Work Sheet Iit Based

B3.2-R3: BASIC MATHEMATICS NOTE: 1. 2.

Answer question 1 and any FOUR questions from 2 to 7. Parts of the same question should be answered together and in the same sequence.

Time: 3 Hours 1. a) b)

Total Marks: 100

Find the square root of (3+4i). Find the real eigenvalue and associated eigenvector of the following 0 ½ 0

A=

0 0 ⅓

6 0 0

x

∫ sin

2

ydy

0

c)

Find lim

d)

Find the eccentricity of the hyperbola

e)

5 x 2 − 4 y 2 − 10 x − 16 y − 31 = 0 1 1 1 . then find lim S n . Let Sn = + + ........ + x →∞ 1.2 2.3 n(n + 1) Find the area of the region bounded by the curve y = ( x + 2)( x + 1)( x − 1), and the lines

f)

x →0

x3

x = -2 and x = 1. g)

x4 x + 6 Find the derivative of (2 x + 1)(2 x + 3) (7x4)

2. a)

b)

Find the value of k, for which the systems of equations x – ky +z = 0 kx + 3y – kz = 0 3x + y – z = 0, has (i) trivial solution, (ii) non-trivial solutions. If 1 x+1 x+2 f (x) =

c)

2(x + 1)

2(x + 1)

(x + 1) (x + 2)

3x (x + 1)

(x – 1) x (x + 1)

x (x + 1) (x + 2)

then find the value of f (100). Determine the rank of the following matrix, for all values of x . A=

B3.2-R3

5- x 2 1

2 1- x 0

1 0 1-x

(6+6+6)

Page 1 of 3

JULY, 2005

3. a)

0 ≤ x ≤1 x>0

The function f (x) = x3 + 2ax + b, bx2 + 3a ,

is differentiable for all x > 0. Find the values of a and b. Draw the graph of the function f (x ) . π

sin x

∫ 1 − sin x dx

b)

Evaluate

c)

Find all functions f (x ) such that

−π

df ( x) = −( x − 1) 2 dx

and draw the graph of f ( x ) in the x-y plane. Find in particular the function whose graph passes through the point ( x 0 , y 0 ) = (1, 1) . (5+5+8) 4. a) b) c)

Find the equations of the common tangents to the circle (x – 2)2 + y2 = 4 and the parabola y2 = - 4x. The lengths of two non-zero vectors a and b are equal. If the vectors a – 3b and 7a +5b are the right angles, then find the angle between the vectors a and b. AOB is the positive quadrant of the ellipse 9x2 + y2 = 36. Find the area between the arc AB and the chord AB. (6+6+6)

5.

dny and x = 0 dx n

a)

If y = x e

b)

Find the extreme values of the function

c)

f (x) = x x Assume that x1 is a point of maxima and x2 is a point of minima of the function f (x) = 2x3 – 9ax2 + 12a2x + 1

3 2x

, then find

1

2

Find the value of a > 0 for which x1 = x 2 (6+6+6) 6. a)

1− x2 

 3x − x 3 

−1  with respect to tan −1  . Find the derivative of cos  2  2  1+ x   1 − 3x 

∫

(1 + sin x )e x

dx

b)

Evaluate

e)

State the mean-value theorem. For f ( x ) = x - x in [a,b], determine all numbers ξ in the specified interval, such that

1 + cos x

3

f ' (ξ ) =

f (b ) − f (a ) b−a

Take a=0 and b=2. (6+6+6)

B3.2-R3

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JULY, 2005

7. a)

If Zk= cos

π π + i sin k , k = 1, 2, ........ then find the value of k 3 3

Z1 Z2 …….. ∞. b)

From the top of a tower 15 feet high, the angles of depression at two points on opposite sides of it are 30º and 60º. Find the distance between the points.

c)

If a , b , c are three vectors such that a + b + c =0, |a| = 3, |b|= 4, |c| = 5, then find the value of a . b + b . c + c . a

d)

Let P (x1, y1) and Q (x2, y2) be two distinct points on the parabola y = Ax2 + Bx + C, A≠0.Using Lagrange mean value theorem, find a point R on the arc PQ, where the tangent to the curve is parallel to the chord PQ. (6+4+4+4)

B3.2-R3

Page 3 of 3

JULY, 2005

B3.2-R3: BASIC MATHEMATICS NOTE:

1. 2.

Answer question 1 and any FOUR questions from 2 to 7. Parts of the same question should be answered together and in the same sequence.

Time: 3 Hours

Total Marks: 100

1.

r

r

r r

r

r

a)

Show that the points whose position vectors are i − 2 j + 3k ,2i + 3 j − 4k

b)

Show that the roots of the equation x2+x+1 =0 are of the form w and w2. Also, find the sum of the roots.

Ans

x2 + x +1 = 0

r r 7 j + 1 0k are collinear.

2

and

2

1 1 1 2 ⇒ ( x ) + 2.x. +   + 1 −   = 0 2 2 2 2

1 1  ⇒  x +  = −1 2 4  2

3 1 3  ⇒  x +  = − = i2. 4 2 4  1 3 3  ⇒  x +  = i2. = i 2 4 2  1 3 ⇒x=− + i 2 2 1 3 Let ω = − + i 2 2  1 3  ∴ ω =  − + i  2 2  

2

2

2  1  1   3   3  =  −  + 2. − . i + i  2  2   2   2 

2

1 3 3 − i− 4 2 4 1 3 =− − i 2 2 ∴ ω andω 2 are the roots of the given equation x 2 + x + 1 = 0 1 3  1 3 ω +ω2 = − + i + −  − i = −1 2 2  2 2 =

c)

Let A and B be 2x2 matrices such that AB=0. Justify.

B3.2-R3

Page 1 of 11

Is it always true that A=0 and B=0?

JANUARY, 2005

d)

Ans

log(1 + x 3 ) x →0 sin 3 x log(1 + x 3 ) log(1 + x 3 ) log(1 + x 3 ) lt lt 3 log(1 + x ) 1 1 x →0 x 3 →0 x3 x3 x3 = = = = 3 = =1 lt lt 3 3 3 3 → x →0 x 0 1 1 Sin x Sin x Sin x Sin x   lt lt   3 3 x →0 x x  x →0 x 

Evaluate lt

e)

Show that the line y=x+2 is a tangent to the parabola y2=8x. Also, find the point of contact.

f)

Test the Convergence of the series

g) h)

find the area included between the circle x2+y2=1 and the lines x=0 and x=1. Find points of maxima/minima of the function

∞

3n

∑ 2(n + 1) . n =1

f(x)=x+sin 2x; 0 ≤ x ≤ 2 Π i)

2. a)

If a circle passes through the point (a,b) and touches both the co-ordinate axit, then show that a=b. (3+3+3+3+3+3+3+4+3) State DeMoirre’s Theorem. Express the complex number Z=

(Cosθ + iSinθ ) 3 ( Sinθ + iCosθ ) (Cos 2θ − iSin 2θ )

In the form x-iy, where x and y are real numbers. Also find |z| and arg. z.

Ans

(cosθ + i sin θ )3 (sin θ + i cos θ ) (cos 2θ − i sin 2θ ) (cosθ + i sin θ )3 (sin θ + i cos θ ) = {cos (− 2θ ) − i (− sin (− 2θ ))} 3 ( cosθ + i sin θ ) (sin θ + i cosθ ) = {cos(− 2θ ) + i sin (− 2θ )} 3 ( cosθ + i sin θ ) (− i 2 sin θ + i cosθ ) = −2 (cosθ + i sin θ ) 5 = (cosθ + i sin θ ) i (cosθ − i sin θ ) 4 = i (cosθ + i sin θ ) (cosθ + i sin θ )(cosθ − i sin θ ) 4 = i (cosθ + i sin θ ) (cos 2 θ − i 2 sin 2 θ ) 4 = i (cosθ + i sin θ ) (cos 2 θ + sin 2 θ ) 4 = i (cosθ + i sin θ ) = i (cos (4θ ) + i sin (4θ )) = i cos (4θ ) + i 2 sin (4θ ) = − sin (4θ ) + i cos (4θ ) ∴ x = − sin (4θ ), y = cos (4θ ) z=

B3.2-R3

Page 2 of 11

 Sinθ = − Sin (− θ )  Cosθ = Cos (− θ )   

[i

2

]

= −1 ⇒ −i 2 = 1

JANUARY, 2005

(− sin (4θ ))2 + (cos (4θ ))2 = sin 2 (4θ ) + cos 2 (4θ ) = 1 = 1  cos (4θ )   π   = π − tan −1 (− cot (4θ )) = π − tan −1  tan + 4θ   Arg ( z ) = π − tan −1    2  − sin (4θ )  z = x2 + y2 =

π π π  = π −  + 4θ  = π − − 4θ = + 4θ 2 2 2  b) c)

r

r

r r

r

r

r

r

r

Find the value of λ so that the vectors i − j + k ,2i + j − k and λi − j + λk are coplaner. Which of the following matrices is non-sungular?

1 2  , B = 3 6

A = 

 2 0    2 0

Also, find its inverse.

Ans

1 2 A=  3 6 1 2 A= = 6−6 = 0 3 6 ∴A is a Singular Matrix 2 0 B=  0 2 2 0 B = = 4−0 = 4 0 2 ∴B is a Non-Singular Matrix C11 = (−1)1+1 ⋅ (2) = 2 C12 = (−1)1+ 2 ⋅ (0) = 0 C 21 = (−1) 2+1 ⋅ (0) = 0 C 22 = (−1) 2+ 2 ⋅ (2) = 2 C 21  2 0 = C 22  0 2 2 0  1    Adj ( B) 0 2  2 0  = = = 1 B 4 0   2

C Adj ( B ) =  11 C12 B −1

d)

Solve the following system of equations by Cramer’s rule or by Gauss-elimination method: x1 - 2x2 + x3 = 0, -x2+x3 = -2, 2x1 − 3x3 = 10.

B3.2-R3

Page 3 of 11

JANUARY, 2005

Ans

x1 − 2 x 2 + x3 = 0 LLL (1) − x 2 + x3 = −2 LLL (2) 2 x1 − 3x3 = 10 LLL (3) By Cramer’s Rule 1 −2 1 D = 0 − 1 1 = 1(3 − 0) + 2(0 − 2) + 1(0 + 2) = 3 − 4 + 2 = 1 2 0 −3 0 −2 1 D1 = − 2 − 1 1 = 0(3 − 0) + 2(6 − 10) + 1(0 + 10) = 0 − 8 + 10 = 2 10 0 − 3 1 0 1 D3 = 0 − 2 1 = 1(6 − 10) − 0(0 − 2) + 1(0 + 4) = −4 − 0 + 4 = 0 2 10 − 3 1 −2 0 D3 = 0 − 1 − 2 = 1(−10 − 0) + 2(0 + 4) + 0(0 + 2) = −10 + 8 = −2 2 0 10 D1 2 = =2 D 1 D 0 x2 = 2 = = 0 D 1 D −2 x3 = 3 = = −2 D 1

∴ x1 =

By Gauss Elimination Method x1 − 2 x 2 + x3 = 0 LLL (1)

0 1 − 2 1 0 − 1 1 − 2   2 0 − 3 10 

− x 2 + x3 = −2 LLL (2) 2 x1 − 3 x3 = 10 LLL (3) Step-1: Elimination of x1 (1) ⇒ x1 − 2 x 2 + x3 = 0 LLL (1) ⇒

− x 2 + x3 = −2 LLL (2)

(3) = (3) − (1) × 2 ⇒

4 x 2 − 5 x3 = 10 LLL (3)

(2)

B3.2-R3

Page 4 of 11

0 1 − 2 1 0 − 1 1 − 2    0 4 − 5 10 

JANUARY, 2005

Step-2: Elimination of x 2 (1) ⇒ x1 − 2 x 2 + x3 = 0 LLL (1) ⇒

(2)

0 1 − 2 1 0 − 1 1 − 2    0 0 − 1 2 

− x 2 + x3 = −2 LLL (2)

(3) = (3) + (2) × 4 ⇒

− x3 = 2 LLL (3)

∴ x3 = −2 x 2 = x3 + 2 = −2 + 2 = 0 x1 = 2 x 2 − x3 = (2 × 0) − (−2) = 0 + 2 = 2 e)

2 + 3i in the form x+iy where x 3 − 2i

Find the multiplicative inverse of the complex number and y are real numbers.

Ans

2 + 3i (2 + 3i )(3 + 2i ) 6 + 4i + 9i + 6i 2 6 − 6 + 4i + 9i 13i = = = = =i 2 3 − 2i (3 − 2i )(3 + 2i ) 9+4 13 3 2 − (2i ) ∴ x = 0, y = 1 (x + iy )−1 = 2 x 2 − i. 2 y 2 = 0 − i 1 = 0 − 1 i = −i 0 +1 1+ 0 x +y x +y z=

(4+3+3+5+3) 3. a)

Without expanding, show that the determinant

5 7 9 Ans

2 3 4

3 4 5

vanishes.

5 2 3 D= 7 3 4 9 4 5 5 5 3 = 7 7 4 =0 9 9 5

b) Ans

If f(x) =

[C 2 = C 2 + 1.C3 ]

25 − x 2 , prove that lt

x →3

f ( x) − f (3) − 3 . = x−3 4

f ( x) = 25 − x 2 f (3) = 25 − 3 2 = 25 − 9 = 16 = 4 lt

x →3

f ( x) − f (3) 25 − x 2 − 4 = lt = lt x →3 x →3 x−3 x−3

= lt

x →3

B3.2-R3

25 − x − 4

(x − 3)(

2

2

25 − x 2 + 4

)

= lt

x →3

( 25 − x

−4

(x − 3)(

9− x

(x − 3)(

2

2

25 − x 2 + 4

Page 5 of 11

)( 25 − x

+4

2

25 − x + 4 2

)

= lt

x →3

)

(

)

− x2 − 9

(x − 3)(

)

25 − x 2 + 4

)

JANUARY, 2005

(x − 3)(x + 3) = − lt ( x + 3) = − x →3 (x − 3)( 25 − x 2 + 4) x→3 ( 25 − x 2 + 4)

= − lt =−

3+3 25 − 3 2 + 4

=−

6 16 + 4

6 6 3 =− =− 4+4 8 4

c) d)

find the distance between the parallel lines 2x+4y = 7 and x+2y = 3. Using the concept of ‘rank’ of a matrix, list for consistency the following system of equation 2x + 8y + 5z = 5, x + 2y – z = 2, x + y +z = -2.

e)

If y = A cos mx + B Sin mx, then show that

Ans

y = A Cos mx + B Sin mx dy d = ( A Cos mx + B Sin mx) dx dx d d = A Cos mx + B Sin mx dx dx d d = A Cos mx + B Sin mx dx dx = − Am Sin mx + Bm Cos mx = Bm Cos mx − Am Sin mx

d2y + m2 y = 0 2 dx

d 2 y d  dy  d =  = ( Bm Cos mx − Am Sin mx) dx 2 dx  dx  dx d d = Bm Cos mx − Am Sin mx dx dx d d = Bm Cos mx − Am Sin mx dx dx = − Bm . m Sin mx − Am . m Cos mx = − Bm 2 Sin mx − Am 2 Cos mx m 2 y = m 2 ( A Cos mx + B Sin mx ) = Am 2 Cos mx + Bm 2 Sin mx d2y + my 2 = − Bm 2 Sin mx − Am 2 Cos mx + Am 2 Cos mx + Bm 2 Sin mx = 0 dx 2 (Proved) f)

Ans

Find the value of k so that the function f(x) defined below is continuous at x =

Π . 2

π  kCosx , when x ≠ π − 2 x 2 f(x) =  π k , when x = 2  The function f (x) is defined as: k cos x f ( x) = , when x ≠ π 2 π − 2x =k , when x = π 2

B3.2-R3

Page 6 of 11

JANUARY, 2005

At the point x = π 2 π  f =k 2 k cos x 1 k cos x lt f ( x) = lt = lt x →π 2 x →π 2 π − 2 x 2 x →π 2 π 2 − x Put x − π 2 = θ ∴θ → 0 as x → π 2 k cos(π 2 + θ ) 1 k cos x 1 k (− sin θ ) k k sin θ k 1 ∴ lt = lt = lt = lt = .1 = −θ 2 θ →0 − θ 2 θ →0 θ 2 2 2 x →π 2 π 2 − x 2 θ →0 As the function in continuous at x = π 2 ∴ f (π 2 ) = lt f ( x) x →π 2

k 2 ⇒ 2k = k ⇒ 2k − k = 0 ⇒k =0 ⇒k=

(2+3+3+4+3+3) 4. a)

Find the characteristics roots of the matrix

 2 3 4    0 −1 5   0 0 3  

b)

Find local maximum/minimum value (if any) for the function f(x) = x3 – 12x2 + 36x + 17, 1 ≤ x ≤ 10.

c)

Evaluate

Ans

I =∫

dx 4 + 1)

∫ x( x

x dx dx =∫ 2 4 4 x( x + 1) x ( x + 1)

Put x 2 dx = u ⇒ 2 x dx = du 1 ⇒ x dx = du 2 1 1 (u 2 + 1) − u 2 du ∴I = ∫ = du 2 u (u 2 + 1) 2 ∫ u (u 2 + 1) =

1 (u 2 + 1) 1 u2 du = J say du − 2 ∫ u (u 2 + 1) 2 ∫ u (u 2 + 1)

Put u 2 + 1 = v

B3.2-R3

Page 7 of 11

JANUARY, 2005

⇒ 2u du = dv 1 ⇒ u du = dv 2 1 u du 1 1 dv 1 dv 1 1 = × ∫ = ∫ = log v = log (u 2 + 1) 2 ∫ 2 u +1 2 2 v 4 v 4 4 1 1 u du 1 1 1 1 J = log u − ∫ 2 = log u − log(u 2 + 1) = log x 2 − log x 2 2 2 u +1 2 4 2 4 1 1 1 1  = log x 2 − log( x 4 + 1) = log x 2 − log( x 4 + 1) + C 2 4 2 2 

{(

)

2

}

+1

d)

Find the asymptotes of the Curse x2y – xy2 + xy + y2 + x – y =0.

Ans

x 2 y − xy 2 + xy + y 2 + x − y = 0 The given equation is of 3rd degree where the term y 3 and x 3 are absent. So, it is possible to exist an asymptotes parallel to y-axis and x-axis. 2 Equating the co-efficient of x with 0 , we get: y = 0 is a required asymptotes Again the above equation can be written as:

x 2 y − ( x − 1) y 2 + xy + x − y = 0 2 Equating the co-efficient of y with 0 , we get: x = 1 is another required asymptotes. Now, the given equation can be written as:

xy ( x − y ) + xy + y 2 + x − y = 0 The asymptotes parallel to x − y = 0 is: 2x 2 xy + y 2 + x − y x2 + x2 + x − x = x − y + lt = x − y + lt = x− y+2 x − y + lt x →∞ x →∞ x →∞ x 2 xy x2 y=x

∴ The required asymptotes is x − y + 2 = 0 ⇒ y = x+2 Thus, y = 0, x = 1, y = x + 2 are the required asymptotes. e)

Verify the thpothesis and the conclusion of the Rolle’s theorem for the function f(x) = (x – 2)

x on [0,2]. (3+4+3+4+4)

5. a)

b)

Write the equation of the ellipse 3x2 + 4y2 = 12 in standard form and sketch it. Clearly indicate its; center and vertices.

4 1 + ) x →2 x − 4 2− x 1  1  4− x −2  4 − (x + 2)   4  4 lt  2 + −   = lt  2  = lt  2  = lt  2 x →2 x − 4 2 − x  x → 2 x − 4 x − 2  x → 2 x − 4  x → 2 x − 4  

Evaluate:

B3.2-R3

lt (

2

Page 8 of 11

JANUARY, 2005

 − (x − 2)  1 1  2− x   1   = − lt = lt  2 = −  = lt  =− x→2 x − 4 x→2 x + 2 4   x→2 (x + 2)( x − 2 )  2+ 2

c)

Show that the conic 9x2 – 24xy + 16y2 – 18x – 101y + 19 = 0 represents a parabola. 1

d)

Evaluate

∫ 0

1

x x2 + 3

dx

x

∫

dx = I Say 2 x + 3 0 Let x 2 + 3 = u ⇒ 2 x dx = 2u du ⇒ x dx = du u du x ∫ x 2 + 3 dx = ∫ u = ∫ du 1

I =∫ 0

e)

2

x x +3 2

x

0

z

1 2

3

2

dx = ∫ du = z ] = 2 − 3 3

3

Applying Leibnitz’s test to show that the series 1 -

1 2

1

+

3

1

−

4

+ ...... is convergent (4+4+4+3+3)

6. a) Ans

3+i in polar form. 1− i 3+i 3 + i (1 + i ) 3 + 3i + i + i 2 3 − 1 + 3i + i = z= = = 2 2 (1 − i )(1 + i ) 1− i 1 − (− 1) 1 −i

Express the complex number

(

=

(

)

)

3 −1+ i 3 +1 3 −1  3 +1  = + i  2 2  2  3 −1 3 +1 and y = 2 2

∴x =

{( 3 ) + 1 } =

2 2 2  3 −1  3 +1  +  =  r = x + y =       2   2   2

=

2

2

4

 

8 = 2 4

 3 +1     1+   3 +1 −1  y  −1  −1 −1  2    = tan  = tan θ = tan   = tan  3 −1   3 − 1   x    1−   2 

B3.2-R3

2(3 + 1) 4

2

Page 9 of 11

1   3 1   3

JANUARY, 2005

π π    tan + tan  4 6  = tan −1 tan π + π  = tan −1 tan 10π = tan −1       π π   4 6     4 − 1 tan . tan   4 6  5π 5π   + i sin ∴ z = 2  cos  12 12   b) c) Ans

d) Ans

 10π 5π =  = 4 2 

Find the equation of the circle whose center is (1, 2) and which touches the line 3x+4y=1

 1 − 1 1 2   and B =   , then is it true that (AB)’ = -A’ B’?  2 − 1  4 −1 1 − 1 1 2  A= and B =    2 − 1 4 − 1 1 − 1 1 2  1 − 4 2 + 1  − 3 3 AB =   = =  2 − 1 4 − 1 2 − 4 4 + 1 − 2 5 − 3 − 2 ( AB )/ =  5  3 2 1 A/ =   − 1 − 1  − 1 − 2 − A/ =  1  1 1 4  B/ =   2 − 1  − 1 − 2  1 4   − 1 − 4 − 4 + 2   − 5 − 2  = = − A/ B / =  1  2 − 1  1 + 2 4 − 1   3 3  1 / ∴ ( AB ) ≠ − A / B / If A = 

Use DeMoivre’s Theorem to show that

Cos3θ = 4Cos 3θ − 3Cosθ cos 3θ = 4 cos 3 θ − 3 cosθ By Demoivre’s Theorem: (cos 3θ + i sin 3θ ) = (cosθ + i sin θ )3 ⇒ cos 3θ + i sin 3θ = cos 3 θ + 3 cos 2 i sin θ + 3i 2 sin 2 θ cosθ + i 3 sin 3 θ ⇒ cos 3θ + i sin 3θ = cos 3 θ + 3 cos 2 i sin θ − 3 sin 2 θ cosθ − i sin 3 θ ⇒ cos 3θ + i sin 3θ = cos 3 θ − 3 sin 2 θ cosθ + i 3 cos 2 sin θ − sin 3 θ ∴ cos 3θ = cos 3 θ − 3 sin 2 θ cosθ [If , x + iy = a + ib, then x = a] 3 2 ⇒ cos 3θ = cos θ − 3 cosθ 1 − cos θ

(

) (

(

)

)

⇒ cos 3θ = cos θ − 3 cosθ + 3 cos θ ⇒ cos 3θ = 4 cos 3 θ − 3 cosθ (Pr oved ) 3

3

(4+4+5+5)

B3.2-R3 2005

Page 10 of 11

JANUARY,

7. a) b) c) Ans d)

Sketch the graph of the function y = Sin3x in [0,Π]. Find the area enclosed between the parabola y = 4x2 , the x axis and the lines x=1 and x =2.

x− | x | x →0 + 2 x− x 0−0 0 lt = = =0 x →0 + 2 2 2

Find

lt

 x = x, when x ≥ 0     x = − x, when x ≤ 0

Assuming the validity of the Macularin’s series expansion, find the first four terms of the function f(x) = ex Cosx. (5+5+4+4)

B3.2-R3 2005

Page 11 of 11

JANUARY,

B3.2-R3: BASIC MATHEMATICS Question Papers July, 2004 NOTE:

1. 2.

Answer question 1 and any FOUR questions from 2 to 7. Parts of the same question should be answered together and in the same sequence. Time: 3 Hours Total Marks: 100 1. a.

b.

c.

If 1, ω, ω2 are the cube roots of unity, then find the value of (1+ω)(1+ω2)(1+ω4)(1+ω5)  2 1 2 Find the inverse of the matrix 1 2 2 using the Guass-Jordan 2 2 2 method

Using the binomial theorem, find the coefficient of x4 in the 10

x 3  expansion of  − 2  3 x 

d.

lim  x + 5  x +3 Find   x → ∞  x +1 

e.

Derive a reduction formula for I m,n = ∫ x m (log x) n dx m, n are integers relating Im,n and Im,n-1. π

f.

Evaluate the definite integral

xdx

∫ 1 + cos 0

g.

Test for convergence, the series

∑[ ∞

n =1

h.

2

x

n4 + 1 − n4 −1

]

Find the equation of the tangent to the parabola y2=4(x+1) which is parallel to the line x+y+1=0.

i. 2. a.

b. c. d.

3. a.

b. c. d. e.

4. a.

b.

Find the projection of the vector 2i+3j-k along the vector 4i+j+2k. (3+4+3+3+2+4+3+4+2) Find all the characteristic roots (elgen values) and the corresponding  1 2 2 characteristic vector (elgen vectors) of the matrix  0 2 1  − 1 2 2 Show that the length of the segment of the tangent line to the curve x=acos3t, y=asin3t, cut off by the coordinate axis is constant. Find the area of the region bounded by y=|x+5|, x=-1, x=-6 and the xaxis. Obtain the first four terms of the Taylor series of f ( x) = x about x=2. Estimate the error if this series is used in the interval[2,3]. (5+5+4+4) Find the complex numbers, which satisfy both the equations z−6 5 z−2 = and =1 z − 4i 9 z−4 x y z If xyz = 1 and y z x = 1, then find the value of x3+y3+z3. z x y Find the sides of a rectangle of greatest area that can be inscribed in the ellipse 4x2+9y2=36 Find the area of the region bounded by {( x, y ) : x 2 + y 2 ≤ 16 and x + y ≥ 4} . Find a unit vector perpendicular to both the vectors 2i − 3 j + 6k , i + j + k (4+2+5+4+3) If z=x+iy where x and y are variables, then find the locus represented z −1 =1 by the equation z +1 Find the values of the parameters ka and a such that the system of x1 − x2 + 2 x3 = 3 equations 2 x1 − 3 x2 + x3 = −2 2 x1 + x2 + kx3 = a has (i) unique solution, (ii) infinite number of solutions, (iii) no solution. 2 ∫ [x ]dx where [x ] denotes the greatest integer function 2

c.

Evaluate the integral

2

0

2

at x . d.

Test for convergence, the series

n2 −1 n x for all values of x. 2 +2

∑n

e.

5. a.

A stone is dropped in quiet water. The water moves in circles. The radii of the circles are increasing at the rate of 0.2 cm/sec. Find the rate at which the area of a circle is increasing when radius is 5 cm. (2+5+4+5+2) If a i > 0, i = 1,2,...,9and they form a geometric progression, then find the value of the determinant log a1 log a 4 log a 7

log a 2 log a5 log a8

log a3 log a 6 log a9

b.

Find the equations of the tangents to the ellipse 16x2+3y2=1, which are perpendicular to the line 3x=4y+1

c.

Using the DeMoivre's theorem, find the values of 1 − 3i

d.

Examine whether the vectors i+2j+3k, 3i+4j+5k, 6i+7j+8k are linearly dependent or linearly independent.

e.

Find a point on the curve y = x which is nearest to the point(2,0). (4+4+3+3+4)

(

)

1/ 4

.

6. lim e sin 3 x − 1 . x→0 x

a.

Find the limit

b.

Find the conic, which is represented by the equation 9x2-4y2+36x+8y-4=0 Hence, find its (i) centre, (ii) vertices, (ii) eccentricity

c.

Find the rank of the matrix − 2 − 1 3 − 2  − 3 4 1 5   1 2 7 1     11 − 8 11 − 13

d.

Using vectors, find the unit normal to the plane containing the points A(1,2,3), B(2,1,0), C(3,2,1).

e.

Evaluate the integral ∫ log[ 1 − x + 1 + x ]dx (2+4+4+3+5)

7. a.

Find the values of a and b such that the function f(x)=x-3, for x ≥ 2 =ax+b, for 0 x ≤ 2

=-2x-1, for x < 0 is continuous for all x. b.

Prove that the feet of perpendiculars from the foci of the ellipse x2 y2 + =1 a2 b2 upon any tangent to this ellipse lie on the auxiliary circle.

c.

The following vectors are given: a = i + j + k , b = 2i − j + k and c = 3i + 2 j + 2k . Determing a vector d such that d • a = 0, and d × b = c × b.

d.

Find the intervals in which f ( x) = sin x + sin x ,0 < x ≤ 2π , is increasing or decreasing or neither increasing nor decreasing. π /2

e.

Evaluate the integral

∫ sin(2 x) log(tan x)dx 0

(3+4+4+4+3)

B3.2-R3: BASIC MATHEMATICS NOTE:

1. 2.

Answer question 1 and any FOUR questions from 2 to 7. Parts of the same question should be answered together and in the same sequence.

Time: 3 Hours

Total Marks: 100

1. a) Ans

Using DeMoivre's theorem, find all the values of z = 1 +

(

z = 1 + 3i = 1 + 3i We have, 1 +

)

1

2

1 π π 3   3i = 2 + i  = 2 cos + i sin  3 3  2 2 

  π π   = 2cos 2kπ +  + i sin  2kπ +  3 3    

(

3i .

)

1

2

1

  π π  2  2 Hence, 1 + 3i = 2 2 cos 2kπ +  + i sin  2kπ +  3 3     1  1 1 π π  = 2 2 cos  2kπ +  + i sin  2kπ +  , where k = 0, 1, 2 2 3 2 3   b) Ans

1

1

 2 3  as the sum of a symmetric and a skew-symmetric matrix. 1 4 a b  0 x Let S =  be s 2x2 symmetric matrix and K =    be a 2x2 skewb c  − x 0 Write the matrix A= 

symmetric matrix.

A=S+K  2 3  a b   0 x  ⇒  + = 1 4   b c   − x 0  b + x  2 3  a = ⇒  c  1 4 b − x ⇒ a = 2, b + x = 3, b − x = 1, c = 4 b + x + b − x = 3 +1 = 4 ⇒ 2b = 4 ⇒b=2 b+x =3 ⇒ x = 3−b = 3− 2 =1  2 2  0 1 ∴S =  and K =     2 4  − 1 0 A=S+K B3.2-R3

Page 1 of 9

JANUARY, 2004

c)

 2 3  2 2  0 ⇒ + = 1 4   2 4   − 1 − 1 − 2  For the matrix A= − 1 1   0 1

1 0 0 1 , determine A3 and hence A-1. 0

 − 1 2 0 A = − 1 2 2  0 1 0 3 A = A2 . A  − 1 2 0  − 1 A = − 1 1 1.− 1  0 1 0  0 − 1 0 3 2 A = A . A =  0 0 − 1 1 2

2 0 1 − 2 + 0 − 2 + 2 + 0 0 + 2 + 0 − 1 0 2 1 1 = 1 − 1 + 0 − 2 + 1 + 1 0 + 1 + 0  =  0 0 1 1 0 0 − 1 + 0 0 + 1 + 0 0 + 1 + 0  − 1 1 1 2   − 1 2 0  1 + 0 + 0 − 2 + 0 + 2 0 + 0 + 0  1.− 1 1 1 = 0 + 0 + 0 0 + 0 + 1 0 + 0 + 0 1  0 1 0  1 − 1 + 0 − 2 + 1 + 1 0 + 1 + 0 

1 0 0 = 0 1 0 0 0 1  − 1 2 0 A = − 1 1 1  0 1 0  − 1 2 0 1 0 0 [A I ] = − 1 1 1 0 1 0  0 1 0 0 0 1 − 1 2 0 1 0 =  0 − 1 1 − 1 1  0 1 0 0 0 − 1 2 =  0 − 1  0 0 1 − 2 = 0 1 0 0

B3.2-R3

0 0 R 2 = R 2 − R1 1

0 1 0 0 1 − 1 1 0 1 − 1 1 1 R3 = R3 + R 2 0 − 1 0 0 − R1 − 1 1 − 1 0 − R3 1 − 1 1 1

Page 2 of 9

JANUARY, 2004

1 − 2 0 − 1 0 0 = 0 1 0 0 0 1 R 2 = R 2 − R3 0 0 1 − 1 1 1 1 0 0 − 1 0 2 R1 = R1 + 2 R 2 = 0 1 0 0 0 1  0 0 1 − 1 1 1   − 1 0 2 Thus A =  0 0 1  − 1 1 1  tan − 1 x lim Find , if it exists. x− > 0 x −1

d) e)

Find the equation of the tangent to the parabola x

2 = 4( y + 1) , which is parallel to the

line x + y + 1 = 0 . f)

Let the curve C be defined by x = a cos θ , y = a sin θ , 0 ≤ θ ≤ π 3

3

2

. Find the

coordinates of a point P, on the curve C where the tangent to the curve C is parallel to the chord joining the points A(a,0) and B (0, a ) . g)

( x + 1)e x Evaluate the integral I = ∫ dx . cos 2 ( xe x ) ( x + 1)e x dx I =∫ cos 2 ( xe x )

h) i) j)

Let xe x = u ⇒ ( xe x + e x ) dx = du ⇒ e x ( x + 1) dx = du du I =∫ = ∫ sec 2 u du = tan u = tan( xe x ) + c 2 cos u 2 Find the area of the region bounded by the curves y = x and y = 8 x .  1  Discuss the convergence of the sequence {a n }, where a n = n sin  .  2n  If a . i = (i + j) = a . (i + j + k) = 1, then determine the vector a. (3+3+3+2+3+3+3+3+2+3)

2. a) Ans

If A and B are symmetric matrices, then show that the matrix A B - B A is a skewsymmetric matrix.

 x y a b  and B =    y z b d  a b   x y  ax − by ay − bz  AB =   =  b d   y z  bx − dy by − dz 

Let, A = 

B3.2-R3

Page 3 of 9

JANUARY, 2004

y  a b  ax − by bx − dy  = z  b d   ay − bz by − dz  ax − by ay − bz  ax − by bx − dy  AB − BA =   − bx − dy by − dz   ay − bz by − dz  0 ay − bz − bx + dy   =  0  bx − dy − ay + bz 0 ay + dy − bx − bz   =  , which is a skew-symmetric matrix. 0  − (ay + dy − bx − bz ) x BA =  y

b)

Without expanding, find the value of the determinant.

a −b b−c c −a D= x − y y − z z − x p−q q−r r− p Ans

a−b b−c c−a D = x− y y−z z−x p−q q−r r− p a−b+b−c+c−a = x− y+ y−z+z−x p−q+q−r+r− p

b−c c−a y−z z−x q−r r− p

(C1 = C1 + C 2 + C3 )

0 b−c c−a = 0 y−z z−x =0 0 q−r r− p c)

Determine a and b such that the function f ( x ) = x + ax + bx has an extremum at 3

3

x = 1 and f (1) = −3 . d) e)

Find the value of a . b if |a| = 6, |b| = 4 and |a x b| = 12. If z1=2+i, z2=3-4i, z3=-3+2i, then find the principal value of

Ans

z z  z = Re  1 2  .  z3  z1 = 2 + i, z 2 = 3 − 4i, z 3 = −3 + 2i

arg(z ) , where

z z   (2 + i )(3 − 4i )   6 − 8i + 3i − 4i 2   − 5i + 10  z = Re  1 2  = Re   = Re   = Re  3 − 2i  3 − 2i     − 3 + 2i   z3   − 15i − 10i 2 + 30 + 20i   (− 5i + 10i )(3 + 2i )   40 + 5i  40 = Re  = Re  =  = Re   2 2 3 − (2i )  9 + 4  13  (3 − 2i )(3 + 2i )   

(

Here x =

B3.2-R3

)

40 , y = 0. 13

Page 4 of 9

JANUARY, 2004

2

40  40  Arg ( z ) = x + y =   + 0 2 = 13  13  2

2

(5+2+3+4+4) 3. a)

Find the rank of the matrix

 1 2 3   A= 2 4 6 .   − 1 5 4

b) c) d)

Does A-1 exist? Examine whether the vectors i - j+ 2 k, 3 I + j - 3 k, 2i - 5j + k are linearly independent or linearly dependent. Find the value of p for which the equation 2pxy+4x-6y+9=0, p ≠ 0 represents a pair of straight lines. It is given that the Rolle's theorem hold for the function f(x)=x3+bx2+cx, at

1 ≤ x ≤ 2 at x = 4 . Find the values of b and c. 3 e)

Evaluate the integral 1

I = ∫ x(1 − x) n dx 0

(3+3+4+4+4) 4. 2

a)

Evaluate the integral I =

∫ | x − 1 | dx.

−2

b)

Find the angle of intersection between the curves C2 : x2 = -4(y - 1); C1 : x2 = 4(y + 1);

c) d)

If 1, ω ,ω are the cube roots of unity, then find the roots of (z – 2)3 + 27 = 0. Discuss the convergence of the series

x>0

2

∞

∑a n =1

n2 +1 n2 n =1 ∞

n

=∑

dy , where y = x + x + x + ... dx

e)

Find

Ans

y = x + x + x + ... = x + y ⇒ y = x+ y ⇒ y2 = x + y d 2 d (x + y ) ⇒ y = dx dx dy dy ⇒ 2y = 1+ dx dx dy dy ⇒ 2y − =1 dx dx

B3.2-R3

Page 5 of 9

JANUARY, 2004

dy (2 y − 1) = 1 dx dy 1 ⇒ = dx (2 y − 1)

⇒

(4+4+4+2+4) 5.

 1 + cos 2 x     1 − cos 2 x dx .    2 cos 2 x   1 + cos 2 x   dx = cot −1 cot 2 x dx  dx = ∫  Ans I = ∫ cot −1  ∫   2 sin 2 x  − 1 cos 2 x     x2 = ∫ cot −1 (cot x ) dx = ∫ x dx = +c 2

a)

Evaluate I =

∫ cot

−1

(

b)

)

Find the value of k so that the function

sin(kx) /(5 x), x ≠ 0 3/5 , x=0 is continuous at x = 0. The function f ( x ) is given as: sin(kx) /(5 x), x ≠ 0 f ( x) = 3/5 , x=0 At x = 0 3 f (0) = 5 sin kx sin kx  sin kx kx  k = lim lim f ( x ) = lim .  = lim x →0 x →0 5 x x →0  kx 5 x  5 x→0 kx Put θ = kx θ → 0 as x → 0 k k sin kx k sin θ k ∴ lim = lim = .1 = 5 x→0 kx 5 θ →0 θ 5 5 As the function is continuous at x = 0 f (0) = lim f ( x ) f ( x) =

Ans

x →0

3 k = 5 5 15 ⇒k = =3 5

⇒

c)

Find the conditions on a for which the system of equations

ax + 2 y + 3z = 4 4 x + 5 y + 6az = 3 7 x + 8 y + 9az = 6

B3.2-R3

Page 6 of 9

JANUARY, 2004

d) e)

has a unique solution. Find the area of region bounded by y = x( x − 2)( x − 3), x = 0, x = 3 and the x axis. Find the coordinates of the vertex, the coordinates of the focus and the equation of the directrix for the parabola.

4 x − y 2 + 2 y − 13 = 0 (2+2+4+5+5) 6. 1

a)

Evaluate the integral I =

Ans

I =∫

x2 + x −1 ∫0 ( x + 1)( x + 2) dx

1

1

1

x2 + x −1 x2 + x −1 x2 + x −1 dx dx = ∫ 2 dx = ∫ 2 ( x + 1)( x + 2) 0 0 x + x + 2x + 2 0 x + 3x + 2

1

=∫ 0

(x

2

)

(

)

+ 3x + 2 − 2 x − 3 x 2 + 3 x + 2 − (2 x + 3) = dx dx ∫0 x 2 + 3x + 2 x 2 + 3x + 2

(

1

1

)

(

1

)

1

1

2x + 3 2x + 3 x 2 + 3x + 2 dx = I Say dx == ∫ dx − ∫ 2 dx − ∫ 2 2 3 2 3 2 + 3 + 2 + + x x + + x x x x 0 0 0 0 2x + 3 ∴∫ 2 dx x + 3x + 2 Let x 2 + 3 x + 2 = z ⇒ (2 x + 3) dx = dz dz 2x + 3 2 ∫ x 2 + 3x + 2 dx = ∫ z = log z = log x + 3x + 2 1 1 1 2x + 3 1 dx = x ]0 − log x 2 + 3 x + 2 0 I = ∫ dx − ∫ 2 0 0 x + 3x + 2 =∫

(

)

)]

(

{ (

)

(

)}

= ( x − 1) − log 12 + 3.1 + 2 − log 0 2 + 3.0 + 2 = x − (log 6 − log 2 ) = x − log = x − log 3 b)

Solve the system of equations

Ans

using Cramer’s rule. The system equation is given as:

6 2

x− y =2

2 x + 3 y = −1 x− y =2 2 x + 3 y = −1 1 −1 D= = 3 − (− 2) = 3 + 2 = 5 2 3 2 −1 D1 = = 6 −1 = 5 −1 3 1 2 D2 = = −1 − 4 = −5 2 −1

B3.2-R3

Page 7 of 9

JANUARY, 2004

D1 5 = =1 D 5 D −5 x2 = 2 = = −1 D 5

∴ x1 =

c)

Find the absolute maximum value of

f ( x) = (sin x)(a + cos x), d)

0≤ x≤

π 2

Find the eccentricity of the ellipse

4 x 2 + 9 y 2 − 32 x + 54 y + 109 = 0 e)

Given a = I + 3j – k, b = 2j – 3k,

find the value of (a – b) . (a + b). (4+3+5+4+2)

7. a)

Find the value of x so that

[1

Ans

1

1 [1 1 x] 0 2

1 0 2  1  x ] 0 2 1 2 = 0 2 1 0 1 0 2  1  2 1 2 = 0 1 0 1

1  ⇒ [1 + 0 + 2 x 0 + 2 + x 2 + 1 + 0] 2 = 0 1 1  ⇒ [2 x + 1 x + 2 3] 2 = 0 1 ⇒ [2 x + 1 + 2 x + 4 + 3] = 0 ⇒ 2x + 1 + 2x + 4 + 3 = 0 ⇒ 4x + 8 = 0 ⇒ 4 x = −8 −8 ⇒x= = −2 4

b)

Evaluate the integral π /2

I=

1 + sin 2 x

∫ 1 − cos 2 x dx

π /4 π /2

Ans

I=

1 + sin 2 x

∫ 1 − cos 2 x dx

π /4

1 + sin 2 x sin 2 x + cos 2 x + 2 sin x cos x dx Let I 1 = ∫ dx = ∫ 1 − cos 2 x 2 sin 2 x

B3.2-R3

Page 8 of 9

JANUARY, 2004

1 sin 2 x 1 cos 2 x 1 2 sin x cos x + dx dx + ∫ dx 2 2 ∫ ∫ 2 sin x 2 sin x 2 sin 2 x 1 1 1 1 = ∫ dx + ∫ cot 2 x dx + ∫ cot x dx = x − cos ecx + log(sin x ) 2 2 2 2

=

π 2

π /2

I=

1 + sin 2 x x cos ecx  + log(sin x ) dx = − ∫ 1 − cos 2 x 2 2 π π /4

4

π 2 cos ec(π 2)  π  π 4 cos ec(π 4)  π  = − + log sin  −  − + log sin  2 2   2 2 4     2 2 1 2 1  π π 1 π 1  π = − − + =  − + log 1 −  − + log + log 1 − log  2 2 2 4 8 2 2 4 2  8 = c)

π 2 −1 1 − − log 8 2 2

Obtain the second-degree Taylor’s polynomial approximation to

f ( x) = x about

x = 1. d)

Find the angle between the tangents to the circle x + y + 6 y + 7 = 0 at the points of 2

2

intersection with the line x = 1 e)

Discuss the convergence of the series

∑a

n

where a n =

n4 +1 − n4 −1 . (3+4+4+4+3)

B3.2-R3

Page 9 of 9

JANUARY, 2004