix} \) and \( Q = \begin{bmatrix} -4 & 3 \\ 7 & -5 \end{bmatrix} \) are inverses. Find \( x \) if:

\[P\begin{bmatrix} 5 \\ x \end{bmatrix}=3\begin{bmatrix}7\\0\end{bmatrix}, Q=\begin{bmatrix}-4 & 3\end{bmatrix}\]

14. Show that lines \(2x + 3y = 7\) and \(3x - 2y - 5 = 0\) are perpendicular.

15. If \(\sin \theta = \frac{1}{2}(y+\frac{1}{y})\), find \(\cos 2\theta\).

16. Prove: \(\frac{1+\cos \theta}{\sin \theta}=\cot \frac{\theta}{2}\).

17. Find angle between \(\vec{p} = 4\hat{i}-3\hat{j}\) and x-axis.

18. Continuous series: \(Q_1=25, Q_3=40\). Find quartile deviation & coefficient.

19. If \(f=(x,3x-8)\), \(g=(x,\frac{4x+5}{7})\), find \(gof^{-1}(3)\).

20. Garden has 10 flower varieties; each variety is double another. First variety has 2 flowers; find last variety and total flowers.

Group "C" [11 × 3 = 33]

21. Check continuity of \(f(x)=5x+9\,(x \leq 3), f(x)=8x\,(x>3)\) at \(x=3\).

22. Solve by matrix method: \(4x - 3y = 11, 3x + 7y = -1\).

23. Given circle center \(C(4,3)\), diameter \(PQ\), points \(P(3,2), Q(5,n)\), find circle equation.

24. Prove:

\[\frac{\sin 7\theta-5\sin 5\theta-3\sin 3\theta-\sin \theta}{\cos 7\theta-\cos 5\theta-\cos 3\theta+\cos \theta}=\tan 2\theta\]

25. If \(A+B+C=\pi\), prove:

\[\sin^2A+\sin^2B+\sin^2C=2(1+\cos A\cos B\cos C)\]

26-29. (Partially visible: Enlargement, reflection, mean-deviation, standard deviation problems.)

Group "D" [4 × 4 = 16]

30. Minimize \(B(x,y)=5x+4y\) under: \(2x+5y \leq 16, 2x+y \leq 8, x,y \geq 0\).

31. Show \(x^2 - xy - 6y^2 -3x +14y -4=0\) represents two lines inclined 45°. Find equations.

32. Prove diagonals of rhombus PART bisect perpendicularly using vectors.

33. Transform unit square by \(\begin{bmatrix}3 & 2\\1 & 1\end{bmatrix}\) followed by enlargement \(E[(0,0);3]\). Write image coordinates.

THE END