## Diploma Engineering Mathematics

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### Hari B

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SHORTS 1 .short 1. Resolve (x 2. Resolve (x 3. Resolve (x 1st year questions in papers into partial fractions. 1 into partial fractions. 1 into partial fractions. -11) 7x-6 4. Resolve into partial fractions. 3x-20 5. Resolve into partial fractions. 1 6. Resolve into partial fractions. 7x-1 7. Resolve into partial fractions. 8. Resolve 2 .short o 1. If A 3x into partial fractions. 21 X + 1) 1 3 o 1 1 2 4 2 3 1 find 2A+3B. 2 1 3 2 o and B — 4 4 find A2. 2. 4. 2 2 2 1 1 3. If A 3 Show that (A -k = AT + BT and B — Define skew symmetric matrix .Give an example.
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2 5. If A 7 2 6. If A 0 1 7. If A = 2 3 5 6 3 -1 3 and B 2 1 and B 5 1 5 8 and B = 4 -1 3 1 0 2 5 8 2 —5 2 —1 3 6 o 4 7 verify that (A + BY 4 —6 Find 2A-3B. 3 verify that (A + BY 7 1 and B — 2 8. Find AB if A 1 3 .short 3 2 o 3 1 a Find the value of h 1. 2. 3. 4. 5. 6. If A If A I 2 2 1 1 3 1 1 4 3 then find A2-3A+21,where I is a 2 1 find the matrix A2 2 unit matrix of order 2. Show that b a 4 a b —3 6 -2 = 4abc Find the value of 1 Find the value of 5 7 4 short 1. Show that tan8A—tan5A—tan3A = tan8Atan5Atan3A.
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4. 2. If A+B=45 then showthat (1+ + tanB) 2. 1 3. Prove that cos70cos10+sin70sin10 — -1 9. If 10 Prove that sin2 52— 2 2. 10 — sin2 22 5 405 1 5. If tan A = — and tan B = then showthat tan(A + B) 6 11 6. Prove that tan(45+A)tan(45—A)=1. cosl 1+ sinl 1 7. Prove that = tan 56. cosl I— sinl 1 5 short 1. 2. 3. 4. 5. 6. 7. 8. sin 20 = cote. Prove that I —cos20 sin 30 = sin 0 Provethat 1+ 2cos20 cot A — cot 2A = cosec2A Prove that -k COS 20 = cote. Provethat sin 20 sin Asin(60 — + A) = — sin 3A. Provethat (cosA + sin A) Show that Provethat tan — + O 4 Prove that sin 20 Prove that I + cos20 = I + Sin 214. = 2 tan 20. —tan — _ O 4 = tan 0. sin x = sin(60 + x)then show that sin(60 1 4
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6 short Find the real and imaginary part of 1-2i 1. 3 + 4i Find the modulus of the complex number 2. Find the additive and multiplicative inverse of 2+ 3i 3. (I + i 12 -k i) . In a+ib form. Express 4. (3 - 41) Find the modulus of 5+71' 5. Find the additive and multiplicative inverse of 4+ 3i 6. in a+ib form. Express 7. 1 8. If x 1 = 2cos0 then show that x 2 + 2 x 9 = 2cos20 . 9. Find the additive inverse of 2 + 7Short 1) Find the perpendicular distance from the point (3,2) to the line 4x+5y+6=0. 2) Find the equation of the straight line passing through the point (3,-4) and perpendicular to the line 3x+5y-2=0. 3) Find the equation of the straight line passing through the points (-5,2) and (3,-2). 4) Find the distance between parallel lines 2x+3y+5=0 and 2x+3y+9=0. 5) Find the equation of the circle with (2,3) and (6,9) as ends of a diameter. 6) Find the distance between parallel lines 3x+2y-9=0 and 3x+2y+12=0. 3 7) Find the equation of the line passing through the points 5 8) Find the locus of the points which is at a distance of 5 units from (2, 1 2 -3).
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9) Find the equation of the straight line passing through the point (2,-5) & perpendicular to the line 7x+2y-1=0. 8Short Find the equation of the circle with center(2,-3) and radius 4. 1) 2) Find the equation of the circle having (a,0) & (b,0) as the extremities of the diameter. 3) Find the equation to the circle whose center is 1,2) and radius 3. 4) Find the equation of the circle with (-5,3) and (6,-7)as ends of a diameter. 5) Find the equation of the circle with center(l and radius 5. 6) Find the centre and radius of the circle x2+y2+4x-6y=0. 7) Find the centre and radius of the circle 3x2+3y2-5x-6y+4=0. 8) Find the centre and radius of the circle 3x2+3y2-1 Ix-7y+1=0. 9Short 1 — cos x 1) Evaluate hm Evaluate 1m 2) Evaluate 1m 3) Evaluate 1m 4) Evaluate 1m 5) Evaluate 1m 6) sin x x 125 1 x x 243 12 +22 +32 3 n 1+2+3+ 2 n 2
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Evaluate 1m 7) Evaluate 1m 8) Evaluate 1m 9) IOShort sin 3x sin 5x sin 39 sin 59 1 — cose 92 1) Differentiate x2ex with respect to x. 2) Find the derivative of log(log(logx)). 3) Find the derivative of 3tanx-410gx-7x3+9 with respect to x. sin x 4) Find the derivative of with respect to x. 1 -k COS X dy 5)If x=at2, y=2 , Then find dx dy 6)Find dx , if y=xsinx. dy 7) Find , if derivative of e4x 8) dy 9) Find dx , if x=at2 dy 10) Find dx Essays Il(a) 2 2 1 , if y=x o 1 1 1 3 0 , y=2at. 6-6X5+3X2+1. computeA +61,1 is the unit of matrix of order 3.
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1 S.T 1 2) 1 P.T a 3) a a b 2 a 2 1 (a 1 2 c)(c a) (a 4) Express the matrix matrix. 2 5 c)(c 7 3 0 a) 1 4 as sum of symmetric & skew 5 symmetric 1 2 6 b b 1 3 2 1 3 1 5) Find the adjoint of the matrix a 6. Show that 3 T=BTAT 1 3 4 8. If A = 2 o 5 2 2 1 1 2 o 1 1 and 1 a 1 2 o o 1 1 3 compute A o 0 verify that (AB) 3 2 -5A+61. Il(b) essay questions 1 .Solve the following system of equations by using matrix inversion method x +2y+3z=6; 2x+4y+z=7; 3x+2y+3z=8. 2. solve the equations x+2y-z=0;3x-y-2z=5;x-y-3z=0 by cramer's rule. 1 1 2 2 1 1 2 find the adjoint of the matrix A. 2
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4. solve the equations 3x+y+2z=3;2x-3y-z=-3;x+2y+z=4 by determinate method. 5. solve the equations x+2y+3z=6;2x+4y+z=7;3x+2y+9z=14 by the cramer's rule. 6. solve the equations x+y+z=6;x+2y+3z=14; x+4y+9z=36 by using matrix inversion method. 7. solve the system of equations by Gauss-Jordan method x+y+z=6; x- y+z=2; 2x+y-z=1. 8. solve the following system of equations by using cramer'srule x+2y-z=-3; 3x+y+z=4; x-y+2z=6. 9. solve the equations 3x+y+2z=3; 2x-3y-z=-3; x+2y+z=4 by the inverse of matrix method. 12(a)essay question 1. A+B+C=II ;Prove that cos2A+cos2B+cos2C= -1-4cosAcosBcosC. 2 ; Prove that sin2A+sin2B+sin2C= 4cosAcosBcosC. 3. A+B+C=1800 ; prove that sin2A+sin2B+sin2C= 4sinAsinBsinC. 4. A+B+C=II ; Prove that sin2A+sin2B-sin2C= 4cosAcosBcosC. 5. If Sina +sinß & cosa+ cosß= b; then Show that Tan 2 12(b) essay questions. 1. Tan ix+Tan ly+Tan Iz=ll; Show that xyz = x+y+z. 2. Show that Tan +Tan = cot 3. Prove that Tan +Tan 4.Prove that Tan -Tan-I(n2 + n + l)+C0t-l (11+1) = O. 5 .Prove that Tan + Tan 6. If Tan I(z) = ; Show that xy+yz+zx=l. 2 7 .Prove that Tan + Tan 8. If sin x Sin 1 y + Sin • Show that x2 + +z2 +2xyz=1. 2 13(a) essay question 1 .solve sin5 0 + sin sin3 0. 2.solve the equation + sine 0. 3.solve cose + US Sin 9 = I a b
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4.solve the equation 4+cose 6sin 2 e 5 .solve cos 9 + Sin 9 = I 6.solve 2cos2e 3cos6+1=0 7. solve Sin6 0 cos 20 sin 50 cos o. 8.solve 2sin2e + Sin e —1=0. 9.Solve 2cos2 + 3sin e —0 13(b) essay questions 1.1n any A ABC Prove that if a cosA= b cosB then the triangle is either isosceles or right angled. 2.Solve the A ABC with a=2, b=J5 & c= US +1. b 3.1f any A ABC , If A=600then Show that -1. 4. solve the AABCwith & c= 2. a +122 + c 5. In a A ABC, Prove that cotA+cotB+cotC= 4A 6.1n a A ABC , prove that E2bccosA a2 +192 +c2 7.solve the A ABC with a=2, C=JÄ +1 8.1n A ABC , Prove that sinA+sinB+sinC=S/R. 9.Solve the A ABC with a=l, b=2& c=JÄ 14aESSAY 1. Find the the equation of the parabola whose axis is parallel to x-axis and which passes through the points (2, 2 . Find the the equation of the parabola with focus (5, 0) and directrix 3 . Find the vertex, focus, equation of the axis, directrix and latus rectum of the parabola x2=36 4. Find the the equation of the of rectangular hyperbola whose focus is (-3, 4) and directrix is 4x+3y+1=0 5. Find the the equation of the of rectangular hyperbola whose focus is (1,-5) and directrix isx+y+3=0 6. Find the the equation of ellipse which passes through the points (2, 2) and (3, 1) with axes as Coordinate axes 7. Find the the equation of the parabola whose focus is at (1,-1) and directrix is the line 3x+4y+5=0
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8. Find the centre, vertices, the eccentricity, foci, length of the latus rectum of latera recta and Directrics of ellipse 9x2+9y2-18x-100y-116=0 9. Find the the equation of parabola passing through the points (-1, 2), (1,-1), (2, 1) and having its axis parallel to the-axis 1 .Find the equation of the ellipse whose focus is (1,-1), directrix is the line x- 1 y+3=0 and eccentricity is-. 2 2. Find the lengths of the major and minor axis, length of latusrectum (LLR), 2 eccentricity and foci of the ellipse — + 25 16 3. Find the equation of the ellipse whose focus is (3, 1), eccentricity is - and directrix is x-y+6=0 4. Find the eccentricity vertices and foci of ellipse 9x2+16y2=1 5. Find the centre, vertices, eccentricity, foci, LLRand equation of the directrices of 2 the ellipse — + — 25 9 6. Find the equation of the hyperbola whose axes arte axes of coordinate latus rectum is 8, Eccentricity is 3 15. (a)ESSAY dy 1. If y=sinx (logx) find — dx a—bcosx 2. Find the derivative of w.r.t x a+bcosx dy 3. Find — if y=sin-l (3x — 4x3) dx 2 4. Differentiate log w.r.t x I—x . ,if x3 + y3 — 3axy = 0 5 1 6. If x=t4-5, y=t7+6 find at t= - 2 7.Find if y=xx dx ,ax2+2hxy+by2+2gx +2fy+c=0 8. Find dx 3 dy — if y=log (24) 9. Find dx I—x sinx 10. Differentiate x w.r.t x 15b essay x.... ...Tterms ,if y=xx 1 .Find dx dy I—cosx 2.Find — if y=tan I dx I cosx
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dy 3. Find — ify= cosx+ cosx+ cosx + dx sinx 4.Find the derivative of tan I wrt x I + COSX 5.1f y=aeX+be-X then show that — y — 0 dy 6.Find — if x3+Y3 = 3axy dx dy 7.1f y=tan-l (cosxfi) find — dx 8. Ify= sinx+ sinx+ sinx+ sinx + cosx + 9.x= a(9 — sine ) and y=a(1-cos9) dy x + find— dx x 1. Differentiate sinx w.r.t e dy 2.Find — if x=a cos39 ,y= bsin39 dx 2 3 .If u=sm prove that x— + y — log x dy 4.1f xY=eX-Y then show that — = dx (1+10gx)2 = tanu Show that =— dx dy cosx T,Show that — =— 2y—1 1 2y—1 51f u= log (x+y+z) prove that x— + y— + z — ôx 16b) ESSAY 1 .If y=log(x+ 1+x2 ),then prove that ô2u ô2u 2.1f u=log(x2+y2),then prove that ôxôy ôyôx 3.1f y=sin(logx) then prove that x2Y2+xY1+y=0 ôu 4.1f u=log(x+y+z) then prove that x — +y— + ôx ôy ôz ô2u 5.u(x,y)=x2+xy+y2,find and — ôx2 ôy2
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dy — if x3+y3=9xy 6.Find dx dy — if x=a cos39 ,y=b sin 39 7 .Find dx 2 Du Du 8.1f u=sin show that x— + y— Ox Dy 17a)ESSAY tan u 1 .find the equation of tangent & normal to the curve y=x2+4x-10 at (2,2) 2.find the equation of tangent& normal to the curve y=4ax at the point (at2,2at) 3.A circular meter plate expands by heat so that its radius is increasing at the rate of 0.02cm per second .At what rate its area is increasing when the radius is 20cm? 4.Find the length of the tangent,normal,sudtangent and subnormal to the curve + Sin 9), y a(l — cos 9) at 9 3 5 .find the length of the tangent,normal,subtangent & subnormal for the curve Y2=4x at (1,2) 6.Find the angle between the curve y2=4x & x2=4y 7.Find the equation of tangent and normal to the curve y=x2+2x-1 at (1,2) 8.Find the equathion of the tangent and normal to the curve 3y=x2-6x+17 9.Find the length of the tangent,normal,subtangent and subnormal to the x2+y2-6x- 2y+5=0 at the point (2,-1) 10.Find the equation to the tangent and normal to the curve y=2x2-4x 17b)ESSAY 1 .A circular plate of metal expands by heat so that its radius increases at the rate of 0.0 lcm/sec.what rate is the surface area increasing when the radius is 2cm? 2.Find the length of the tangent,normal,subtangent& subnormal for the curve Y2=4x at(l ,2)
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3. The volume of sphere is increasing at the rate of 1m3/min.Find the rate at which 32m 3 the radius and surface area are increasing when the volume is m 3 4.A light is long 8m directly above a straight horizontal floor . A man 2m tall is walking away from the lamp at the rate of 5.4m/min.Find the rate at which his shadow is lightening. 5. The volume of the cube increases at the rate of 0.3 cubic cm/sec.Find the rate at which the surface area changes when the edge is 20cm? 6. The side of equilateral triangle is increasing at a rate of 2cm/sec.Find the rate of increase of its area given its side is 25cm. 7. The volume of a sphere is increase at the rate of 400cm3/sec.Find the rate of increase of its radius and its surface area at the in stant radius of the sphere is 4cm. 18A. ESSAY 1. A wire of length 40cm is bent so as to form a rectangle. Find the maximum area that can be enclosed by the wire 2. Find the dimensions of a rectangle maximum area having a perimeter of 36 feet. 3. Show that maximum rectangle that can be inscribed in a circle is a square 4. Find the maximum &minimum values of 2x3-9x2+12x+15 5. Find the maximum &minimum values of the function x3-6x2+9x +1 6. The sum of two numbers is 24.find the numbers so that the sum of their squares is a minimum 7. A right circular cylinder is inscribed in a sphere of radius R. S.T. the maximum when its height is 8. A cricket field is to be in the form of a rectangle with a semicircular area & the perimeter of field is 400m.find the dimensions of the field so that rectangular area is maximum 18b.ESSAY 1. If there is an error of 1% in measuring the side of a square plate, find the % error in its area.
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2. The circumference of a circle is measured as 28cm with an error of 0.04cm.find the approximate %error in the area of the circle. 3. The time of oscillation of a simple pendulum of length I is given by T 27t if the length is increased by 2%. Find the approximate % increase in its time of oscillation where g is constant. 4. The radios of a spherical balloon is increased by 1% .find the the approximate % increase in its surface area 5. If the radius of a spherical balloon is increased by 0.1 % find the approximate % increase in its volume 6. An error of 0.05cm is committed in measuring a length of 10cm if so, absolute error relative error &% error. 7. The circumference of a circle is measured as 28cm and with an error of 0.01cm.find the approximate %error in the area of circle

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