Mathematics

Which one of the following is an identity property at group \( (G, \circ) \)

 A)\( a^{-1} \circ a = a \circ a^{-1} \) = e for all \( a, a^{-1}, e \in (G, \circ) \)

B) \( a \circ (b \circ c) \) =\( (a \circ b) \circ c for all a, b, c \in (G, \circ) \)

 C) \( e \circ a \) = a \circ e = \( a for all a, e \in (G, \circ) \)

 D) \( b \circ a \) = \( c \circ a implies b = c for all a, b, c \in (G, \circ) \)

Which one of the following is standard form of the quadratic equation having one root \( 3 - 2\sqrt{2} \)?

 A) \( x^2 - 3x - 2\sqrt{2} = 0 \)

B) \( x^2 - 2\sqrt{2}x + 3 = 0 \)

C) \( x^2 + 6x - 1 = 0 \)

 D) \( x^2 - 6x + 1 = 0 \)

Which one of the following is general solution for \( \cos \theta = 0 \)

A)\( n\pi; n = 0, \pm 1, \pm 2, \dots \)

B) \( (2n + 1)\frac{\pi}{2}; n = 0, \pm 1, \pm 2, \ \)

C)\( n\pi + (-1)^n \frac{\pi}{2}; n = 0, \pm 1, \pm 2, \dots \)

D) \( 2n\pi; n = 0, \pm 1, \pm 2, \dots \)

Which one of the following is the value of\( \sin(\cos^{-1} \frac{1}{\sqrt{2}}) \)?

 A)0     B) \( \frac{1}{\sqrt{2}} \)    C)\( \frac{1}{2} \)   D) \( \frac{\sqrt{3}}{2} \)

In the ellipse,\( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) has foci \( (\pm c, 0) \)   When will the ellipse become a circle?

A)\( \sqrt{a^2 + b^2} = 0 \)

B)\( \sqrt{b^2 - a^2} = 0 \)

C)\( \sqrt{a^2 - b^2} = 0 \)

 D) c = a = b

Which one of the following is equal to\( (\vec{a} \times \vec{b})^2 \)

A)\( a^2b^2 - (\vec{a} \cdot \vec{b})^2 \)

 B) \( (\vec{a} \cdot \vec{b})^2 \)

C) \( (\vec{a} \cdot \vec{b})^2 - a^2b^2 \)

D) \( 1 - (\vec{a} \cdot \vec{b})^2 \)

A   and B are events with P(A) = 0.5, P(B) = 0.7 and \( P(A/B) = 0.4 \)  What is \( P(A \cap B) \)?

A) 0.20      B) 0.28       C) 0.35   D) 0.40

If\( \frac{dy}{dx} = -\frac{2}{5} \) what is the slope of normal at (1, 1) for \( y = f(x) \)?

 A) \( -\frac{2}{5} \)

 B)\( -\frac{5}{2} \)

C) \( \frac{2}{5} \)

  D) \( \frac{5}{2} \)

Which one of the following is the integral of\( \int \frac{1}{\sqrt{4x^2 - 9}} dx \)?

 A) \( \frac{1}{2} \log |x + \sqrt{x^2 - \frac{9}{4}}| + c \)

B)\( 2 \log |x + \sqrt{x^2 - \frac{9}{4}}| + c \)

 C) \( \frac{1}{2} \log |x - \sqrt{x^2 - \frac{9}{4}}| + c \)

 D) \( 2 \log |x - \sqrt{x^2 - \frac{9}{4}}| + c \)

Which one of the following is the system of linear equations in two variables?

A) \( x + y = 3; y + z = 5, z + x = 4 \)

B) \( x = 4; 2x = 8 \)

 C) \( x + y = 5; x - y = 1 \)

 D) \( x = y; y = z; z = x \)

A particle is thrown from top of a house to get maximum range, 10m. If \( g = 10 \text{ms}^{-2} \), what is the velocity of projection?

 A)\( 100 \text{ms}^{-1} \)

 B) \( 10 \text{ms}^{-1} \)

C)\( 5\sqrt{2} \text{ms}^{-1} \)

D) \( 5 \text{ms}^{-1} \)

  OR 

  The demand function for a particular commodity is y = 20 - 4x; where 'x' represents the quantity. What is the revenue function?

a) \( 20 - 4x^2 \)

b) \( -4x^2 + 20x \)

c)\( 20x + 4x^2 \)

d) \( 4x^2 + 20 \)

a) If \( z_1 = r_1(\cos \theta_1 + i\sin \theta_1) and z_2 = r_2(\cos \theta_2 + i\sin \theta_2) \)

i) What is the argument of \( z_1z_2? \) 

ii) What is the modulus of \( z_1z_2? \)

iii) What is the polar form of\( z_1z_2? \)

b) In the expansion of \( (1 + 2x + x^2)^{50} \) answer the following:

 i) Write the total number of terms in the expansion. 

 ii) Write the middle term in the expansion. 

a) Find the number of ways of arranging the letters in the word "objective". 

 b) Show that : \( 2 \left[ \frac{1}{2!} + \frac{1 + 2}{3!} + \frac{1 + 2 + 3}{4!} + \dots \right] = e \)

a) Show that\( 2\tan^{-1}(\frac{1}{2}) + \tan^{-1}(\frac{1}{7}) = \tan^{-1}(\frac{31}{17}) \)

 b) Find \( \vec{a} \times \vec{b} if \vec{a} = 2\vec{i} + \vec{k} and \vec{b} = \vec{i} + \vec{j} + \vec{k} \)

The supply and price of a commodity for the last seven years is given below.

Estimate the price when supply is 110kg.

a) Write the derivatives of \( y = \sinh^{-1}x \) and \( y = \tanh x \)

b) Write the meaning of integral of f(x). 

c) Write the degree and order of the differential equation \( \left[ 1 + \left( \frac{dy}{dx} \right)^2 \right]^3 = \left( \frac{d^2y}{dx^2} \right)^2 \)

a) Solve the differential equation : \( xy (y + 1) dy - (x^2 + 1) dx = 0 \)

 b) Using partial fraction integrate : \( \int \frac{8}{(2x - 1)(2x + 1)} dx \)

Applying Simplex method, solve the following linear programming problem.

Maximize \( P(x, y) = 3x + 5y subject to x + 2y \leq 40; 2x + y \leq 50, x, y \geq 0 \)

Two like parallel forces of magnitudes\( S_1 \) and \( S_2 \) are acting at the endpoints M and N of a rod MN of length 'k'. If two opposite forces each of magnitude P are added to \( S_1 \) and \( S_2 \), then prove that the line of action of new resultant will be displaced through a distance \( \frac{Pk}{S_1 + S_2} \) 

  OR ,

Consider a very simple hypothetical economy of two industries A and B as given in the table.

Determine the output vector of the economy if the final demand changes 90% of initial value of A and 105% of initial value of B.

a) Use determinant method to solve the equations \( x = 2y \) and \( 3x + 2y = 8 \)

b) A box contains 8 blue socks and 6 red socks. Find the number of ways two socks can be drawn from the box at random if they be of same colour. 

 c) Use mathematical induction and prove that \( 2^{3n} 1$ \) is divisible by 7.

a) Find the latus rectum of the conic section \( 9y^2 - 4x^2 = 36 \).

b) Find the direction cosines of the line joining the points A(3, 2, 2) and B(2, 1, 2). 

c) Find the equation of the plane passing through the intersection of the planes x - 2y + 3z + 4 = 0 and 2x - 3y + 4z - 7 = 0 and containing the point (1, -1, 1).

a) L-Hospital's rule is useful to evaluate the limit while we get indeterminate form. Justify the statement with suitable example. 

b) Are there two integrals obtained by \( \int \frac{dx}{a + b\cos x} \)? Justify. Obtain the result when a = 3, b = 2. 

c) Find the solution of differential equation \( x \frac{dy}{dx} + y - x \log x = 0 \)