(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= IR.
Question 2 ( 1.0 marks)
A function
f:
Z→
A, where
![](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sNy77NUcANIwAaJACPccELCzNlF7UOKSMTrVwBKc43FiXUz60x3GgrF8Qh2HQyEenhG3bkf8eT_wJEgYxPAjetA0TkCEFzS1fr87S5-G3ctIY2mvvSf2olHMR2IyyLdacrxH-vFvuV7n8jdDuFNp1lcQ=s0-d)
is defined as
![](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uBoNkeo4l1cmMvI6bxm2o41SO4SdUmtU-z4zQDtouA3mOLZG6dW6SXXuAkaCqZSQurwWuh2Br5a3Fxbzv38yRckqFs0lRYZNr949afSdN5BLARhzwhwghJfoOX1pAQ-fBZI8SE4qqUQxkoFi8g9-0K=s0-d)
.
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 ‘*’, ‘
![](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uTvvtreWP4LB3YZOwtQhm8NgcHNHp8oZFCZ7BWyDZNXKCLJBrCHawUm9px9IOiP7JeRFfXypCeFZ8hxpJg0hY9sbh1NW_s_J2CdsaXro6UqZ1MkVNDZPLZfzfZdv7x_eL3nVdPGeEgR7YalN5deNwEy4w=s0-d)
’ and ‘Δ’ on
Xare defined as follows:
a*
b= LCM (
a, b)
a
b= HCF (
a, b)
aΔ
b=
![](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tHTdLw0pPew68Bmd9Z0nZj0QGrZ0dVv5NMpRyofIGXG1735MN_5rEUv-DQa21q_eI5iFDXNuMelwTZ0CUi263UNKIBNNDPvtmt9Yz4JdXtj4Y-Ir9KjTBUYo8jFifZzcslageUjMc39piTpXPfQ9IS9Lc=s0-d)
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
![](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tlScjd2n-3Zi5cQouIqZgJDjjQQaK57pVhRK4s2yWYRAldgNvdbYHLg3GADtfeoLqUJ16DwFKXQNJSoLTIWd1IPMFo9abBzPzcyDH89eXYjsMNWeVwG29C-rFEDMe4sGigACq3I2X-rPGLnaXKHyxe7Q=s0-d)
is defined as
f(
x) =
x2− 3
x+ 2.
- By not calculating the actual inverse of f show that f is invertible.
- 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) =
![](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uOD_JmA4_GMB0McsuY-XID4s_pL_7Zcf0tF6ERGzRT5sWZOg8TMYnqw9Ed6p90aKfkOaitXClp7iOovkwAwZK7tqndkOJaDV9Pf4mFOZCDXeL4e8Fj9_sLP7dLPQOF3-5ezTFedCholvNeT3V8u29vjQ=s0-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
![](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sgYiUTU4yI6NTO9HI6sWzayhPw9zTBraBLseBBrPR0HUizp31D34gHA0XYNUU7mnD7cyVJKkPAw6FNi-3cOTW4_UCIsT6bsrDfSqplIcCecCT-awIegHroH-ItAtI9GCfxwquUWPmR34r9FwYqIUCcdVs=s0-d)
.
Show that
Ris an equivalence relation.
Question 11 ( 6.0 marks)
Two real valued functions fand gare defined as f(x) = e2xand g(x) = x− 3. Show that the functions
are one-to-one and onto, and thus, find their inverse.