Download DBATU B-Tech 1st Year 2017 Dec Engineering Mathematics I 3 Question Paper

FirstRanker.com

DR. BABASAHEB AMBEDKAR TECHNOLOGICAL UNIVERSITY, LONERE - RAIGAD - 402 103

Semester Examination: December - 2017

--- Content provided by‌ FirstRanker.com ---

Branch: All Courses Semester: I

Subject with Subject Code: Engineering Mathematics-I Marks: 60 (MATH101)

Date: 11/12/2017 Time: 3 Hrs.

Instructions to the Students:-

1. Each question carries 12 marks.

--- Content provided by‍ FirstRanker.com ---

2. Attempt any five questions of the following.
3. Illustrate your answers with neat sketches, diagram etc., wherever necessary.
4. If some part or parameter is noticed to be missing, you may appropriately assume it and should mention it clearly.

Q.1. (a) For what value of ? the following system of linear equations is consistent and solve it completely in each case: (06)

x+y+z=1, x+2y+4z= ?, x+4y+10z= ?²

--- Content provided by⁠ FirstRanker.com ---

(b) Find the eigen values and the corresponding eigen vectors for the matrix (06)

A= | 0 5 4 | | -4 4 3 |

Q.2. (a) If y=sin^px+cos^px , then prove that y_n=pⁿ[1+(-1)ⁿsin{2px}]/2 . (04)

(b) If y=e^{m sin-1x} , then prove that (1-x²)y_n+2 - (2n+1)xy_n+1 - (m²+n²)y_n =0 (04)

(c) Expand y = log|cos x| about the point X = ?/3 up to third degree by using Taylor’s series. (04)

--- Content provided by‌ FirstRanker.com ---

FirstRanker.com

Q3. Attempt Any Three: (12)

(a) If x^ay^bz^c=c , then prove that at point x=y=z

(b) If u=tan^-1((x³+y³)/(x-y)) , prove that x²(?²u/?x²)+2xy(?²u/?x?y)+y²(?²u/?y²) = sin⁴u-sin²u

(c) If x²=au+bv, y²=au-bv and z=f[u,v] , then prove that x(?z/?x)+y(?z/?y)=2(u(?z/?u)+v(?z/?v)) .

--- Content provided by​ FirstRanker.com ---

(d) If z=sin(xy) where x=e^t, y=t² , then find dz/dt

Q4. Attempt Any Three: (12)

(a) If u=yz, v=zx, w=xy , then prove that J.J* = 1 where J=?(u,v,w)/?(x,y,z) and J*=?(x,y,z)/?(u,v,w)

(b) If the sides and angles of a plane triangle vary in such a that its circum-radius remains constant, then prove that da/cosA + db/cosB + dc/cosC = 0

(c) A rectangular box open at the top is to have volume of 32cubic units. Find the dimensions of the box requiring the least material for its construction by Lagrange’s method of undetermined multipliers.

--- Content provided by‍ FirstRanker.com ---

(d) Expand f(x,y)=x^y as for as second degree in the powers of (x—1) and (y-1) using Taylor's theorem.

Q5. Attempt Any Three: (12)

(a) Change the order of integration and evaluate I= ?₀¹ ?_x^2-x xy dy dx

(b) Use elliptical polar form to evaluate I = ?_R xy v(x²/2 + y²/3) dxdy , where R is the region of ellipse in positive quadrant.

(c) Use spherical polar transformation to evaluate I= ?₀^a ?₀^v(a2-x2) ?₀^v(a2-x2-y2) 1/v(x²+y²+z²) dz dy dx

--- Content provided by‍ FirstRanker.com ---

FirstRanker.com

(d) Find the centroid of the positive loop of the curve r²=acos2? .

Q.6. (a) Test the convergence of the series S ((n+1)/(n+2))ⁿ (04)

(b) Test the convergence of the series S v(n+1)-vn (04)

(c) Test the absolute convergence of the series S n((ln(n))²/n³) (04)

--- Content provided by‍ FirstRanker.com ---