Sample question for class 12 2013 - Recruitment 2022-23 Notification

## Feb 18, 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: RR be a function defined by f (x) = 2x + 3. Find a function g: R R satisfying the condition gof = fog = IR.
Question 2 ( 1.0 marks)
A function f: ZA, where is 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: NN 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: xN 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: RR 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) = e2x and g(x) = x − 3.
Show that the functions are one-to-one and onto, and thus, find their inverse. 