Download GTU BE/B.Tech 2019 Summer 1st And 2nd Sem (New And SPFU) 2110014 Calculus Question Paper

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GUJARAT TECHNOLOGICAL UNIVERSITY
- SEMESTER-I & II (NEW) EXAMINATION — SUMMER-2019

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Instructions:

1. Question No.1 is compulsory. Attempt any four out of remaining six questions.

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2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.

Q.1 Objective Question (MCQ) Marks

1. (a) 07
   1. For the Jacobian J, value of the J - J^-1 is
      (a)1 (b) -1 (c)0 (d)2

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   2. Value of dy/dx for ax²+ 2hxy + by²=1 is
      (a) (hx+by)/(ax+hy) (b) (ax+hy)/(hx+by) (c) (ax+hy)/(-hx-by) (d) (-hx-by)/(ax+hy)
   3. u=sin^-1(x/y) is a homogeneous function of degree
      (a)1/2 (b)0 (c)1 (d) -1
   4. The curve r = 2 is
      

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      (a) straight line (b) point at distance ‘2’ on initial line
      (c) circle with centre origin and radius 2 (d) cardioid
   5. If x = rcos?, y = rsin? then which is correct?
      (a) r=v(x²+y²), ?=tan^-1(x/y) (b) r=v(x²+y²), ?=tan^-1(y/x)
      (c) r=x²+y², ?=tan^-1(y/x) (d) r=x²+y², ?=tan^-1(x/y)

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   6. Infinite Sequence {1,1,1, ...} is
      (a) convergent (b) divergent (c) oscillatory (d) None of these
   7. Infinite Series 1+ 1+ 1+ ... is
      (a) convergent (b) divergent (c) oscillatory (d) None of these

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2. (b) 07
   1. Infinite series 1-1/2+1/3-1/4+1/5--- is
      (a) convergent (b) divergent (c) oscillatory (d) None of these
   2. Curve (y — 1)² = (x — 5) is symmetric to
      (a) X-axis (b) line y = —x (c) line y = x (d) Y- axis
   3. lim_x?0 (tan mx)/x is
      

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      (a) 1/m (b) 0 (c) m (d) m
   4. The sum of the series ?_n=0⁸ xⁿ/n! is
      (a) e^x (b)1/2 (c) 2 (d) 1
   5. The Maclaurin series for the function (x + 1)^x is
      (a) 1+x+x² (b) 1+2x+x² (c) 1+x (d) x+x²

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   6. The straight line y = 2 is revolved about x- axis between 0 = x = 4. The generated solid is
      (a) cone (b) sphere (c) cuboid (d) cylinder
   7. For a series ? a_n, if lim_n?8 a_n ? 0, then
      (a) series is convergent (b) series is divergent
      (c) sum of series is finite number (d) series is conditionally convergent

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Q.2 (a) Find the Taylor series for f(x) = e^x at a = 2. 03

(b) Is the series absolutely convergent or conditionally convergent? 04
1 - 1/v2 + 1/v3 - 1/v4 + ...

(c) (i) Discuss the convergence of the series 04

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(ii) Find the Radius of convergence for the series ?_n=1⁸ n² xⁿ .

Q.3 (a) Evaluate lim_x?0 xlogx 03

(b) Trace the curve y²(a+ x) = x²(a —x), a> 0. 04

(c) Prove that the series ? 1/n^p is convergent if p > 1 and divergent if p < 1 07

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Q.4 (a) Evaluate ? x³ e^x dx 03

(b) Find the equation of the tangent plane and normal line to the surface x²+ y²+z²-9=0 at (1,2,4). 04

(c) (i)Evaluate ? sinⁿ x dx 04
(ii) Evaluate lim_x?0 (1 —cos x)/x² 03

Q.5 (a) If u=f(x—y,y—z,z—x) prove that u_x+u_y+u_z =0 03

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(b) Find maximum and minimum values. 04
f(x,y) =(x² - y²) —x³ +y³

(c) If u=tan^-1((x³+y³)/(x+y)) 07
(i) xu_x + yu_y = sin 2u
(ii) x²u_xx + 2xyu_xy + y²u_yy =2 sinucos3u

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Q.6 (a) The region between the curve y = vx, 0 = x = 4 and the x-axis is revolved about the x-axis to generate a solid. Find its volume. 03

(b) Using volume by-slicing method, find the volume of a cylinder with radius ‘r” and height ‘h’ . 04

(c) Evaluate ?_R (y/x) dxdy, R is triangle (0,0),(1,0),(1,1) using transformations x = u, y = uv. 07

Q.7 (a) Evaluate ? r³ drd? over the area bounded between the circles r = 2cos? and r = 4cos?. 03

(b) Evaluate 04

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(c) Change the order of integration and evaluate. 07
?₀¹ ?_x^2-x xy dy dx

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