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Questions:

Generic 5

22 x 5 = 110

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

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

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.

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')

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')

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)

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)

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

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

10.

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

11.

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

12.

Using binomial theorem, determine which number is larger (1.2)⁴⁰⁰⁰or 800?

13.

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 ?

14.

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

15.

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

16.

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

17.

Solve for \(x,|x+1|+|x|>3\)

18.

Solve \(1\le |x-2|\le 3\).

19.

Find the complex number satisfying the equation \(z+\sqrt { 2 } \left| (z+1) \right| +i=0\)

20.

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

21.

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

22.

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

Generic 2

3 x 2 = 6

23.

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.
If A={John promoted or Gurpreet promoted}, then find P(A)

24.

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

25.

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Answers Key:

Generic 5

22 x 5 = 110

\(1\over8\)

\([{100^o,{5\pi\over9}}]\)

We have, A and B are mutually exclusive events. Also, it is given that P(A)=0.35 and P(B)=0.45
P(A'\(\cap\)B')=P((A\(\cup\)B)')=1-P(A\(\cup\)B)=1-0.8=0.2

We have, A and B are mutually exclusive events. Also, it is given that P(A)=0.35 and P(B)=0.45
P(A\(\cap\)B') = P(A) - P(A\(\cap\)B)
=P(A) - 0 = P(A) = 0.35

We have, A and B are mutually exclusive events. Also, it is given that P(A)=0.35 and P(B)=0.45
P(A\(\cap \)B) = 0

We have, A and B are mutually exclusive events. Also, it is given that P(A)=0.35 and P(B)=0.45
P(A\(\cup \)B)=P(A)+P(B)=0.35+0.45=0.8

We have, A and B are mutually exclusive events. Also, it is given that P(A)=0.35 and P(B)=0.45
P(B')=1-P(B)=1-0.45=0.55

We have, A and B are mutually exclusive events. Also, it is given that P(A)=0.35 and P(B)=0.45
P(A') = 1 - P(A) = 1 - 0.35 = 0.65

10.

a₂ - a₁ = a₃ - a₂ =....= a_n - a_n-1 = d (say)
If a₂ - a₁ = d, then \(\left( \sqrt { { a }_{ 2 } } \right) ^{ 2 }\) - \(\left( \sqrt { { a }_{ 1 } } \right) ^{ 2 }\) = d
\(\Longrightarrow \)\(\left( \sqrt { { a }_{ 2 } } -\sqrt { { a }_{ 1 } } \right) \) \(\left( \sqrt { { a }_{ 2 } } +\sqrt { { a }_{ 1 } } \right) \) = d
\(\Longrightarrow \) \(\frac { 1 }{ \sqrt { { a }_{ 1 }+\sqrt { { a }_{ 2 } } } } \) = \(\frac { \sqrt { { a }_{ 2 }-{ a }_{ 1 } } }{ d } \)
On adding, we get
\(\frac { 1 }{ \sqrt { { a }_{ 1 }+\sqrt { { a }_{ 2 } } } } \) + \(\frac { 1 }{ \sqrt { { a }_{ 2 } } +\sqrt { { a }_{ 3 } } } \) + .....+ \(\frac { 1 }{ \sqrt { { a }_{ n-1 } } +\sqrt { { a }_{ n } } } \)
= \(\frac { 1 }{ d } \) [\(\sqrt { { a }_{ 2 } } \) - \(\sqrt { { a }_{ 1 } } \) + \(\sqrt { { a }_{ 3 } } \) -\(\sqrt { { a }_{ 2 } } \) +...+ \(\sqrt { { a }_{ n } } \) - \(\sqrt { { a }_{ n-1 } } \) ]
= \(\frac { 1 }{ d } \) [\(\sqrt { { a }_{ n } } \) - \(\sqrt { { a }_{ 1 } } \) ].......(i)
Also, a_n - a₁ = (n - 1) d
\(\Longrightarrow \) \(\left( \sqrt { { a }_{ n } } \right) ^{ 2 }\) - \(\left( \sqrt { { a }_{ 1 } } \right) ^{ 2 }\)= (n - 1) d
\(\Longrightarrow \) \(\sqrt { { a }_{ n } } \) - \(\sqrt { { a }_{ 1 } } \) = \(\frac { (n-1)d }{ \sqrt { { a }_{ n }+\sqrt { { a }_{ 1 } } } } \)

11.

2⁴ⁿ⁺⁴ - 15n - 16 = (2)⁴ⁿ⁺¹ - 15(n+1) - 1
= 16ⁿ⁺¹- 15(n+1) - 1
= (1+15)ⁿ⁺¹- 15(n+1) - 1
={ⁿ⁺¹C₀ + ⁿ⁺¹C₁ (15) + ⁿ⁺¹C₂ (15)²+ ⁿ⁺¹C₃ (15)³ +........+ ⁿ⁺¹C_n+1 (15)ⁿ⁺¹} - 15(n+1) - 1
= 1 + 15ⁿ⁺¹+ ⁿ⁺¹C₁ (15) + ⁿ⁺¹C₂ (15)²+ ⁿ⁺¹C₃ (15)³ +........+ ⁿ⁺¹C_n+1 (15)ⁿ⁺¹} - 15(n+1) - 1
= 225 {ⁿ⁺¹C₂+ ⁿ⁺¹C₃ (15) +....+ ⁿ⁺¹C_n+1 (15)ⁿ⁺¹}
= 225 \(\times \) a natural number.

12.

(1.2)⁴⁰⁰⁰ = (1 +0.2)⁴⁰⁰⁰
= ⁴⁰⁰⁰C₀ + ⁴⁰⁰⁰C₁(0.2) + sum of positive terms
= 1 + 4000(0.2) + a positive number
= 1 + 800 + positive number
= 800.

13.

Number of four different digits formed =⁷C₄
(i) Even number should be in unit's place and remaining any there digits, any different number can be placed.
\(\therefore \)Total number of ways = ⁶P₄ x ³C₁
(ii) Any four digit number is divisible by 25, if last two digits should be divisible by 25 i.e. 25,75.
(iii) Any number is divisible by 4, if last two digits is divisible by 4 i.e. last digits can be 12, 16, 24, 32, 36, 52, 56, 54, 72, 76
\(\therefore \) Total number of ways = ⁵P₂ x ¹⁰C₁
\(\therefore \)Total number of ways = ⁵P₂x ²C₁
(iv) Total number of words = \(\frac { 10 }{ 3!\times 2!\times 4! } \)

14.

In automobile licence plates, two different letters can be arranging im ²⁶P₂ways and three next different digits can be arranging in ¹⁰P₃ways.
\(\therefore \) Total number of ways = ²⁶P₂ x ¹⁰P₃
= 26 x 25 x10 x 9 x 8 = 468000

15.

Required number of ways= ⁿC₁ +ⁿC₂+....+ⁿC_n
=ⁿC₀ +ⁿC₁ +ⁿC₂+....+ⁿC_n-ⁿC₀=2ⁿ-1

16.

(0, 1] \(\cup \) [3, 4)

17.

Firstly, put \(x+1=0\) and \(x=0\Longrightarrow x=-1\) and \(x=0\)
\(\therefore \) \(x=-1\), 0 are critical points. So, we will consider three intervals (-\(\infty \), -1), [-1, 0), [0, \(\infty \))
Ans. (-\(\infty \), -2) \(\cup \) (1, \(\infty \) )

18.

Write the given inequalities as \(|x-2|\ge 1\) and \(|x-2|\le 3\) Solve each inequality separately and then take the intersection of their solution sets
Ans. [-1, 1] \(\cup \) [3, 5]

19.

\(z+\sqrt { 2 } \left| (z+1) \right| +i=0\)
Let \( z=x+iy\)
Then, \( (x+iy)+\sqrt { 2 } \left| (x+iy+1 \right| +i\ =0\)
\( \Rightarrow x+i(y+1)+\sqrt { 2 } \left| (x+1)+iy \right| =0\)
\(\Rightarrow x+i(y+1)+\sqrt { 2 } \sqrt { { (x+1) }^{ 2 }+{ y }^{ 2 } } =0\)
\([if\quad z=a+ib,\quad then\quad \left| z \right| =\sqrt { { a }^{ 2 }+{ b }^{ 2 } } ]\)
\(\Rightarrow x+\sqrt { 2 } \sqrt { { x }^{ 2 }+1+2x+{ y }^{ 2 } } +i(y+1)=0+0i\)
On equating real and imaginary part, we get
\( x+\sqrt { 2 } \sqrt { { x }^{ 2 }+1+2x+{ y }^{ 2 } } =0\quad .......(i)\)
\(and\quad y+1=0\quad ......(ii)\)
From Eq. (ii), we get
y = -1
Now, on substituting y = - 1 in Eq. (i), we get
\( x+\sqrt { 2 } \sqrt { { x }^{ 2 }+1+2x+{ +1 }^{ } } =0\)
\(\Rightarrow x=-\sqrt { 2 } \sqrt { { x }^{ 2 }+2x+2 } \)
On squaring both sides, we get
\({ x }^{ 2 }=2({ x }^{ 2 }+2x+2)\)
\(\Rightarrow { x }^{ 2 }=2{ x }^{ 2 }+4x+4\Rightarrow { x }^{ 2 }+4x+4=0\)
\(\Rightarrow { (x+1) }^{ 2 }=0\Rightarrow x+2=0\Rightarrow x=-2\)
\(Hence,\ z=x+iy=-2-i\)

20.

Consider, \(\frac { 1+i }{ 1-i } =\frac { 1+i }{ 1-i } \times \frac { 1+i }{ 1+i } \)
[by rationalising the denominator]
\(=\frac { { \left( 1+i \right) }^{ 2 } }{ 1-{ i }^{ 2 } } =\frac { 1+{ i }^{ 2 }+2i }{ 1+1 }\)
\(\Rightarrow \ \frac { 1+i }{ 1-i } =\frac { 1-1+2i }{ 2 } =i\quad \left[ \because \ { i }^{ 2 }=-1 \right] ...(i)\)
\( Now,\ \frac { 1-i }{ 1+i } =\frac { 1 }{ \left( \frac { 1+i }{ 1-i } \right) } =\frac { 1 }{ i } \)
\(=\frac { 1 }{ i } \times \frac { i }{ i } =\frac { i }{ { i }^{ 2 } } =\frac { i }{ (-1) } =-i\quad \left[ \because \ { i }^{ 2 }=-1 \right] ...(ii)\)
\(Hence,\ { \left( \frac { 1+i }{ 1-i } \right) }^{ 3 }-{ \left( \frac { 1-i }{ 1+i } \right) }^{ 3 }={ i }^{ 3 }-{ \left( -i \right) }^{ 3 }\)
\(={ i }^{ 3 }+{ i }^{ 3 }=2{ i }^{ 3 }=2(-i)=0-2i\ \left[ \because \ { i }^{ 3 }=-i \right] \)
\(\therefore \ x+iy=0-2i\)
On comparing real and imaginary parts on both sides, we get x=0 and y=-2
\(\therefore \) (x,y) = (0,-2)

21.

\(cos^{ 4 }\frac { \pi }{ 8 } +cos^{ 4 }\frac { 3\pi }{ 8 } +cos^{ 4 }\frac { 5\pi }{ 8 } +cos^{ 4 }\frac { 7\pi }{ 8 } \)
\(=cos^{ 4 }\frac { \pi }{ 8 } +cos^{ 4 }\frac { 3\pi }{ 8 } +cos^{ 4 }\left( \pi -\frac { 3\pi }{ 8 } \right) +cos^{ 4 }\left( \pi -\frac { \pi }{ 8 } \right) \)
\(= cos^{ 4 }\frac { \pi }{ 8 } +cos^{ 4 }\frac { 3\pi }{ 8 } +cos^{ 4 }\frac { \pi }{ 8 } \)
\(= 2\left[ cos^{ 4 }\frac { \pi }{ 8 } +cos^{ 4 }\frac { 3\pi }{ 8 } \right] =2\left[ cos^{ 4 }\frac { \pi }{ 8 } +cos^{ 4 }\left( \frac { \pi }{ 2 } -\frac { \pi }{ 8 } \right) \right] \)
\( =2-\left( sin\frac { 2\pi }{ 8 } \right) ^{ 2 }=\frac { 3 }{ 2 } \)

22.

\(f(x)=\begin{cases} 2x\quad 2\le x\le 3 \\ 4,\quad -2\le x<2 \\ -2x,\quad -3\le x<-2 \end{cases}\)

Generic 2

3 x 2 = 6

23.

\(P(A)\frac { 1 }{ 4 } \)

24.

0.57

25.

0.51