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Seat No.: Enrolment
GUJARAT TECHNOLOGICAL UNIVERSITY
DIPLOMA ENGINEERING SEMESTER I/II • EXAMINATION SUMMER- 2017
Subject Code: 310034 Date: 03 06- 2017
Subject Name: Mathematics I
Time: 02:30 PM TO 05:00 PM Total Marks: 70
Instructions:
1. Attempt all questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.
4. Each question carry equal marks (14 marks)
Q.1



Filling the blanks

If

If for G.P.

(3)|200820092010201120122013201420152016|=______.
(4)The total number of terms in the expansion of

If then

If [1234] then adj
[1234] adj
If A=i−4j+3k then
A=i−4j+3k
07

Define Radian and find the length of chord and area of sector of circle whose angle and radius of sector of circle are 1200 and 4 cm respectively.
. 4 1200
(2)Prove that


03
Q.2



Filling the blanks


If tanθ= 1√8 thencotθ=
tanθ= 1√8 cotθ=
07
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The principal period of Sin x
Sin x sin 280cos170+cos In ΔABC,if a=3 cm.,b=3 cm.and c=4 cm.then
ΔABC a=3 c=4
sin−1x+cos−1x=


Prove that log(1+2+7)=log1+log2+log5
Find sum 1+3+9+.....+2187 .

04
OR


Find the middle term of the expansion of by using Binomial theorem.
By using properties of Determinat ,prove that |1ab+c1bc+a1ca+b|=0 |1ab+c1bc+a1ca+b|=0
03
04
Q.3



Find the value of x and y from the matrix equation y
Solve by using Matrx method∶ 3x−2y=8 and 4y+5x=6 3x−2y=8 and 4y+5x=6

04


Find the unit vector in the direction of a−b+2c where a=i+j−k ,b=3i+4k−2j and c=2j−i+4k
a−b+2c a=i+j−k b=3i+4k−2j c=2j−i+4k
If x i j k and y 2i j k then prove that x is perpendicular
to y . Also find a vector which is perpendicular to both x and y .
x i j k y 2i j k x y x y

04
OR
Q.3


If [123456] and [122112] then find AB and BA. [123456] B=[122112] AB BA If [2−231]and [−154−3] then prove that (AB
03
04

Simplify

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The constant forces 3i−k+2j and i−3j+2k acting on a particle displace it from the point i+4j−3k to the point 3i+j+4k.Find the total work done during the displacement . 3i−k+2j i−3j+2k i+4j−3k 3i+j+4k
04
Q.4



Evaluate Prove that
03
04


Prove that tan570=cos120+sin120cos120−sin120 tan570=cos120+sin120cos120−sin120 Evaluate sin2(3712)0− sin2(712)0 sin2(3712)0− sin2(712)0

04
OR
Q.4



Prove that

In
03
04


Prove that
tan500= tan400+2 tan100 (2)sin4x+sin6𝑥cos4x+cos6x=tan5x
03
04
Q.5



Draw the Graph of y=sinx y=sinx Prove that 1+sin2θ−cos2θ1+sin2θ+cos2θ=tanθ 1+sin2θ−cos2θ1+sin2θ+cos2θ=tanθ
03
04


In usual notation ,in ΔABC if a=4 cm cm and c cm then find R ΔABC a=4 c R Prove that


04
OR
Q.5



In usual notation ,in ΔABC ,Prove that
ΔABC
Evaluate sin(2212)0 and tan(2212)0. sin(2212)0 tan(2212)0.
03
04
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Solve ΔABC where b=√3 and 300 ΔABC b=√3 300 Prove that tan−1(23)=12tan−1(125) tan−1(23)=12tan−1(125)

04