NEB Class 12 Science Mathematics (Optional) Question Paper 2080 (Set A)

In how many ways can $r$ letters be posted in $n$ letter box $(n \ge r)$.

Let $(G, O)$ be a group such that $aOb = 2a + 3b$ for all integers $a, b$. What is $2O3$?

Which one of the following is the general solution for $x$ satisfying the equation $\tan^2 x = \tan^2\theta$? ($n$ represents the integer.)

What is the range of $y = \sin^{-1}x$?

What is the value of $\vec{i}\cdot(\vec{j}\times\vec{k}) + \vec{j}\cdot(\vec{k}\times\vec{i}) - \vec{k}\cdot(\vec{i}\times\vec{j})$? Where $\vec{i}, \vec{j}, \vec{k}$ are unit vectors along x-axis, y-axis and z-axis respectively.

From a well shuffled pack of 52 cards, two cards are drawn at random successively without replacement. What is the probability of getting both ace cards?

For what value of c, planes $2x + 3y + 5z + 11 = 0$ and $4x + 6y + cz + d = 0$ are parallel?

Using L-Hospital's rule, what is the value of $\displaystyle\lim_{x\to 0}\dfrac{2(e^{2x}-1)}{\log(1+x)}$?

What is the integral of $\displaystyle\int\dfrac{dx}{x^2 - 36}$?

The Gaussian forward elimination step ends a system of equation with matrix $\begin{bmatrix} a_{11} & a_{12} & : & c_1 \\ 0 & a_{22} & : & c_2 \end{bmatrix}$ such that $a_{22} \ne 0$. The system has...

A body of mass 250 gm, initially at rest is subjected to a force of 3N for 1 second. The velocity acquired during the second is

Or

The demand equation of a certain commodity is $P = 60 - 2Q - Q^2$, where P is price and Q is quantity. If the demand is 6, what is the consumer surplus?

a) What is the sum of coefficient of even terms in the expansion of $(1+x)^n$? (1)

b) How many terms are there in the expansion $\left(x + \dfrac{1}{x}\right)^6$? (1)

c) Express $e^{-x}$ in an infinite series. (1)

d) Write the condition that $ax^2 + bx + c = 0\ (a \ne 0)$ has equal roots. (1)

e) What does $\theta$ represent in $z = r(\cos\theta + i\sin\theta)$? (1)

a) Evaluate $(1 - \sqrt3 i)^6$ by using De-Moivre's theorem. (2)

b) Using principle of mathematical induction, show that $n(n+1)(2n+1)$ is divisible by 6 for all $n \in N$. (3)

a) Find the value of $\sin\left(\cos^{-1}\dfrac{1}{2} + \sin^{-1}\dfrac{3}{5}\right)$ (3)

b) Find the area of parallelogram determined by the vectors $-\vec{i} - \vec{j} + \vec{k}$ and $3\vec{i} - \vec{j} + 2\vec{k}$ (2)

a) Calculate the coefficient of rank correlation between age (in yrs.) and weight (in kg) of the following observations. (3)

b) If 20% of the bulbs produced by a machine are defective, determine the probability that out of 4 balls chosen at random two are defective. (2)

a) What does $\dfrac{dy}{dx}$ represent geometrically at particular point of a curve? (1)

b) Write the integral of $\int\dfrac{1}{a^2 + x^2}dx$ (1)

c) What is the difference between $\Delta y$ and $dy$? (1)

d) What is the derivative of $\sinh x$? (1)

e) Write a differential equations in linear form. (1)

a) Solve: $\dfrac{dy}{dx} = \dfrac{1 + y^2}{1 + x^2}$ (2)

b) Verify Lagrange's mean value theorem for $f(x) = (x+1)(x-2)$ in $[-1, 2]$ (3)

Using Simplex method, maximize $Z(x,y) = 10x + 15y$ subject to $x + 3y \le 21,\ 2x + 3y \le 24,\ x, y \ge 0$.

Two like parallel forces of magnitudes A and B are acting at the end points M and N of a rod MN of length $l$. If two opposite forces each of magnitude T are added to A and B, then prove that the line of action of the new resultant will be displaced through a distance $\dfrac{Tl}{A+B}$. (5)

Or

P and Q be the input-output coefficient matrix and the demand vector are $P = \begin{bmatrix} 0.1 & 0.4 \\ 0.2 & 0.2 \end{bmatrix}$ and $Q = \begin{bmatrix} 3200 \\ 1800 \end{bmatrix}$ respectively, find the total output. Comment in your results.

a) Using Row-equivalent matrix method, solve the following system of linear equations: $2x + y - z = 9,\ 3x - y + 2z = -1,\ 4x + y - 3z = 17$ (4)

b) The sum of roots of a quadratic equation is 5 and the sum of their square is 13. Find the equation. (2)

c) Show that: $\dfrac{1}{2!} + \dfrac{1+2}{3!} + \dfrac{1+2+3}{4!} + \dots = \dfrac{e}{2}$ (2)

a) Find the intercepts of plane $3x + 4y + 6z - 24 = 0$ on axes. (2)

b) Find the equation of the ellipse in standard form with its length of the major axis 12 and eccentricity $\dfrac{2}{3}$. (2)

c) Find the direction cosines of two lines which satisfy the relations $4l + 3m - 2n = 0$ and $lm - mn + nl = 0$. Also find the angle between the lines. (4)

a) Integrate $\displaystyle\int\dfrac{x^2}{(x+2)^2(x+3)}dx$. What concept is used to integrate the above integral. (2)

b) A function $f(x)$ is continuous in $[a, b]$ and differentiable in $(a, b)$. If $f(a) \ne f(b)$, does Rolle's theorem exist in $[a, b]$? Give reason. (3)

c) Write an example of homogeneous differential equation of first order and solve it. (3)