**Question 1 is Compulsory**

**Attempt Five Question From Question 2 to Question 9**

**Attempt Any two Question From Question 10 to Question 15**

Find the value of k if M = `[(1,2),(2,3)]` and `M^2 - km - I_2 = 0`

Concept: Order of a Matrix

Find the equation of an ellipse whose latus rectum is 8 and eccentricity is `1/3`

Concept: Area Under Simple Curves

Solve `cos^(-1)(sin cos^(-1)x) = pi/2`

Concept: Derivatives of Inverse Trigonometric Functions

Using L'Hospital's rule, evaluate : `lim_(x->0) (x - sinx)/(x^2 sinx)`

Concept: L' Hospital'S Theorem

Evaluate: `int (2y^2)/(y^2 + 4)dx`

Concept: Methods of Integration - Integration by Substitution

Evaluate: `int_0^3 f(x)dx` where f(x) = `{(cos 2x, 0<= x <= pi/2),(3, pi/2 <= x <= 3) :}`

Concept: Methods of Integration - Integration by Substitution

The two lines of regressions are 4x + 2y- 3 = 0 and 3x + 6y + 5 =0. Find the correlation co-efficient between x and y.

Concept: Regression Coefficient of X on Y and Y on X

A card is drawn from a well-shuffled pack of playing cards. What is the probability that it is either a spade or an ace or both?

Concept: Conditional Probability

If 1, `omega` and `omega^2` are the cube roots of unity, prove `(a + b omega + c omega^2)/(c + s omega + b omega^2) = omega^2`

Concept: Basic Concepts of Differential Equation

Solve the differential equation `sin^(-1) (dy/dx) = x + y`

Concept: Solutions of Linear Differential Equation

Using properties of determinants, prove that:

`|(1+a^2-b^2, 2ab, -2b),(2ab, 1-a^2+b^2, 2a),(2b, -2a, 1-a^2-b^2)| = (1 + a^2 + b^2)^3`

Concept: Properties of Determinants

Given two matrices A and B

`A = [(1,-2,3),(1,4,1),(1,-3, 2)] and B = [(11,-5,-14),(-1, -1,2),(-7,1,6)]`

find AB and use this result to solve the following system of equations:

x - 2y + 3z = 6, x + 4x + z = 12, x - 3y + 2z = 1

Concept: Types of Matrices

Solve the equation for x: `sin^(-1) 5/x + sin^(-1) 12/x = pi/2, x != 0`

Concept: Basic Concepts of Differential Equation

A, Band C represent switches in 'on' position and A', B' and C' represent them in off position. Construct a switching circuit representing the polynomial ABC + AB'C· + A'B'C. Using Boolean Algebra, prove that the .given 'polynomial can be simplified to C(A + B'). Construct an equivalent switching circuit

Concept: Anti-derivatives of Polynomials and Functions

Verify Lagrange's Mean Value Theorem for the following function:

`f(x ) = 2 sin x + sin 2x " on " [0, pi]`

Concept: Mean Value Theorem

Find the equation of the hyperbola whose foci are `(0,+- sqrt10)` and passing through the point (2,3)

Concept: Solve Population Based Problems on Application of Differential Equations

if `y = e^(mcos^(-1)x)`, prove that `(1 - x^2) (d^2y)/(dx^2) - x dy/dx = m^2y`

Concept: Solve Population Based Problems on Application of Differential Equations

Show that the rectangle of the maximum perimeter which can be inscribed in the circle of radius 10 cm is a square of side `10sqrt2` cm.

Concept: Area Between Two Curves

Evaluate: `int (sec x)/(1 + cosec x) dx`

Concept: Methods of Integration - Integration by Substitution

Given that the observations are: (9, -4), (10, -3), (11, -1), (12, 0), (13, 1), (14, 3), (15, 5), (16, 8). Find the two lines of regression and estimate the value of y when x = 13·5.

Concept: Lines of Regression of X on Y and Y on X Or Equation of Line of Regression

In a contest, the competitors are awarded marks out of 20 by two judges. The scores of the 10 competitors are given below. Calculate Spearman's rank correlation.

Competitors | A | B | C | D | E | F | G | H | I | J |

Judge A | 2 | 11 | 11 | 18 | 6 | 5 | 8 | 16 | 13 | 15 |

Judge B | 6 | 11 | 16 | 9 | 14 | 20 | 4 | 3 | 13 | 17 |

Concept: Regression Coefficient of X on Y and Y on X

An urn contains 2 white and 2 black balls. A ball is drawn at random. If it is white, it is not replaced into the urn. Otherwise, it is replaced with another ball of the same colour. The process is repeated. Find the probability that the third ball is drawn is black.

Concept: Conditional Probability

Three persons A, B and C shoot to hit a target. If A hits the target four times in five trials, B hits it three times in four trials and C hits it two times in three trials, find the probability that:

1) Exactly two persons hit the target.

2) At least two persons hit the target.

3) None hit the target.

Concept: Random Variables and Its Probability Distributions

if z = x + iy, `w = (2 -iz)/(2z - i)` and |w| = 1. Find the locus of z and illustrate it in the Argand Plane.

Concept: Plane - Intercept Form of the Equation of a Plane

Solve the differential equation:

`e^(x/y)(1-x/y) + (1 + e^(x/y)) dx/dy = 0` when x = 0, y = 1

Concept: General and Particular Solutions of a Differential Equation

Using Vectors, prove that angle in a semicircle is a right angle

Concept: Basic Concepts of Vector Algebra

Find the volume of a parallelopiped whose edges are represented by the vectors:

`vec a = 2 hat i - 3 hat j - 4 hat k`, `vec b = hat i + 2 hat j - hat k` and `vec c = 3 hat i + hat j + 2 hatk`

Concept: Scalar Triple Product of Vectors

Find the equation of the plane passing through the intersection of the planes: x + y + z + 1 = 0 and 2x -3y + 5z -2 = 0 and the point ( -1, 2, 1 ).

Concept: Intersection of the Line and Plane

Find the shortest distance between the lines `vec r = hat i + 2hat j + 3 hat k + lambda(2 hat i + 3hatj + 4hatk)` and `vec r = 2hat i + 4 hat j + 5 hat k + mu (4hat i + 6 hat j + 8 hat k)`

Concept: Shortest Distance Between Two Lines

Box I contains two white and three black balls. Box II contains four white and one black balls and box III contains three white ·and four black balls. A dice having three red, two yellow and one green face, is thrown to select the box. If red face turns up, we pick up the box I, if a yellow face turns up we pick up box II, otherwise, we pick up box III. Then, we draw a ball from the selected box. If the ball is drawn is white, what is the probability that the dice had turned up with a red face?

Concept: Conditional Probability

Five dice are thrown simultaneously. If the occurrence of an odd number in a single dice is considered a success, find the probability of maximum three successes.

Concept: Conditional Probability

Mr. Nirav borrowed Rs 50,000 from the bank for 5 years. The rate of interest is 9% per annum compounded monthly. Find the payment he makes monthly if he pays back at the beginning of each month.

Concept: Application of Calculus in Commerce and Economics in the Profit Function and Breakeven Point

A dietician wishes to mix two kinds ·of food X· and Y in such a way that the mixture contains at least 10 units of vitamin A, 12 units of vitamin B arid 8 units of vitamin C. The vitamin contents of one kg food is given below:

Food |
Vitamin A |
Vitamin.B |
Vitamin C |

X | 1 unit | 2 unit | 3 unit |

Y | 2 unit | 2 unit | 1 unit |

Orie kg of food X costs Rs 24 and one kg of food Y costs Rs 36. Using Linear Programming, find the least cost of the total mixture. which will contain the required vitamins.

Concept: Graphical Method of Solving Linear Programming Problems

A bill for Rs 7,650 was drawn on 8th March 2013, at 7 months. It was discounted on 18th May 2013 and the holder of the bill received Rs 7,497. What is the rate of interest charged by the bank?

Concept: Application of Calculus in Commerce and Economics in the Profit Function and Breakeven Point

The average cost function, AC for a commodity is given by AC = `x + 5 + 36/x` in terms of output x. Find

1) The total cost, C and marginal cost, MC as a function of x.

2) The outputs for which AC increases

Concept: Application of Calculus in Commerce and Economics in the Average Cost

Calculate the index number. for the year 2014, with 2010 as the base year by the weighted aggregate method from the following data:

Commodity | Price in Rs | Weight | |

2010 | 2014 | ||

A | 2 | 4 | 8 |

B | 5 | 6 | 10 |

C | 4 | 5 | 14 |

D | 2 | 2 | 19 |

Concept: Application of Calculus in Commerce and Economics in the Profit Function and Breakeven Point

The quarterly profits of a small scale industry (in thousands of rupees)· is as follows:

Year |
Quarter 1 |
Quarter 2 |
Quarter 3 |
Quarter 4 |

2012 | 39 | 47 | 20 | 56 |

2013 | 68 | 59 | 66 | 72 |

2014 | 88 | 60 | 60 | 67 |

Calculate four quarterly moving averages. Display these and the original figures graphically on the same graph sheet.

Concept: Application of Calculus in Commerce and Economics in the Profit Function and Breakeven Point