1 1 1 A If A. If A A and B. - Page No -1 XII STD. 1. Matrix and Determinants and its Applications-1

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1 - Page No - XII STD. Matrix and Determinants and its Applications- MATHS THIRU TUITION CENTRE Solve by matrix inverse method x y z 7,3x y 5z 3, x y z 5. Solve by matrix inverse method x 3y 8z 0 0,3x y 4,x 5y 6z 3 3. Solve by Matrix inverse method x y 3z 9, x y z 6, x y z 4. Solve by matrix inverse method x y z 9,x 5y 7z 5,x y z 0 5. If A 3 3 A( adja) ( adja) A A I then, verify that If A then,find inverse matrix and verify A 3 A 7. If A 3 then,prove that A T A 5 8. A and B 7 3 then prove that i. ( AB) B A ii. ( AB) T B T A T

2 - Page No -. Matrix and Determinants and its Applications Discuss the solutions of the system of equations for all values of x y z, x y z, x y 4z. For what values of k, the system of equations kx y z, x ky z, x y kz have i. unique solution ii. more than one solution iii.no solution. Verify whether the given system of equations is consistent.if it is consistent, solve them 4. Solve x 5y 7z 5, x y z 9,x y z 0 Data : Let x, y, zbetheno. of red, blue, and greenchairs x y z 00 40x 60y 300z Solve by determinant method (Cramer s method) 4 3, 5, 0 x y z x y z x y z 6. Solve Data : Let x, y, zbetheno. of Rs., Rs. and Rs.5 coins. x y z 30 x y 5z Solve the non-homogeneous system of linear equations by determinant method x y z 6, 3x y z, 4x y z 8

3 - Page No -3 THIRU TUITION CENTRE VECTOR ALGEBRA - SECTION-B. If a b c 0, a 3, b 5, c 7, find the angle between a and b.. Show that the vectors 3i j k, i 3 j 5 k, and i j 4k triangle. form a right angled 3. Find the vectors whose length 5 and which are perpendicular to the vectors a 3i j 4 k, and b 6i 5 j k. 4. Prove by vector method a b, b c, c a a, b, c. 5. Show that the lines their point of intersection. x y z x y z and 3 intersect and find 6. Derive the equation of the plane in the intercept form. 7. Find the coordinates of the centre and the radius of the sphere whose vector equation is r r.(8i 6 j 0 k) The volume of a paralleopiped whose edges are represented by i k, 3 j k, i j 5k is 546. Find the value of. a b 9. If, are any two vectors, then a b ( a. b) a b. 0. Find the area of the triangle whose vectors are (3,,),(,, 3), and (4, 3,). of a. Find the magnitude and direction cosines of the moment about the point (,, 3) force i 3 j 6k whose line of action passes through the origin. a b c. Prove by vector method. sin A sin B sin C 3. Diagonals of a rhombus are at right angles. Prove by vector methods 4. Angle in a semi-circle is a right angle. Prove by vector method. 3

4 - Page No -4 THIRU TUITION CENTRE. VECTOR ALGEBRA Find the vector equation and Cartesian equation a 3i 4 j k b i j k c 7i k. Find the vector equation and Cartesian equation A,, x y z The planecontaining theline 3 3. Find the vector equation and Cartesian equation x y z The planecontaining theline 3 3 x y z parallel totheline 3 4. If a i j k b i k c i j k d i j k Then prove that ( a b) ( c d) a bd c a b c d 5. Prove by vector method i. Cos( A B) CosACosB SinASinB ii. Cos( A B) CosACosB SinASinB iii. Sin( A B) SinACosB CosASinB iv. Sin( A B) SinACosB CosASinB 4

5 - Page No COMPLEX NUMBERS Find all the values of.solve the equations 3 i 3 4 and hence prove that the product of the values is. 3.If i x x x ii x x x x cos and y cos then Prove that x y m n x y i. cos( m n) n m y x m n x y ii. isin( m n). n m y x 4. If a cos i sin, b cos i sin, c cos i sin then Prove that i. abc cos( ) abc a b c ii. cos( ) abc 5. If and are the roots of x n n n x 4 0 Prove that isin 3 and deduct 9 9. n n n 3 i 3 i cos. 6 n 6. If n is a positive integer, Prove that 7. If and are the roots of show that n n ( y ) ( y ) sinn. n sin x x and y 0 cot,. 8.If P represents the variable complex number z.find the locus of P arg z z 3 5

6 - Page No Analytical Geometry Find the axis, vertex, focus, directrix, equation of the latus rectum, length of the latus rectum for the following parabolas and hence draw their graphs. i y x y ii y x y iii x x y iv x x y Find the eccentricity, centre, foci, and vertices of the following ellipses and also trace the curve. i x y x y ii x y x y iii x y x y Find the eccentricity, centre, foci and vertices of the following hyperbolas and also trace the curve. i x y x y ii x y x y iii x y x y iv x y x y Best wishes by M.THIRUPATHYSATHIYA M.Sc., M.Phil., CCA., KUNICHI MOTTUR(VILLAGE),KUNICHI(POST),TIRUPATTUR(TK) Mobile no: id: thirumath0@gmail.com 6

7 - Page No ANALYTICAL GEOMETRY Find the axis, vertex, focus, directrix, equation of the latus rectum, length of the latus rectum for the parabola and draw the graph, y x y Find the eccentricity, centre, foci, and vertices of the ellipse 36x 4y 7x 3y Find the eccentricity, centre, foci, and vertices of the Hyperbola x y x y Find the equation of the asymptotes to the hyperbola 8x 0xy 3y x 4y 0 5. Find the equation of the asymptotes to the rectangular hyperbola 6x 5xy 6y x 5y A comet is moving in a parabolic orbit around the sun which is at the focus of a parabola. When the comet is 80 million kms from the sun, the line segment from the sun to the comet makes an angle of 3 radians with axis of the orbit. Find i. The equation of the comet s orbit, ii. How close does the comet come nearer to the sun? (Take the orbit as open right ward ). 7. A cable of a suspension bridge is in the form of a parabola whose span is 40mts.The road way is 5mts below the lowest point of the cable. If an extra support is provided across the cable 30 mts above the ground level. Find the length of the support if the height of the pillars is 55mts. 8. Find the equations of the two tangents that can be drawn from the point (5,) to the ellipse x 7y 4 7

8 - Page No Differential calculus and its applications. Trace the curve. Trace the curve 3. Trace the curve y 3 x y x y 3 x 3 4. Discuss the curve a y x a x a ( ), 0 for i. Existence ii. Symmetry iii. Asymptote iv. Loops 5. If x u tan y then Prove that u u xy yx 6. Using Euler s theorem, prove that x y u sin x y u u x y tan u x y if u u 7. Using Euler s theorem, prove that x y sin u x y if 3 3 x y u tan x y 8. If axby V ze and z is a homogeneous function of degree n in x and y Prove V V that x y ( ax by n) V x y 9. Verify EULER S theorem for f ( x, y) x y 0. i. If u log(tan x tan y tan z) then prove that sin x u x ii. If U ( x y)( y z)( z x) then prove that U U U 0 x y Z 8

9 - Page No INTEGRAL CALCULUS Answer any 0 questions. Find the area between the curves x = and x = 4. y x x, x-axis and the lines. Find the area between the line y=x + and the curve y = x 3. Compute the area between the curve y sin the lines x 0and x xand y cos x and 4. Find the area of the curve y x x ( 5) ( 6) i. between x = 5 and x = 6 (ii) between x = 6 and x = 7 5. Find the area of the loop of the curve 3 ay x( x a). 6. Find the area bounded by x-axis and an arch of the cycloid x = a (t sin t), y = a ( cos t) 7. Derive the formula for the volume of a right circular cone with radius r and height h. 8.Find the length of the curve 3 3 x y a a 9. Find the surface area of the solid generated by revolving the cycloid x = a(t + sin t), y = a( + cos t) about its base (x-axis). 0. Find the perimeter of the circle with radius a by using integral calculus method.. Find the length of the curve x = a(t sin t), y = a( cos t) between t = 0 and π.. Find the surface area of the solid generated by revolving the arc of the parabola y 4ax, bounded by its latus rectum about x-axis. 3. Prove that the curved surface area of a sphere of radius r intercepted between two parallel planes at a distance a and b from the centre of the Sphere is r( b a) and hence deduct the surface area of the sphere (b > a). 9

10 - Page No -0 கண தம 9.த ல கணக க னல ( Z, ) ஒர ம ட யற எ னன க ம எக க ட ட க. இங க ஆத ab a b எ யரபனற க கப ட ட ள த. xx. ; xr{0} என அரநப ல உள அண கள ன வ ம அடங க ன கணம G x x ஆத அண ப பர க கல ஒர க ம எக க ட ட க. 3. எ னன க ம எக க ட ட க. i) G Q {}, wherea, b, and ab Define a b a b ab, a, bg ii) G Q { }, where a, b, and ab Define a b a b ab, a, bg iii) G { f f, f, f } or G C {0}, 3 4 Define f( z) z, f( z) z, f3( z), f 4( z) z z a 0 a R {0} iv) G, 00 a 0 To prove : ( G,.) is abelian group. n v) G { ; n z} 4.( Z, ) n To prove : ( G,.) is abelian group. n ஒர க ம எக க ட ட க. 5. ( Z7 {[0]},. 7) ஒர க ம எக க ட ட க. 6. இன நட ட க க க ணப பற பர க க ன க ழ {[],[3],[4],[5],[9]} என கணம ஒர எ னன க ம க ட ட க இன 4 ஆம ட ம ங கள பர க க ன க ழ ஒர எ னன க ம க ட ட க இன 3 ஆம ட ம ங கள பர க க ன க ழ ஒர எ னன க ம க ட ட க ,,, அண ப பர க க ன க ழ ஒர எ னன க ம க ட ட க. 0. i) க கல ய த கர (Cancellation law) எழ த, ர. ஆக ன ன க அண கள ம அடங க ன கணம 0

11 - Page No -.. ii) ன த ர ப ப ரக ய த (Reversal law) எழ த, ர. 0. கழ தகவ ப பயல கள ( ) x x f x ce, X எ ல 4 ( ) x x f x ke, X எ ல c,, இயற ரக க ண க. k,, இயற ரக க ண க. 3.இனல ர ந X இன ச ப சர 6, த ட டவ க கம 5 ஆக ம. ( i) P(0 X 8) ( ii) P( X 6 0) இயற ரக க ண க. 4. ய ஸ ன பய ன எட த த க க ட ட கள ன ரய? 5.இனல ரப பய ன ண ப கள ன ரய? 6.பயல ச ர ன ண ப கள ன ரய? 7. ன யர ம கழ தகவ அடர த த ச ர ன சப சர,த ட டவ க கம க ண க. 3 ( ),0, x x x x xe x 0 ( i) f ( x) 4 ( iv) f ( x) 0 elsewhere 0 elsewhere, 3 3x 0 x e x ( ii) f ( x) 4 ( v) f ( x) 0 elsewhere 0 elsewhere e ( iii) f ( x) 0 x x 0, elsewhere 8.ஒர சநய ய ப ப ந x இன கழ தகவ அடர த த ச ர ப x kx e x,, 0 f( x) 0 elsewhere எ ல ( ik ) இன நத ப ரக க ண க. ( ii) P( X 0) க ண க. 9.Refer : Page Number Question Number 7 Exercise 0.: 4,7,8,0 8 Examples: 0.3, Exercise 0. :,,6 38 Exercise 0.3: 5,6 40 and 4 Examples: 0.3, 0.4 and Exercise0.4: 3,4,5,6 50 Examples: 0.30, Exercise0.5: 4,5,6,8

12 - Page No - THIRU TUITION CENTRE FIRST MID-TERM MODEL TEST-0 XII STD MATHS 7. Find the rank of the matrix. Section-A 6 6. Solve by determinant method x y 3;x 4y 8 3. The rank of an m n matrix A cannot exceed the minimum of m and n. that is Find the d.c.s of a vector whose direction ratios are,3, If a 3, b 5 and a. b 60 then find a b 6. Solve the fourth root of unity. Section-B (Compulsory question-0) Find the rank of the matrix If A then prove that A A. 9. Show that the points whose position vectors 4i 3 j k,i 4 j 5 k, i j triangle. form a right of a Find the magnitude and direction cosines of the moment about the point (,, 3) force i 3 j 6k whose line of action passes through the origin. a b c. Prove by vector method. sin A sin B sin C. Find the square root of ( 7 4 i). n n n 3 i 3 i cos. 6 n 3. If n is a positive integer.prove that

13 - Page No -3 a b b c c a a bc 4. Prove by vector method 5. For any two complex numbers zand z then prove that i. z z z z ii z z z.arg arg( ) arg( ) z OR 6. Solve the equation x x x 0 Section-C (Compulsory question-4) If a cos i sin, b cos i sin, c cos i sin then prove that i. abc cos( ) abc a b c ii. cos( ) abc. Find the vector and Cartesian equation of the plane containing the line x y z 3 -, -. and passing through the point Prove by vector method Sin(A-B)=SinACosB-CosASinB 4. Verify whether the given system of equations is consistent. if it is consistent,solve them 5. 4x+3y+6z=5,x+5y+7z=3,x+9y+z= OR Solve by Cramer s rule x y z 6,3x 3y z 3,x y z 3 Best wishes by M.THIRUPATHYSATHIYA M.Sc., M.Phil., CCA., Mobile no: id: thirumath0@gmail.com 3

14 - Page No -4 கண தம த ர ட ய சன சசன டர நத ப சண கள : 00 ஆம வக ப ப நபம :.30hrs SECTION-A என அண ன ன தபம க ண க A 0 என ம ரய ட ட அண ன ன தபம க ண க. எ ல, T AA இன தபம க ண. 4. A 3 எ ல, T AA இன தபம க ண என அண ன ன தபம எ ல இ ன நத ப ப க ண. 6. ஒர த ரசன அண ன ன யர ரச 3, த ரசன k 0 எ ல A என த A 34 என அண க க ( adja) A ஒர சத ப அண A இன யர ரச n எ ல adja என த A எ ல, A என த Aஎன அண ன ன யர ரச 3 எ ல det( ka ) என த... 4

15 - Page No -5. u a( bc) b( c a) c( ab) எ ல. a b c 0, a 3, b 4, c 5 எ ல a க க ம b க க ம இரடப ட டகக ணம 3. a b a b எ ல 4. PR i j k, QS i 3 j k 5. a b, bc, c a 64 எ ல a, b, c 6. i j, j k, k i இ ன நத ப ப 7. a b, b c, c a 8 எ ல a, b, c 8. x y z 6x 8y 0z 0 9. r si t j எ ல, ற கபம PQRS இன பப ப... இ ன நத ப ப இ ன நத ப ப என கக த த ன ரநனம நற ற ம ஆபம... என சநன ட க ப த i 3 j 4 k, ai b j ck ஆக ன பயக டர கள பசங க த த பயக டர க ன ன... SECTION-B A இ ன கசர ப ப அண ரனக க ண க. இ ன அண ன ன தபம க ண க. 3. அண க கக ரய ம ரன ல த ர க க: x y z 5, x y z,3x y z 4 4.பயக டர ம ரன ல ற வ க: 5. i k, 3 j k, i j 5k a b c. sin A sin B sinc என கயக டர கர ம ரப ப ள கரக பக ண ட இரணகபத த ண நத த ன க அவ 546 எ ல இன நத ப ரக க ண க. 6. r r. (4i j 6 k) 0 என கக த த ன ரநனம நற ற ம ஆபம க ண க. (அல த ) 5

16 - Page No -6 x y z 3 நற ற ம க ட ட க.கநல ம அரய பயட ட ம ப ள ரனக க ண க. x y z என கக ட கள பயட ட க பக ள ள ம எக SECTION-C ஒர ரன ல ர. நற ற ம ர. நற ற ம ர.5 ணனங கள உள.ர ய 00 நத ப ற க பந த தம 30 ணனங கள உள.அவ ய ன ன ஒவ பய ர யரகன ல ம உள ணனங க ன எண ண க ரகரனக க ண.. k இன எம நத ப ப கள க க ன யர ம சநன ட ட த பத க ப ப kx y z, x ky z, x y kz have i ) ஒகப ஒர த ர வ ii ) ஒன ற க க கநற ட ட த ர வ iii ) த ர வ இல ரந பற ம. 3. பயக டர ம ரன ல ற வ க: Cos( A B) CosACosB SinASinB 4. ஒர ம க கக ணத த ல க த த பக ட கள ஒகப ப ள ன ல சந த க க ம என தர பயக டர ம ரன ல ற வ க. 5. a i j k, b i k, c i j k, d i j k ( ab) ( c d) [ abd] c [ a b c] d எ ற வ க. எ ல 6. a i 3 j k, b i 5 k, c j 3k எ ல a( bc) ( a. c) b ( a. b) c எ ற வ க. 7. (,, ), (3,4,), and (7,0,6) ஆக ன ப ள கள யம ச பசல ல ம தத த ன பயக டர நற ற ம க ர ட ச னன சநன ட கரக க ண க. (அல த ) x 6 y 7 z 4 3 நற ற ம x y 9 z 3 4 என ஒர தத த ல அரநன த கக ட க ன இரடப ட ட ந ச ச ற த த பத ரதக க ண க. Best wishes by M.THIRUPATHYSATHIYA M.Sc., M.Phil. CCA., Mobile no: id: thirumath0@gmail.com 6

K.S ACADEMY, SALEM- POLYTECHNIC TRB QUESTION PAPER PG TRB,ENG & POLYTECHNIC TRB & TNSET COACHING CENTRE FOR PHYSICS UNIT 4 :ELECTROMAGNETIC THEORY

K.S ACADEMY, SALEM- POLYTECHNIC TRB QUESTION PAPER PG TRB,ENG & POLYTECHNIC TRB & TNSET COACHING CENTRE FOR PHYSICS UNIT 4 :ELECTROMAGNETIC THEORY PG TRB,ENG & POLYTECHNIC TRB & TNSET COACHING CENTRE FOR PHYSICS UNIT 4 :ELECTROMAGNETIC THEORY TIME: 1/2HOURS Marks: 30 ***************************************************************************** ANSWER