Diff Cal

Differential Calculus and Trigonometry Semester-I

Code: 2K6M1:1/U03MA1:M1 Level : KUnit: 1.1 Type : MCQ

1. Dn { (ax + b)-1 } is:

(a) (b)

(c) (d)

2. Dn {log (ax+b)} is:

(a) (b)

(c) (d)

3. Dn { sin (ax + b) } is:

(a) (b) an cos (ax + b)

(c) an cos (d) an sin

4. Dn {cos (ax + b) } is:

(a) (b) an sin (ax + b)

(c) an cos (d) an cos

5. Dn {eax cos (bx + c)} is

(a)

(b)

(c)

(d)

6. Dn {eax sin (bx+c) } is:(a) (a2+ b2)n/2 eax sin (n + bx + c), tan = b/a(b) an/2 eax cos (n + bx + c), = tan-1 b/a(c) (a2+b2)n/2 eax cos (n + bx + c), = tan-1 a/b(d) bn/2 enax sin (bx + c)

7. Dn {eax} is:(a) enax (b) neax (c) an eax (d) aneax

Code: 2K6M1:1 Level : UUnit: 1.1 Type : MCQ

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1. D4 {(ax+b)-1} is:

(a) (b) (c) (d)

2. D3 {log (3x+4)} is:

(a) (b) (c) (d)

3. D5 {ex sin x} is:(a) 2e5 sin 5x (b) 25 (ex sinx)5

(c) 25/2 ex sin (5/4 + x) (d) 25/2 ex cos (5/4 + x)

4. D7 {ex cos x} is (a) e7x cos 7x (b) 27/2 ex cos (7/4+x)

(c) 27ex cos 7x (d) 27/2 ex sin ( +x)

5. If y1 = 0, y2 = 6, then is:

(a) 0 (b) 6 (c) (d)

6. If x1 = 0, x2 = 3, then is:

(a) (b) 3 (c) (d)

Code: 2K6M1:1 Level : KUnit: 1.1 Type : VSA

1. Write down the nth derivative of y = (ax + b)m w.r. to x when m < n.2. Write down the nth derivative of y = (ax+b)m w.r. to x when m = n.3. What is Dn {log (ax + b) }?4. What is the nth derivative of sin (ax+b)? 5. What is the nth derivative of cos (ax+b)?6. What is the nth derivative of eax cos (bx+c)? 7. What is the nth derivative of eax sin (bx+c)?8. What is the nth derivative of eax?

Code: 2K6M1:1 Level : UUnit: 1.1 Type : VSA

1. Write down the nth derivative of y = log (5x + 6).2. What us D10 {sin (5x + 3) }?3. What is D15 {cos (6x + 7) }?4. Write down the 20th derivative of e3x cos (2x + 1).5. Write down the 6th derivative of e2x sin (7x + 6).6. Write down the 50th derivative of (2x + 3)50.

Code: 2K6M1:1 Level : UUnit: 1.1 Type : PARA

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1. If y = (ax+b)m, find the nth derivative yn of y w.r. to x Deduce the result for m = -1 and m = -2.

2. If y = sin (ax + b), find yn. Hence or otherwise find y15 if y = sin (11x + 10).3. Find Dn {eax cos (bx+c). Hence find D5 {ex cos (3x+1) }.

4. Find the nth differential coefficient of .

5. Find the nth differential coefficient of .

6. Find .

7. If y3 – 3ax2 + x3 = 0, prove that .

8. If x = a (t – sint), y = a (1 + cost), find as a function of t.

9. If ax2 + 2hxy + by2 = 1, show that D2y = .

10. If x3+y3-3axy = 0, prove that D2y = .

11. If y = , prove that (1 + x2)y2 + xy1 – m2y = 0.

12. If y = ax cos mx, prove that .

13. If y = , prove that (1 – x)y3 = 3y2.

14. If y = Aex + , prove that (2x – 1)y2 – (4x2+1)y1 + 2 (2x2 – x + 1)=0.

Code: 2K6M1:1 Level : UUnit: 1.1 Type : ESSAY

1. a. Find the nth differential coefficient of e4x sin2x

b. Find Dn

2. a. Find Dn

b. Find Dn {sin3x Cos5x}

3. a. Find Dn {log (4-x2)}

b. Find Dn

4. Find (a) Dn {e5x sin3ax} and *(b) Dn {tan-1 }

5. Find the nth differential coefficients of

(a) cosxsin3x (b)

6. Find (a) Dn {Cos4x} and (b) Dn

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7. Find (a) Dn and (b) Dn {sinx sin2x sin3x}

8. Find (a) Dn {exsinx} and (b) Dn {cosx cos2x cos3x}

9. Find (a) Dn {ex sinx sin2x} and (b) Dn {Cos5x sin2x}


1. State Leibritz theorem for the nth derivative of a product of two function.


1. Write down the nth derivative of x e5x by applying Leibnitz theorem. 2. Write down the nth derivative of x2 sinx by applying Leibnitz theorem. 3. Write down the nth derivative of x cosx by applying Leibnitz theorem.4. Write down the nth derivative of x logx by applying Leibnitz theorem.5. Write down the nth derivative of ex sinx by applying Leibnitz theorem.6. Write down the nth derivative of ex cosx by applying Leibnitz theorem.


1. If y1 = 1, y2 = 2, then equals:

(a) ½ (b) 2 (c) (d)


1. Define curvature of curve at a point.2. Define radius of curvature of a curve at a point.3. Write down the formula for the radius of curvature in Cartesian co-ordinates.

4. Write down the formula for when .

5. Write down the formula for when .


1. What is when y1 = 0 and y2 = 1?2. What is when y1 = 2 and y2 = 6?3. What is when x1 = 0 and x2 = 2?


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1. Find the radius of curvature at any point (x, y) of the curve y = c cos h x/c.2. Find the radius of curvature at (a, 0) for the curve xy2 = a3 – x3.

3. Find the radius of curvature for the curve x3 + y3 = 3axy at .

4. Find the radius of curvature of the curve at (3, 4).

5. Find the radius of curvature of the curve y3 = x (x+2y) at (1, - 1).6. Find the radius of curvature of the curve y = 4 sin x – sin 2x at the point where x=

/2.


1. a. Find the radius of curvature for the curve y2=x3+8 at (-2, 0) b. Find for the curve (x2+y2)2 = a2 (y2-x2) at (0,a)2. a. Find the radius of curvature for the curve 4ay2 = (2a-x)2 at (a, a/2) b. Find for the curve xy3 = a4 at (a,a).3. Find for the curves

(a) 2y=x (1-x+x2) at (1,1/2) and (b) x2y=a (x2+y2) at (-2a, 2a)

Code: 2K6M1:1 Level : KUnit: 1.4 Type : MCQ

1. If x = x(t), y = y(t), then the curvature is :

(a) (b) (c) (d)

2. If x1 = 0, x11 = 2, y1 = 1, y11 = 0, then is:

(a) –2 (b) (c) 2 (d)


1. Write down the formula for the radius of curvature when the equation of the curve is given in the form x = X(t), y = Y(t).


1. What is when y1 = tan and y2 = ?

2. Find when x1 = 2a, x11 = 0, y1 = 0 and y11 = a.


1. Find at any point ‘t’ for the curve x = a (cost + sin t), y = a (cos t – sin t).2. Find for the curve x = a ( - sin ), y = a (1 – cos ) at any point .3. For the curve x = 6t2 – 3t4, y = 8t3, show that at any point ‘t’ is bt (1 + t2)2.

4. Find for the curve x = log t, y = at any point ‘t’.

Code: 2K6M1:1 Level : U

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Unit: 1.4 Type : ESSAY

1. a. Find for the curve x=3t2, y=3t – t3 at t=1 b. Prove that the radius of curvature at a point (acos3, asin3) on the curve

x2/3 + y2/3=a2/3 is 3a sincos.2. a. Prove that the radius of curvature at any point on the cycloid x=a ( + sin),

y = a (1-cos) is 4acos /2 b. Prove that the radius of curvature at any point ‘’ on the curve x=csin2 (1+cos2),

y=C Cos2 (1-cos2) is 4ccos3.3. a. Find the radius of curvature at the point ‘’ on the curve x=a log sec, y=a (tan-). b. Prove that at any point ‘’ on the curve x=3acos - acos3, y=3asin - asin3 is

3asin.


1. The formula for the radius of curvature in polar co-ordinates is :

(a) (b)

(c) (d)


1. Write down the formula for the radius of curvature when the equation is given in polar co-ordinates.


1. Find when r = a, r1 = a, 4r11 = 0.2. Find when r = a, r1 = a, r11 = 0.


1. Find the radius of curvature of the cardioids r = a (1 – cos ).

2. Show that the radius of curvature of the curve rn = an cos n is .

3. Find the radius of curvature for the curve r2 = a2 sec 2.

4. Show that in the cardioid r = a(1 + cos ), is constant.

5. Show that the curvatures of the curves r = a and r = a at their intersecting point are in the ratio 3:1.

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1. a. Find the radius of curvature for the curve =1+e cos.

b. Show that for the curve r2=a2cos2 is

2. a. Find for the curve r=a (1-cos)

b. Show that the radius of curvature of the curve rn=ancosn is

Code: 2K6M1:1 Level : KUnit: 1.6 Type : MCQ1. X co-ordinate of the centre of curvature at (x, y) is:

(a) (b)

(c) (d)

2. Y co-ordinate of the centre of curvature at (x, y) is:

(a) (b)

(c) (d)


1. Write down the x co-ordinate of the centre of curvature of the curve y = f(x) at (x, y).2. Write down the y co-ordinate of the centre of curvature of curve y = f(x) at (x, y).3. Define evolute of curve.

Code: 2K6M1:1 Level : UUnit: 1.6 Type : VSA1. What is the x co-ordinate of the centre of curvature of the curve xy = 2 at (2, 1) if

y1= ½ and y2 = ½? 2. Write down the y co-ordinate of the centre of curvature of the curve if y1 = cot /2 and

y2 = - 1/4a cosec4 /4 at (a ( - sin ), a (1 – cos ).


1. Find the coordinate of the centre of curvature of the curve xy = c2 at (c, c).2. Find the co-ordinates of the centre of curvature of the curve y = log sec x at

(/3,log2).3. Show that for the curve x2/3 + y2/3=a2/3, X=a cos3t+3a cost sin2t, Y=a sin3t+3asint cos2t

where (X, Y) is the centre of curvature.4. Find the evolute of the parabola y2 = 4ax.5. If the centre of curvature of a curve at a variable point ‘t’ on it is (2a + 3at2, - 2at3),

find the evolute of the curve.

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1. a. Find the centre of curvature of the curves x2=4ay at (2at, at2)

b. Find the evolute of the curve

2. a. Find the centre of curvature of the curve x2/3 + y2/3 = a2/3.

b. Find the evolute of the ellipse =1

3. a. Find the centre of curvature of the curve x=a(-sin), y=a(1-cos) b. Show that the equation of the evolute of the curve 2xy=a2 is (x+y)2/3 + (x-y)2/3=2a2/3


1. If V = (x2 + y2 + z2)-1/2 , then is:

(a) - x (x2 + y2 + z2)-3/2 (b) - y (x2 + y2 + z2)-3/2

(c) - z (x2 + y2 + z2)-3/2 (d) yz (x2 + y2 + z2)1/2

2. If u = (y – z) (z – x) (x – y), then equals :

(a) x + y + z (b) 3 (x + y + z) (c) 0 (d) 2 (x + y + z)

3. If u = log , then equals

(a) (b) (c) (d)

4. If u is a function of x and y, then

(a) (b) (c) (d)

5. If u = e y/x, then equals :

(a) (b) (c) (d) none of these


1. What is , if u = sin ?

2. What is if u = ?

3. What is if u = log ?

4. What is , if u = exy?

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5. What is if u = sin (ax + by + cz)?


1. If u = tan-1 , prove that .

2. If prove that

3. If u = sin-1 show that .

4. Show that for u = sin-1 .

5. Show that for u = xy

6. If u = rn, where r2 = x2 + y2 + z2, prove that

7. If f(x, y) = log , find the value of .

8. If r2 = (x – a)2 + *y – b)2 + (z – c)2, prove that .

9. If v = ea cos (a log r), show that

10. If u = exy, show that .

11. If u = tan-1 prove that .

12. If v = (x2 + y2 + z2)-1/2, show that .

13. If u = log (x3 + y3 – x2y – 2y2), show that

(i)

(ii) .

14. If u = log (x2+y2), prove that .

15. If z = f (x2+y2), show that


1. (a) xx yy zz = C, show that when x = y = z.

(b) If z = sin (x + y) + log (x + y), show that .

2. (a) If u = x2 tan-1 prove that .

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(b) If z = ex (x cos y – y sin y), show that

3. (a) If u = sin-1 show that

(b) If x = r cos , y = r sin , prove that

4. (a) If v = , prove that .

(b) If z = ( y – x)2 + sin (x +y), prove that

5. (a) If x = e-t cos , y = e-t sin prove that :

(i) and (ii) .

(b) If v = x2 + y2 + (xy)+ f prove that:

6. (a) If u = tan-1 prove that

(b) If z = x tan-1 , show that .


1. If u is a function of x, y, z, then du equals :

(a) dx + dy + dz (b)

(c) (d) none of these

2. If f (x, y) = C, where C is a constant, then equals:

(a) (b) (c) (d)

3. If us is a homogenous function of x, y of degree n, then

(a) (b)

(c) (d)

4. If u = f (x, y) is a homogeneous function of x, y, of degree n then u equals:

(a) (b) (c) (d)


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1. If equals

(a) (b) (c) (d)

2. If u = x2 + y2 and y = equals:

(a) 2x + (b) –2x + (c) –2x (d) 2x

3. If u = x3 + y3 + z3, then equals:

(a) 3u (b) u (c) 0 (d) 2u

4. If u = xy2 f , then by Euler’s theorem on homogeneous equals

(a) 0 (b) u (c) 2u (d) 3u

5. If u = x2 y3 f , then by Euler’s theorem on homogeneous function,

equals (a) 4u (b) 2u (c) 3u (d) 5u


1. If u = x2 + y2 + z2, what is du?

2. If u = xyz, what is ?

3. If u = x3 + y3 + z3, what is ?

4. If xy = c2, what is ?

5. If y2 = 4ax, what is ?

6. Define homogeneous function.7. State Euler’s theorem on homogeneous function.8. Verify Euler’s theorem for u = x3 + y3.


1. If u = tan-1 where y = tan2x, find

2. If u = xyz where x = e-t y = e-t sin2t, z = sint, find

3. If u = sin (x2 + y2) where

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4. If u = log (x + y + z) where x = cos t, y = sin2 t and z = cos2t, find .

5. If u = show that (i) and

6. If u = f prove that: .

7. If u = (x + y) f , prove that:

Code: 2K6M1:1 Level : KUnit: 2.2 Type : ESSAY1. (a) State and prove Euler’s theorem on homogeneous function.

(b) If u is a homogeneous function of x, y of degree n, prove that:


1. (a) If u = sin (x y2), x = log t, y = et, find .

(b) Verify Euler’s theorem for u =

2. (a) If u = x3 y4 z2 where x = t2 y=t3 and z = t4, find

(b) Verify Euler’s theorem for u = xy2 f .

3. (a) Find if u = x log (x, y) where x3 + y3 – 3axy = 0.

(b) If u be a homogeneous function of x and y of degree n, prove that:

4. (a) If u = , verify Euler’s theorem.

(b) If u = (x + y) f , prove that

5. (a) If z is a function of x and y and x = eu sinv, y = eu cosv, prove that:

(b) If u = x prove that


1. If V = F(u,v), u=f1(x,y) and =f2(x,y) then equals

(a) (b)

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(c) (d)

2. If V=f (u,v), u=f1(x,y) and =f2(x,y), then equals

(a) (b)

(c) (d)


1. If V = F (x, y), x=r cos , y=r sin , then is equivalent to

(a) (b)

(c) (d)

2. If Z is a function of x and y and x = eu sinv, Y = eu cos V, then equals

(a) (b) (c) (d)

ode: 2K6M1:1 Level : KUnit: 2.3 Type : VSA1. If V = f(u, v), where u and v are functions of x and y, write down the equivalent

expression for .

2. If V = f(u, v) where u and v are functions of x and y, write down the equivalent

expression for .


1. If u = f (x – y, y – z, z – x), what is

2. If Z = f ( x, y), x = r cos , y = r sin , what is ?

Code: 2K6M1:1 Level : UUnit: 2.3 Type : PARA1. If z is a function of x & y and if x = u – v, y = uv, prove that :

(i) (u + v) and

(ii) (u + v)

2. If z is a function of x and y and if x = eu sin v, y = eu cos v, prove that :

.

3. If x = X cos - Y sin , y = X sin + Y cos and if u is any function of x and y,

prove that : .

4. If u = x+ y, v = and z is a function of x and y show that :

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.

5. If z = f(x, y) where x = eu + e-v, y = e-u – ev, prove that

6. If u is a function of x and y and x = r cos , y = r sin prove that :

(i)

and (ii) .

7. If z be a function of x and y and u and v be two variables such that u = lx + my and

v = ly – mx, prove that

8. If v = f (xz, , prove that


1. Transform into polar co-ordinates.

2. (a) If v = f prove that

(b) If z is a function of x and y and x = eu sinv, y = eu cosv, prove that :

.

3. (a) If v = f (x,y), x = r cos , y = r sin , prove that .

(b) if v = f (x, y) and if x = u2 – v2, y = 2uv, prove that :

4. (a) If z is a function of x and y and if x = eu cos v, prove that :

(b) If v = f(x, y) and if x = u2 – v2, y = 2uv prove that :


1. If u is a function of x, y, then is:

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(a) (b) (c) (d)

2. If u, v are functions of x & y and x, y are functions of , , then

equals:

(a) (b) 1 (c) (d) 0


1. If x + y = u, x – y = v, then equals:

(a) (b) –2 (c) 0 (d) –1

2. If x = r cos , y = r sin , then equals:

(a) 1 (b) r2 (c) 0 (d) r

3. If u = x + y, v = xy, then equals:

(a) x – y (b) x + y (c) 0 (d) xy


1. Define Jacobian of functions of two variables. 2. Define Jacobian of functions of three variables.


1. If

2. If u = x2 – y2, v = x2 + y2 what is


1. If x + y = u, x – y = v, show that

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2. If x2 – y2 = u, xy = v, show that

3. If x = r sin cos , y = r sin sin , z = r cos find

4. If x + y = u, y = uv verify that

5. Prove that where u = x + y + z, v = xy + yz + zx and w = x2 + y2 + z2.

6. If u = show that


1. (a) If x2 – y2 = u, xy = v, show that

(b) If u = show that

2. (a) If x+y=u, x-y = v, show that

(b) If x = r sin cos , y = r sin sin , z = rcos find

3. (a) Verify that given x + y =u, y = ur.

(b) If u = x + y + z, v = xy + yz + zx and w = x2 + y2 + z2, prove that

4. (a) If x = r cos, y = r sin , verify that

(b) If x + y + z = u, y + z = ur, z = urw, find .


1. f (x, y) attains a maximum or minimum if

(a)

(b)

(c)

(d)

Code: 2K6M1:1 Level : K

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Unit: 2.5 Type : VSA

1. State the conditions of f(x, y) to attain a maximum or minimum at (a, b)2. State the condition for f (x, y) to attain its maximum at (a, b).3. State the condition for f (x, y) to attain its minimum.4. Does f(x, y) attain its minimum or maximum at (a, b), if

? If not when?


1. Find the maximum or minimum values of the function u = 2 (x2 – y-2) – x4 + y4.2. Discuss the maxima and minima of the function u = x3 y2 (6 – x – y)3. Discuss the maxima and minima of x2 + 3xy2 – 15x2 – 15y2 + 72x. 4. Discuss the maxima and minima of 3x2 y2 + 6xy3 – 2xy2.5. Find the maximum or minimum values of

(i) u = x2 + 3xy2 – 15x2 + 72x(ii) u = 3x2y2 + 6xy3 – 2xy2.


1. (a) Discuss the maxima and minima of xy (3x + 2y+1)(b) Show that, if the perimeter of a triangle is constant, the triangle has maximum

area when it is equilateral. 2. Discuss the maxima and minima of

(a) y2 + 2yx2 + 4x – 3(b) 2 (x3 + y3) – 3 (x2 + y2 ) +1.

3. Find the maximum or minimum values of (i) u = 2 (x2 – y2 ) – x4 + y4.(ii) u = x3y2 (6-x-y)


1. Expansion of Cos n is(a) nC1 Cosn-1 Sin + nC3 Cosn-3 Sin3 + nC5 Cosn-5 Sin5 +………(b) Cosn + nC2 Cosn-2 Sin2+nC4 Cosn-4 Sin4+……..(c) nC1 Cosn-1 Sin - nC3 Cosn-3 Sin3 + nC5 Cosn-5 Sin5 +………(d) Cosn -nC2 Cosn-2 Sin2+nC4 Cosn-4 Sin4+……..


1. The coefficient of Cos in the expansion of Cos7 as a series of powers of cosis (a) –7 (b) 3 (c) –3 (d) 7

2. Expansion of Cos7 is(a) Cos7 + 21 Cos5 Sin2 + 35 Cos3 Sin4 + 7 Cos Sin6(b) Cos7 - 21 Cos5 Sin2 + 35 Cos3 Sin4 - 7 Cos Sin6(c) Cos7 Sin -21 Cos5 Sin3 + 35 Cos3 Sin5 - 7 Cos Sin7

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(d) Cos7 Sin + 21 Cos5 Sin3 + 35 Cos3 Sin5 + 7 Cos Sin7

3. Expansion of Cos5 is (a) Cos5 + 10Cos3 Sin2 + 5 Cos Sin4 (b) 5Cos5 -10Cos3 Sin2 + Cos Sin4 (c) Cos5 -10Cos3 Sin2 + 5 Cos Sin4 (d) 5Cos5 + 10Cos3 Sin2 + Cos Sin4


1. Write the expansion of cosn as a series in powers of sin and cos .


1. Write the expansion of cos 7 in powers of sin and cos .2. Determine A, B, C if cos 5 = A cos 5 + B cos3 sin2 + C cos sin4 3. Write the expansion of cos 4 in powers of sin and cos.4. Write the expansion of cos6 in powers of sin and cos.

Code: 2K6M1:1 Level : KUnit: 3.1 Type : PARA

1. Derive the expansion of Cos n in powers of Sin and Cos .


1. Express Cos 6 in powers of sin ?2. Express Cos 5 in terms of powers of Cos ?3. Expand Cos 7 in terms of powers of Cos ?4. Determine A, B, C if Cos 5= A Cos5 + B Cos3 + C Cos5. Determine A,B,C, D if Cos 6= A+BSin6 +C Sin4 +D Sin26. Express Cos 6 as a polynomial in Cos7. Prove that Cos 7=64 Cos7-112 Cos5+56 Cos3-7 Cos8. Show that Cos 5 =16 Cos15-20 Cos3+ 5 Cos

Code: 2K6M1:1 Level : KUnit: 3.1 Type : ESSAY

1. (a) Express sin6 in powers of cos (7 Marks)(b) Derive the expansion of cos n. where n is a positive integer. (8 marks)


1. (a) Express sin5 in powers of sin ( 7 Marks)(b) Prove that cos8 = 1 – 32 sin2 + 160 sin4 - 256 sin6 + 128sin8 ( 8 Marks)

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2. (a) Prove that . (7 Marks)

(b) Express cos8 in powers of sin (8 Marks)


1. Expansion of Sin n is(a) nC1 Cosn-1 Sin + nC3 Cosn-3 Sin3 + nC5 Cosn-5 Sin5 +………(b) Cosn - nC2 Cosn-2 Sin2+nC4 Cosn-4 Sin4+……..(c) Cosn + nC2 Cosn-2 Sin2+nC4 Cosn-4 Sin4+……..(d) nC1 Cosn-1 Sin - nC3 Cosn-3 Sin3 + nC5 Cosn-5 Sin5 +………


1. The coefficient of Sin in the expansion of sin5 as a series of powers of sin is (a) 15 (b) 5 (c)10 (d) 1

2. Expansion of Sin 7 is(a) Cos7 - 21 Cos5 Sin2 + 35 Cos3 Sin4 -7 Cos Sin6(b) 7Cos6 Sin -35 Cos4 Sin3 + 21 Cos2 Sin5 -Sin7(c) Cos6 Sin -21 Cos4 Sin3 + 35 Cos2 Sin5 - 7 Sin7(d) 7Cos7-35 Cos5 Sin2 +21 Cos3 Sin4 - Cos Sin6

3. Expansion of Sin 5 is (a) 5Cos4 Sin + 10Cos2 Sin3 + Sin5 (b) Cos5 -10Cos3 Sin2 + 5 Cos Sin4 (c) 5Cos4 Sin-10Cos3 Sin3 + Sin5 (d) Cos5 + 10Cos3 Sin2 + 5Cos Sin4


1. Write the expansion of sin 5 in powers of sin and cos.2. Write the expansion of sin 6 in powers of sin and cos.3. Write the expansion of sin 7 in powers of sin and cos.4. Write the expansion of sin4 in powers of sin and cos .5. Determine A, B, C if sin 5 =A cos 4 sin - B cos2 sin 3 + C sin5 .6. What is the expansion of sin n?


1. Derive the expansion of sin n in powers of sin and cos .


1. Prove that

2. Show that

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3. Express in terms of Cos .

4. Determine A,B,C if Sin5 = A sin + B sin3+ C Sin5

5. Determine A,B,C

6. Express in Powers of Cos

7. Express Sin5 in Powers of Sin.


1. (a) Express cos 6 in powers of sin . (7 Marks)(b) Derive the expansion of sin, n being a positive integer. (8 Marks)


1. (a) Express cos5 in powers of cos . (7 Marks) (b) Prove that sin7 = 7 sin - 56sin3 + 112 sin5 - 64sin7. (8 Marks)

2. (a) Express cos4 as a polynomial in cos . (5 Marks)

(b) Express in powers of cos . (10 Marks)


1. The expansion of tan n is

(a)

(b)

(c)

(d)


1. The expansion of tan5 is

(a) (b)

(c) (d)


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(a) (b)

(c) (d)


(a) (b)

(c) (d)


1. Write the expansion of tan n in powers of tan.


1. Write down the expansion of tan7 in terms of tan.2. Write the expansion of tan5 in terms of powers of tan.3. Write the expansion of tan4 in terms of tan.4. Write the expansion of tan6 in terms of tan.


1. Derive the expansion of sin n and cos n and hence obtain that of tan n.2. Find the expansion of tan (1 + 2 + …… + n) and hence deduce the expansion of

tan n.


1. If x=cos + i sin then

(a) 2cos4 (b) 2i Sin4 (c) (2i sin)4 (d) (2cos)4

2. If x=cos + i Sin then

(a) 2cos5 (b) 2i Sin5 (c) (2i sin)5 (d) (2cos)5


1. If x=Cos + i sin then is

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(a) (2i Cos)6 (b) 2 Cos6 (c) 26 cos6 (d) 2 Sin 6

2. If x = Cos + i Sin then is

a) (2i sin)10 (b)210cos10 (c) 2i Sin10(d) 2i Cos10

Code: 2K6M1:1 Level : UUnit: 3.4 Type : MATCH

1. If x=cos + i Sin then

1) 2i Sin 3 a)

2) (2 Cos)3 b)

3) (2i Sin)3 c) 2x3

4) 2 Cos 3 d)

e)

(a) 1A, 2E, 3D, 4B (b) 1C, 2d, 3B, 4E (c) 1B, 2A, 3E, 4D (d) 1D, 2E, 3A, 4B


1. If x = Cos + i Sin, What is xn + ?

2. If x = Cos - i Sin, what is Xn - ?


1. Express Sin3 in a series of sines of multiples of

2. If x = Sin + i Cos what is xn + ?


1. Prove that cos3sin4 =

2. Prove that 64 sin2 cos5 = 5 cos - cos3 - 3 cos5 - cos 7.3. Show that 28 sin5 cos4 = sin9 - sin 7 - 4sin5 + 4 sin 3 + 6 sin .4. Prove that 64 sin5cos2 = sin 7 - 3 sin 5 + sin3 + 5 sin.5. Prove that 27 sin3 cos5 = 6 sin2 + 2 sin4 - 2 sin6 - sin8 .6. Expand cos8 in a series of cosines of multiples of .7. Prove that 32 cos6 = cos6 + 6cos 4 + 15 cos 2 + 10.8. Prove that 32 Sin6=10-15Cos2+6Cos4-Cos6

Code: 2K6M1:1 Level : UUnit: 3.4 Type : ESSAY1. (a) Prove that 16 sin5 = sin5 - 5 sin3 + 10 sin. (5 Marks)

(b) Show that 512 sin7 cos3 = 14sin2 - 8sin4 - 3 sin6+4Sin8-Sin10(10 Marks)

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2. (a) Prove that 16 cos5 = cos5 + 5 cos3 + 10 cos. (5 Marks)(b) Show that 29 cos4 sin6 = 6– 2 cos2 - 8 cos4 + 3 cos6 (10 Marks)

+ 2cos8 - cos103. (a) Show that 32sin2 cos4 = 2 + cos2 - 2 cos 4 - cos6. (7 Marks)

(b) Show that 256 sin9 = sin9 - 9sin7 + 36 sin5 - 8 sin3 + 126sin. (8 Marks)4. (a) Show that 32 sin4 cos2 = Cos6 - 2 cos4 - cos2 +2. (7 Marks)

(b) Show that 64 (cos8 + sin8) = cos8 + 28cos4 + 35. (8 Marks)

Code: 2K6M1:1 Level : KUnit: 3.5 Type : MCQ1. The expansion of Cos in powers of is

(a) (b)

(c) (d)

2. The expansion of Sin in powers of is

(a) (b)

(c) (d)

3. The expansion of tan is

(a) (b)

(c) (d)


1. is equal to

(a) 0 (b) 1 (c) n (d)

2. is

(a) (b) - (c) (d)

3. is

(a) 1 (b) (c) (d) –1

4. is

(a) 0 (b) 3 (c) 1 (d)


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1. Write down the expansion of Sin in terms of ?2. What is the series for tan in powers of ?3. What is the expansion of Cosx in powers of x?


1. Find


1. Find .

2. Evaluate .

3. Show that .

4. If show that is 1° 58’ approximately.

5. If show that = 3° nearly.

6. Evaluate

7. Show that .

8. Determine a and b so that as 0

9. If prove that a = -5/2 and b = -3/2.

10. Evaluate .

11. Show that the error involved in replacing [8 sin /2 - sin] by is numerically less

than if is small.

12. Evaluate

13. Evaluate

14. Find


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1. (a) Find . (5 Marks)

(b) Determine a, b, c such that . (10 Marks)

2. (a) Evaluate (5 Marks)

(b) If is small and positive, show that differs from by about .

Code: 2K6M1:1 Level : APUnit: 3.5 Type : ESSAY

1. (a) Solve approximately the equation tan (/4 - ) = 1.001. (5 Marks)(b) Prove that the length of a small circular arc is approximately (10 Marks)1/3 (8C' -C) where C is the chord of the arc and C1, the chord of half the arc.

2. (a) Find (5 Marks)

(b) Show that the length of a circular arc is approximately where a is

the length of the chord corresponding to the arc, b, corresponding to half the arc and C, corresponding to one – fourth of the arc. (10 Marks)


1. Sinh (x-y) is equal to a) Sinhx Coshy – Coshx sinhy b) Sinhx Sinhy + Cosh x Coshy c) Sinhx Coshy + CoshxSinhy d) SinhxSinhy – Coshx Coshy

2. tan h2x is

a) b) c) d)

3. Sin (i) is a) Sin h b) i sin h c) Cos h d) i cos h

4. Sin hx is equal to

a) b) c) d)

5. tanh3x is

a) b)

c) d)

6. ei is a) Cos - isin b) Sin + iCos c) Sin - icos d) Cos + isin

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7. Cosh2x is

a) b) c) 1 – Cosh2x d) 1 + Cosh2x

8. tan h (i) is a) i tan b) i tan h c) tan d) tan h

9. Sec h (ix) is a) i Secx b) i Sechx c) Sec x d) –i Secx.

10. Sin h2x is

a) b)

c) d) cos h2 x+1

Code: 2K6M1:1 Level : KUnit: 4.1 Type : MATCH

1. 1. Cos (ix) A) iSinx2. Cosh (ix) B) iSinhx3. Sin (ix) C) Cosx4. Sin (ix) D) Coshx

E) iCosxa) 1D, 2C, 3B, 4A b) E, 2C, 3B, 4A c) 1D, 2C, 3A, 4B d) 1C, 2D, 3B, 4A

2. 1. Cosh2x – Sinh2x A) 2Sinhx Coshx2. Cosh2x +Sinh2x B) Sinhx Coshy – Coshx Sinhy3. Sinh 2x C) Sinhx Coshy + Coshx Sinhy4. Sinh (x+y) D) 1

E) Cosh2xa) 1E, 2D, 3C, 4B b) 1D, 2E, 3A, 4B C) 1D, 2E, 3A, 4C D) 1E, 2D, 3A, 4B


1. Write the relation between tanh x and sech x.2. Expand Cosh (x+y)3. What is tanh (i) in terms of tan?4. Write the relation in hyperbolic functions corresponding to 1+tan2 = Sec2.

5. Prove that tan h2x =

6. Write down the expansion of Sinh in powers of .7. Write down the expansion of cos h in powers of .8. Write the expression for tan h (+i).


1. Expand Cos h (x+iy)2. What is the value of tanh3x if tanh x=1/3?

26

3. Prove that sinh2x = 2sinhx cosh x.4. Show that cosh (x+y) = coshx coshy + sinhx sinhy5. Write the expansion of sinh8 in a series of hyperbolic cosines of multiples of .6. Write down the infinite series for sin h2x.7. Prove that cosh2x – sinh2x = 1.8. Write the series expansion of cosh3.


1. Derive the expression for sinh (x+y) and cosh (x-y)2. Derive the expression for Sinh (x-y) and Cosh (x+y)

3. Prove that tanh3x =

4. Express sinh7 in terms of hyperbolic sines of multiples of .


1. If Sin (A+iB) = x+iy, prove that

2. If Coshu=Sec, show that u=log tan (/4 + /2)3. If tanh x/2 = tan x/2, show that cosxcoshx=14. Express cosh6 in terms of hyperbolic cosines of multiples of .

5. Prove that =Cosh2x + Sinh 2x

6. Expand Sinh8 in a Series of hyperbolic cosines of multiples of .


1. a) If tanA = tan tanh, tanB = Cot tanh, prove that tan (A+B) = Sinh2 Cosec2 (7 marks)b) Express Sinh7 in terms of hyperbolic sines of multiples of ( 8 marks)

2. a) If tanhx = Sin, show that sinhx = tan and Coshx = Sec (7 marks)

b) If tan = tanhx coty and tan = tanhx tany show that =


1. Sinh-1x is

a) log (x- ) b) log (x+ c) log (x+ d) log(x- )

2. Cosh-1x is a) log b) log c) log d) log (x-

3. tanh-1x is

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a) b) c) d)

4. cot h-1 x is

a) b) c) d)

Code: 2K6M1:1 Level : KUnit: 4.2 Type : MATCH

1. 1. Sinh-1x A) log (x+

2. Cosh-1x B)

3. Coth-1x C) log (x-

4. tanh-1x D) log (x -

E)

a) 1C, 2D, 3E, 4B b) 1D, 2A, 3E, 4B c) 1C, 2D, 3B, 4E d) 1D, 2A, 3B, 4E


1. Express tanh-1x in terms of logarithmic function.2. Express cosh-1x in terms of logarithmic function.3. Write the expression for sinh-1x in terms of logarithmic function.4. Express coth-1x in terms of logarithmic function.


1. Prove that Sinh-1x = log (x+

2. Prove that Cosh-1x = log (x+3. Find the expression for tanh-1x in terms of logarithmic function.

4. Prove that Coth-1x = log


1. Prove that tanh-1x = Sinh-1

2. Prove that Cosh-1x = Sinh-1 (

3. Prove that Coth-1 = log x, x>0


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1. a) Prove that tanh-1 = logx, x>0 ( 7 marks)

b) Derive the expression for sinh-1x in terms of logarithmic function (8 marks)


1. a) Derive the expression for tan-1hx in terms of logarithmic function (7 marks)b) If tan x/2 = tanh y/2, prove that Sinhy = tanx and y = log tan (/4 + x/2) (8 marks)

Code: 2K6M1:1 Level : UUnit: 4.3 Type : MCQ1. The real part of Sin (+i) is

a) SinCosh b) SinSinh c) -SinCosh d) -sinSinh

2. The imaginary part of Cos (x+iy) isa) Sin x Cos hy b) Sin hx Sin hyc) –Sin x Sin hy d) -Sin x Cos hy

3. The real part of cos (x+iy) is a) cos x sin hy b) sin x cos hy c) cos x coshy d) sin x sin hy

4. The imaginary part of Sin (+i) is a) -Coshsin b) -Cossinh c) Coshsin d) Cossinh

5. The real part of Sinh (+i) is a) SinhCos b) SinCosh c) SinhCosh d) SinCos

6. The imaginary part of sinh (+i) isa) Cosh.Sin h b) CoshSin c) CosSinh d) CosSin

7. The real part of Cosh (+i) is a) CoshCosh b) - CoshCosh c) CoshCos d) -CoshCos

8. The imaginary part of Cosh(+i) is a) CoshCos b) -SinhSin c) -CoshSin d) SinhSin

Code: 2K6M1:1 Level : UUnit: 4.3 Type : VSA1. What is the real part of cos(x+iy)?2. What is the imaginary part of cos(x-iy)?3. What is the real part of sin (x-iy)?4. What is the imaginary part of Sin (x+iy)?5. Separate into real and imaginary parts ex+iy.


1. Separate tan-1 (x+iy) into real and imaginary parts.2. Separate tanh (+i) into real and imaginary parts.3. Separate Cosec. (x+iy) into real and imaginary parts.

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4. Separate Sech (+i) into real and imaginary parts.


1. If cos + isin = Cos (+i) prove that Sin2 = sin.2. If sin (+i) = cos + i sin , prove that cos2 = sin .3. Find real and imaginary parts of tanh (1+i)4. Find real and imaginary parts of tan-1 (2+i)5. If cos (x+iV) = Cos + iSin, prove that cos2x + cosh2y = 26. If cos (u + iv) = x+iy where x,y,z,u,v are real, prove that (1+x)2 + y2 = (cosu +

coshv)2.

7. If sin ( + i) = tan (x+iy), show that =

8. If tan (+i) = Cos + isin, prove that = n/2 + /4 and = 1/2 log tan (/4 + /2)

9. If tan (x+iy) = u + iv, prove that =


1. a) If sin (x+iy) = (Cos + isin), show that 22 = cosh2y – cosh2x (7 marks)b) Separate sinh (+i) into real and imaginary parts (8 marks)

2. a) If sin ( + i) = tan + isec, prove that cos2 cosh2 = 3 ( 7 marks)b) Separate tan (+i) into real and imaginary parts (8 marks)


1. a) If cos (x+iy) = r(cos + isin), prove that y=1/2 log (7 marks)

b) If tan (A+iB) = x=iy, prove that x2 + y2 + 2xcot 2A=1 (8 marks)


1. log is a) log2 + i (/6 + 2n) b) log2 - i/6c) log2 – i(/6 + 2n) d) log2 + i/6

2. log (iy) is a) logy + i/2 b) logy + i (2n+1/2)c) ½ logy + i (2n+1) d) 1/2 logy + i/4

3. log (1+i) is a) log2+i/4 b) 1/2 log2 c) 1/2 log2 + i/2 d) 1/2 log2+i

4. Log (1-i) is a) log2-i/4 b) log -i/4 c) log +i/2 d) log -i/2

Code: 2K6M1:1 Level : K

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Unit: 4.4 Type : VSA

1. Define logarithm of a complex number x+iy.


1. Show that log (1+itan) = logsec +i.2. Write the value of log (iy).3. Write the value of log (1+i)4. Write the value of logi.5. Write the value of log (4+3i).


1. If log Sin (+i) = A + iB, prove that 2e2A = Cosh2 - Cos22. Express logcos (x+iy) in the form A+iB.3. If log Sin (-i) = A+iB, Prove that Cos (-B) = e2 Cos (+B)

4. If +i = bx+iy prove that =

5. Prove that log


1. a) Deduce the expansion of tan-1 x in powers of x from the expansion of log (a+ib)(7 marks)

b) If tan (log (x+iy) = a+ib where a2+b2 1 prove that tan (log (x2+y2)) =

(8 marks)2. a) Reduce (+i)x+iy to the form A+iB (7 marks)

b) If = A+iB, prove that A2+B2 = e-B and tan . (8 marks)


1. Imaginary part of Log (x+iy) is a) tan-1 (y/x)+2n b) 1/2 log(x2+y2) c) i(tan-1(y/x)+2n) d) 1/2 log(x2-y2)

2. Log x is a) log x b) tan-1 x c) log x + i2n d) itan-1x


1. Log (-x1) is a) logx1 + i b) logx1 + i(2n+1) c) –logx1 d) –logx1+i

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2. Log (iy) is a) logy + i (2n+1) b) logy+i (n+1/2) c) logy + i(2n+1/2) d) logy + i (n+1/2) /2

3. Log (1-i) isa) log +i/2 b) 1/2 log2+i(2n-/4) c) log +i(2n + /4) d) log +i(2n+/2)

4. Real part of Log (1-i) is a) log2 b) log (-2) c) log d) log (1/2)

5. Imaginary part of Log (1-i) is a) 2n+/2 b) 2n+/4 c) 2n-/2 d) 2n - /4

Code: 2K6M1:1 Level : UUnit: 4.5 Type : MATCH

1. 1. Log 2 A) log2 + i (2n+1)2. Log 2i B) 1/2 log5 + i tan-11/2 + i2n3. Log (-2) C) log 2 + i2n4. Log(2+i) D) log2 – i (1-2n)

E) log2 + i (2n+ ½) a) 1E, 2A, 3B, 4C b) 1E, 2D, 3C, 4B c) 1C, 2D, 3B, 4A d) 1C, 2E, 3A, 4B

Code: 2K6M1:1 Level : KUnit: 4.5 Type : VSA1. Write the general value of Log (x+iy).


1. Write the imaginary part of Logi.2. Write the value of Log (iy)3. Write the value of Log (1+i).4. Write the value of Log (-1+i)5. Write the value of Log (i1/2).6. Write the value of Log (7. Write the value of Log (-8. Write the value of Log (-9. Write the value of Log ( 1-i)10. Write the value of Log (4+3i)11. Write the value of Log ( +i)


1. Prove that ii = e-(4n+1)/2

2. Prove that Log

3. Prove that Log (1+cos2 + isin2) = log(2cos) + i(+2n), n is an integer.

32

4. Prove that ai = e-2n (cos (loga) + isin (loga))

5. Show that logi i = where m and n are integers.


1. a) Prove that ii = e-(4n+1)/2 where n is an integer. ( 7 marks)b) If ix+iy = A+iB, show that A2+B2 = e-(4n+1) y (8 marks)


1. tan can be written as: (a) cot 2 - cot (b) cot - cot2 (c) sec - sec2 (d) sec2 - sec

2. tan /2 sec can be written as:(a) tan/2 - tan (b) tan/2-tan/4 (c) tan-tan/2 (d) tan/4 - tan/2

3. tan-1 can be written as:

(a) tan-1 (r+1) – tan-1 (r) (b) tan-1 (r+1)2 – tan-1 r2

(c) tan-1r –tan-1(r-1) (d) tan-1r2 – tan-1 (r-1)2


1. Express Tr = as the difference of two functions.

2. Express Tr = tan-1 as the difference of two functions.



5. Express Tr = sin as the difference of two functions.


1. Find the sum of the series cosec + cosec 2 + cosec22 + ……… + cosec2n-1. 2. Sum the series upto n terms tan/2 sec + tan/4 sec/2 + tan/8 sec/4 +……..

3. Sum the series tan-11 + tan-1 to n

terms.4. Find the sum of the series, tan + 2 tan 2 + 22 tan (22) + …. + 2n-1 tan(2n-1).

5. Sum the series, to n terms.

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6. Sum the series, sin sin 3 + sin/2 sin 3/2 + sin /22 sin 3/22 + ….. to n terms.7. Sum the series, sec sec 3 + sec 2 sec 4 + ……. to n terms.8. Sum the series, sec sec (+) + sec (+) sec(+2) + sec ( + 2) sec (+3) + …

to n terms.9. Sum the series, cos3 - 1/3 cos33 + 1/32 cos3 32 - 1/33 cos333 + ….. to n terms.


1. (a) Find the sum of the series, (7 Marks)

.

(b) Sum the series, to n terms (8 Marks)

2. (a) Sum the series, (7 Marks)

to n terms .

(b) Sum the series, (8 Marks)


1. sin + sin (+) + sin ( +2 ) + ……….. + sin ( + is

(a) (b)

(c) (d)


1. Sin + sin2 + sin3 + …….. to n terms is:

(a) (b)

34

(c) (d)

2. Sin - sin (+) + sin (+2) -……….. to n terms is :

(a) (b)

(c) (d)


1. What is the sum of sin + sin (+) + …… + sin (+(n-1))?2. What is the sum of the series sin hx + sinh (x + y) +sinh (x+2y) + …… to n terms?


1. Write the sum to n terms of the series sin + sin 2 + sin 3 + ……….2. Find the sum of n terms of the series sin - sin ( + ) + sin ( + 2 ).


1. Sum the series, sinhx + sinh (x+y) + sinh (x+2y) + …… to n terms.2. Sum the series, sin sin 3+ sin 3 sin 5 + sin 5 sin 7 + …. to n terms.


1. (a) Sum the series upto n terms, cos2 + cos22 + cos23+….. (7 Marks)(b) Find the sum of the series, sin+sin (+) + ….. + sin (+ ). (8 Marks)

2. (a) Sum the series, cos x cos2x + cos2x cos3x + ….. to n terms (7 Marks)(b) Find the sum of the series, sin3 + sin3 2 + sin33 +…. to n terms (8 Marks)


1. Cos + cos2 + cos3 + …………. to n terms is

35

(a) (b)

(c) (d)

2. cos - cos ( + ) + cos (+2)-…….. to n terms is :

(a) (b)

(c) (d)

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1. What is the sum of cos + cos (+) +……… +cos (+ ?2. What is the sum of the series, cos hx + cosh (x+y) + cosh (x+2y) + …. to n terms?


1. Write down the sum of n terms of the series cos + cos2 + cos3 + ………2. Prove that the sum of n terms is zero for the series cos + cos (+) + cos(+2) +…

when

3. Sum to n terms the series cos - cos (+) + cos (+2)--………..


1. Find the sum of the series, cos2x + cos2(x+y) + Cos2(x+2y) + ….. to n terms.2. Sum the series, cos3 + cos32 + cos33 + ……. to n terms.3. Sum the series, cos cos3+ cos3 cos5 + cos5 cos7 + ….. to n terms.4. Find the sum of the series, coshx + cosh (x+y) + cosh (x+2y) + …… to n terms.


1. (a) Show that tan n = (7 Marks)

(b) Sum the series cosx sin2x + cos2x sin3x +….. to n terms. (8 Marks)2. (a) Sum the series sin2 + sin2 2 + sin23 +…… to n terms. (7 Marks)

(b) Sum the series, coshx + cosh (x+y) + cosh (x+2y) + … to n terms. (8 Marks)


1. Gregory’s series is:

(a) = tan + (b) = tan +

(c) = tan - (d) = tan -

2. Euler’s series is

(a) (b)

(c) (d)

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3. Gregory’s series for tan-1x with x 1 is given by

(a) (b)

(c) (d)


1. Write down Gregory’s series.2. Write down Euler’s series.3. Write down Gregory’s series for tan-1x.

4. Prove that using Gregory’s series.


1. Derive the Gregory’s series, tan-1x = x – x3/3 + x5/5 -………..


1. Sum to the series,

2. Sum to infinity the series,

3. If , find the sum to of the series,

4. If find the sum to of the series,

5. Sum to infinity the series, Cos + ½ Cos (+) +

6. Sum to infinity the series, sin

7. Sum to the series, C cos - C3/3 cos 3 +C5/5 cos 58. Sum to the series, sin + ½ sin 2 +1/22 sin 3+ ........

9. Sum the series, cos + .

10. Sum to the series,

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1. (a) Find the sum to of the series cos sin +

(b) Sum to n terms of the series, sin + Csin (+) + C2sin (+2) +….. and hence find the sum of the series to when C < 1. (10 Marks)


1. (a) Sum to the series, C sin + C2/2! sin 2 C3/3! sin 3 +….. (7 Marks)

(b) Show that the sum of the series, to

is cos (/4 - /2) (2 sin /2)-1/2.

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Diff Cal

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