NEB Class 12 Science Mathematics Question Paper 2077

Show that $\dfrac{1}{2!}+\dfrac{2}{3!}+\dfrac{3}{4!}+\dots = 1$.

Find the ratio in which the line joining the points $P(-2,4,7)$ and $Q(3,-5,-1)$ is divided by the ZX-plane.

If $\vec{a}=\hat{i}+2\hat{j}-\hat{k}$ and $\vec{b}=\hat{i}-\hat{j}+\hat{k}$, find the projection of $\vec{b}$ on $\vec{a}$.

Solve: $\dfrac{dy}{dx}+\dfrac{1+\cos 2y}{1-\cos 2x}=0$.

Calculate the mean deviation from mean of the data: $3, 5, 9, 11, 7, 6$.

Define abelian group. If $(G,*)$ is an abelian group, prove that $(a*b)^{-1}=a^{-1}*b^{-1}\ \forall\, a,b\in G$.

Find the condition that a line $ax+by+c=0$ may be normal to the parabola $y^2=4mx$.

Or

Find the vertices and foci of the ellipse $\dfrac{(x+2)^2}{16}+\dfrac{(y-5)^2}{9}=1$.

Evaluate: $\displaystyle\int \dfrac{dx}{1+\sin x+\cos x}$.

From definition, find the derivative of $e^{\tan x}$.

Or

State Mean Value theorem. Verify it for the function $f(x)=2x^2-10x+29$ in $[2,7]$.

A ball is thrown vertically upwards at a rate of $40\,\text{ms}^{-1}$. Find the time taken to attain the maximum height. $(g=10\,\text{ms}^{-2})$

A body slides down from rest from the top of a smooth plane of height $44.1\,\text{m}$ and inclination $30^\circ$ with the horizon. Divide the plane into three parts so that the body at the top of the plane may describe each part in equal interval of time. $(g=9.8\,\text{ms}^{-2})$

Or

A stone is dropped into a well and the sound of its striking the water is heard in $4\tfrac{2}{9}$ seconds. If the velocity of the sound is $352.8\,\text{ms}^{-1}$, find the depth of the well. $(g=9.8\,\text{ms}^{-2})$

Deduce the resultant of two parallel forces.

Or

Define Moment geometrically. Also state and prove the Varignon's theorem for two intersecting forces.

Examine whether the system of equations $3x+12y-z=28$, $x+4y+7z=2$ and $10x+4y-2z=20$ is diagonally dominant.

Use the Bisection method to find solutions accurate to within $10^{-2}$ for $x^3-7x^2+14x-6=0$ in $(0,1)$.

By Simplex method maximize $F=15x_1+10x_2$ subject to $2x_1+x_2\le 10$, $x_1+3x_2\le 10$; $x_1,x_2\ge 0$.