Madurai Kamraj University (MKU) 2007 B.Sc Mathematics CCA - Question Paper

CCA

(6 pages)

6246/Ml 1    OCTOBER 2007

Paper I CALCULUS AND CLASSICAL ALGEBRA

4.    x¹ -2x²y + 3xy² - 4y³ + 5x² - 6xy + ly² - By = 0

<oT<offTJp QJ{GTT61JQDIJl]6tfT 6U6tf)<oTT6l_|    3jrT6tflfl'5>.

Find the radius of curvature of the curve given by x³ -2x¹y + 3xy'¹ -4y³ + 5x²-6xy + 7y²-8y = 0.

ft/ 4

5.    Jlog(l f dQ ^-61flyF LDlu6DLJS 5)n"feOOlSa.

0

tf/4

Evaluate jlog(l + tanB)dO.

6. f: RR -S5T suentrujaop fix) = x|jc| crgsflsb e0 fD6S)(Dij uaoiff ffrriTL| srasrp iflpaja.

Prove that the function f: R > R defined by fix) = jcIjcI is an odd function.

Prove that lim ⁺ = 1.

n> 2 71

8. - -6t 60rEj(gLb dSfflfraoLDaoujff Gffirlaaeiib. / v

Jlⁿ

Test the convergence of -.

(11 1 'l

9. lim 1+++-----logn ereisrug) 0 mibpLb 1-jt

2 3 . Tt y

n>oo

J)6ff>l_(Suj    CTOTTp jlpO|.

riii

Prove that lim 1 + + +-----logn. exists and

n- \ 2 o n    ,

lies between 0 and 1.

10.    (LpffiGffiireror ffiasjfluL) Qrn_(flssT suaDrriusnn) ujrrg]? 60

Define a trigonometric series. Give an example.

11.    (l-2x) ² cT6ifru6STSlifl6i51ia)Qurrg] aL-puciDuaffiirassTffi.

Find the general term in the expansion of (1-2 x)~².

io    ^{1 14} 1-4-7    .

1Z. 1 + +-+-+    6T65T]T)    (olffirrL_irl65T

5 5 10 5 10 15

-6U6!S)i7iiSla)rr6OT 3.(i)laj6i06O<$ airioOTrffi.

a    1 1-4 1-4-7

. Sum to the series 1 + +-+-+ ,

Answer any SIX questions.

13.    y = cos(log;c)

^x2y_n+2 ⁺ (2n. + l)ry_n+1 + (n² + l)y_n = 0 OT6brp fglpia.

If    y = cos (log a:)    prove    that

x²y_n+2 + (2n + l)*y₊i + (² + l)y = 0

14.    y²=4ax, erarrD ff(njuLD(T6iD60uSla) g<,|pLb 60 HeTTGifluSleorrOT <$Lp    Q,5rr6S)a>eSl!T ||0

LDi_rEjrr 0@ld croitpLb Jsy} Qffi5jGir@ ir>irrlaSlujir

0@ld OT65rpii) jSlpaja.

Prove that for the parabola y² = 4ax, sub-tangent

at any point is double the abscissa and sub-normal is a constant.

15.    a²+b²=k² cresflo), + = 1 erGmirCSrr(5><seiflOTi

a b

0ii)uls5T (LpaSlanujffi rr6ror66.

Find the envelope of the family of straight lines

+ ~ = 1 where a² + b² = k². a b

6246/Ml 1

r-f-* m y~v -1

16.    m,n> 1 ersBfleb jsin^m x cos'¹ x dx -sirr 0

eurrLULJUfrtl0Df_& ffifriswi".

Establish a reduction formula for jsin^m x cos x dx where m,n> 1.

17.    (0, ;r)-u51sb f(x) = n-x otqjtjd fftriri51tr y,iflujir Qsjfri_sou" iSrrtooot<s.

Find the Fourier cosine series for the function fix) = n-x in (0,n).

18.    a > 0 srsisTugi    QixiujQujann- srssfld) i_

limaⁿ =1 ersup (Slpeij.

n

3.

Prove that lima" =1 where a>0 is any real

n

number.

19.    V-L grajrfD    Qrriir, p > 1 crssflsb 60i4j@ih

n^p

ff,6t5\QS)l> 2_S)L_lUrr<561JLh p < 1 <5T6cfleb 6)S1lfllL(Lb 0j0ST6ff>LD 2_s5)i_iuFrffi6i(Lb l0ffi@Lb srajrp rlp&j<$.

Prove that the harmonic series j converges if p > 1 and diverges if p< 1.

20. sSusjflusm Gffir65)6T65)UJu uiu6tu@    OT63Tp

Qm_rflrr 60rj@ii> fB6ffrDiiD6S)iiju urDjfil QSl6un\<56i|ii>.

Using Leibnitzs test, discuss the convergence of

(-lY¹

the series T-

21. FF0ipUL| Gn)|D0DU UUJ6fffUffil ~r_ "0ST 1>{0U0OU

V128

ffflujiTffi gpgl ulsisiu@Ljqa6TT J6J>p5 ffirrajsTffi.

1

Using Binomial theorem, find the value of

Vl28

correct to five places of decimals.

1

   r = a (l-cos#) srsiirD eu65)OT6U65>(r<$@ (r< lOTffiueifr mrfipLb QirtTJlCan' eurr)r5l65T 65)l_Guj EL-errerT Gffi(r6B5r65).s

Find the angle between the radius vector and the tangent for the curve r = a (l - cos 6).

Attachment: madurai-kamraj-university--mku--2007-b-sc-mathematics-21871-1.pdf

Earning:   Approval pending.