NEB Class 11 Mathematics Question Paper 2078

Which of the following is a statement?

The value of $\sqrt{-16}\times\sqrt{-25}$ is

If $\angle C=60^\circ$, $b=5$ cm and $a=4$ cm of $\triangle ABC$, what is the value of $c$?

In a triangle ABC, $B=120^\circ$, $a=1$, $c=1$, then the other angles and sides are

The cosine of the angle between the vectors $\vec{a}=\hat{i}-2\hat{j}+3\hat{k}$ and $\vec{b}=\hat{i}+3\hat{j}+3\hat{k}$ is

The equation of parabola with the vertex at the origin and the directrix $y-2=0$ is

A mathematical problem is given to three students Sumit, Sujan and Rakesh whose chances of solving it are $\dfrac{1}{2}$, $\dfrac{1}{3}$ and $\dfrac{1}{a}$ respectively. The probability that the problem is solved is $\dfrac{3}{4}$. The possible value of $a$ is

$\displaystyle\lim_{\theta\to0}\dfrac{\sin\theta}{\theta}$ is equal to

The derivative of $\dfrac{4x^2+3}{3x^2-2}$ is

By Newton's Raphson method, the positive root of $x^3-18=0$ in $(2,3)$ is

Two forces acting at an angle of $45^\circ$ have a resultant equal to $\sqrt{10}\,N$. If one of the forces is $\sqrt{2}\,N$, what is the other force?

OR

The total cost function of a producer is given as $C=500+30Q+\dfrac{1}{2}Q^2$. What is the marginal cost (MC) at $Q=4$?

A function $f(x)=x^2$ is given. Answer the following for the function $f(x)$.

(i) What is the algebraic nature of the function?

(ii) Write the name of the locus of the curve.

(iii) Write the vertex of the function.

(iv) Write any one property for sketching the curve.

(v) Write the domain of the function.

Compare the sum of $n$ terms of the series: $1+2a+3a^2+4a^3+\dots$ and $a+2a+3a+4a\dots$ up to $n$ terms.

a) In any triangle, prove that: $(b+c)\sin\dfrac{A}{2}=a\sin\left(\dfrac{A}{2}+B\right)$. $[3]$

b) Express $\vec{r}=(4,7)$ as the linear combination of $\vec{a}=(5,-4)$ and $\vec{b}=(-2,5)$. $[2]$

Calculate the appropriate measure of Skewness for the data below.

Define different types of discontinuity of a function. Also write the condition for increasing, decreasing and concavity of a function. $(2+3)$

Evaluate: $\displaystyle\int \dfrac{x^2\,dx}{\sqrt{a^2-x^2}}$.

Define Trapezoidal rule. Evaluate using Trapezoidal rule for $\displaystyle\int_0^1 \dfrac{dx}{1+x}$ with $n=4$.

State sine law and use it to prove Lami's theorem.

OR

A decline in the price of good X by Rs. 5 causes an increase in its demand by 20 units to 50 units. The new price of X is 15.

(i) Calculate elasticity of demand.

(ii) The elasticity of demand is negative, what does it mean?

a) The factors of expression $\omega^3-1$ are $\omega-1$ and $\omega^2+\omega+1$. If $\omega^3-1=0$:

(i) Find the possible values of $\omega$ and write the real and imaginary roots of $\omega$. $[2]$

(ii) Prove that $\begin{vmatrix}1&\omega^n&\omega^{2n}\\\omega^{2n}&1&\omega^n\\\omega^n&\omega^{2n}&1\end{vmatrix}=0$, where $n$ is a positive integer (not a multiple of 3). $[4]$

b) Verify that $|x+y|\le|x|+|y|$ with $x=2$ and $y=-3$. $[2]$

a) The single equation of pair of lines is $2x^2+3xy+y^2+5x+2y-3=0$.

(i) Find the equation of the pair of straight lines represented by the single equation. $[4]$

(ii) Are the lines represented by the given equation passing through the origin? Write with reason. $[1]$

(iii) Find the point of intersection of the pair of lines. $[2]$

b) If three vectors $\vec{a},\vec{b}$ and $\vec{c}$ are mutually perpendicular unit vectors in space, write a relation between them. $[1]$

(i) Distinguish between derivative and anti-derivative of a function. Write their physical meanings and illustrate with example in your context. Find the differential coefficient of $\log\sin x$ with respect to $x$. $(1+2+2)$

(ii) Find the area bounded by the y-axis, the curve $x^2=4(y-2)$ and the line $y=11$. $[3]$