Sample question for class 12 2013

Subjective Test

(i)         All questions are compulsory.

(ii)        Questions 1 to 7 are very short answer type questions. These questions carry one mark each.

(iii)       Questions 8 to 10 are short answer type questions. These questions carry four marks each.

(iv)       Question 11 is long answer type question. This question carries six marks.

Question 1 ( 1.0 marks)

Let f: R → R be a function defined by f (x) = 2x + 3. Find a function g: R → R satisfying the condition gof = fog = I_R.

Question 2 ( 1.0 marks)

A function f: Z → A, whereis defined as.
Without finding the actual inverse of f, show that f is bijective.

Question 3 ( 1.0 marks)

Let X = {1, 2, 3, 5, 6, 10, 15, 30}
Operations ‘*’, ‘’ and ‘Δ’ on X are defined as follows:
a * b = LCM (a, b)
a b = HCF (a, b)
a Δ b =
Which of the given operations is (are) binary operation(s)?

Question 4 ( 1.0 marks)

If f: N → N is one-one and onto function, then find the value of the following expression.

Question 5 ( 1.0 marks)

Let f be any non-zero real valued function and let g be a function given by g(x) = (k + 2)x. Find the value of k such that gof = 2f.

Question 6 ( 1.0 marks)

If Y = {x: x ∈ N and x ≤ 4}, then show that the relation R on Y defined by R = {(a − b): |a − b| is a multiple of 3} is not a trivial relation.

Question 7 ( 1.0 marks)

A function f: R → R is defined as f(x) = 3x −5.
Find α such that f(α) = f⁻¹(α).

Question 8 ( 4.0 marks)

A function is defined as f (x) = x² − 3x + 2.

1. By not calculating the actual inverse of f show that f is invertible.
   
2. Find the inverse of f.
   

Question 9 ( 4.0 marks)

Let A = R − {0} and B = A × R.
Let the binary operation * on B be defined as (a, b) * (c, d) = .
(i) Find the identity element of B with respect to the operation *.
(ii) Show that set B is invertible with respect to the operation *.
Also, find the inverse of (−5, 3).

Question 10 ( 4.0 marks)

Let Q⁺ denote the set of all positive rational numbers.
A relation R on Q⁺ × Q⁺ is defined as .
Show that R is an equivalence relation.

Question 11 ( 6.0 marks)

Two real valued functions f and g are defined as f(x) = e^2x and g(x) = x − 3.
Show that the functions are one-to-one and onto, and thus, find their inverse.