B.Sc (Pass) Nautical & Engineering Model Question 2 Mathematics Second Paper (Differential Calculus, Integral Calculus, Two-dimensional & Three- dimensional Geometry)

1. MARINE FISHERIES ACADEMY CHITTAGONG
B.Sc (Pass) Nautical & Engineering Model Question 2016
Mathematics Second Paper
(Differential Calculus, Integral Calculus, Two-dimensional & Three- dimensional Geometry)
Full mark: -80 Time: - 3 hours.
[N.B.-The figures in the right hand margin indicate full marks. Answer any two from group A,
any three from group B, one from group C and one from group D]
Group A -- Differential Calculus (Answer any two of the following)
1. (a) Test the continuity at = of the following function
( ) = 3 − 2 when 0 ≤ <
= −3 − 2 ,, x ≥ 3/2
(b). Differentiate log( ) from the first principle (w.r.to. )
(c). Evaluate lim →
( )
2. ( )Differentiate any three of the following. 3 × 3 = 9
( ). { ( )} ( ). 2 √1 − , ( )( ) . ( ) ( ) =
(vi) y =sin[2 √( ) ]
(b). Find the equation of tangent at (2,-2) of the curve = − 3 + 2 06
3. (a) Y= ( ℎ ) show that (1 + ) +(2n+1) + = 0 [10 + 5]
(b) Show that the function sinx (1+cosx)=0 is maximum at =
Group B -- Integral Calculus (Answer any three of the following)
4. Integrate any four of the following 4 × 2.5 = 10
(a). ∫ (1 + ) , (b). ∫ (c∫ ( )√
, (d) ∫
( ) ∫ . ( ). ∫

2. 5. Evaluate any four of the following 4 × 2.5 = 10
(a). ∫ , (b). ∫ √
( ) ∫ 慜 ; ( ) ∫ ( )
(e) ∫ √1 + 3 ( ) ∫ tan
√
6. Solve any two of the following equation 5 × 2 = 10
(a) sin
(
) = + (b) − = ( + )
(c) + = 0
7. Answer any two of the following equation 5 × 2 = 10
(a) Define Beta & Game function and show that ( ) = ( )
(b). , ( , ) =
( ) ( )
( )
Prove that ( ) ∫ ( 1 − ) =
Group C -- Two-dimensional Geometry (Answer any two of the following) 10 × 1 = 10
8. (a) Show that the equation + 2ℎ + + 2 + 2 + = 0 represents two parallel lines if
a;h=h:b=g:f Also show that the distance between is = ( )
(b). ). find the equation of circle whose centre is (4,5) and passing though the center of the given circle +
+ 4 − 6 − 12 = 0
9.(a). Find the angle between the lines represent by the equation
+ 2ℎ + + 2 + 2 + = 0 and also find the Condition of perpendicularity
(b) Find length of the chord intercept by the circle + = on the line y=mx+c
Group D -- Three- dimensional Geometry (Answer any two of the following) 10 × 1 = 10
10.(a) Find the equation of the plane through the point (2,3,1), (1,1,3), and (2,2,3) . Also find the perpendicular
distance from the point (5,6,7) to the plane
(b). Find the equation of the plane through the lines of intersection of the planes
2 − = 0 3 − = 0 and perpendicular to the plane 4 + 5 − 3 + 1 = 0
11. (a) Find the distance of the point (-1,-5,-10) from the point intersection of the line = = and
the plane − + = 5
(b) = = and 3 + 2 + − 2 = 0 = − 3 + 2 − 13 are coplanar and find the equation to the
plan in which they lie.