MATHEMATICS EXAM TIPS: AP SET

100 SOLVED QUESTIONS FOR CSIR,GATE, SET AND OTHER COMPETITIVE EXAMS-PART 3

Question 10: Let $H=\{e,(12)(34)\}$ and $K=\{e,(12)(34),(13)(24),(14)(23)\}$ be subgroup of $S_4$ , where $e$ denotes the identify element of $S_4$ . Then

$H$ and $K$ are normal subgroup of $S_4$
$H$ is normal in $K$ and $K$ are normal in $A_4$
$H$ is normal in $A_4$ but not in $S_4$
$H$ is normal in $S_4$ but $H$ is not

100 SOLVED QUESTIONS FOR CSIR,GATE, SET AND OTHER COMPETITIVE EXAMS-PART2

100 SOLVED QUESTIONS FOR CSIR,GATE, SET AND OTHER COMPETITIVE EXAMS-PART 2

Question 6: The number of elements in the set $\{m:1\leq m \leq 1000, m \text{ and } 1000 \text{ are relatively prime} \}$ is
$1.~100 \quad 2.~250 \quad 3.~300 \quad 4.~400$
Answer: The question asks for $\phi(1000)$ .

And $\phi(n)$ can be calculated using following formula.
1. $\phi(p_1^{m_1}p_2^{m_2} \cdots p_r^{m_r}) = \phi(p_1^{m_1})\phi(p_2^{m_2}) \cdots \phi(p_r^{m_r})$ .
2. $\phi(p^m) = \begin{cases}2^{m-1} &\text{ if } p= 2 \\ p^m - p^{m-1} &\text{ if } p \neq 2\end{cases}$
Or equivalently it can be calculated by single formula as follows
$\phi(n) = n \prod_{p|n}\left(1-\frac{1}{p} \right)$

$\phi(1000)=\phi(2^3)\cdot \phi(5^3) = 2^2(5^3-5^2)=400$ .

100 SOLVED QUESTIONS FOR CSIR,GATE, SET AND OTHER COMPETITIVE EXAMS-PART1

100 SOLVED QUESTIONS FOR CSIR,GATE, SET AND OTHER COMPETITIVE EXAMS-PART 1

Question 1: The number of surjective maps from a set of $4$ elements to a set of $3$ elements is
$1.~36 \quad 2.~64 \quad 3.~69 \quad 4.~81$
Answer: Let $A$ be a $4$ elements set and $B$ be a $3$ elements set. Here we need to find number of surjective maps between $A$ onto $B$ . The idea is to partition the set $A$ into $3$ non-empty subset and assign each subset to a element of $B$ .

AP-TS SET 2014 MATHEMATICS SOLUTIONS

PAPER-2 SOLUTIONS

Q 1

$S=\left\{ 1-\frac { { \left( -1 \right) }^{ n } }{ n } \quad :\quad n=1,2,3,... \right\} \\ \quad =\left\{ 2,1-\frac { 1 }{ 2 } ,1+\frac { 1 }{ 3 } ,1-\frac { 1 }{ 4 } ,... \right\} \\ \alpha \quad =\quad inf\quad S\quad =\quad \frac { 1 }{ 2 } \\ \beta \quad =\quad sup\quad S\quad =\quad 2\\ \\ \therefore \quad \left( \alpha ,\beta \right) \quad =\left( \frac { 1 }{ 2 } ,2 \right)$

Q 2

$Let\quad { x }_{ n }\rightarrow l\\ { x }_{ n+1 }\quad =\frac { 1 }{ 2 } \left( { x }_{ n }+\frac { a }{ { x }_{ n } } \right) ;\quad n\ge 1.\\ \Rightarrow \quad l=\frac { 1 }{ 2 } \left( l+\frac { a }{ l } \right) \\ \Rightarrow { l }^{ 2 }=a\quad \Rightarrow l\quad =\sqrt { a } \\ By\quad AM-GM\quad inequality,\\ { x }_{ n+1 }=\frac { 1 }{ 2 } \left( { x }_{ n }+\frac { a }{ { x }_{ n } } \right) \quad \ge \quad \sqrt { { x }_{ n }.\frac { a }{ { x }_{ n } } } =\sqrt { a } \quad \forall n\\ and\quad { x }_{ n }\rightarrow \sqrt { a } \\ \therefore \quad { x }_{ n }\quad is\quad dicreasing.\\$

Q 3

$f:{ R }\longrightarrow { R };f(x)=\begin{cases} 1 & \quad x\in { Q } \\ 0 & \quad x\in { Q^{ c } } \end{cases}\\ we\quad know\quad f\quad is\quad continuous\quad at\quad x\in { R }\\ \quad\quad\quad\quad\quad iff\quad { x }_{ n }\longrightarrow x\quad impies\quad that\quad f\left( { x }_{ n } \right) \longrightarrow f\left( x \right) .\\ let\quad x\in { Q },\quad then\quad \exists \quad ({ s }_{ n })\quad in\quad { Q }^{ c }\quad :\quad { s }_{ n }\longrightarrow x,\quad but\quad f\left( { s }_{ n } \right) =\quad 0\nrightarrow f\left( x \right) =1.\\ so\quad f\quad is\quad not\quad continuous\quad at\quad x\in { Q. }\\ similiarly\quad f\quad is\quad not\quad continuous\quad at\quad x\in { Q }^{ c }\quad as\quad \exists \quad { r }_{ n }\longrightarrow x\quad in\quad { Q }\\ but\quad f\left( { r }_{ n } \right) =1\nrightarrow f\left( x \right) =0.\\ threfore\quad set\quad of\quad continuity\quad points\quad is\quad empty.$

Q 4

$Given f:\mathbb{R}\longrightarrow\mathbb{R} \quad is\quad monotonically\quad decreasing\quad\\ \\ f\left(x\right)\leq f\left(0\right)=a \quad \forall x\in\mathbb{R} \\ \therefore f\left((0,\infty\right))\subseteq (-\infty,a)\\$

Q 5

$f:\left[ a,b \right] \longrightarrow \quad { R }.\quad Given\quad f\quad is\quad H\ddot { o } lder's\quad function\quad with\quad exponent\quad \delta \quad >0.\\ \quad \quad \left| f\left( x \right) -f\left( y \right) \right| <{ \left| x-y \right| }^{ 1+\delta },\quad \forall \quad x,y\in \left[ a,b \right] \\ \Rightarrow \frac { \left| f\left( x \right) -f\left( y \right) \right| }{ \left| x-y \right| } <\quad \left| x-y \right| ^{ \delta }\\ \Rightarrow \quad \lim _{ y\rightarrow x }{ \frac { \left| f\left( x \right) -f\left( y \right) \right| }{ \left| x-y \right| } } <\quad \lim _{ y\rightarrow x }{ \left| x-y \right| ^{ \delta } } \\ \Rightarrow \quad \left| f^{ ' }\left( x \right) \right| =\quad 0\quad \forall x\in \left[ a,b \right] \\ \Leftrightarrow \quad f^{ ' }\left( x \right) =\quad 0\quad \quad \forall x\in \left[ a,b \right] \\ \Leftrightarrow \quad f\left( x \right) =\quad k\quad constant\quad \forall x\in \left[ a,b \right]$

Q 6

$\left( \begin{matrix} 1, & -2, & 5 \end{matrix} \right) =\quad -6\left( \begin{matrix} 1, & 1, & 1 \end{matrix} \right) +3\left( \begin{matrix} 1, & 2, & 3 \end{matrix} \right) +2\left( \begin{matrix} 2, & -1, & 1 \end{matrix} \right) \\ \therefore \quad \left( \begin{matrix} \lambda _{ 1 }, & -\lambda _{ 2 }, & \lambda _{ 3 } \end{matrix} \right) =\left( \begin{matrix} -6 & 3 & 2 \end{matrix} \right).$

Q 7

${ B }=\left\{ \alpha _{ 1 },\alpha _{ 2 },...,\alpha _{ n } \right\} \quad is\quad a\quad basis\quad for\quad { V }\quad a\quad vector\quad space.\\ i.e.\quad { B }\quad is\quad a\quad maximal\quad linearly\quad independent\quad set\quad in\quad { V }\\ { S }=\left\{ \beta _{ 1 },\beta _{ 2 },...,\beta _{ m } \right\} \quad is\quad a\quad linearly\quad independent\quad set\quad in\quad { V }.\\ \therefore \quad m\le n.\\ \\ Note:\quad { S }{ \subseteq B }\quad need\quad not\quad be\quad true.$

Q 8

$T:{ { R } }^{ 3 }\longrightarrow { { R } }^{ 3 },\quad T(\begin{matrix} x, & y, & z \end{matrix})=(\begin{matrix} x+2y, & y-z & z+2z \end{matrix})\\ The\quad matrix\quad of\quad transformation\quad T\quad with\quad respect\quad to\quad standard\quad basis\quad is\\ \therefore \quad M=\begin{pmatrix} 1 & \quad 0 & \quad 1 \\ 2 & \quad 1 & \quad 0 \\ 0 & \quad -1 & \quad 2 \end{pmatrix}\\ by\quad row\quad operations\quad on\quad M\\ \begin{pmatrix} 1 & \quad 0 & \quad 1 \\ 2 & \quad 1 & \quad 0 \\ 0 & \quad -1 & \quad 2 \end{pmatrix}\xrightarrow { R_{ 2 }-2R_{ 1 } } \begin{pmatrix} 1 & \quad 0 & \quad \quad 1 \\ 0 & \quad 1 & \quad -2 \\ 0 & \quad -1 & \quad 2 \end{pmatrix}\xrightarrow { R_{ 3 }+R_{ 2 } } \begin{pmatrix} 1 & \quad \quad 0 & \quad \quad 1 \\ 0 & \quad 1 & \quad -2 \\ 0 & \quad 0 & \quad 0 \end{pmatrix}\\ \therefore \quad Rank\quad of\quad T\quad is\quad 2.$

Q 9

$T:{ { R } }^{ 3 }\longrightarrow { { R } }^{ 3 },\quad T(\begin{matrix} x, & y, & z \end{matrix})=\begin{pmatrix} x+y+z,&y+z,&z\end{pmatrix}.\\ Let \quad T\begin{pmatrix}x,&y,&z\end{pmatrix}=\begin{pmatrix}\alpha&\beta&\gamma\end{pmatrix}\\ \Rightarrow\begin{pmatrix}x+y+z,&y+z,&z\end{pmatrix}=\begin{pmatrix}\alpha&\beta&\gamma\end{pmatrix}\\ \Rightarrow z=\gamma;\quad y+z=\beta \quad \Rightarrow y=\beta-\gamma;$
$\\x+y+z=\alpha \quad \Rightarrow x=\alpha-\beta.\\$
$ T^{-1} \begin{pmatrix}\alpha,&\beta,&\gamma\end{pmatrix}$
$= \begin{pmatrix}\alpha-\beta,&\beta-\gamma,&\gamma\end{pmatrix}$

Q 10

$3t^2+4t+5=a+b(1+t)+c(1+t^2)$
$ =(a+b+c)+bt+c+t^2;\\ $
$ \Rightarrow \quad a+b+c=5,\quad b=4, \quad c=3,$
$ \Rightarrow a=5-3-4=-2;\\$
$\therefore$$\begin{pmatrix}a,&b,&c\end{pmatrix}=\begin{pmatrix}-2,&4,&3\end{pmatrix}$

Q 11

$Let\quad f(z)=zcosec(z)=\frac{z}{\sin(z)}.\\ $
Observe that f is analytic on

$\therefore \oint_Cf(z)dz=0\\ $

Q 12

The path $ C: |z-1|=1/2,\quad f(z)=\frac{1}{z(z-1)}.\\$ f has simple polesat $z=0$ and $ z=1\\$
The $z=1$ lies inside the closed path C.
$\therefore \oint_Cf(z)dz=2\pi i*Res(f;z=1)\\ $
$=2\pi i*lim_{z\rightarrow 1}(z-1)f(z)\\$
$=2\pi i*lim_{z\rightarrow1}\frac{1}{z}=2\pi i$

Q 13

$f(z)=u+iv$ is an entire function with $u^2\leq v;\\$
$\Rightarrow f $ is bounded,
$\Rightarrow$ by Liovelli's theorem f is constant.
Therefore $ f(2014)=f(0)=1.$

Q 16

Given the order of Group o(G)=143=11$\times$13;
$let\quad n_{13}=\quad number\quad of\quad Syllow-3\quad subgroups\quad in\quad G\\$
by Syllow's theorem
$n_{13}\mid11\quad \Rightarrow n_{13}\in \{1,11\},\\$
$n_{13}\equiv1mod(13)\quad \Rightarrow n_{13}=1.\\$
also we know that unique is normal.
Therefore answer is 1.

Q 19

Given set $X=\{a,b,c,d,e\},A=\{a,c,d,e\}\subseteq X;\\$
topology on X is $T=\{\phi,X,\{a\},\{a,b\},\{a,b,e\},\{a,c,d\},\{a,b,c,d\}\}.\\$
Interior$(A^o)$ of A is the largest open in A.
Therefore $ A^o=\{a,c,d\}. $

Q 21

Given differential is $(x^2-y^2)dx+2xydy=0;\\$
$\Rightarrow \frac{dy}{dx}=\frac{y^2-x^2}{2xy}=\frac{(\frac{y}{x})^2-1}{2(\frac{y}{x})}.\\$
$Let\quad y=zx\quad \Rightarrow \frac{dy}{dx}=z+x\frac{dz}{dx},\\$
$x\frac{dz}{dx}=\frac{z^2-1}{2z}-z=\frac{-1-z^2}{2z}\\$
$\Rightarrow \frac{1}{x}dx+\frac{2x}{1+z^2}dz=0\\$
$\Rightarrow\quad \log{x(z^2+1)}=C.\\$
$\Rightarrow\quad x(z^2+1)=C.\\$
Therefore the general solution is $ x^2+y^2=Cx.$

Q 33

$P(X=5) =\quad P(X=5|Y=1)+P(X=5|Y=2)+P(X=5|Y=3) \\$
$\quad=\quad \frac{3}{20}+\frac{1}{5}+\frac{6}{20} \\$
$\quad = \quad \frac{13}{20}\\$

>$P(X=6) =\quad P(X=6|Y=1)+P(X=6|Y=2)+P(X=6|Y=3) \\$
$\quad=\quad \frac{2}{20}+\frac{1}{5}+\frac{1}{20} \\$
$\quad = \quad \frac{7}{20}\\$

Q 34

$\varphi_X(t)\quad = \quad E(e^{itX})\\$
$=\int_{-\infty}^{\infty} e^{itx}.f_X(x) dx\\$
=$\int_{-\infty}^{\infty}e^{itx}.\frac{1}{\sqrt{2}\pi}e^{-\frac{x^2}{2}}dx\\$
=$\frac{1}{\sqrt{2}\pi}\int_{-\infty}^{\infty}e^{itx-\frac{x^2}{2}}dx\\$
=$\frac{1}{\sqrt{2}\pi}\int_{-\infty}^{\infty}e^{\frac{(it)^2}{2}}e^{-\frac{(x-it)^2}{2}}dx\\$
$=e^{-\frac{t^2}{2}}$

Q 22
Given $y.f(x)=\tan{\left( \frac{\pi}{4}-\frac{1}{4}\cos^4{x}\right)} $; and
$y\left(\frac{\pi}{2}\right)=1$
$\Rightarrow \quad y\left(\frac{\pi}{2}\right).f\left(\frac{\pi}{2}\right)=\tan{\left(\frac{\pi}{4}-\frac{1}{4}\cos^4{\frac{\pi}{2}}\right)}\\$

$\Rightarrow f\left(\frac{\pi}{2}\right)=\tan{\left(\frac{\pi}{4}-0\right)}=1.$

AP SET 2013 MATHEMATICAL SCIENCE ANSWER KEY

AP SET ANSWERS KEY –MATHEMATICAL SCIENCES(2013)

Subject: Mathematical Science

Paper-II

Q.No. Key Q.No. Key
1 A 34 D
2 D 35 B
3 C 36 A
4 A 37 B
5 C 38 C
6 B 39 D
7 B 40 B
8 B 41 C
9 D 42 D
10 C 43 A
11 A 44 C
12 C 45 A
13 D 46 A
14 B 47 C
15 C 48 C
16 D 49 B
17 B 50 A
18 A
19 C
20 B
21 C
22 B
23 B
24 D
25 D
26 D
27 A
28 A
29 D
30 A
31 B
32 B
33 A

Paper-III

Q.No. Key Q.No. Key Q.No. Key
1 A 36 C 71 B
2 A 37 A 72 A
3 B 38 B 73 B
4 A 39 C 74 A
5 D 40 A 75 B
6 D 41 C
7 A 42 B
8 C 43 B
9 B 44 A
10 B 45 D
11 C 46 C
12 C 47 D
13 A 48 B
14 C 49 A
15 B 50 B
16 A 51 A
17 A 52 B
18 D 53 B
19 C 54 C
20 B 55 C
21 D 56 B
22 C 57 D
23 A 58 C
24 B 59 B
25 C 60 D
26 D 61 C
27 B 62 B
28 A 63 D
29 B 64 B
30 B 65 A
31 B 66 C
32 A 67 B
33 C 68 A
34 A 69 A
35 B 70 C

AP SET 2012 MATHEMATICAL SCIENCE ANSWER KEY

AP SET ANSWERS KEY –MATHEMATICAL SCIENCES(2012)

PAPER – II

Q.No KEY Q.No KEY

1 D 26 A

2 C 27 C

3 C 28 B

4 A 29 A

5 C 30 C

6 B 31 A

7 B 32 D

8 C 33 A

9 C 34 C

10 A 35 A

11 A 36 D

12 D 37 D

13 C 38 C

14 D 39 B

15 B 40 D

16 C 41 A

17 D 42 B

18 D 43 B

19 C 44 A

20 B 45 A

21 A 46 A

22 D 47 B

23 A 48 A

24 B 49 B

25 C 50 C

PAPER – III

Q.No KEY Q.No KEY Q.No KEY

1 B 26 All 51 C

2 B 27 B 52 D

3 D 28 C 53 D

4 A 29 D 54 D

5 C 30 All 55 B

6 C 31 D 56 C

7 C 32 A 57 A

8 C 33 A 58 C

9 D 34 C 59 C

10 B 35 C 60 B

11 B 36 B 61 A

12 C 37 B 62 D

13 B 38 A 63 C

14 A 39 B 64 A

15 B 40 A 65 D

16 A 41 D 66 A

17 B 42 C 67 C

18 B 43 All 68 B

19 B 44 All 69 C

20 B 45 C 70 C

21 B 46 B 71 A

22 A&C 47 B 72 A

23 A 48 A 73 C

24 C 49 C 74 C

25 B 50 B 75 D

MATHEMATICS EXAM TIPS

100 SOLVED QUESTIONS FOR CSIR,GATE, SET AND OTHER COMPETITIVE EXAMS-PART 3

100 SOLVED QUESTIONS FOR CSIR,GATE, SET AND OTHER COMPETITIVE EXAMS-PART 3

100 SOLVED QUESTIONS FOR CSIR,GATE, SET AND OTHER COMPETITIVE EXAMS-PART2

100 SOLVED QUESTIONS FOR CSIR,GATE, SET AND OTHER COMPETITIVE EXAMS-PART 2

100 SOLVED QUESTIONS FOR CSIR,GATE, SET AND OTHER COMPETITIVE EXAMS-PART1

100 SOLVED QUESTIONS FOR CSIR,GATE, SET AND OTHER COMPETITIVE EXAMS-PART 1

AP-TS SET 2014 MATHEMATICS SOLUTIONS

AP-TS SET 2014 MATHEMATICS SOLUTIONS

PAPER-2 SOLUTIONS

AP SET 2013 MATHEMATICAL SCIENCE ANSWER KEY

AP SET 2012 MATHEMATICAL SCIENCE ANSWER KEY

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