GTU Calculus - June 2014 Exam Question Paper

GTU First Year Engineering (Semester 1)
Calculus
June 2014

Total marks: --
Total time: --

INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

1 (a) (i) The value of

lim_{x \to 0} x \sin \frac{1}{x} i s

$\lim_{x\rightarrow 0}x \sin \dfrac {1}{x}\ is$ (A) 1
(B) π
(C) 0
(D) ∞

1 M

1 (a) (ii)

y = \log (\sin x) t h e n v a l u e o f \frac{d^{2} y}{d x^{2}} =

$y=\log (\sin x )\ then \ value \ of \ \dfrac {d^2y}{dx^2}=\$
(A) sec² x
(B) -cosec² x
(C) -cosec xcotx
(D) secxtanx

1 M

1 (a) (iii) Curve: x=t²-1, y=t²-t. For which value of t tangents of the curve are parallel to the x-axis?

(A) t = 0 (B) t = \frac{1}{\sqrt{3}} (C) t = \frac{1}{2} (D) t = - \frac{1}{\sqrt{3}}

$(A) \ t=0 \\ (B)\ t=\dfrac {1} {\sqrt{3}} \\ (C)\ t=\dfrac {1}{2}\\ (D) \ t=-\dfrac {1}{\sqrt{3}}$

1 M

1 (a) (iv)

\int_{- 1}^{1} x | x | d x = . . . . . . .

$\int^{1}_{-1}x|x| dx= .......$ (A) 2 (B) 1 (C) 0 (D) None of these

1 M

1 (a) (v)

\begin{aligned} I f f (x) & = 4 - 2 x, x < 1 \\ = 6 x - 4, x \geq 1 t h e n f i n d lim_{x \to 1} f (x) \end{aligned}

$\begin {align*} If \ f(x) &=4-2x,x<1 \\ &=6x-4, x \ge 1 \ then \ find \ \lim_{x\rightarrow 1} f(x) \end{align*}$ (A) -2 (B) 2 (C) 0 (D) doesn't exist.

1 M

1 (a) (vi) If f:R → R, f(x) = 3x+2, then find f^-1
(A) Not possible
(B) 3x-2
(C) 2x-3

(D) \frac{x - 2}{3}

$(D) \ \dfrac {x-2}{3}$

1 M

1 (a) (vii) What is the solution for following equations ?
2x+3y-5=0, 4x+6y-7=0
(A) No solution
(B) Infinitely many solution
(C) unique solution
(D) { (0,0), (-5, -7) }

1 M

1 (b) (i)

lim_{n \to \infty} \frac{1 - n^{2}}{\sum n} =

$\lim_{n\to \infty}\dfrac {1-n^2}{\sum n}=$ (A) -2 (B) 1 (C) 2 (D) doesn't exist

1 M

1 (b) (ii)

\int \frac{1}{\sqrt{x}} \cos \sqrt{x} d x = . . . . + c (A) \frac{2}{3} \cos^{\frac{3}{2}} x (B) 2 \sqrt{x} (C) 2 \sin \sqrt{x} (D) \frac{2}{3} \cos \sqrt{x}

$\int \dfrac {1}{\sqrt{x}}\cos \sqrt{x}dx=....+c \\(A) \ \dfrac {2}{3}\cos^{\frac {3}{2}}x \\ (B) \ 2 \sqrt{x} \\ (C) 2 \sin \sqrt{x} \\ (D) \ \dfrac {2}{3} \cos \sqrt{x}$

1 M

1 (b) (iii) Find the area of the curve y=x² + 1 bounded by x-axis and lines x=1 and x=2.

(A) \frac{3}{10} (B) \frac{10}{3} (C) 6 (D) \frac{1}{6}

$(A) \ \dfrac {3}{10} \\ (B) \ \dfrac {10}{3} \\ (C) 6 \\ (D) \ \dfrac {1}{6}$

1 M

1 (b) (iv) f(x) = [x], x ∈ R then f(x) is
(A) Continuous for all real numbers.
(B) Continuous for all integers
(C) discontinuous for all integers
(D) none of these

1 M

1 (b) (v) Find the vertical asymptotes of the curve

y = \frac{2 x^{z}}{x^{2} - 1}

$y=\dfrac {2x^z}{x^2-1}$ (A) x=1
(B) x=-1
(C) x= ± 1
(D) y=2

1 M

1 (b) (vi) f(x) =x³-3x²+3x-100 is ....... function.
(A) Increasing
(B) decreasing
(C) constant
(D) none

1 M

1 (b) (vii) Find the value of c using Lagrange's mean value theorem for f(x)=e^x, x ∈ [0,1]
(A) log 1
(B) log c
(C) log (1-e)
(D) log (e-1)

1 M

2 (a) (i)

E v a l u a t e lim_{x \to 0} \frac{e^{x} + e^{- x} - x^{2} - 2}{\sin^{2} x - x^{2}}

$Evaluate \ \lim_{x\rightarrow 0}\dfrac {e^x+e^{-x}-x^2-2}{\sin^2 x-x^2}$

2 M

2 (a) (ii) Test for convergence the series

\frac{1}{1.2} + \frac{1}{2.3} + \frac{1}{3.4} + . . . . .

$\dfrac {1}{1.2}+ \dfrac {1}{2.3}+ \dfrac {1}{3.4}+ .....$

2 M

2 (a) (iii) Expand log sin x in powers of (x-2)

3 M

2 (b) (i) Test for convergence the series

2 + \frac{3}{2} x + \frac{4}{3} x^{2} + \frac{5}{4} x^{3} + . . . . . .

$2+\dfrac {3}{2}x+\dfrac {4}{3}x^2+\dfrac {5}{4}x^3+......$

4 M

2 (b) (ii) Trace the curve r=a sin 3θ

3 M

3 (a) (i)

(i) E v a l u a t e lim_{x \to 0} (a^{x} + x)^{\frac{1}{x}}

$(i) \ Evaluate \ \lim_{x\rightarrow 0}(a^x+x)^{\frac {1}{x}}$
(ii) Test the convergence of

\sum_{n = 0}^{\infty} \frac{3^{2 n}}{2^{3 n}}

$\sum^{\infty}_{n=0} \dfrac {3^{2n}}{2^{3n}}$

4 M

3 (a) (ii) Trace the curve x³ + y³ = 3axy.

3 M

3 (b) (i) Test the convergence of

\sum_{n = 1}^{\infty} \frac{[(n + 1) x]^{n}}{n^{n} + 1}

$\sum^{\infty}_{n=1} \dfrac {[(n+1)x]^n}{n^n+1}$

4 M

3 (b) (ii) Evaluate improper integral

\int_{0}^{3} \frac{1}{\sqrt{3 - x}} d x

$\int^{3}_0 \dfrac {1}{\sqrt{3-x}}dx$

3 M

4 (a) (i)

(i) I f u = l n (\frac{x^{7} + y^{7} + z^{7}}{x + y + z}) s h o w t h a t x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} + z \frac{\partial u}{\partial z} = 6

$(i)\ If \ u=ln \left ( \dfrac {x^7+y^7+z^7}{x+y+z}\right ) \ show \ that \ x\dfrac {\partial u}{\partial x}+ y\dfrac {\partial u}{\partial y}+z\dfrac {\partial u}{\partial z}=6$
(ii) State modified Euler's theorem and find the degree of homogeneous function

u (x, y) = \frac{1}{x^{2}} + \frac{1}{x y} + \frac{\log x - \log y}{x^{2}}

$u(x,y)=\dfrac {1}{x^2}+\dfrac {1}{xy}+\dfrac {\log x-\log y}{x^2}$

4 M

4 (a) (ii) z=f(x,y), xe^u + e^-v, y=e^-u - e^v then prove that

\frac{\partial z}{\partial z} - \frac{\partial z}{\partial v} = x \frac{\partial z}{\partial x} - y \frac{\partial u}{\partial y}

$\dfrac {\partial z}{\partial z}-\dfrac {\partial z}{\partial v}=x\dfrac {\partial z}{\partial x} -y\dfrac {\partial u}{\partial y}$

3 M

4 (b) (i)

I f u = e^{3 x y z} s h o w t h a t \frac{\partial^{3} u}{\partial x \partial y \partial z} = (3 + 27 x y z + 27 x^{2} y^{2} z^{2}) e^{3 x y z}

$If \ u=e^{3xyz} \ show \ that \ \dfrac {\partial^3 u}{\partial x \partial y \partial z}= (3+27xyz+27x^2y^2z^2)e^{3xyz}$

4 M

4 (b) (ii)

\begin{aligned} s h o w t h a t f (x, y) & = \frac{x y}{\sqrt{x^{2} + y^{2}}}; & (x, y) e (0, 0) \\ = 0, & (x, y) = (0, 0) \end{aligned}

$\begin {align*} show \ that \ f(x,y)&=\dfrac {xy}{\sqrt{x^2+y^2}}; \ &(x,y)e (0,0) \\ &=0, \ &(x,y)=(0,0) \end{align*}$ is continuous at the origin.

3 M

5 (a) (i) Find the extreme values of the function
f(x,y) = x³+y³-3x-12y+20.

4 M

5 (a) (ii) Show that the surface z=xy-2 and x²+y²+z² =3 have the same tangent plane at (1,1,1).

3 M

5 (b) (i) Find the minimum value of x² y z³ subject to the condition 2x+y+3z=a using Lagrange's method of undetermined multipliers.

4 M

5 (b) (ii) Expand sin xy in powers of (x-1) and (y- π/2) upto second degree terms.

3 M

6 (a) (i) Sketch the region of integration, reverse the order of integration and evaluate

\int_{0}^{2} \int_{0}^{4 - x^{2}} \frac{x e^{2 y}}{4 - y} d y d x .

$\int^{2}_0\int^{4-x^2}_0 \dfrac {xe^{2y}}{4-y}dydx.$

4 M

6 (a) (ii) Evaluate the integral

\int_{0}^{\frac{π}{2}} \int_{0}^{1 - \sin θ} r^{2} \cos θ d r d θ

$\int^{\frac {\pi}{2}}_0\int^{1-\sin\theta}_0 r^2 \cos\theta drd\theta$

3 M

6 (b) (i) Evaluate the integral

\int_{0}^{2} \int_{1}^{2} \int_{0}^{y z} x y z d x d y d x .

$\int^2_0\int^2_1\int^{yz}_0 xyz \ dxdydx.$

4 M

6 (b) (ii) Evaluate the integral

\int_{0}^{a} \int_{0}^{\sqrt{a^{2} - y^{2}}} y^{2} \sqrt{x^{2} + y^{2}} d y d x .

$\int^a_0\int^{\sqrt{a^2-y^2}}_0 y^2\sqrt{x^2+y^2}dydx.$ by changing into polar co-ordinates

3 M

7 (a) (i) Find the volume bounded by the cylinder x² + y² = 4 and the planes y+z=3, z=0.

4 M

7 (a) (ii) Evaluate ∬_R x²+y² dA by changing the variables ,where R is the region lying in first quadrant and bounded by the hyperbola
x² - y²=1, x² - y²=9, xy=2 and xy=4.

3 M

7 (b) (i) The region between the curve y=√x, 0≤x≤4 and the x-axis is revolved about the x-axis to generate a solid. Find its volume.

4 M

7 (b) (ii) Test the convergence of the series

\sum_{n = 1}^{\infty} \frac{(- 1)^{n + 1}}{\log n + 1}

$\sum^{\infty}_{n=1}\dfrac {(-1)^{n+1}}{\log n+1}$

3 M

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