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 $\sin\left(2\cos^{-1}\dfrac{1}{2}\right)$ 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 $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = -1$ ?

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 $\left(\dfrac{d^2 y}{dx^2}\right)^3 + \left(\dfrac{dy}{dx}\right)^2 + x + 4 = 0$ is

Solving a system of equations by Gauss eliminations method, a student obtained the following three equations $x_1 + x_3 - x_2 = 1$, $x_3 - x_2 = 1$, $0\cdot x_2 = -5$. 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.

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

b) Write down its first four coefficients.

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

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

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

d) If n is even, write the middle term.

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

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

Prove by mathematical induction, $1^2 + 2^2 + 3^2 + \dots + n^2 = \dfrac{n(n+1)(2n+1)}{6}$, for every natural number $n$.

a) If $\tan^{-1}x + \tan^{-1}y + \tan^{-1}z = \dfrac{\pi}{2}$, prove that $xy + yz + zx = 1$.

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

a) 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.

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

a) Write the statement of mean value theorem.

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

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

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

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 $\dfrac{f(x)}{g(x)}$ is called ?

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

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

a) 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. [2]

b) 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

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]

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 ?

a) 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 ?

b) Find the sum of the first $n$ terms of the natural numbers using mathematical induction.

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

a) 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.

b) Prove by vector method $\sin(A - B) = \sin A\cos B - \cos A\sin B$.

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

b) Evaluate: $\displaystyle\lim_{x \to 5} \dfrac{x^2 - 25}{5 + 4x - x^2}$, using L-Hospital's rule.