Mathematics Question Banks

The NEB Class 12 Mathematics course (Mat. 402) is a comprehensive extension of the Grade 11 curriculum, designed to provide the analytical "backbone" required for higher education in fields like engineering, medicine, and finance.

It is given that $ (r_1, \theta_1) $and $ (r_2, \theta_2) $ are polar co-ordinates at $z_1$ ₁and $z_2$ What is the polar co-ordinates represented by the image of the complex number $z_1 \cdot z_2$ ?

A) $ (r_1 r_2, \theta_1 - \theta_2) $ $(r_1 r_2, \theta_1 - \theta_2)$ B)$ (r_1 r_2, \theta_1 + \theta_2) $

C) $ (r_1 + r_2, \theta_1 + \theta_2) $ $(r_1 + r_2, \theta_1 + \theta_2)$ D)$ (r_1 - r_2, \theta_1 + \theta_2) $

How many numbers are there in between 999 and 10,000?

A) $ 9 \times 10^3 $ $9 \times 10^3$ B) $ 8 \times 10^3 $ C) $9^4$ D) 8⁴

In any triangle $ABC$ , $a, b, c$ are sides of the triangle $ABC$ and $s$ is semi-perimeter of the triangle. Which one of the following is equal to $ \cos \frac{A}{2} $?

A) $ \sqrt{\frac{s(s-a)}{bc}} $ $\sqrt{\frac{s(s-a)}{bc}}$ B) $ \sqrt{\frac{(s-b)(s-c)}{s(s-a)}} $ C) $ \sqrt{\frac{s(s-a)}{(s-b)(s-c)}} $ $\sqrt{\frac{s(s-a)}{(s-b)(s-c)}}$ D) $ \sqrt{\frac{(s-b)(s-c)}{bc}} $

A parabolic reflector is $20\text{cm}$ in diameter and $5\text{cm}$ deep. Which one of the following is co-ordinates of its focus?

A) $(0, 5)$ B) $(0, -5)$ C) $(5, 0)$ D) $(-5, 0)$

$ \vec{a} $ and $ \vec{b} $ are two non-zero vectors. Which of the following results is the smallest number?

A)$ (\vec{a} \times \vec{b}) \cdot (\vec{b} \times \vec{a}) $ $(\vec{a} \times \vec{b}) \cdot (\vec{b} \times \vec{a})$ B) $ (\vec{a} \times \vec{a}) \cdot \vec{a} $ C) $ (\vec{b} \times \vec{b}) \cdot \vec{b} $ D) $ (\vec{a} \times \vec{a}) \cdot \vec{b} $

Let $A$ and $B$ are two dependent events. If $ P(A \cap B) = \frac{1}{18} $ , $ P(B) = \frac{1}{18} $ and $ 2 \cdot P(A) = 5 \cdot P(B) $. What is the value of$ P(B/A) $?

A) $ \frac{1}{324} $ $\frac{1}{324}$ B) $ \frac{1}{18} $ C) $ \frac{5}{36} $ $\frac{5}{36}$ D)$ \frac{2}{5} $

It is given that $f'(x) = 3x^2$ and $g'(x) = 2x$ where $f'(x)$ and $g'(x)$ are first order derivatives of $f(x)$ and $g(x)$ . What is the value of$ \lim_{x \to 1} \frac{f(x)}{g(x)} $ by using L-Hospital's rule.

A) $ \frac{3}{2} $ $\frac{3}{2}$ B) $ \frac{1}{3} $ C) $ \frac{1}{2} $ $\frac{1}{2}$ D) $ \frac{1}{6} $

Which one of the following is the result of $ \int \frac{1}{x^2 + 4} dx $?

A) $ \frac{1}{2} \tan^{-1} \left( \frac{2}{x} \right) $ $\frac{1}{2} \tan^{-1} \left( \frac{2}{x} \right) + \text{constant}$ B) $ \frac{1}{2} \tan^{-1} \left( \frac{x}{2} \right) $ $\frac{1}{2} \tan^{-1} \left( \frac{x}{2} \right) + \text{constant}$

C) $ 2 \tan^{-1} \left( \frac{x}{2} \right) $ $2 \tan^{-1} \left( \frac{x}{2} \right) + \text{constant}$ D) $ \tan^{-1} \left( \frac{x}{2} \right) $ $\tan^{-1} \left( \frac{x}{2} \right) + \text{constant}$

What is the degree of ordinary differential equation $ \frac{d^3s}{dt^3} = \left( 4 + \left( \frac{d^2s}{dt^2} \right)^2 \right)^{\frac{1}{2}} $

A) 1 B) 2 C) 3 D) 4

Which one of the following is an example of homogeneous differential equation of first order?

A) $3x \, dy + 2y \, dx = 4$ B) $x \, dy - y \, dx + 1 = 0$

C) $x \, dx + y \, dy = 2$ D) $ (x^2 + xy) \, dy - (xy - y^2) \, dx = 0 $

It is given that $4x + 3y = 26 \dots (i)$ and

$3x - 2y = 11 \dots (ii)$

A student eliminates $y$ from the above equations. Which of the following operations does the student apply?

A)$ 4 \times \text{eq}^n(ii) + 3 \times \text{eq}^n(i) $

B) $ 4 \times \text{eq}^n(ii) - 3 \times \text{eq}^n(i) $

C) $ 2 \times \text{eq}^n(i) + 3 \times \text{eq}^n(ii) $

D) $ 2 \times \text{eq}^n(i) - 3 \times \text{eq}^n(ii) $

What is the greatest height attained by a projectile, whose initial velocity and angle of elevation are $ 160 \text{ ms}^{-1} $ and$ 30^\circ $ respectively? ($ g = 10 \text{ms}^{-2} $)

A) $203\text{m}$ B) $230\text{m}$ C) $320\text{m}$ D) $640\text{m}$

a) Write the sum of the series $ 1^2 + 2^2 + 3^2 + \dots + n^2 $

b) State principle of mathematical induction.

c) For what condition $ \log_e \left( x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \dots \right) $necessarily exist.

d) Write $e^x$ in expanded form. $ (2 < e < 3) $

Use row-equivalent matrix method to solve the following system of linear equations.

$2x + 3y + 5z - 23 = 0, \quad 3x - 3y + 8z - 21 = 0; \quad x + 4y - z - 6 = 0$

a) Find the equation of the hyperbola with vertices $(0, -4)$ and $(0, 4)$ and foci $(0, 6)$ and $(0, -6)$ .

b) Find the dot product of the vectors $ 2\vec{i} - \vec{j} + 2\vec{k} $ and $ 3\vec{i} - 2\vec{j} + 6\vec{k} $.

a) Show that $ 2 \left( a \sin^2 \frac{C}{2} + c \sin^2 \frac{A}{2} \right) $ is less than the perimeter of $\Delta ABC$ .

b) The condition of a straight line $y = mx + c$ to a circle $ x^2 + y^2 = a^2 $ is $ c^2 = a^2(1 + m^2) $. Justify it with an example.

Consider the following data for supply (x) and price (y) of a commodity for the last seven years.

Year	2017	2018	2019	2020	2021	2022	2023
x	80	84	86	88	92	96	97
y	12	14	15	16	18	20	22

Find the likely price (y) when the supply (x) is 120.

a) Write the derivative of $\sinh x$ with respect to x.

b) Under which necessary condition, $ \frac{d}{dx}(\text{sech}^{-1}x) = -\frac{1}{x\sqrt{1-x^2}} $

c) Write the meaning of $dy$ and $ \Delta y $ in percentage error in the estimate$ = \frac{\Delta y - dy}{y} \times 100\% $

d) Write the integral of $ \int \frac{1}{\sqrt{x^2 - a^2}} dx $

e) Write the integrating factor (I.F.) in a linear differential equation of the form $ \frac{dy}{dx} $ $\frac{dy}{dx} + Py = Q$ where P and Q are functions of x. [1]

a) Find the equation of tangent to the curve $y = 2x^3 + x^2 - 7x - 2$ at $(2, 5)$ .

b) Evaluate: $\int \frac{2x^2 - x + 4}{x^3 + 4x} dx$

Using simplex method, maximize $Z(x, y) = 25x + 35y$ subject to$ 2x + 3y \le 15, \,\, 3x + y \le 12, \,\, x \ge 0, \,\, y \ge 0 $

a) A body weight of $78\text{ N}$ is suspended by two strings of length $4\text{m}$ and $3\text{m}$ attached to two points in the same horizontal line whose distance apart is $5\text{m}$ . Find the tensions of the strings.

b) A rocket is moving upward in the air. If it loses its mass at the rate of $ 0.2\text{ kgs}^{-1} $ and its velocity changes from $ 90\text{ ms}^{-1} $ to $ 10\text{ ms}^{-1} $, find the force retarding pressure the motion of the rocket.

a) Prove that the coefficient of$ x^n $ in the expansion $ (1+x)^{2n} $ is twice the coefficient of $x^n$ in the expansion $ (1+x)^{2n-1} $.

b) Use De-moivre's theorem to find the square root of $ -(2 + 2\sqrt{3}i) $

c) By using principles of mathematical induction, prove that $n! < 2^n$ .

a) If $ a = 3 + \sqrt{3}, \,\, b = 2 $ and $ c = \sqrt{3} $ then solve the triangle.

b) Find the co-ordinates of the foci and the length of the latus rectum of the conic$ \frac{x^2}{25} + \frac{y^2}{100} = 1 $

c) Find the area of the parallelogram whose diagonals are represented by the vectors$ \vec{a} = \vec{i} - 2\vec{k}$ and $\vec{b} = 4\vec{i} + 3\vec{j} + \vec{k}$. $

a) A curve$ y^2 $ $y^2 = 8x$ changes its abscissa and ordinate at the same rate. The ordinate is double of its abscissa. Justify it with calculation.

b) The integral of $ \int \frac{dx}{a + b\cos x} $ depends upon the values of $a$ and $b$ . Explain.

c) An exact differential equation have always the mathematical form like $M(x, y) dx + N(x, y) dy = 0$ . Take a simple example to justify the statement with characteristics of $M(x, y)$ and $N(x, y)$ .