Mathematics NEB 2081 question paper

Group B

?(8 × 5 = 40)

1. 12.
   • a) Write the total number of permutations of set of n objects arranged in a circle. [1]
   • b) Write the general term in the expansion $(a+x{)}^n$. [1]
   • c) Write the series of ${\log }_e(1-x)[x<1]$ [1]
   • d) Write the complex number $(\cos \theta +i\sin \theta )$ in Euler's form. [1]
   • e) Write the sum of cube of first n natural numbers. [1]

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2. 13.
   • a) Show that $\frac{a+b\omega +c{\omega }^2}{b+c\omega +a{\omega }^2}=\omega$, where, $\omega$ and ${\omega }^2$ are cube root of unity. [2]
   • b) Solve the following system of equations by row equivalent matrix method.
     $4x-5y+2z=1$, $3x=4z-10$, $2y=3z-6$. [3]

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3. 14.
   • a) If $\sin C.\sin (A-B)=\sin A\sin (B-C)$, prove that $a^2$, $b^2$, $c^2$ are in A.P. [3]
   • b) Find the equation of ellipse whose major axis is twice the minor axis and passes through the point $(1,0)$. [2]

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4. 15.
   • a) Does a conic, $y^2=12x$ have two tangents from the point $(6,9)$ ? Justify it with calculation. [3]
   • b) The dot product of two non-zero vectors gives a positive real number. Justify it with example. [2]

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5. 16.
   • a) Write the slope of tangent to the curve $y=f(x)$ at $(x_1,y_1)$. [1]
   • b) Write the derivative of $\mathrm{cosech}x$ with respect to $x$. [1]
   • c) A differential equation is in the form $\frac{dy}{dx}+Py=Q$, where $P$ and $Q$ are functions of $x$ only. Name the differential equation. [1]
   • d) Write the integral of $\int \frac{1}{a^2-x^2}dx$ [1]
   • e) Write a characteristic of L-Hospital's rule. [1]

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6. 17.

   The following table gives the age and weight of school children in a locality

   age in year	4	5	7	9	10	11
   weight in kg.	20	25	28	30	32	33

   • a) Find the co-efficient of correlation between age and the weight. [2]
   • b) Estimate the weight when the age is 12 years. [3]

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7. 18.
   • a) Integrate : $\int \frac{dx}{x^2+9}$. [2]
   • b) Solve: $\frac{dy}{dx}=\frac{1-y}{2x+1}$. [3]

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8. 19.

   Use the Simplex method and maximize $z=15x+12y$ subject to $2x+3y\le 21$, $3x+2y\le 24$, $x,y\ge 0$. [5] OR

   • a) Write any one example that satisfies triangle of forces. [2]
   • b) The velocity of a particle when at its greatest height is $\frac{\sqrt{10}}{5}$ of its velocity when at half its greatest height. Find the angle of projection . [3] GROUP: C | ? | 3 × 8 = 24

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