CBSE Guess > Papers > Important Questions > Class XII > 2010 > Maths > Mathematics By Mr. M.P.Keshari

CBSE CLASS XII

Q.3. If f : R→ R defined as f(x) = (2x – 7)/4 is an invertible function, find f–1.

Solution :

We have, f(x) = (2x – 7)/4.
Let f(x) = y , then y = (2x – 7)/4
=> 2x – 7 = 4y
=> 2x = 4y + 7
=> x = (4y + 7)/2
Therefore, f –1(y) = (4y + 7)/2
=> f –1(x) = (4x + 7)/2. [Ans.]

Q.4. If f : R→ Rdefined by f(x) = (3x + 5)/2 is an invertible function, find f–1.

Solution :

We have, f(x) = (3x + 5)/2 = y (say)
Therefore, 3x + 5 = 2y
=> 3x = 2y – 5
=> x = (2y – 5)/3
=> f–1(y) = (2y – 5)/3
=> f–1(x) = (2x – 5)/3. [Ans.]

1.4 Binary Operations.

Q.1.

1. Is the binary operation *, defined on set N, given by a*b = (a + b)/2, for all a, b ε N, commutative ?
2. Is the above binary operation * associative ?

Solution :

1. We have, a * b = (a + b)/2
   = (b + a)/2 = b * a [As, + is commutative in the set of N]
   Hence, * is commutative. [Proved]
   
   
2. (a * b) * c = {(a + b)/2}* c
   = [{(a + b)/2} + c]/2
   = (a + b + 2c)/4.
   And a * (b * c) = a * {(b + c)/2}
   = [a + (b + c)/2]/2
   = (2a + b + c)/4.
   We see that (a + b + 2c)/4 ≠ (2a + b + c)/4.
   Hence, * is not associative. [Proved.]

Q.2. Let * be a binary operation defined by a*b = 2a + b – 3. Find 3*4.

Solution :

We have, a*b = 2a + b – 3, a = 3 and b = 4.
Therefore, 3*4 = 2(3) + 4 – 3
= 6 + 4 – 3
= 7. [Ans.]

Paper By Mr. M.P.Keshari
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Ph No. : 09434150289