The document is a set of model examination questions for a B.Sc. (Pass) degree in Nautical & Engineering at the Marine Fisheries Academy in Chittagong, focusing on mathematics, specifically calculus and geometry. It includes various groups of questions covering differential and integral calculus, two-dimensional and three-dimensional geometry, with specific instructions on how many questions to answer from each group. Overall, it serves as a guide for students to prepare for their mathematics examination.

1. MARINE FISHERIES ACADEMY, CHITTAGONG
B.Sc (Pass) Nautical & Engineering Model Question n 2016
Mathematics Second Paper (Calculus & Geometry )
Full mark: -80 Time: - 3hours.
N.B.-The figures in the right hand margin indicate full marks. Answer any two from group A,
any three from group B, two from group C and two from group D.
Group A Differential Calculus
1. (a).Examine differentiability at = /2of the following function. 3 × 5 = 15
( ) = 1 + sin when 0 ≤ < /2
= 2 + − ,, ≥ /2
(b). Differentiate from the first principle (w.r.to )
(c). Evaluate lim →
2. ( )Differentiate any three of the following. 3 × 3 + 6 = 15
( ). = tan , ( ). = 10 , ( ) = sin[2 tan , ( ) =
(b). Show that the equation of tangent at ( , ) of the parabola = 4 is
y = 2 ( + )
3. (a) If = sin( log ) Show that (1- ) – (2 + 1) − ( + ) =0 [10,5]
(b) Discuses the maximum and minimum value for the function ( ) = − 3 − 45 + 20
Group B Integral Calculus
Integrate any four of the following 4 × 2.5 = 10
(a). ∫
( )
, (b). ∫ (c∫
( )√
, (d) ∫
( ) ∫ . ( ) ∫
√

2. 5. Evaluate any four of the following 4 × 2.5 = 10
(a). ∫ , (b). ∫ √
( ) ∫ ; ( ) ∫ ( )
(e) ∫ √1 + 3 ( ) ∫ cos( )
6. Solve any two of the following equation 5 × 2 = 10
(a) + = 0 (b) 1 − + √1 − = 0
(c) + = 0
7. Answer any two of the following equation 5 × 2 = 10
(a) Prove that ( ) = ( )
(b). Prove that ∫ ( 1 − ) =
(c) Define Beta & Game function, Prove that ( , ) =
ℾ ℾ
ℾ( )
Group C --- Two-dimensional Geometry 5 × 2 = 10
7. (a) Show that the general equation of second degree + 2ℎ + + 2 + 2 + = 0 may
represents a pair of straight lines if ∆= + 2 ℎ + − − ℎ = 0
(b). Find the equation to the circle passing through the point (1,2), (3,4) and center in the line
3 + − 3 = 0
(c). Prove that the straight lines represented by the equation + 2ℎ + + 2 +
2 + = 0 will be equidistant from the origin if − = ( − )
(d) . Find the equation to the circle which touches the x-axis at the point (4,0) and cuts off a
chord of length 6 unit from the y-axis
Group D -- Three- dimensional Geometry 5 × 2 = 10
9.(a) Find the equation of the plane passing throw the intersection of the planes
+ 2 + 3 + 4 = 0 and 4 + 3 + 2 + 1 = 0 and the point (1,2,3)
(b). Find the point where the line + 2 + 4 − 2 = 0 = 2 + 3 − 2 + 3 cuts plane
2 − + 4 + 8 = 0
(c) (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
(d). Find the distance of the point (-1,-5,-10) from the point intersection of the line
= = .