Mathematics NEB 2081 question paper

Group A

Very short answer questions(11 × 1 = 11)

1. 1.

   It is given that $(r_1,{\theta }_1)$ and $(r_2,{\theta }_2)$ are polar co-ordinates at $z_1$ and $z_2$. What is the polar co-ordinates represented by the image of the complex number $z_1\cdot z_2$? a) $(r_1{r}_2,{\theta }_1-{\theta }_2)$ b) $(r_1{r}_2,{\theta }_1+{\theta }_2)$ c) $(r_1+r_2,{\theta }_1+{\theta }_2)$ d) $(r_1-r_2,{\theta }_1+{\theta }_2)$

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2. 2.

   How many numbers are there in between 999 and 10,000? a) $9\times 10^3$ b) $8\times 10^3$ c) $9^4$ d) $8^4$

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3. 3.

   In any triangle ABC, a,b,c are sides of the triangle ABC and s is semi-perimeter of the triangle. Which one of the following is equal to $\cos \frac{A}{2}$? a) $\sqrt{\frac{s(s-a)}{bc}}$ b) $\sqrt{\frac{(s-b)(s-c)}{s(s-a)}}$ c) $\sqrt{\frac{s(s-a)}{(s-b)(s-c)}}$ d) $\sqrt{\frac{(s-b)(s-c)}{bc}}$

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4. 4.

   A parabolic reflector is 20cm in diameter and 5cm deep. Which one of the following is co-ordinates of its focus? a) (0, 5) b) (0, -5) c) (5, 0) d) (-5, 0)

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5. 5.

   $\vec{a}$ and $\vec{b}$ are two non-zero vectors. Which of the following results is the smallest number? a) $(\vec{a}\times \vec{b})\cdot (\vec{b}\times \vec{a})$ b) $(\vec{a}\times \vec{a})\cdot \vec{a}$ c) $(\vec{b}\times \vec{b})\cdot \vec{b}$ d) $(\vec{a}\times \vec{a})\cdot \vec{b}$

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6. 6.

   Let A and B are two dependent events. If $P(A\cap B)=\frac{1}{18}$, $P(B)=\frac{1}{18}$ and $2.P(A)=5.P(B)$. What is the value of P(B/A) a) $\frac{1}{324}$ b) $\frac{1}{18}$ c) $\frac{5}{36}$ d) $\frac{2}{5}$

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7. 7.

   It is given that $f^{\prime }(x)=3x^2$ and $g^{\prime }(x)=2x$ where $f^{\prime }(x)$ and $g^{\prime }(x)$ are first order derivatives of $f(x)$ and $g(x)$. What is the value of ${\lim }_{x\to 1}\frac{f(x)}{g(x)}$ by using L-Hospital's rule. a) $\frac{3}{2}$ b) $\frac{1}{3}$ c) $\frac{1}{2}$ d) $\frac{1}{6}$

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8. 8.

   Which one of the following is the result of $\int \frac{1}{x^2+4}dx$? a) $\frac{1}{2}{\tan }^{-1}\left(\frac{2}{x}\right)+\text{constant}$ b) $\frac{1}{2}{\tan }^{-1}\left(\frac{x}{2}\right)+\text{constant}$ c) $2{\tan }^{-1}\left(\frac{x}{2}\right)+\text{constant}$ d) ${\tan }^{-1}\left(\frac{x}{2}\right)+\text{constant}$

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9. 9.

   What is the degree of ordinary differential equation $\frac{d^{3}s}{d{t}^3}={\left(4+{\left(\frac{d^{2}s}{d{t}^2}\right)}^2\right)}^{\frac{1}{2}}$? a) 1 b) 2 c) 3 d) 4

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10. 10.

    Which one of the following is an example of homogeneous differential equation of first order? a) $3xdy+2ydx=4$ b) $xdy-ydx+1=0$ c) $xdx+ydy=2$ d) $(x^2+xy)dy-(xy-y^2)dx=0$

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