NEB Class 12 Science Mathematics (Optional) Question Paper 2082 (Set A)

What is the number of combinations of $n$ object taken $r$ at a time?

Which one of following is the argument of a complex number $z = -1 + \sqrt{3}\,i$?

In a triangle ABC, $2\cos A = \sin B : \sin C$, what type of triangle is ABC?

What is the length of the tangent to the circle $x^2 + y^2 = 25$ from a point $(3, 5)$?

If $|\vec{a}| = 5$, and $|\vec{b}| = 6$ and $\vec{a} \cdot \vec{b} = 15$, which of the following is the angle between $\vec{a}$ and $\vec{b}$?

For two events A and B, $P(B) = 0.32$, $P(A \cap B) = 0.2$ and $P(B/A) = 0.5$, then what is the probability of $P(A)$?

Which one of the following is the derivative of $\coth x$?

What is the slope of the curve of the function $f(x) = x^2 - 2x$ at $x = 5$?

Which one of the following is equal to $\lim_{x \to 0} \dfrac{1 - \cos x}{6x^2}$?

A circular copper plate is heated so that its radius increases from 5 cm to 5.06 cm then what is approximate increase in area?

Which of the following system of linear equation is diagonally dominant?

A)

B)

C)

D)

Or

A particle starts from rest and moves with a uniform acceleration of $10\,\text{cm/sec}^2$. What will be its velocity at the end of 20 seconds?

A) $200\,\text{cm/sec}$ B) $100\,\text{cm/sec}$ C) $2\,\text{cm/sec}$ D) $0.5\,\text{cm/sec}$

Write the total number of permutations $P(n, n)$ of a set of $n$ different objects taken $n$ at a time.

In the expansion of $(1 + x)^n$, what is the sum of the binomial coefficients?

Write the general term in the expansion of $(1 + x)^n$.

Write the series for $\log_e(1 + x)$, $|x| < 1$.

Write series representing $e^{-1}$.

Sum to $n$ terms of the series $1^2 \cdot 1 + 2^2 \cdot 3 + 3^2 \cdot 5 + \cdots$.

Solve the following system of linear equations by using matrix inversion method.

Solve the triangle ABC if $a = 1$, $b = \sqrt{3}$ and $C = 30°$.

Prove that the equation $9x^2 - 16y^2 + 18x - 152y - 151 = 0$ represents equation of hyperbola. Also find eccentricity and foci of the given equation of hyperbola.

Find the condition that a line $\ell x + my + c = 0$ may be normal to the parabola $y^2 = 4mx$.

Prove that the area of a plane quadrilateral ABCD is $\dfrac{1}{2}\left|\vec{AC} \times \vec{BD}\right|$, where AC and BD are its diagonals of the quadrilateral ABCD.

Following are the marks in physics and chemistry of six students:

Find the coefficient of correlation between X and Y.

Estimate the marks in Physics whose marks in Chemistry is 30.

What is the derivative of $\cosh^{-1} x$?

What is the integral of $\displaystyle\int \dfrac{dx}{a^2 - x^2}$, $|x| < a$?

Define L'Hospital's rule for the form $\dfrac{0}{0}$.

Write the order of differential equation $\dfrac{d^2 y}{dx^2} + \dfrac{dy}{dx} + 5 = 0$.

If the differential equation $y\,dx - x\,dy = 0$ is not exact differential equation then how can you make exact differential equation?

Evaluate: $\displaystyle\int \dfrac{dx}{5 + 4\cos x}$.

Solve the differential equation $\sqrt{1 - x^2}\,dy + \sqrt{1 - y^2}\,dx = 0$.

Solve the following system of linear equations by Gauss elimination method.

Or

A bullet of mass 0.006 kg travelling at $120\,\text{ms}^{-1}$ penetrates deeply into a fixed target and is then brought to rest in 0.01 sec. Find the distance of penetration of the target.

Using simplex method to maximize $(Z) = 6x - 9y$ subject to the constraints $x + y \le 20$; $2x - 3y \le 6$; $x \ge 0$, $y \ge 0$.

Or

A ball is thrown with the velocity of $29.4\,\text{m/sec}$, find the two directions in which the ball may be thrown so as to give a range of $44.1\,\text{m}$. ($g = 9.8\,\text{m/s}^2$)

In how many ways can 8 boys and 6 girls be arranged in a straight line so that no two girls are together?

If $y = \dfrac{x}{1!} + \dfrac{x^2}{2!} + \dfrac{x^3}{3!} + \cdots$ to $\infty$, prove that $x = y - \dfrac{y^2}{2} + \dfrac{y^3}{3} - \dfrac{y^4}{4} + \cdots$ to $\infty$.

Prove by the method of mathematical induction that:

Find the equation of the ellipse whose major axis is twice its minor axis and passes through the point $(0, 1)$.

The position vectors of the vertices of $\triangle ABC$ are $7\vec{j} + 10\vec{k}$, $-\vec{i} + 6\vec{j} + 6\vec{k}$ and $-4\vec{i} + 9\vec{j} + 6\vec{k}$. Prove that the triangle is isosceles right angled triangle.

In any triangle ABC, prove that: $a^2 + b^2 + c^2 - 2(bc\cos A + ca\cos B + ab\cos C) = 0$.

Water is poured into a right circular cylinder of radius 8cm at the rate of $18\,\text{cu.cm/min}$. Prove that the rate which the level of water is rising in the cylinder is $\dfrac{9}{32\pi}\,\text{cm/min}$.

Evaluate: $\displaystyle\int \dfrac{x^2 - 1}{x^4 + x^2 + 1}\,dx$.

$\dfrac{dy}{dx} = \dfrac{x^2 + y^2}{2xy}$ gives a solution. Is this solution represents a polynomial? Give reason.