Class 12 Mathematics

Which one is the highest Common Factor (HCF) of \( n!, (n+1)!, and (n+2)! \) ?

A) (n+2)!        B) (n+1)!        C) n!        D) n

a) How many word can be formed from the letter of the word "DAUGHTER" so that vowels always come together ? 

b) Find the cube roots of : \( -\frac{1}{2} + \frac{\sqrt{3}}{2}i \) 

a) A Committee of 5 members is to be formed from 6 boys and 4 girls. In how many ways can this be done so as to include at most three girls. 

b) Prove that \( 1 + 3 + 5 + \dots + (2n-1) = n^2 \) using principle of mathematical induction. 

c) The total cost of 2 pen and 3 book is Rs. 1,100 and the total cost of 4 pen and 1 book in Rs. 700. Find the cost of a pen and a book using Cramer's rule. 

Which one is the relation between permutation and combination of \( n \)things taken \( r \) things at a time?

A) \( C(n, r) = r! \times P(n, r) \)          B)\( C(n, r) \times r! = P(n, r) \)
C) \( C(n, r) \times r = P(n, r) \)           D) \( C(n, r) = P(n, r) \times r \)

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. 

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

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) State the principle of mathematical induction. 

b) Write the general term of \( (x + a)^n \)

c) When does given linear equations become inconsistent ? 

d) State Demoivre's theorem for positive integral power. 

e) Write any one property of combination. 

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

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

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) Write the expansion of \( \log_e(1-x); \ |x| < 1 \). 
b) Write the total number of permutations of a set having $n$ elements. 
c) State De-Moivre's theorem. 
d) Write the sum of cubes of first n natural numbers. 
e) Write the augmented matrix of the system of equation 3x+2y-1=0 and 4x+y=3. 

a) A committee is to be chosen from a boys and 6 girls and is to consist of 2 boys and 3 girls. If 120 committees are formed, what is the number of boys represented by a? 

b) The square roots of any complex number are \( (\sqrt{3}+i) \) and \( (-\sqrt{3}-i) \). Write the complex number in polar form. 

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. \( (2 < e < 3) \) 

Which one of the following represents \( \sum n^3 \)?

A) \( \frac{n(n+1)}{2} \)     B) \( \frac{n^2(n+1)}{4} \)      C) \( \frac{n^2(n+1)(n+2)}{4} \)        D)\( \frac{n^2(n+1)^2}{4} \)

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) If the middle term in the expansion \( (\frac{a}{2} + 2)^8 \) is 1120, find the value of a. 

b) Using mathematical induction, prove that \( 1 + 7 + 13 + 19 + ... + (6n - 5) = n(3n - 2) \)

c) Solve the following linear equations by using matrix method

\( 7x - 2y = 18 \), \( 3x + 7z = 33 \), \( x + y + z = 12 \)

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) \( \sqrt{\frac{(s-b)(s-c)}{bc}} \)

 In any triangle ABC, a=3, b=4 and c=5, then which is equal to \( Sin\frac{A}{2} \) ?

A) \( \frac{3}{\sqrt{10}} \)       B)\( \frac{1}{\sqrt{10}} \)      C) \( \frac{1}{10} \)        D) \( \frac{3}{10} \)

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. 

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\( \vec{a} = \vec{i} - 2\vec{k} and \vec{b} = 4\vec{i} + 3\vec{j} + \vec{k}. \)

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) \( \frac{\sqrt{3}}{2} \)

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

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

a) Find the solution of triangle given by a = 2, \( b = \sqrt{6} and A = 45^\circ \) 

b) Two vectors \( \vec{a} = 2\vec{i} - 3\vec{j} + 4\vec{k} \) and \( \vec{b} = -\vec{i} + 5\vec{j} + 2\vec{k} \) are given. Is the area of parallelogram formed by\( \vec{a} and \vec{b} \), a whole number ? Find it by calculation. $[3]$

If the system of equations \( p_1x + q_1y = r_1 \) and \( p_2x + q_2y = r_2 \), what is the value of y?

A) \( \frac{\begin{vmatrix} p_1 & r_1 \\ p_2 & r_2 \end{vmatrix}}{\begin{vmatrix} p_1 & q_1 \\ p_2 & q_2 \end{vmatrix}} \)                       B) \( \frac{\begin{vmatrix} p_1 & q_1 \\ p_2 & q_2 \end{vmatrix}}{\begin{vmatrix} r_1 & q_1 \\ r_2 & q_2 \end{vmatrix}} \)

C) \( \frac{\begin{vmatrix} r_1 & q_1 \\ r_2 & q_2 \end{vmatrix}}{\begin{vmatrix} p_1 & q_1 \\ p_2 & q_2 \end{vmatrix}} \)                 D) \( \frac{\begin{vmatrix} r_1 & p_1 \\ r_2 & p_2 \end{vmatrix}}{\begin{vmatrix} p_1 & q_1 \\ p_2 & q_2 \end{vmatrix}} \)

Two simultaneous equations are given as 3x + 4y = 13 and x - 2y = 1. What is the equation after eliminating x?

A) 10y = 10     B) 10y = 16         C) y = 10            D) 2y = 10

Or

What is the maximum height attained by a particle in a projectile motion if initial velocity and angle of inclination are \( 40\text{m/sec} \)   and\( 30^\circ \)?\( [g = 10\text{ms}^{-2}] \)

A) \( 20\text{m} \)      B) \( 40\text{m} \)    C) \( 80\text{m} \)    D) \( 160\text{m} \)

a) In any triangle PQR, if \( p \sin^2 \frac{R}{2} + r \sin^2 \frac{P}{2} = \frac{q}{2} \), prove that the sides are in A.P. 

b) If\( \vec{a} = 4\vec{i} - 3\vec{j} + 2\vec{k} \)  and \( \vec{b} = 3\vec{i} - 2\vec{j} + 4\vec{k} \)  are two vectors, find the projection of \( \vec{b} \) on \( \vec{a} \)

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)

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

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

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

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).

 Which one of the following is equation of circle which touches x-axis and whose centre is at (3,4) ?

A) \( x^2 + y^2 + 6x + 8y + 9 = 0 \)                       B) \( x^2 + y^2 - 6x - 8y + 9 = 0 \)

C) \( x^2 + y^2 + 6x + 8y + 16 = 0 \)                      D) \( x^2 + y^2 - 6x - 8y + 16 = 0 \)

In which condition the line y = mx + c will be tangent to the circle \( x^2 + y^2 = a^2 \)?

A) \( a = c\sqrt{1+m^2} \)    B) \( a = \pm c\sqrt{1+m^2} \)           C) \( c = \pm a\sqrt{1+m^2} \)          D) \( c = a\sqrt{1+m^2} \)

Which one of the following is the slope of normal to the curve \( y = 3x^2 - x at (-1, 1) \)?

A) -7      B) \( -5 \)   C) \( \frac{1}{7} \)    D) \( \frac{1}{5} \)

a) Find the eccentricity of conic\( 3x^2 - 4y^2 - 6x = 0 \). 

b) Find the eccentricity of ellipse whose major axis is four times its minor axis and passes through the point (4, 2) 

\( \vec{a} \)   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) \( (\vec{a} \times \vec{a}) \cdot \vec{b} \)

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

 If \( \vec{a} \) and \( \vec{b} \) be two vectors then the projection of the vector \( \vec{a} \) on \( \vec{b} \) is :

A) \( \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|} \)      B) \( \frac{\vec{a} \cdot \vec{b}}{|\vec{a}|} \)  C) \( \vec{a} \cdot \vec{b} \)       D)\( \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|} \)

a) Prove that :\( (c + a - b)\left( \cot\frac{C}{2} + \cot\frac{A}{2} \right) = 2b\cot\frac{B}{2} \)

b) Find vertex, focus, and equation of directrix of conic \( 9x^2 + 4y^2 = 36 \)

c) Find the angle between \( \vec{a} = \vec{i} + 2\vec{j} - \vec{k} and \vec{b} = \vec{i} - \vec{j} + \vec{k} \)

What is \( \vec{a} \times \vec{b} \) if \( \vec{a} = (0, 0, 1) \) and \( \vec{b} = (0, 1, 0) \)?

 A) (-1, 0, 0)         B) (1, 0, 0)          C) (0, 1, 0)           D) (0, 0, -1)

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.

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

Estimate the price when supply is 110kg.

The height in inches (X) and weight in kg (Y) of 7 students are given below:

Find the correlation coefficient between X and Y and also find regression coefficient of X on Y.

Consider the following data for supply (X) and the price (Y) of a commodity for last six years. 

a) Find the correlation coefficient between X and Y. 

b) Calculate the supply when the price of commodity is Rs. 150.

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

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

A and B are dependent events. \( P(A) = \frac{1}{2} \), \( P(B) = \frac{1}{3} \) and \( P(A \cap B) = \frac{1}{5} \), which one of the following is P(A/B) ?

A) \( \frac{1}{6} \)     B) \( \frac{2}{5} \)    C) \( \frac{3}{5} \)      D) \( \frac{19}{30} \)

If\( P(A) = \frac{1}{3} \), \( P(B) = \frac{2}{3} \) and\( P(A \cap B) = \frac{1}{5} \) which one of the following is \( P(B/A) \)?

 A) \( \frac{3}{5} \)        B) \( \frac{2}{5} \)         C) \( \frac{2}{15} \)    D) \( \frac{1}{15} \)

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) \( \frac{1}{6} \)

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) 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 \( \left[ 1 + \left( \frac{dy}{dx} \right)^2 \right]^3 = \left( \frac{d^2y}{dx^2} \right)^2 \)

Which one is the derivative of \( log\left(cosh \frac{x}{a}\right) \) ?

A) \( \frac{1}{a}tanh\frac{x}{a} \)   B)\( tanh\frac{x}{a} \)    C)\( \frac{1}{a}sech\frac{x}{a} \)     D) \( \frac{1}{a}coth\frac{x}{a} \)

What is the value of \( \lim_{x \to 1} \left( \frac{x^3 - 1}{x^2 - 1} \right) \) ?

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

a) A container is made up of a hollow inverted right circular cone whose height is \( 24\text{ cm} \) and the radius at the top is \( 16\text{ cm} \) Water is flowing in the container at the rate of \( 8\text{ cm}^3\text{/sec} \)and find rate of change of depth of water when depth is \( 12\text{ cm} \). 

b) Integrate : \( \int \frac{y^2 + 1}{(y^2 + 49)(y^2 + 4)} \, \)

c) Solve : \( xdy + ydx = \sqrt{x^2 + a^2} \, dx \)

Which one of the following is the value of \( \sec \frac{B}{2}? \)

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

a) Write the derivative of\( cosech^{-1}(x) \)

b) Define L-Hospital's rule. 

c) Write the condition where the curve y=f(x) has tangent parallel to y-axis. 

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

e) Write the standard form of first order linear differential equation. 

a) Find the derivative of \( coth^{-1}(sin zx) \). 

b) Integrate : \( \int \frac{dx}{x^3 - x^2 - 2x} \).

a) Find the rate of change of volume of a sphere with respect to its surface area when radius is 7 cm. 

b) Integrate: \( \int \frac{dx}{5 - 3\cos x} \)

c) Solve : \( (1 + x^2)dy - (1 + y^2)dx = 0 \)

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

a) Find the equation of tangent to the curve \( y^2 = 4ax at (t_1, t_2) \). 

b) Find the equation of hyperbola with focus at \( (-7, 0) \) and\( \text{eccentricity} = \frac{7}{4} \)

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

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).

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

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 \( x \frac{dy}{dx} + y - x \log x = 0 \)

What is the integral of \( \int \frac{dx}{9x^2 + 1} \)?

A) \( \frac{1}{3}\tan^{-1}(3x) + C \)     B) \( \frac{1}{27}\tan^{-1}(3x) + C \)    C) \( \frac{1}{27}\tan^{-1}\left(\frac{x}{3}\right) + C \)      D) \( \frac{1}{27}\tan^{-1}\left(\frac{3}{x}\right) + C \)

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

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

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

Which one is the solution of \( ydx - \frac{x}{2}dy = 0 \)?

A) \( y = c\sqrt{x} \)   B) \( x = c\sqrt{y} \)

C) \( y = cx \)             D) \( x = cy \)

 The integrating factor for the differential equation \( \frac{dy}{dx} - \frac{y}{x} = x^2 \)is

A) \( \log \frac{1}{x} \)    B) \( e^{-\frac{1}{x}} \)      C) \( \frac{1}{x} \)               D) \( -\frac{1}{x} \)

a) If a particle moves in a path given by S = f(t), write the velocity. 

b) Write the derivative of \( \sinh^{-1}x \)with respect to \( x \). 

c) Write the integral of \( \int \cosh x \, dx \). 

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

e) Write the order of\( \frac{d^2y}{2x^2} + y\frac{dy}{dx} = 5x \)

a) Using L. Hospital's rule, find the limit of \( \lim_{x \to 0} \left( \frac{x - \sin x}{x^2} \right) \) 

b) Solve:\( \sqrt[3]{\frac{dy}{dx} + \frac{1}{x}y} = x \) 

Which one of the following is the homogeneous differential equation?

 A) \( \frac{dy}{dx} = \frac{x^2 + y^2}{x + y} \)   B) \( \frac{dy}{dx} = \frac{x^2 + y^2}{x - y} \) C) \( \frac{dy}{dx} = \frac{x^2 + y^2}{x^2 - y^2} \)    D) \( \frac{dy}{dx} = \frac{x + y}{x^2 + y^2} \)

Which is the integrating factor of differential linear equation\( \cos^2 x \frac{dy}{dx} = 1 - y \)?

A) \( \tan x \)       B) \( e^{\tan x} \)  C) \( e^{\sec^2 x} \)     D)\( \sec^2 x \)

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

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) 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.

After forward row reduction in Gauss elimination method, if the augmented matrix of given system of three linear equations reduces to following form, then the system of equations has

A) Unique solutions      B) Infinitely many solutions        C) No solutions         D) Only one solution

OR

In a projectile, initial velocity is \( 40\text{ms}^{-1} \) and angle of projection is \( 30^\circ \), what is its time of flight ?

A) \( \frac{10}{g} \)      B) \( \frac{20}{g} \)     C) \( \frac{40}{g} \)       D) \( \frac{80}{g} \)

   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.

Applying Simplex method, solve the following linear programming problem.

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

Use simplex method to solve maximize Z = 8x + 36y subject to constraints

OR

a) A mass of \( 10\text{ kg} \) is acted on by a constant force which in \( 5\text{ seconds} \), produced a velocity of\( 20\text{m per second} \). Find the force if the mass was initially rest. 

b) A particle of mass \( 5\text{ kg} \) slides down ache inclination of the plane. If smooth inclined plane to the horizontal is \( 30^\circ \). Find the acceleration of the particle and reaction between the particle and the plane.

Using simplex method, maximize \( P(x, y) = 10x + 3y \) subject to constraint \( 6x + 7y \le 42 \)  ,\( x + 3y \le 42 \), \( x + 3y \le 9, x \ge 0, y \ge 0 \). 

OR 

Two forces A and B acting parallel to the length and base of an inclined plane respectively, would each of them singly support a weight 'R' on the plane, prove that \( \frac{1}{A^2} = \frac{1}{B^2} + \frac{1}{R^2} \)

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.

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 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) \( 4x^2 + 20 \)