2024 Sample Question Set I

2024 SAMPLE QUESTION SET I
MATRICULATION EXAMINATION
DEPARTMENT OF MYANMAR EXAMINATIONS

MATHEMATICS Time: 3 hours

Answer ALL Questions. Write your answers in the answer booklet.

Section A (Each question carries 1 mark.)

Choose the correct or the most appropriate answer for each question. Write the letter of the correct or the most appropriate answer.

1. $x+3+(y-2)i = 5+2i$, find the values of $x$ and $y$.
   A. $x=8$ and $y=4$ B. $x=2$ and $y=4$ C. $x=2$ and $y=0$ D. $x=8$ and $y=0$
2. The Cartesian form of the equation of the plane $\vec{r}(2\hat{i}+3\hat{j}-\hat{k})=10$ is
   A. $2x+3y-z=10$ B. $2x+3y-z=\sqrt{14}$ C. $2x+3y-z=-\sqrt{14}$ D. $2x+3y+z+10=0$
3. After a sport tournament, each player shakes hands with every other player once. If there are 36 handshakes in total, the number of players at the tournament is
   A. 18 B. 8 C. 10 D. 9
4. For $y^2=-4px$, the directrix is
   A. $x=-p$ B. $x=p$ C. $y=-p$ D. $y=p$
5. The period of the function $y = 3\cos \pi(x+1)+2$ is
   A. 1 B. 2 C. 3 D. $\pi$
6. $x$-axis reflection for the graph of $y=\log_b x$ is
   A. $y=\log_b x$ B. $\log_{\frac{1}{b}} x$ C. $y=\log_{10}x$ D. $y=\log_b 2$
7. The range of $y=-2e^{-x+1}+3$ is
   A. $\{3\}$ B. $\{y:y>3\}$ C. $\{y:y<3\}$ D. $\{y:y\ge3\}$
8. How many inflection points are there in the graph of $x^4+2x^2+5$?
   A. 0 B. 1 C. 2 D. 3
9. $\int 2^{3x+1}dx =$
   A. $\frac{1}{\ln 2} 2^{3x+1}+C$ B. $\frac{1}{3} 2^{3x+1}+C$ C. $\frac{1}{3 \ln 2} 2^{3x+1}+C$ D. $\frac{1}{6 \ln 2}2^{3x+1}+C$
10. Area of the region bounded by the curve $y=\cos x$ between $x=0$ and $x=\pi$ is
    A. 2 sq. units B. 4 sq. units C. 3 sq. units D. 1 sq. units

Section B (Each question carries 2 marks.)

Write only the solution of each question. (There is no need to show your working.)

11. Simplify $\frac{\overline{2+3i}}{\overline{-4-5i}}$.
12. Find the unit vector of $-2\hat{i}+3\hat{j}-7\hat{k}$.
13. In how many ways can a president, a treasurer and a secretary for a committee be selected from a group of 15 people?
14. Find the center and radius of the circle $x^2-2x+y^2+4y-4=0$.
15. If the point $(x,y)$ is on the graph of $y=\cos x$, then find the respective point on the graph of $y=f(3x-3)+2$.
16. Points $(0,1)$ and $(1,b)$ are on the graph of $y=b^x$. Find the corresponding points on the graphs of $y=ab^x$ and $y=ab^{x-h}+k$.
17. Determine the open intervals on which the graph $\frac{x}{x^2+1}$ is concave up or concave down.
18. Evaluate $\int (2x+1)\cos x \,dx$.
19. Find $\int \left(-\frac{2}{x}+3e^{x}\right) dx$.
20. Find the shaded area.
    

Section C (Each question carries 3 marks.)

21. Find the cube roots of $z=2-2i$.
22. Use the mathematical induction principle to prove that $1+3+3^2+\dots+3^{n-1} = \frac{3^n-1}{2}$.
23. Let $\vec{p} = \begin{pmatrix} 1 \\ -1 \\ 4 \end{pmatrix}$ and $\vec{q} = \begin{pmatrix} -2 \\ 0 \\ 2 \end{pmatrix}$. Find
    (a) $\frac{3}{2}\vec{p}-\vec{q}$ (b) $\vec{p}\cdot\vec{q}$ (c) $\vec{p}\times\vec{q}$
24. A group of 15 friends go on a trip in 3 cars, which respectively can contain 6, 5 and 4 people (each a driver). The owners of cars are members of the group and want to drive their own car. In how many ways can the remaining 12 members be divided between 3 cars?
25. Find the new coordinates of the point $(1,2)$ if the coordinates axes are rotated angle of $\theta=30^\circ$.
26. Show that $y=a\sin(bx)$ is an odd function.
27. Draw the graph of $y=2\log_2 x$.
28. Find the range of $f(x)=x-e^x$.
29. Evaluate $\int (xe^x+2\sin x)dx$.
30. Find the area of the region enclosed by $y=x^2+1$, the x-axis, $x=1$ and $x=2$.

Section D (Each question carries 5 marks.)

31. Let $z=-\sqrt{3}-i$. Using trigonometric form of $z$, find $z^{-1}$. Use your answer to show that $z^2(z^{-1})^2=1$.
32. Use the mathematical induction principle to prove that $4n<2^n$ for all-natural numbers $n\ge5$.
33. The points A(3, 1, 2), B$(-1, 1, 5)$ and C(7, 2, 3) are vertices of a parallelogram ABCD.
    1. Find the coordinates of D.
    2. Calculate the area of the parallelogram.
34. From the letters of the word CELESTIAL, 7 letters are to be chosen and arrange them in a line. In how many ways can it be done
    1. if there is no other restriction?
    2. if there is at least one E in the arrangements?
    3. if there is at least one E and at most one L in the arrangements?
35. Write the standard form and sketch the graph of $x^2-2x+8y-23=0$, showing the vertex, focus, directrix and end points of the latus rectum.
36. Draw the graph of $y=3\sin 2(x-1)+4$.
37. Differentiate the following functions with respect to $x$.
    (a) $\tan 3x+e^{7-2x^2}$ (b) $\frac{\ln 7x}{\sin (x^2+5)}$ (c) $\frac{x^2}{\log_{10} x}$
38. Find the values of $\int_2^6 \left(\frac{1}{x} + \frac{4}{2x} + \frac{3}{3x-2}\right)dx$ and give your answer in the form $p \ln p + q \ln q$, where $p$ and $q$ are prime numbers to be formed.

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