Find the simplified form of \(f(x)=|x-2|+|2+x|,\quad if-3\le x\le 3.\)

6

(a)

Find the value of the expression
\({ cos }^{ 4 }\frac { \pi }{ 8 } +{ cos }^{ 4 }\frac { 3\pi }{ 8 } +{ cos }^{ 4 }\frac { 5\pi }{ 8 } +{ cos }^{ 4 }\frac { 7\pi }{ 8 } .\)

6

(a)

If \({ \left( \frac { 1+i }{ 1-i } \right) }^{ 3 }-{ \left( \frac { 1-i }{ 1+i } \right) }^{ 3 }=x+iy\), then find (x,y).

6

(a)

If z₁ =\(\sqrt { 3 } +i\sqrt { 3 } \) and \({ z }_{ 2 }=\sqrt { 3 } +i\)  then find the quadrant in which \((\frac { { z }_{ 1 } }{ { z }_{ 2 } } )\)  lies.

6

(a)

Solve \(1\le |x-2|\le 3\).

6

(a)

Solve for \(x,|x+1|+|x|>3\)

6

(a)

Solve for \(x,\frac { |x-2|-1 }{ |x-2|-2 } \le 0\)

6

(a)

Find the number of combinations , whwn some or all of n objects are taken at a time.

6

(a)

How many automobile lincene plates can be made, if each plate contains two different letters followed by three different digits?

6

(a)

A student has to answer 10 questiion choosing atleast 4 from each part A and B . If there are 6 question in part A and 7 question from part B, in how many ways can the student choose 10 queations?

6

(a)

Find the number of arrangements of the letters of the word 'INDEPENDENCE' . In how many of these arrangements
(i) do the words start with P?
(ii) do all the vowels always occur together?
(iii) do all the vowels never occur together ?
(iv) do the words begin with I and end in P?

6

(a)

Using the digits 1,2,3,4,5,6,7,a number of different digit is formed . Find how many numbers are formed ? Further , Find how many of these are
(i) exactly divisible by 2 ?
(ii) exactly divisible by 25 ?
(iii) exactly divisible by 4 ?

6

(a)

Show that 2⁴ⁿ⁺⁴ - 15n - 16, where n\(\in\) N, is divisible by 225.

6

(a)

Find the term independent of x in the expansion of (1+x+2x³)\(\left( \frac { 3 }{ 2 } x^{ 2 }-\frac { 1 }{ 3x } \right) ^{ 9 }\)

6

(a)

If a₁, a₂, a₃, ....,a_n are in AP, where a_i > 0 for all i, Show that \(\frac { 1 }{ \sqrt { { a }_{ 1 } } +\sqrt { { a }_{ 2 } } } \) + \(\frac { 1 }{ \sqrt { { a }_{ 2 } } +\sqrt { { a }_{ 3 } } } \) + .....+ \(\frac { 1 }{ \sqrt { { a }_{ n-1 } } +\sqrt { { a }_{ n } } } \)  = \(\frac { n-1 }{ \sqrt { { a }_{ 1 } } +\sqrt { { a }_{ n } } } \)

6

(a)

Find the equation of the lines passing through the point (1,0) and at a distance \(\frac { \sqrt { 3 } }{ 2 } \) from the origin.

6

(a)

A team of medical students doing their internship have to assist during surgeries at a city hospital.The probabilities of surgeries rated as very complex,routine,simple rated as very complex,complex,complex, routine,simple or very simple respectively 0.15,0.20,0.31,0.26 and 0.08.Find the probabilities that a particular surgery will be rated
neither very complex nor very simple

6

(a)

A team of medical students doing their internship have to assist during surgeries at a city hospital.The probabilities of surgeries rated as very complex,routine,simple rated as very complex,complex,complex, routine,simple or very simple respectively 0.15,0.20,0.31,0.26 and 0.08.Find the probabilities that a particular surgery will be rated
routine or complex

6

(a)

A team of medical students doing their internship have to assist during surgeries at a city hospital.The probabilities of surgeries rated as very complex,routine,simple rated as very complex,complex,complex, routine,simple or very simple respectively 0.15,0.20,0.31,0.26 and 0.08.Find the probabilities that a particular surgery will be rated
routine or simple

6

(a)

One of the four persons John, Rita, Aslam or Gurpreet will be promoted next month.Consequently, the sample space consists of four elementary outcomes S={Johnpromoted, Rita promoted, Gurpreet promoted} You are told that the chances of John's promotion is same as that of Gurpreet, Rita's Chances of promotion are twice as likely as Johns.Aslam's chances are four times that of John.
Determine P(John promoted)  P(Rita promoted);  P(Aslam promoted)
P(Gurpreet promoted)

6

(a)

If A and B are mutually exclusive events, such that P(A)=0.35 and P(B)=0.45, then find P(A\(\cup \)B)

6

(a)

If A and B are mutually exclusive events, such that P(A)=0.35 and P(B)=0.45, then find P(A\(\cap \)B)

6

(a)

Draw the Venn diagrams to illustrate the following relationship among sets, E, M and U, where E is the set of students studying English in a school, M is the set of students studying Mathematics in the same school, U is the set of all students in that school.
There is no student who studies both Mathematics and English.
 

6

(a)

Find the angle in degrees and radians between minutes hand and hour hand of a clock at 7 : 20.

6

(a)

Find the value of \([1+cos{\pi\over 8}][1+cos{3\pi\over 8}][1+cos{5\pi\over 8}][1+cos{{7\pi\over8}}]\)

6

(a)