DE5/DC5 ENGINEERING MATHEMATICS I DEC π π Q.. Prove tht cos α + cos α + + cos α + L.H.S. π π cos α + cos α + + cos α + + α + cos α + cos ( α ) + cos ( ) cos α + cos ( 9 + α + ) + cos(8 + α + 6 ) cos α - sin ( α + )- cos ( α + 6 ) cos α - (sin α cos + cos α sin ) cos α - (sin α cos + cosαsin ) (cosα cos6 sin αsin 6 ) cos α - ( sin α + cosα) ( cosα sin α ) cos α - sin α + cosα - cosα + sin α cos α - cos α b. Prove tht L.H.S sec8a sec A tn8a tn A sec8a sec A cos8a cos A cos A( cos8a) cos8a( cos A) cos A sin. cos8a sin A A sin A cos A.sin A cos8a.sin A tn8a R.H.S. tn A IETE
DE5/DC5 ENGINEERING MATHEMATICS I DEC Q.. Find the coefficient of 8 in the epnsion of Given epression is 5 + 5 + Let 8 occurs in the given epnsion in T r+ T r+ 5 Cr ( ) 5-r r n r r ( Tr + n C ) r 5! r!(5 r)! r () r r 5! r!(5 r)! r r () (i) Since 8 occurs in T r+ r 8 or -r 8 or -r - or r Putting r in eqution (i), in get 5! T 5 ().!! 5! 8 8! (65)(8 ) 8 Required coefficient of 8 in the given epnsion is 565 b. If the first term of n A.P is nd the sum of first five terms is equl to one fourth of the sum of the net five terms, then (i) show tht T (ii) find the sum of first terms. Let the A.P. be, +d, +d ------------ T + T + T + T + T 5 [ T6 + T7 + T8 + T9 + T] 5 5 ( T + T5 ) ( T6 + T ) IETE
DE5/DC5 ENGINEERING MATHEMATICS I DEC (sum n (first term + lst term) 5 5 8 [ + ( + d) ] [( + 5d) + ( + 9d) ] + d ( + d) +8 + 7d d - d -() -6 ( is geven) & d -6 (i) T + (-)d +9(-6) - - (ii) S [(() + (-)(-6)] 5[ - 7] 5 (-7) -55 Q.. Show tht b + c c + b c L.H.S b c b + c c + + b b b c c + b (b c)(c )( b)( + b + c) Operting R R + R + b + c b + c + b + c b c + + b + c c + b ( + b + c) b + c b c + c + b Operting C C C, C C C we hve IETE
DE5/DC5 ENGINEERING MATHEMATICS I DEC ( + b + c) b + c b b c c ( + b + c) ( + b + c) ( b)(c ) b + c ( + b) ( b)(c ) ( + b) c + c + ( + b + c) ( b)(c ) + b c ( + b + c) ( b)(b c)(c ) b. Using determinnts solve the following system of equtions: y z + y + y The given eqution re. + y y + y +. - + y +. Here (epnding by C ) ( 9) 5 Becuse therefore, system hs unique solution ( ) ( 6 9) IETE
DE5/DC5 ENGINEERING MATHEMATICS I DEC 75 ( + ) 5 ( + ) 75 5 5 5 y 5 z 5 5, y, z Q.5. Find the eqution of the right bisector of the segment joining A(, ) nd B(, ) Let AB be segment joining the points A(,),B(,) & CD be its right bisector. Then CD psses through the midpoint of AB is + +,, This CD psses through, IETE 5
DE5/DC5 ENGINEERING MATHEMATICS I DEC Also slope of AB Now CD AB eqution of CD which psses through nd hs slop -/ is y-y m(- ) y- y-8 - + +y- Which is the required eqution., b. Find the eqution of the lines through the origin nd mking n ngle of 6 with the line + y + Let m be the slope of ny one of the required lines through (,) Then the eqution is y- m(-) y m (i) lso slop of + y + is Angle between the two line is given to be 6 Tn 6 m + + m m m (Using tnθ + m m ) IETE 6
DE5/DC5 ENGINEERING MATHEMATICS I DEC ± m + m Tking + ve sign, we here - m m + - m - m from (i), y Tking ve sign, we here + m m + Here m does not here finite vlue, then the line is. ( The line psses through the orgin). Q.6. Find the eqution of the circle which psses through the points (5, 8), (, 9) nd (, ). Find lso the co-ordintes of its centre nd rdius. Let the required eqution of the circle be + y + g + fy + c (i) (i) Psses through (5, -8) (5) + (-8) + g (5) + f (-8) + c 5 + 6 + g 6f + c g -6f + c -89 (ii) (i) Psses through (, -9) () + (-9) + g() +f(9) + c g 8f + c -85 (iii) (i) Psses through (, ) () + () + g() + f() + c g + f + c -5 (iv) Solving (ii) & (iii) g -6f + c 89 g -8f + c -85 6g + f - (v) Solving (iii) & (iv) g 8f + c - 85 g + f + c -5 IETE 7
DE5/DC5 ENGINEERING MATHEMATICS I DEC -f - 8 f Substituting the vlue of f in (v) 6g + () - 6g - -8 - g - Substituting the vlue of g nd f in eqution (iv) (-) + () + c c -5 substituting the vlue of g, f, c in eqution (i) (-) + () + c -5 c -5 Substituting the vlues of g, f, c in eqution (i) + y - + 8y -5 c ( -g, -f) c (, -) & r g + f c + 6 + 5 5 5 b. Find the length of mjor nd minor is, eccentricity, the co-ordintes of vertices nd foci, directrices nd the length ltus rectum of the ellipse +y 6. The given ellipse is + y 6 y Compring it with + b, b, < b u < b Length of the mjor is b Length of the minor is b (-e ) ( e ) e e - e co- ordintes of vertices re (, ± b) ie, (, ± ) co-ordintes of foci re (, ± be) ie (, ± ) (, ± ) b Eqution of the directices re y ± e IETE 8
DE5/DC5 ENGINEERING MATHEMATICS I DEC y ± ± length of ltus here b () Q.7. If y d y og ( + + ), Prove tht ( + ) + siny sin(+y) sin y sin( + y) Differentiting both sides w.r.t y sin( + d y) sin y sin y sin ( + y) sin( + y)cos y cos( + y)sin y sin ( + y) sin( + y y) sin sin ( + y) sin ( + y) d sin( + y) sin sin ( + y) sin ( + y) sin 7 b. Find the eqution of the tngent to the curve y t the ( )( ) point where it cuts -is. 7 The given curve is y (i) ( )( ) This curve cut is; y We get 7 ( )( ) IETE 9
DE5/DC5 ENGINEERING MATHEMATICS I DEC 7 This (i) cuts t is t (7, ) Diff. (i) w.r.t both sides d d ( 5 + 6) ( 7) (X 7) ( 5 + 6) ( 5 + 6) ( 5 + 6)() ( 7)( 5) ( 5 + 6) ( 5 + 6) ( 5 + 5) ( 5 + 6) 5 + 6 + 9 5 ( 5 + 6) + 9 ( 5 + 6) (7) + (7) 9 9 + 98 9 (7,) ((7) 5(7) + 6) (9 5 + 6) 78 + 98 () () New eqution of tngent t the point (7, ) shring slop is y- (-7) (Using y-y m ( )) y 7 - y - 7 + Q.8. Evlute + + + Let I + + d + λ ( + + ) + µ IETE
DE5/DC5 ENGINEERING MATHEMATICS I DEC + λ + λ + µ Compring the coefficients. λ, λ + µ λ, () + µ µ -6-5 Substituting the vlues of λ & µ in eqution (i) + ( + ) -5 I ( + ) 5 + + + 5 + + + + + 5 + + 9 9 + + + + 5 + + + + og + + 5 og + c + + + og + + 5 og + c + + og + + 5 og + c + π b. Evlute log( + tn ) IETE
DE5/DC5 ENGINEERING MATHEMATICS I DEC Let I og( + tn ) (i) (Using I f () f ( ) π og( + tn( ) tn og + + tn + tn + tn og + tn og + tn ( og og( + tn ) ) (ii) Adding (i) & (ii) I og og π 8 og Q.9. Solve the initil vlue problem + cot y, when y( ) π / + cot y + (Using vrible seprted) cot y Integrting both sides tn + og sec y + og og c Tking ntilog both sides X sec y c (i) Also y ( ) π / IETE
DE5/DC5 ENGINEERING MATHEMATICS I DEC u. when, y π put in (i) π sec c. c Put this vlue in (i) sec y b. Solve ( + y) ( + y ) ( + y) ( + y ) + y (i) + y Clerly it is homogeneous eqution Put y v (ii) Differentiting both sides w.r.t. () dv v + (iii) Put the vlues of (ii) nd (iii) in (i) weget dv + v + v v + + (v) + v dv + v v + + v dv + v + v v v v v + v + v + v v dv + v Integrting both sides + v dv v + dv v og v -v + og + c y y og og c y y og. c IETE
DE5/DC5 ENGINEERING MATHEMATICS I DEC y og y + og ( y) ( y) c c Tet Books I. Applied Mthemtics for Polytechnics, H. K. Dss, 8th Edition, CBS Publishers & Distributors. II. A Tet book of Comprehensive Mthemtics Clss XI, Prmnnd Gupt, Lmi Publictions (P) Ltd, New Delhi. III. Engineering Mthemtics, H. K. Dss, S, Chnd nd Compny Ltd, th Edition, New Delhi. IETE