NEB Class 12 Science Mathematics Question Paper 2080

In how many ways the letter of word 'ALGEBRA' be arranged?

If polygon has 44 diagonals, then the number of its side is

$5^{th}$ term from the end of the expansion of $\left(\dfrac{x^3}{2} - \dfrac{2}{x^2}\right)^{12}$ is

The value of $\displaystyle\sum_{n=2}^{\infty} \dfrac{1}{(n-1)!}$ is

The value of $(\cos 60^\circ + \sin 60^\circ)^6$ is

The value of $\dfrac{a + b\omega + c\omega^2}{a\omega + b\omega^2 + c}$ equals to

The condition that the line $lx + my + n = 0$ should be normal to the circle $x^2 + y^2 = a^2$ is

If the straight line $x+y-k=0$ tangent to the parabola $y + x^2 - x = 0$ then the value of k is

The derivative of $\sin^{-1}\dfrac{x}{2}$ is

If the radius of a sphere changes from 2 to 2.1 cm then approximate increase in the surface area is

The equation of the tangent to the curve $x^2 - y^2 = 7$ at $(4,3)$ is

In how many ways can the letters of the word 'INTERVAL' be arranged so that

a. all vowels are always together? (2)

b. The vowels may occupy only the odd positions? (3)

A committee of 5 is to be formed out of 6 gents and 4 ladies. In how many ways this can be done when

a. at least two ladies are included? (2)

b. at most two ladies are included? (3)

a. State binomial theorem. Find the middle term in the expansion of $\left(x^2 + \dfrac{1}{2}\right)^{12}$ (1+2)

b. If $(1+x)^n = C_0 + C_1 x + C_2 x^2 + \dots + C_n x^n$, prove that $C_1 + 2C_2 + 3C_3 + \dots + nC_n = n\cdot 2^{n-1}$. (2)

a. Prove that $\dfrac{1}{1\cdot3} + \dfrac{1}{3\cdot3^3} + \dfrac{1}{5\cdot3^5} + \dfrac{1}{7\cdot3^7} + \dots$ to $\infty = \ln\sqrt{2}$ (2)

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

a. Use De'moivre's theorem to evaluate $(\sqrt3 - i)^4$ (3)

b. Use De'moivre's theorem to find cube root of 1. (2)

a. Find the equation of the tangents to circle $x^2 + y^2 = 4$ which are parallel to the line $3x + 4y + 7 = 0$. (3)

b. If the line $lx + my = 1$ touches the circle $x^2 + y^2 = a^2$, prove that the point $(l, m)$ lies on a circle whose radius is $\dfrac{1}{a}$. (2)

a. Find the coordinates of the centre, vertices, the eccentricity, foci and equation of directrix of $\dfrac{(x+3)^2}{9} + y^2 = 1$ (3)

b. Find the equation of ellipse in standard form satisfying foci at $(0, \pm 4)$, eccentricity $\dfrac{2}{3}$. (2)

a. The distance S in meters travelled in t seconds by a particle moving in a straight line is given by $s = t^3 - 2t^2$. Find the velocity and acceleration of the particle when $t = 2$ seconds. (3)

b. The radius of the circular plate is increasing at 0.20 cm/sec. At what rate is the area increasing when the radius of the plate is 25 cm? (2)

a. In how many ways can 10 girls be arranged in round table? (2)

b. In how many ways can 7 boys be arranged at a round table so that two particular boys can be together? (3)

c. In how many ways 4 girls and 4 boys be arranged alternatively at a round table? (3)

a. Find the equation of common tangent of the parabola $y = 4ax$ and $x^2 = 4by$. (4)

b. Find the equation of the tangents to the parabola $y^2 = 8x$ passing through the point $(2,5)$. And find the point of contacts. (4)

a. Find derivatives of $\tan^{-1}(\sin hx)$ (2)

b. Evaluate by using L'Hospital rule, $\displaystyle\lim_{x\to 0} \dfrac{x - \sin x\cos x}{x^2}$. (3)

c. If the radius of the sphere change from 3 cm to 3.01 cm, find the approximate increase in its volume. (3)