﻿ Write the element  of the matrix  whose elements  are given by  from Class 12 CBSE Previous Year Board Papers | Mathematics 2015 Solved Board Papers

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# CBSE Class 12 Mathematics Solved Question Paper 2015

1.

Show that:

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2.
Show that f is invertible and find the inverse of f. Here, W is the set of all whole numbers.

Let f: W→W be defined as

We need to prove that 'f' is invertible.
In order to prove that 'f' is invertible it is sufficient to prove that f is a bijection.
A function f: A→B is a one-one function or an injection, if
f(x) = f(y) ⇒ x = y for all x, y ∈ A.
Case i:
If x and y are odd.
Let f(x) = f(y)
⇒x − 1 = y − 1
⇒x = y
Case ii:
If x and y are even,
Let f(x) = f(y)
⇒x + 1 = y + 1
⇒x = y
Thus, in both the cases, we have,
f(x) = f(y) ⇒ x = y for all x, y ∈ W.
Hence f is an injection.

Let n be an arbitrary element of W.
If n is an odd whole number, there exists an even whole number n − 1 ∈ W such that
f(n − 1) = n − 1 + 1 = n.
If n is an even whole number, then there exists an odd whole number n + 1 ∈ W such that f(n + 1) = n + 1 − 1 = n. Also, f(1) = 0 and f(0) = 1
Thus, every element of W (co-domain) has its pre-image in W (domain).
So f is an onto function.

Thus, it is proved that f is an invertible function.
Thus, a function g: B→A which associates each element y ∈ B to a unique element x ∈ A such that f(x) = y is called the inverse of f.
That is, f(x) = y ⇔ g(y) = x.
The inverse of f is generally denoted by f -1.

Now let us find the inverse of f.
Let x, y ∈ W such that f(x) = y
⇒x + 1 = y, if x is even
And

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3.

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4.

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5.

Solve the following for x:

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6.

Let A = Q × Q, where Q is the set of all rational numbers, and * be a binary operation on A defined by (a, b) * (c, d) = (ac, b+ad) for (a, b), (c, d)  A. Then find
(i) The identify element of * in A.
(ii) Invertible elements of A, and write the inverse of elements (5, 3) and

Let A = Q x Q, where Q is the set of rational numbers.
Given that * is the binary operation on A defined by (a, b) * (c, d) = (ac, b + ad) for
(a, b), (c, d) ∈ A.
(i)
We need to find the identity element of the operation * in A.
Let (x, y) be the identity element in A.
Thus,
(a, b) * (x, y) = (x, y) * (a, b) = (a, b), for all (a, b) ∈ A
⇒(ax, b + ay) = (a, b)
⇒ ax = a and b + ay =b
⇒ y = 0 and x = 1
Therefore, (1, 0) ∈ A is the identity element in A with respect to the operation *.

(ii) We need to find the invertible elements of A.
Let (p, q) be the inverse of the element (a, b)
Thus,

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7.

Using the properties of determinants, solve the following for x:

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8.

Find the absolute maximum and absolute minimum values of the function f given by

Of these values, the maximum value is 1, and the minimum value is −1.
Thus, the absolute maximum and absolute minimum values of f(x) are 1 and −1, which it attains at x = 0 and
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9.

If

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