Q24 of 45 Page 1

Determine the product and use it to solve the system of equations x – y + z = 4, x – 2y – 2z = 9, 2x + y + 3z = 1.




= 8I3


Now,


we have-


x - y + z = 4


x - 2y - 2z = 9


2x + y + 3z = 1


Writing these equations in matrix form-



⇒ AX = B


where,



If |A| ≠ 0


Then the solution of the equations by matrix method is given by-


X = A-1B


Now,


Expanding along C1


|A| = [(1)(-6+2)-(1)(-3-1)+(2)(2+2)] = [(1)(-4)-(1)(-4)+(2)(4)]


= [-4+4+8]


= 8


we know that-


A(adj A) = |A|In = (adj A)A


Comparing this with



we get-



.


Thus,


.


Hence,


x = 3, y = -2, z = -1


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22

Of the students in a school, it is known that 30% have 100% attendance and 70% students are irregular. Previous year results report that 70% of all students who have 100% attendance attain A grade and 10% irregular students attain A grade in their annual examination. At the end of the year, one student is chosen at random from the school and he was found to have an A grade. What is the probability that the student has 100% attendance? Is regularity required only in school? Justify your answer.

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Maximise z = x + 2y

Subject to the constraints


x + 2y ≥ 100


2x – y ≤ 0


2x + y ≤ 200


x, y ≥ 0


Solve the above LPP graphically.

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Consider f: given by Show that f is bijective. Find the inverse of f and hence find f-1(0) and x such that f-1(x) = 2.

 25

Let A = Q × Q and let * be a binary operation on A defined by (a,b)*(c,d) = (ac,b + ad) for (a,b), (c,d) ∈ A Determine, whether * is commutative and associative. Then, with respect to * on A

(i) Find the identity element in A.


(ii) Find the invertible element of A.