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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:                                                                                               [10 × 2]

(i) The binary operation is defined as Find .

(ii) If and is symmetric matrix, show that

(iii) Solve :

(iv) Without expanding at any stage, find the value of:

(v) Find the value of constant so that the function defined as:

is Continuous at .

(vi) Find the approximate change in the volume of a cube of side meters caused by decreasing the side by .

(vii) Evaluate : 

(viii) Find the differential equation of the family of Concentric Circles

(ix) If and are events such that and ,  then find:

(a) 

(b) 

(x) In a Race, the probabilities of A and B winning the race are and respectively.  Find the probability of neither of them winning the race.

Answer:

(i) Given

(ii)  Since is symmetric, therefore

We can compare corresponding terms. We get

(iii) 

(iv) 

Applying and   we get

  since

(v)  Given is continuous at

Therefore

Therefore

Since

Therefore

(vi)  Volume of a cube

Therefore

Hence change in volume

Hence change in volume decrease by

(vii)  

(viii)  Family of concentric circles is

Therefore Differential w.r.t.

Therefore

(ix) 

(x)  Let win the race be    win the race be

Question 2:  If the function is invertible then find its inverse. Hence prove that .               [4]

Answer:

Let

Squaring both sides

Therefore

Now, 

Therefore 

Question 3:  If   , prove that .               [4]

Answer: 

Question 4: Use properties of determinants to solve for :

and                 [4]

Answer:

Given   and

we get

we get

  or

But , therefore

Question 5:                                                                                                             [4]

(a) Show that the function   is continuous at but not differentiable.

OR

(b) Verify Rolle’s theorem for the following function:

Answer:

(a)  Continuity at

Therefore

Therefore is continuous at

Now differentiate at

Therefore

Hence is not differentiable at

(b) 

(i) is continuous on because are continuous function on its domain.

(ii) and is differentiable on

(iii)

(iv) Let be number such that

Therefore

Therefore

Therefore

Therefore

Therefore

Therefore Rolle’s theorem verified

Question 6: If , prove that                  [4]

Answer:

Therefore

differentiating both sides w.r.t.