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)

Letf: R→ Rbe 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 fis bijective.

Question 3 ( 1.0 marks)

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

Question 4 ( 1.0 marks)

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

Question 5 ( 1.0 marks)

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

Question 6 ( 1.0 marks)

If Y= {x: x∈ Nand x≤ 4}, then show that the relation Ron Ydefined by R= {(a − b): |a − b| is a multiple of 3} is nota trivial relation.

Question 7 ( 1.0 marks)

A function f: R→ Ris defined as f(x) = 3x−5.
Find αsuch 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 Bbe defined as (a, b) * (c, d) = .
(i) Find the identity element of Bwith respect to the operation *.
(ii) Show that set Bis 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 Ron Q⁺× Q⁺is defined as .
Show that Ris an equivalence relation.

Question 11 ( 6.0 marks)

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