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MATHEMATICS

(Maximum Marks: 100)

(Time Allowed: Three Hours)

(Candidates are allowed additional 15 minutes for only reading the paper. 

They must NOT start writing during this time)

The Question Paper consists of three sections A, B and C. 

Candidates are required to attempt all questions from Section A and all question EITHER from Section B OR Section C

Section A: Internal choice has been provided  in three questions  of four marks each and two questions of six marks each.

Section B: Internal choice has been provided in two question of four marks each.

Section C: Internal choice has been provided in two question of four marks each.

All working, including rough work, should be done on the same sheet as, and adjacent to, the rest of the answer. 

The intended marks for questions or parts of questions are given in brackets [ ].

Mathematical tables and graphs papers are provided.

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Section – A (80 Marks)

Question 1:                                                                                       

(i)  Find the value of if and

(ii) Find the Equation of an ellipse whose latus rectum is and eccentricity is

(iii) Solve:

(iv) Using L’Hospital’s rule, evaluate: 

(v) Evaluate:

(vi) Evaluate: , where

(vii)  The two lines of regressions are and . Find the correlation co-efficient between .

(viii) 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?

(ix) If are the cube roots of unity, prove that

(x) Solve the differential equation:

Answer:

(i)   

Similarly,

(ii)  

Equation of ellipse:

(iii)  

(iv) 

Apply L’Hospital’s rule

Apply L’Hospital’s rule

Apply L’Hospital’s rule

(v)  

(vi)  , where

Therefore 

(vii)  Let the line of regression of on   be

Let the line of regression of on be

Therefore

Hence   since both  and are negative.

(viii) Probability P(E) = Probability of drawing a spade + Probability of drawing an ace – Probability of drawing ace of spade

(ix)  

Multiplying numerator and denominator by 

We know 

(x)  

Let

Question 2: 

(a) Using properties of determinants, prove that:

    [5]

(b) Given two matrices

and . Find .

Using this result, solve the following system of equation:

    [5]

Answer:

(a)  LHS =

RHS. Hence proved.

(b)  

Now,

          

Hence

Question 3:

(a) Solve the equation for :

     [5]

(b) represent switches in ‘on’ position and represent them in ‘off’ position. Construct a switching circuit representing the polynomial . Using Boolean Algebra, prove that the given polynomial can be simplified to . Construct an equivalent switching circuit.     [5]

Answer:

(a)