Mathematics

It is given that \( (r_1, \theta_1) \)and \( (r_2, \theta_2) \) are polar co-ordinates at z ₁ and z₂ What is the polar co-ordinates represented by the image of the complex number z₁ z₂?

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

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

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

   A) \( 9 \times 10^3 \)     B) \( 8 \times 10^3 \)    C) 9⁴     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) \( \sqrt{\frac{s(s-a)}{bc}} \)      B) \( \sqrt{\frac{(s-b)(s-c)}{s(s-a)}} \)   C) \( \sqrt{\frac{s(s-a)}{(s-b)(s-c)}} \)   D) \( \sqrt{\frac{(s-b)(s-c)}{bc}} \)

A parabolic reflector is 20cm in diameter and 5cm 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)\( (\vec{a} \times \vec{b}) \cdot (\vec{b} \times \vec{a}) \)     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) \( \frac{1}{324} \)        B) \( \frac{1}{18} \)   C) \( \frac{5}{36} \)    D)\( \frac{2}{5} \)

It is given that f'(x) = 3x² 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) \( \frac{3}{2} \)      B) \( \frac{1}{3} \) C) \( \frac{1}{2} \)     D) \( \frac{1}{6} \)

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

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

C) \( 2 \tan^{-1} \left( \frac{x}{2} \right) \)+ constant                                         D) \( \tan^{-1} \left( \frac{x}{2} \right) \) + 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 ......................... (i)   and 

                       3x - 2y = 11 .......................... (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) \( 2 \times \text{eq}^n(i) - 3 \times \text{eq}^n(ii) \)

Or

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) 203m          B) 230m               C) 320m                D) 640m

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, 3x - 3y + 8z - 21 = 0; 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. 

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 \( \frac{dy}{dx} \) + 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\( 2x + 3y \le 15, \,\, 3x + y \le 12, \,\, x \ge 0, \,\, y \ge 0 \) 

                                                                       Or

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

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\( y^2 \) = 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).