Sample Question Paper

Scheme - G
Sample Question Paper
Course Name : All Branches of Diploma in Engineering and Technology.
Course Code : AE/CE/CH/CM/CO/CR/CS/CW/DE/EE/EP/IF/EJ/EN/ET/EV/EX/IC/IE/IS/
ME/MU/PG/PT/PS/CD/CV/ED/EI/FE/IU/MH/MI
Semester
: First
17104
Subject Title : Basic Mathematics
Marks
: 100
Time: 3Hrs.
Instructions:
1. All questions are compulsory.
2. Illustrate your answers with neat sketches wherever necessary.
3. Figures to the right indicate full marks.
4. Assume suitable data if necessary.
5. Preferably, write the answers in sequential order.
Q1. Attempt any TEN of the following
4 3 9
a) Find x if 3 − 2 7 = 0
11 4 x
⎡ 1 2⎤
b) If A = ⎢
⎥ , B=
⎣− 3 4⎦
20 Marks
⎡4 5 ⎤
⎢1 − 3⎥ , Find 3A + B .
⎣
⎦
⎡3 9⎤
2
c) If A = ⎢
⎥ , then show that A is null matrix.
−
1
−
9
⎣
⎦
⎡ 5 4⎤
d) If A = ⎢
⎥
⎣4 3⎦
,
B=
e) Resolve into partial fraction
⎡− 3 4 ⎤
⎢ 4 − 5⎥ verify that AB = BA
⎣
⎦
x +5
x2 − x
f) Prove that sin 2θ = 2 sin θ cos θ
g) Define compound angle.
h) If 2 sin 40 0. cos10 0 = sin A + sin B , then find A & B
tan 660 + tan 690
i) Evaluate without using calculator:
1 − tan 660. tan 690
j) Prove that sin −1 ( − x) = − sin −1 ( x )
3
k) Find the distance between point (– 2 , 3) and the line 3x + 2y + 26 =0
l) Find the range of the following distribution
2, 3, 1, 6, 10, 17, 20, 24, 31
Q2. Attempt any FOUR of the following
a) Solve the following equations by using Cramer’s rule
x + y + z – 6 = 0 , 2x + y – 2z + 2 = 0 , x + y – 3z + 6 = 0
16 Marks
⎡1 2⎤ ⎡ x 5 − 3⎤
⎡5 − 3 7⎤
=
b) Find the value of x and y ,if ⎢
⎥⎢
⎥
⎢7 7 1⎥
⎣3 2⎦ ⎣ 2 y 5 ⎦
⎣
⎦
⎡ 1 2⎤
c) If A = ⎢
⎥ ,B=
⎣− 2 3⎦
Verify that A(B+C) =
⎡2 − 3⎤
d) A = ⎢
⎥
⎣1 5 ⎦
⎡2 1⎤
⎡ − 3 1⎤
⎢2 3⎥ , C = ⎢ 2 0⎥
⎣
⎦
⎣
⎦
AB + AC
⎡3 − 1 2⎤
& B= ⎢
⎥
⎣1 0 1⎦
Verify that (AB)T = BT AT
x −5
x + x 2 − 6x
e) Resolve in to partial fraction
3
x2
( x + 1)( x − 2) 2
f) Resolve in to partial fraction
Q3. Attempt any FOUR of the following
a) Using matrix inversion method , solve the following equations
x + 3y + 3z = 12, x + 4y + 4z = 15, x + 3y + 4z = 13
16 Marks
x 2 + 23x
b) Resolve in to partial fraction
( x + 3 )( x 2 + 1 )
c) Resolve in to partial fraction
tan θ + 1
(tan θ + 2)(tan θ + 3)
π
d) Prove that sin ⎛⎜ + θ ⎞⎟ = cos θ
⎝2
⎠
e) Prove that
sin 4 A + sin 5 A + sin 6 A
= tan 5 A
cos 4 A + cos 5 A + cos 6 A
f) Prove that
tan-1 1/2 + tan-1 1/3 = Π
4
4
Q4. Attempt any FOUR of the following
a) cos (A+B) = cosA cosB - sinA sinB
16 Marks
b) Prove that sin3A = 3sinA - 4 sin3A
3
c) Without using calculator show that cos150 cos300 cos600 cos750 = 16
d) prove that
sin 7 x + sin x
= sin 2 x − cos 2 x cot x
cos 5 x − cos 3 x
e) Prove that tan -1(1) + tan-1(2 ) + tan-1(3 ) = π
f) Prove that
⎛4⎞
⎛ 12 ⎞
⎛ 33 ⎞
cos −1 ⎜ ⎟ + cos −1 ⎜ ⎟ = cos −1 ⎜ ⎟ .( without using calculator)
⎝5⎠
⎝ 13 ⎠
⎝ 65 ⎠
Q5. Attempt any FOUR of the following
cos 3 A sin 3 A
+
= 4 cos 2 A
a) Prove that
cos A sin A
b) Prove that
sin C + sin D = 2 sin
C+D
Cos
2
16 Marks
C-D
2
−1
−1
−1 ⎛ x + y ⎞
⎟⎟
c) Prove that tan ( x) + tan ( y) = tan ⎜⎜
⎝ 1 − xy ⎠
d) Find the perpendicular distance between two parallel lines ax + by +c = 0 ax + by + c1 = 0
e) Find the acute angle between the lines y = 5x +6 and y = x
f) Find the equation of the line passing through the point (3, 2) and through the intersection of
the lines 2x + 3y = 1 and 3x – 4y = 4
Q6. Attempt any FOUR of the following
16 Marks]
a) If m1 and m2 are the slope of the two lines then prove that angle between
m1 − m2
.
1 + m1 m2
b) Find the equation of the line passing through the point of intersection of the lines
4x + 3y = 8 , x + y = 1 and parallel to the line 5x – 7y = 3
two lines is θ = tan −1
c) In the two factories A and B ,engaged in the same industry, the average weekly wages and
standard deviations are as follows .
Factories
Average wages Standard deviation
A
34.5
5.0
B
28.5
4.5
Which factory A & B has greater variability in individual wages?
d) Find the range and the coefficient of range for the following data
5
Temperature
No of days
25-26
2
27-28
11
29-30
12
31-32
10
33-34
4
35-36
1
e) Find mean deviation from mean for the following data .
Marks obtained
No of students
10-20
4
20-30
6
30-40
10
40-50
18
50-60
9
60-70
3
f) Find variance and the coefficient of variance for the following data.
Class
Interval
Frequency
0-30
30-60
60-90
90-120
120-150
150-180
180-210
9
17
43
82
81
44
24
6