ENGINEERING MATHEMATICS –I
(Common to All Engineering Diploma Programmes)
Time: 3
Hours.][Max marks: 100
Note:
(i)
Answer any Ten
questions from section-A, any Eight questions from section-B and any Five questions from section-C.
(ii)
Each question carries 3 marks in section-A.
(iii)
Each question carries 5 marks in section-B.
(iv)
Each question carries 6 marks in section-C.
SECTION – A
2 3 1 4
1.
Find the product of A = [0 −1 3] and B = [−1]
5
2.
If
2 −1 5 1
A = [
3 4
] and B = [
] find adj(AB).
0 −3
]
3. If A + B = [3 −7
0 2
,A − B = [1 5 ]
find A.
4 −6
4. If a⃗ = i + 2j − 3k, b¯⃗ = 3i − 5j + 2k. Find the magnitude of 2a⃗ + 3b¯⃗.
5. If A⃗=(3,-4), B¯⃗= (-5,6) find position
vector of A and B and also find |A¯¯¯¯B¯⃗|
6. Three coins are tossed
simultaneously. List the sample space for event.
7.
If sin 8 = − 8/ and n < 8 < 3n find the
value of4tan8 + 3sec8.
17 2
8. Find the value of sin 75o using standard angles.
9.
Show that cocec(180–Æ)cos(–Æ) = cot2A
sec(180+Æ)cos(90+Æ)
10. Prove thatsin(A + B) sin(A − B) = sin2 A − sin2 B.
11.
Prove thatsin 3Æ − cos 3Æ = 2.
sin Æ cos Æ
12. Express the
product (1 + i)(1 + 2i) in a + ib form and hence
find its modulus.
13.
Evaluate : lim [ s–1 ]
s→3
2s2–7s+5
14.
Evaluate:
lim [3s2+4s+7
2 ]
s→œ
4s +7s–1
SECTION – B
|
|
1 |
x |
0 |
|
1. Find the value of x if | |
2 |
−1 |
3| = 0. |
−2 1 4
2.
]
Find the characteristic equation
and its roots of a square matrix
A = [1 2
2 1
3.
Find the sine of the angle between
the vectors2i − j + 3k and i − 2j
+ 2k.
4.
If vector a⃗ = i + j + 2k, b¯⃗ = 2i − j + k show that a⃗ + b¯⃗perpendiculara⃗ − b¯⃗.
5.
Find the
projection of a⃗ = 2i + j − kon b¯⃗ = 2i − 3i + 4k.
6.
Prove that 1
loga abc
+ 1
logb abc
+ 1 = 1
logc abc
7.
Find the numerical value ofsin (n) . cos (− n) − cos (n) . sin (− 3n)
3 3 4 4
8. Prove that sin(A + B) = sin A cos B + cos A sin B geometrically
9.
IfA + B + C = n, prove that
tan A tan B + tan B tan C + tan C tan A = 1.
2
10. 
Show thatsin 56o– sin 44o = cot 82o
cos 56o+cos 44o
11. Evaluate:lim [√1+s+s2–1]
s→0 s
SECTION – C
1. Solve for x, y & z using
determinant method
x + y = 0, y + z = 1&z + x = 3.
2.
Solve the equationx + y + z = 6, 2x − 3y
+ z = 1&x + 3y
− 2z = 7 using Gauss elimination
method.
3.
A force F⃗ = 2i + j +
k is acting at the point (-3,2,1).
Find the magnitude
of the moment of force F⃗ about the point (2,1,2).
4.
A die is thrown twice and the sum of the numbers
appearing is absorbed tobe. What is the conditional probability that the number
5 has appeared at least once?
5g
5.
Prove that cos( 2 –8) + tan(–8)
= sec28
sin(4n+8) cot(n–8)
6. 
Prove that cos 80o cos 60o cos 40o cos 20o = 1
16
7.
Find the modulus
and argument of the complex
number z = −√3 + i and
hence represent in argand diagram.
8. sin 8
Prove that lim ( ) = 1 where 8 is in radian.
