Download AKTU B-Tech 1st Sem 2015-2016 BT 101 Engineering Mathematics I Question Paper

Printed Pages: 4

BT-101

(Following Paper ID and Roll No. to be filled in your Answer Book)

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Paper ID: 154111

Roll No.

B.Tech.

(SEM. I) THEORY EXAMINATION, 2015-16

ENGINEERING MATHEMATICS-I

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[Time:3 hours] [Total Marks:100]

SECTION-A

Note : Attempt all parts. All parts carry equal marks. Write answer of each part in short. (2×10=20)

1. (a) Evaluate lim _x?0 sin 6x / 5x
2. (b) Find the derivative of 1/tan x + 1/cot x .

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3. (c) State lagrange's mean value theorem.
4. (d) Find the critical points of f(x)=9x²+12x+2.
5. (e) Evaluate: ?(1-x)/x dx.
6. (f) Evaluate: ?₀^p/2 cos²xdx.
7. (g) Find the order and degree of the given differential equation y"+2y'+ sin y = 0.

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8. (h) Form the differential equation representing the family of curves y=mx, where m is the arbitrary constant.
9. (i) If 2/11 is the probability of an event, what is the probability of the event 'not A".
10. (j) If P(A)=7/13, P(B) and P(A') = 4/13, find P(A/B).

SECTION-B

Note: Attempt any five questions from this sections. (10×5=50)

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2. For the function f(x), given by f(x)= b-ax, if x>1
   4, if x = 1
   a+bx if x<1
   lim _x?1 f(x) = f(1), find the value of a and b.
3. If y=a sin t, x=a (cost+logtan(t/2)) find dy/dx .

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4. If y=3 cos(log x)+4 sine(log x), show that x²y2+xy,+y=0.
5. Integrate: e^x (sinx + cos x)
6. Solve dy/dx + 2y = x² log x
7. solve sec² x tan y dx + sec² y tan x dy = 0
8. Bag I contains 3 red and 4 black balls while another bag II contains 5 red and 6 black balls. One ball is drawn at random from one of the bags and it is found to be red. Find the probability that it was drawn from bag II.

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9. Evaluate lim _x?2 (x²-3x+2)/(x³-8)

SECTION-C

Note: Attempt any two questions from this section. (15×2=30)

10. (a) Differentiate the functions (sin x)^x + sin^-1 vx with respect to x.
11. (b) Find the point at which the tangent to the curve y = v(4x-3)-1 has its slope 2/3.

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12. (a) Integrate the function (logx)²/x
13. (b) Solve (x²+x²+x+1) dy/dx = 2x² + x; y = 1 when x = 0.
14. (c) Find the integration, the area of the region bounded by curves, y² = 4ax and x² = 4ay.
15. (a) In class XI of a school 40% of the student study Mathematics and 30% study Biology. If a student is selected at random from the class, find the probability that he will be studying Mathematics or Biology.
16. (b) In a school, there are 1000 students, out of which 430 are girls. It is know that out of 430,10% of the girls study in class XII. What is the probability that a student chosen randomly studies in class XII given that the chosen student is a girl?

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17. (c) Let X denote the no of hours you study during a randomly selected college day, the probability that X can take the value x, has the following form, where k some unknown constant:
    • 0.1, if x = 0
    • P(X = x) = kx if x =1 or x = 2
    • k(5-x), if x = 3 or 4
    • 0, otherwise
    Find the value of k and what is the probability that you study at least two hours? Exactly two hours? At most two hours?

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