Mathematics NEB 2082 question paper

Group A

Group 'A'(?)

Group B

Group 'B'(8×5=40)

1. 12.
   • a) Write the expansion of ${\log }_e(1\text{-}x)$; $|x|<1$. [1]
   • b) Write the total number of permutations of a set having $n$ elements. [1]
   • c) State De-Moivre's theorem. [1]
   • d) Write the sum of cubes of first $n$ natural numbers. [1]
   • e) Write the augmented matrix of the system of equation $3x+2y\text{-}1=0$ and $4x+y=3$. [1]

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2. 13.
   • a) A committee is to be chosen from 'a' boys and 6 girls and is to consist 2 boys and 3 girls. If 120 committees are formed, what is the number of boys represented by 'a' ? [2]
   • b) The square roots of any complex number are $(\sqrt{3}+i)$ and $(-\sqrt{3}-i)$. Write the complex number in polar form. [3]

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3. 14.
   • a) In any triangle PQR, if $p{\sin }^2\frac{R}{2}+r{\sin }^2\frac{P}{2}=\frac{q}{2}$, prove that the sides are in A.P. [3]
   • b) If $\vec{a}=4\vec{i}-3\vec{j}+2\vec{k}$ and $\vec{b}=3\vec{i}-2\vec{j}+4\vec{k}$ are two vectors, find the projection $\vec{b}$ on $\vec{a}$ [2]

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4. 15.
   • a) Find the eccentricity of conic $3x^2\text{ - }4y^2\text{ - }6x=0$. [2]
   • b) Find the eccentricity of ellipse whose major axis is four times its minor axis and passes through the point $(4,2)$ [3]

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5. 16.

   Consider the following data for supply (X) and the price (Y) of a commodity for last six years.

   Year in B.S.	2075	2076	2077	2078	2079	2080
   X	45	50	56	62	65	70
   Y	65	70	75	80	90	100

   • a) Find the correlation coefficient between X and Y. [2]
   • b) Calculate the supply when the price of commodity is Rs. 150. [3]

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6. 17.
   • a) Write the derivative of $cosec{h}^{-1}(x)$. [1]
   • b) Define L-Hospital's rule. [1]
   • c) Write the condition where the curve $y=f(x)$ has tangent parallel to y-axis. [1]
   • d) Write the integral of $\int \frac{1}{x^2-a^2}dx$ [1]
   • e) Write the standard form of first order linear differential equation. [1]

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7. 18.
   • a) Find the derivative of $cot{h}^{-1}(sin2x)$. [2]
   • b) Integrate : $\int \frac{dx}{x^3-x^2-2x}$. [3]

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8. 19.

   Using simplex method, maximize $P(x,y)=10x+3y$ subject to constraint $6x+7y\le 42$, $x+3y\le 42$, $x+3y\le 9$, $x\ge 0$, $y\ge 0$. [5] OR Two forces A and B acting parallel to the length and base of an inclined plane respectively, would each of them singly support a weight 'R' on the plane, prove that $\frac{1}{A^2}=\frac{1}{B^2}+\frac{1}{R^2}$. [5] GROUP: C | Group 'C' | 3×8=24

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