NEB Class 12 Science Mathematics Question Paper 2082

Let $z_1 = \cos\theta_1 + i\sin\theta_1$ and $z_2 = \cos\theta_2 + i\sin\theta_2$ are two complex numbers, then

The nature of the roots of the equation $x^2 - x + 1 = 0$ are.

The equation $\sin x + \cos x = 2$ has

The value of $\sin\left(2\cos^{-1}\frac{1}{2}\right)$ is equal to

What is $\vec{a} \times \vec{b}$ if $\vec{a} = (0, 2, 0)$ and $\vec{b} = (0, 0, 2)$ ?

What is the foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = -1$ ?

A fair coin is tossed ten times. What is the mean of the binomial distribution ?

The first order derivative of $f(x) = 2x^2$ at $x = 1$ ...

The order of the differential equation $\left(\frac{d^2y}{dx^2}\right)^3 + \left(\frac{dy}{dx}\right)^2 + x + 4 = 0$ is

Solving a system of equations by Gauss eliminations method, a student obtained the following three equations $x_1 + x_3 - x_2 = 1$, $x_3 - x_2 = 1$, $0.x_2 = -5$. What relation can be drawn from above about the the system of given equations ?

If two like parallel forces of 5N and 15N act on the light rod at two points P and Q respectively 6m apart. The distance of resultant from the point Q is...

Or

The supply and demand function for particular items are given by $P_s = 160 + 2Q^2$ and $P_d = 240 - 3Q^2$ then the equilibrium quantity is

For binomial expansion of $(1 + x)^n$

a) Write it in expanded form. [1]

b) Write down its first four coefficients. [1]

c) What is the sum of all binomial coefficient when $x = 1$ ? [1]

d) If $n$ is even, write the middle term. [1]

e) If $C(n, r_1) = C(n, r_2)$, then write down the relation of $r_1$, $r_2$ and $n$. [1]

Prove by mathematical induction, $1^2 + 2^2 + 3^2 + \cdots + n^2 = \frac{n(n+1)(2n+1)}{6}$, for every natural number $n$. [5]

If $\tan^{-1}x + \tan^{-1}y + \tan^{-1}z = \frac{\pi}{2}$, prove that $xy + yz + zx = 1$. [3]

Find the equation of a plane passing through $(-3, -2, -1)$ and parallel to the plane $2x + 5y - 2 = 0$. [2]

A factory has two machines P and Q. Machine P produces 70% of the total output and Q produces 45% of the total output. Further 10% of output of machine P and 8% output of machine Q are likely to be defective. If an output selected random is deflective, write the probability of defective separately from P and Q. [2]

Find the most likely price in Pokhara corresponding to Rs. 200 in Chitwan for one kilogram of orange using regression from the following data. [3]

(Correlation coefficient $r = 0.6$ for the pair.)

a) Write the statement of mean value theorem. [1]

b) Write the derivate of $y = \cosh x$? [1]

c) Write the integration of $\sqrt{a^2 + x^2}\, dx$ ? [1]

d) Write the geometrical interpretation of mean-value theorem. [1]

e) If $f(x)$ and $g(x)$ are two function with degree of $f(x) <$ degree of $g(x)$, then what the types of function $\frac{f(x)}{g(x)}$ is called ? [1]

Solve the differential equation by reducing in linear form $\frac{dy}{dx} + \frac{y}{x} - x^2 = 0$. [5]

Using Simplex method, maximize $P(x, y) = 50x + 60y$, subject to constraints $3x + 4y \le 36$, $9x + 4y \le 60$, $x, y \ge 0$. [5]

A straight uniform rod is 3m long when a rod of 5N is placed at one end it balances about a point 25cm from the end. Find the weight of rod.

Or

a) A firm has demand function $P = 108 - 5Q$ and the cost function $C = -12Q + Q^2$. Find the price at which the profit in maximum. [2]

A force equal to 4.9N acting on a body changes its velocity from 3m/s to 5m/s when it covers a distance of 16m. Find the mass of body. [3]

Or

b) A person deposits Rs. 1,00,000 in the bank which pays the compound interest 10% p.a. to its customer. What will be the total value of deposit after 5 years if [3]

i) no extra deposits are made ?

ii) Rs. 20,000 is deposited at the end of each year ?

Rita has $16^{\text{th}}$ birthday party. She has invited 12 friends of whom 7 are relatives. In how many ways can she invite 6 guests so that 4 of them may be relatives ? [3]

Find the sum of the first $n$ terms of the natural numbers using mathematical induction. [2]

Find the values of $x$, $y$ and $z$ by using matrix method of the equations $2x - y + z = -1$, $x - 2y + 3z = 4$ and $4x + y + 2z = 4$. [3]

Find the direction cosines of two lines which satisfy the relation $2l + 2m - n = 0$ and $lm + mn + nl = 0$. Also find the angle between two lines. [5]

Prove by vector method $\sin(A - B) = \sin A\cos B - \cos A\sin B$. [3]

State Rolle's theorem, interpret it geometrically and verify it for $f(x) = x(x - 3)^2$ for $x \in [0, 3]$. [5]

Evaluate: $\lim_{x \to 5} \frac{x^2 - 25}{5 + 4x - x^2}$, using L-Hospital's rule. [3]