### Model Question Paper - ENGINEERING MATHEMATICS –I Diploma 2020

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 = [

0     −3

 ]

3. If A + B = [3             −7

0       2

,A B = [1             5 ] find A.

4     −6

4.   If a = i + 2j 3k, b¯ = 3i 5j + 2k. Find the magnitude of  2a +  3b¯.

5.   If A=(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 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 a = i + j + 2k, b¯ = 2i j + k show that a + b¯perpendiculara b¯.

5.   Find the projection of a = 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 F = 2i + j + k is acting at the point (-3,2,1). Find the magnitude of the moment of force F 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.