NEB Class 12 Science Mathematics Question Paper 2081 (Set W)

The number of combination of n things taken 'r' at a time is...

The polar form of complex number $\dfrac{1-i}{1+i}$ is...

In a triangle ABC, $a = 1$, $b = \sqrt3$ and $\angle C = 30°$. Which one of the following is the type of triangle?

If a conic section has eccentricity $e = \dfrac{\sqrt{a^2+b^2}}{a}$, what is the equation of that conic section?

Which one of the following is the angle between two vectors $\vec{i} + \vec{j}$ and $\vec{j} + \vec{k}$?

Let A and B be two dependent events. If $P(A) = 0.5$, $P(B) = 0.75$ and $P(A\cap B) = 0.4$. What is the value of $P(A/B)$?

Which one of the following is the derivative of $\tanh^{-1}x$?

Which one of the following is equal to $\displaystyle\lim_{x\to \frac{1}{2}}\dfrac{\tan\pi x}{\sec^2\pi x}$?

Which one of the following is the angle made by the tangent to the curve $2y = 2 - x^2$ at $x = 1$?

Which one of the following differential equation gives integrating factor?

After using forward elimination of the variable in solving system of equations by Gauss method, the equation changed into matrix form will be in.

Or

The highest point reached by a projectile is 40m above the horizontal. If the initial velocity is $40\sqrt2$ ms⁻¹, then the angle of projectile is ($g = 10$ ms⁻²)

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)

a) Show that $\dfrac{a + b\omega + c\omega^2}{b + c\omega + a\omega^2} = \omega$, 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)

a) If $\sin C\cdot\sin(A-B) = \sin A\cdot\sin(B-C)$, 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)

a) Does a conic $y^2 = 12x$ have two tangents from the point $(6, 9)$? Justify it with calculation. (3)

c) Prove by the method of principle of mathematical induction that $n^3 + 2n$ is divisible by 3. (3)

a) Write the slope of tangent to the curve $y = f(x)$ at $(x_1, y_1)$. (1)

b) Write the derivative of $\operatorname{cosech} x$ with respect to x. (1)

c) A differential equation is in the form $\dfrac{dy}{dx} + Py = Q$, where P and Q are functions of x only. Name the differential equation. (1)

d) Write the integral of $\int\dfrac{1}{a^2 - x^2}dx$. (1)

e) Write a characteristic of L-Hospital's rule. (1)

(Also: The dot product of two non-zero vectors gives a positive real number. Justify it with example. (2))

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

a) Find the co-efficient of correlation between age and the weight. (2)

b) Estimate the weight when the age is 12 years. (3)

a) Integrate $\displaystyle\int\dfrac{dx}{x^2 + 9}$. (2)

b) Solve: $\dfrac{dy}{dx} = \dfrac{1-y}{2x+1}$. (3)

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

Or

a) Write any one example that satisfies triangle of forces. (2)

b) The velocity of a particle when at its greatest height is $\dfrac{\sqrt{10}}{5}$ of its velocity when at half its greatest height. Find the angle of projection. (3)

a) If $z = \cos\theta + i\sin\theta$, find the value of $z^n + \dfrac{1}{z^n}$ by using De-moivre's theorem. (2)

b) In a group of 12 students, 8 are boys and remaining girls. In how many ways can 5 students be selected for quiz competition so as to include at most three girls. (3)

c) Prove by the method of principle of mathematical induction that $n^3 + 2n$ is divisible by 3. (3)

a) Find the equation of tangents to the circle $x^2 + y^2 = 25$ drawn through the point $(13, 0)$. (3)

b) If three sides of a triangle are proportional to $2 : \sqrt6 : (\sqrt3 + 1)$, find the angles. (2)

c) Find the area of a triangle formed by the points whose position vectors are $2\vec{i} - \vec{j} + 3\vec{k}$, $\vec{i} - \vec{j} - 2\vec{k}$ and $\vec{i} + 2\vec{j} + 3\vec{k}$. (3)

a) Which type of differential equation $\sin x\dfrac{dy}{dx} + y\cos x = x\sin x$ represents? Also solve it. (3)

b) Evaluate: $\displaystyle\int\dfrac{x}{(x-1)(x^2+1)}dx$. (3)

c) Two cars start from certain places at the same instant. One goes east at 60 km/hr and other goes south at 80 km/hr. How fast is the distance between them increasing? Express in symbolic form. (2)