### Guidelines to Question Paper Setting | ENGINEERING MATHEMATICS – II - Diploma karnatak

1.       The question paper must be prepared based on the blue print without changing the weight age of model fixed for each unit.

2.       The question paper pattern provided should be adhered to

Section-A: 10 questions to be answered out of 14 questions each carrying 03 marks. Section-B: 08 questions to be answered out of 11 questions each carrying 05 marks. Section-C: 05 questions to be answered out of 08 questions each carrying 06 marks.

3.       Questions should not be set from the recapitulation topics.

#### UNIT-1: STRAIGHT LINES AND CONIC SECTION:

3 MARK QUESTIONS

1.     Find the equation of the straight line passing through (2,3) and having slope 5.

2.     Find the slope and x-intercept and y-intercepts of the line 2x + 3y 11 = 0.

3.     Find the vertex and focus of the parabola (y 2)2 = 8x.

4.     Show that the lines 3x-2y+2=0, 2x+3y+7=0 are perpendicular.

5.     Find the eccentricity of the ellipse x2 + y2 = 1

5  MARK QUESTIONS

64          9

1.     Find the equation to the line passing through the point (6,-4) and perpendicular to the line 7x-6y+3=0.

2.     Find the equation to the line passing through the point (2,3) parallel to the line joining the points (-8,-6) & (2,-4).

3.     Find the equation of straight line inclined at 135o to the x-axis having y-intercept 2/3.

4.     Find the equation of straight line joining the points (2,3) & (-4,6).

5.      Find the equation of the line passes through (-3,-2) which is perpendicular to x-axis.

6  MARK QUESTIONS

1.     Find the equation to the median of the triangle through the vertex A with vertices A(- 1,3), B(-3,5) &C(7,-9).

2.     The vertices of a quadrilateral taken in order are A(1,2), B(2,1),C(3,4) & D(-1,-2). Find the equation to the diagonal BD.

3.     Obtain the equation of the hyperbola in the form x2 + y2 = 1, whose eccentricity is 8

and distance between the foci is 12.

a2            b2

4.     Find the equation of the ellipse with length of major axis is 8 and minor axis is 3.

5.     Find the equation to the line passing through point (3,-2) and perpendicular to the line joining points (5,2) &(7,-6).

#### UNIT-2:DIFFERENTIATION:

3 MARK QUESTIONS

1.  Find dy, if y = 2x2 − 3x + 1.

dx

2.   Differentiate xtanx with respect to x.

3.   Find dy, if x2 + y2 = 25

dx

4.  Find  dy if  x = ct, y = c,

dx                                       t

5.

 2

Ify = 4ax, find d2y .

dx

5  MARK QUESTIONS:

1.             Differentiate the function xnby method of first principle.

2.             Find dy if y = 6x3 3 cos x + 4 cot x + 2e–x 5.

dx                                                                                                        x

3.             Find dy if y = cosx+sinx

dx                       cosx–sinx

4.             Find dy if y = (cosx)sinx

dx

5.             If y = tan–1 x, provethat (1 + x2)y2 + 2xy1 = 0

6  MARK QUESTIONS:

1.             Find dy if y = xlogx

dx                    1+sinx

2.

 /4

Find dy if x = a cos3 θ , y = a sin3 θ at θ = π                          .

dx

3.

 dx

Find dy if y = xxxx

x...

#### .

4.             Ify = tan–1 (1+x) , find dy.

1–x                    dx

#### 5.             2   Ify=emsin—1x,provethat(1−x2)y

xy1

− m2y = 0

UNIT-3 APPLICATIONS OF DIFFERENTIATION

3 MARK QUESTIONS

1.                           Find the slope of the tangent to the curve x2 + 2y2 = 9 at a point (1, 2) on it.

2.                           Find the slope of the normal to the curve y = 2 3x + x2 at (1, 0).

3.                           The law of motion of a moving particle is S = 5t2 + 6t + 3 where ‘S’ is the distance in metres and ‘t’ time in seconds. Find the velocity when t=2.

4.                           Find the rate of change of area of a circle with respect to its radius.

5.                           Show that the curve 2x3 y = 0 is increasing at the point (1, 2). 5 MARK QUESTIONS

1.  For a moving body vertically upwards, the equation of motion is given by S = 98t 4.9t2 . When does the velocity vanish?

2.  Find the equation to the tangent to the curve y = 2x2 3x 1 at (1,-2).

3.  A circular patch of oil spreads on water and increases its area at the rate of 2 sq.cm/min. find the rate of change of radius when radius when radius is 4 cm.

4.  The volume of the spherical ball is increasing at the rate of 36π cc/sec. Find the rate at which the radius is increasing. When the radius of the ball is 2cm.

5.  Find the max value of the function y = x3 3x + 4.

6 MARK QUESTIONS

1.                            Find the max & min values of the function y = x5 5x4 + 5x3 1.

2.                            Find the equation of normal to the curve y = x2 + 2x + 1 at (1,1).

3.                            If S is the equation of motion where S = t3 2t2 find its acceleration when velocity is 0.

4.                            The volume of sphere is increasing at 3c.c per second. Find the rate of increase of the radius, when the radius is 2cm.

5.                            Water is flowing into a right circular cylindrical tank of radius 50 cms at the rate of 500π cc/min. Find how fast is the level of water going up.

#### UNIT-4: INTEGRATION

3 MARK QUESTIONS

1.   Evaluate:(x2 + x + 1) dx.

2.   Evaluate: cot2 x dx

3.   Evaluate: e5x+8 dx

4.

 ∫

Evaluate:        1 dx

2x+5

5.   Evaluate: sin5 x cos x dx

5  MARK QUESTIONS

1.

 ∫

Evaluate (x4 1 + cosec2x e–2x + cos x) dx.

x

2.   Evaluate: cos3 x dx

3.   Evaluate: sin 6x cos 2x dx

4.   Evaluate: log x dx

5.  Evaluate:

(tan—1 x)3

1+x2             dx

6  MARK QUESTIONS

1.   Evaluate: (tanx + cotx)2dx.

2.   Evaluate:(x + 1)(x  2)(x  3)dx

3.   Evaluate: x2 cos x dx

4.   Prove that

dx     = 1 tan-1Ê x ˆ+ c

Ú x2 + a2    a         Á a ˜

Ë   ¯

 5. Evaluate:∫

 dx

1

9sin2 x+ 4cos2 x

#### UNIT-5: DEFINITE INTEGRATION AND ITS APPLICAITON.

3 MARK QUESTIONS

1.

 2

Evaluate:  3(2x + 1) dx.

2.

 0

Evaluate: π4 sec2 x dx.

3.

 0

Evaluate:2 ex dx

1 (sin—1 x)2

4.   Evaluate: 0

1–x2

dx.

5.

 0

Evaluate: π2 cos x dx.

5  MARK QUESTIONS

1.

 0

Evaluate: π2 sin 3x cos x dx.

2.  Evaluate: π

cos x

dx.

0 1+sin2 x

3.

 0

Evaluate: 1 x(x 1)(x 2) dx.

4.  Find the area bounded by the curve y = x2 + 1  the x-axis and ordinatesx =  1 , x = 3.

5.  Find the volume of the solid generated by the revolving of the curve y2 = x2 + 5x

between the ordinates x=1, x=2 about x-axis.

6  MARK QUESTIONS

1.  Evaluate: 1 cos(tan—1 x) dx.

0          1+x2

2.  Find the area between the curves y = x2 + 5 and y = 2x2 + 1.

3.  Find the volume of ellipsoid generated by revolving x2 + y2 = 1 between the

ordinates x = ±a about x-axis.

4.  Find the centre of gravity of a solid hemisphere.

a2            b2

5.  Determine the moment of inertia of a uniform rod of length 2l,

Cross-sectional area “a” about an axis perpendicular to the rod and passing through the mid-point of the rod.

#### UNIT-6: INTEGRATION

3 MARK QUESTIONS

1.   Write the order and degree of the differential equation(dy)8 + 3 d2y  yex = 0.

dx                   dx2

2.   Form the differential equation by eliminating arbitrary constants in y = m e2x.

3.   Solve xdx + ydy = 0 .

4.

 2

Solve dy

1+y

dx 1+x2

5.

 =

 .

Solve exdx + dy = 0 .

5  MARK QUESTIONS

1.    Form the differential equation by eliminating arbitrary constants A and B iny = Aex + Be–x.

2.    Form the differential equation by eliminating arbitrary constants iny = a cos mx + b sin mx.

3. Solve (1 + y)dx + (1 + x)dy = 0 .

4.  Solvedy + 3y = e2x.

dx

5.  Solve dy + y tan x = cos x

dx

6  MARK QUESTIONS

1. Solve x(1 + y2)dx + y(1 + x2)dy = 0 .

2.  Solve sec2 x tan y dx + sec2 y tan x dy = 0 .

3.  Solve x dy + y = x3 dx

4.  Solvedy + 3y = e2x.

dx

5.  Solve dy + 2y cot x + sin 2x = 0