Q.CODE.
NO: SUB CODE: DNS 101
DIPLOMA IN NAUTICAL SCIENCE SEMESTER – I
APPLIED MATHEMATICS
TIME: 3 HRS MAX.MARKS: 80
PART – A
Answer any Five Questions 5x6=30
1. Find 𝑎⃗x𝑏⃗⃗ and 𝑐⃗x𝑑⃗ if 𝑎⃗ = ⃗i + ⃗j + ⃗⃗
k ,𝑏⃗⃗ = 2i⃗ − ⃗j + ⃗⃗ ⃗⃗ , 𝑑⃗ = ⃗i + ⃗j − 2k
⃗⃗⃗⃗ − ⃗j + 5k
k , 𝑐⃗ = 3i ⃗⃗
𝑑𝑦 2
2. Find if y = x x
𝑑𝑥
3. Define (i) Polar Triangle (ii) Spherical Angle (iii) Spherical Triangle
4. Find the standard deviation for the following data
x 70-80 80-90 90-100 100-110 110-120 120-130 130-140 140-150
y 12 18 35 42 50 45 20 8
1 1
5. If 𝑦 = log(𝑥 2 − 𝑎2 ) P.T 𝑦3 = 2[ + ]
(𝑥+𝑎)3 (𝑥+𝑎)3
1−𝑥 𝑑𝑦
6. If 𝑦 = 𝑠𝑖𝑛−1 ( ) find the value of 𝑑𝑥
1+𝑥)
7. Find ‘K’ so that the point (3,K,-6) may be the centre of a sphere of radius 5√5 unit and
passing through(-2,2,4)
8. In a spherical triangle ABC given AB=50º10´ AC=64º17´ BC=27º37´ find C
PART – B
Answer any Five Questions 5x10=50
9. A random variable x has the following probability function
x 0 1 2 3 4 5 6 7
K 2K 2K 3K 𝐾 2𝐾 2 2 2
P(x) 0 7𝐾 + 𝐾
(i) Find K (ii) Evaluate P(x<5), P(x≥4), P(0<x<6), P(2<x<7)
3th 6 𝑑𝑥
10.Apply Simpson’s rule for ∫0 where n=6.
8 1+𝑥
11.Find the equation of the sphere passing through the points (1,0,-1) (2,1,0) (1,1,-1) and
(1,1,1)
−1
12.(a) If 𝑦 = 𝑒 𝑡𝑎𝑛 𝑥 P. T (1 + 𝑥 2 )𝑦2 + (2𝑥 − 1)𝑦1 = 0
𝑑𝑦 𝑥 2 +𝑒 𝑥 𝑠𝑖𝑛𝑥
(b) Find if 𝑦 =
𝑑𝑥 𝑐𝑜𝑠𝑥+𝑙𝑜𝑔𝑥
𝑥2
13.Integrate 𝑥 5 𝑒 write to X.
14.In spherical right angle triangle PVM, PM=92º 00´ PV=51º 55´ V=90º Calculate P, P and
M.
15.Calculate the correction co-efficient between x and y
x 10 12 13 16 17 20 25
y 19 22 26 27 29 33 37
16.Calculate mean, median, mode for the following data
C.I 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80
f 18 16 15 12 10 5 2 2
Q.CODE.NO: SUB CODE: DNS 101
DIPLOMA IN NAUTICAL SCIENCE
SEMESTER – I
APPLIED MATHEMATICS
TIME: 3 HRS MAX.MARKS: 80
PART – A
Answer any Five Questions 5x6=30
1. Find the value of (i) 𝑎⃗. 𝑏⃗⃗ (ii) (𝑎⃗ + 3𝑏⃗⃗). (𝑎⃗ − 2𝑏⃗⃗) if 𝑎⃗ = 2𝑖⃗ + 2𝑗 − 𝑘, 𝑏⃗⃗ = 6𝑖 + 3𝑗 + 2𝑘
2. Calculate the Arithmetic mean for the following frequency distribution.
Class 10-19 20-29 30-39 40-49 50-59 60-69 70-79 80-89
Frequency 5 9 14 20 25 15 8 4
3. If 𝑦 = 𝐴𝑐𝑜𝑠4𝑥 + 𝐵𝑠𝑖𝑛4𝑥 where A and B are constant show that 𝑦2 + 16𝑦 = 0
4. Define (i) Small Circle (ii) Great Circle (iii) Pole of a great circle
9
5. Integrate (𝑥−1)(𝑥+2)2
6. Find the focus, vertex, directrix, latus rectum length of the latus rectum of the parabola
𝑦 2 = 8𝑥
7. In a spherical triangle LMN, M=33º 14.0´, M=80º 05´, n=70º 12´. Calculate N.
4
8. Compute ∫0 𝑒 𝑥 𝑑𝑥 using simpson’s 1/3rd one-third rule where n=4.
PART – B
Answer any Five Questions 5x10=50
9. Find the line of regression of y on x.
x 1 2 3 4 5 8 10
y 9 8 10 12 14 16 15
10.Find the mean, median & mode of the following frequency distribution.
Class Limits Frequency
130-134 5
135-139 15
140-144 28
145-149 24
150-154 17
155-159 10
160-164 1
11.Find the rank correction co-efficient for the following data.
x 94 98 110 70 65 85 88 59
y 121 138 170 102 90 152 160 85
6 1
12.Evaluate ∫0 𝑑𝑥 where n=6 by using (i) Simpson’s 1/3rd rule (ii) Simpson’s 3/8rd rule.
1+𝑥 2
13.In a spherical triangle CDE, Calculate the angles C,D if C=87º 10´, d=62º 37´ and e=100º
10´
14.Find the equation of the sphere passing through the points (0,0,0) (0,1,-1) (-1,2,0) (1,2,3)
15.Evaluate ∫ 𝑥 2 𝑐𝑜𝑠𝑥𝑑𝑥
16.Differentiate (i) 𝑥 𝑚 𝑦 𝑛 = (𝑥 + 𝑦)𝑚+𝑛 (ii) 𝑥 𝑦 = 𝑦 𝑥 with respect to x.
Q.CODE.NO: SUB CODE: DNS 101
DIPLOMA IN NAUTICAL SCIENCE
SEMESTER – I
APPLIED MATHEMATICS
TIME: 3 HRS MAX.MARKS: 80
PART – A
Answer any Five Questions 5x6=30
1. Show that the sum of three vectors determined by the medians of a triangle directed from
the vertices is zero.
⃗⃗ , 𝑏⃗⃗ = 𝑖⃗ + 2𝑗⃗ − 3𝑘
2. Find the constant 𝜆 so that the vectors 𝑎⃗ = 2𝑖⃗ − 𝑗⃗ + 𝑘 ⃗⃗ and 𝑐⃗ = 3𝑖⃗ +
𝜆𝑗⃗ + 5𝑘⃗⃗ are coplanar.
3. The ages of all members in a salon’s family were received and the following frequency
distribution was obtained.
Age (Years) 0-5 5-10 10-20 20-30 30-40 40-50 50-60 60-70
No.of.Perssson 12 18 16 19 14 11 4 3
Obtain the mean age per person
4. Fit a straight line to the following data:
Year 1991 1995 1999 2003 2007
Production in tons 10 12 8 10 13
Find the estimation on the production in 2005.
5. Find the equation of the circle which passes through the points (1,-2) and (3,-4) and tones
the x-axis.
6. Find the length of semi-axis, co-ordinate of foci for the ellipse 25𝑥 2 + 16𝑦 2 = 400
4
7. Find ∫ 𝑥 3 𝑒 𝑥 𝑑𝑥
𝑑𝑦 𝑎+𝑥 2
8. Find if 𝑦 =
𝑑𝑥 𝑎−𝑥 2
PART – B
Answer any Five Questions 5x10=50
9. Prove that 𝑎⃗x(𝑏⃗⃗x𝑐⃗) + 𝑏⃗⃗x(𝑐⃗x𝑎⃗) + 𝑐⃗x(𝑎⃗x𝑏⃗⃗) = 0
10.The number of visit of ships to a particular point is given below. Find 𝜎 2
No.of.Times 71 76 79 83 86 89 92 97 101 103 107 110
visiting
Frequency 4 3 4 5 6 5 4 4 3 3 3 2
11.Find the centre, the length of axes, the eccentricity and the foci of the hyperbola
9𝑥 2 − 16𝑦 2 − 54𝑥 − 32𝑦 − 79 = 0.
12.Find the area between the curve 𝑦 2 = 4𝑥 and the line 2𝑥 − 3𝑦 + 4 = 0
13.In a quadrantal triangle ABC, C=52º 11´, B=69º 47´ and a=90º. Determine A, C and b.
5𝑥 2 −1
14.Evaluate (i) ∫ 𝑠𝑖𝑛3 𝑥𝑐𝑜𝑠 2 𝑥 𝑑𝑥 (ii) ∫ 𝑑𝑥
𝑥 2 −1
𝑥 𝜋 ⁄2 √𝑠𝑖𝑛𝑥
15.Evaluate (i)∫ 𝑑𝑥 (ii) ∫0 𝑑𝑥
𝑥 2 −3𝑥+2 √𝑠𝑖𝑛𝑥+√𝑐𝑜𝑠𝑥
𝑑𝑦 𝑥+𝑥 2 1
16.Find for (i) 𝑦 = (ii) 𝑦 = 𝑠𝑖𝑛−1 5𝑥 + 𝑡𝑎𝑛−1 ( )
𝑑𝑥 4𝑥−3𝑥 2 𝑥
