GTU Calculus - December 2014 Exam Question Paper

GTU First Year Engineering (Semester 1)
Calculus
December 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

Answer the following questions with most appropriate answer

1 (a) 1 The maximum value of y = 3cos2x is ____
(a) 1
(b)2
(c)-1
(d)3.

1 M

1 (a) 2 If √x+√y=√a then dy/dx_____
a) √(y/a)
(b) √(x/a)
(c) -√(y/x)
(d) -√(y/a).

1 M

1 (a) 3 The ( a ? x ) y² = x² ( a + x ) is symmetric about
(a) X-axis
(b)Y-axis
(c)both X and Y axis (d) line Y = X

1 M

1 (a) 4 Tangents at origin to the curve y²( a + x ) = x²( 3 a ? x ) is ____
a) ±√x
(b)1
(c) ±√2x
(d) none of these.

1 M

1 (a) 5 The parametric equations of x^2/3 + y^2/3 = a^2/3 ( a ≠ 0 ) are
(a) x = a cos θ, y = a sin θ
(b)x = a 3 cos θ , y = a 3 sin θ
(c)x = a cos 3 θ , y = a sin 3 θ
(d)x=1, y=0

1 M

1 (a) 6 The area bounded by the curve y = x³ , x-axis and two ordinates x = 1 to x = 2 equal to
(a) 15/2
(b)15/3
(c)15/5
(d)15/4

1 M

1 (a) 7

$\lim_{x\to 2}\dfrac{x^{2}-x-2}{x^{2}-4}= \ \_\_\_\_\_\_$
(a) 7/4
(b)3/4
(c)2
(d)-3/4

1 M

Answer the following questions with most appropriate answer

1 (b) 1 The slope of the tangent to the curve y = xe^x at ( 0 , 0 ) is___ (a) 0
(b)1
(c)-1
(d)4

1 M

1 (b) 2 The point of inflection of

$\dfrac{x^{3}}{3}-\dfrac{x^{2}}{2}-2x+14$ is______
(a) 1/2
(b)-1/2
(c)-1
(d)-2

1 M

1 (b) 3 Using the matrix method, the solution of x + y = 2 , 4 x + y = 6

$(a)\left ( \dfrac{-4}{3},\dfrac{2}{3} \right ) (b)\left ( \dfrac{4}{3},\dfrac{-2}{2} \right ) (c)\left ( \dfrac{4}{3},\dfrac{2}{3} \right )(d)(1,2)$

1 M

1 (b) 4

$\int e^{2x} \cos 3xdx=\_\_\_\_+c$

$(a)\dfrac{e^{2x}}{13}(2 \cos3x+ 3\sin 3x)(b)\dfrac{e^{2x}}{13}(3 \cos3x+ 2\sin3x) (c)\dfrac{e^{2x}}{13}(2 \cos3x+ 3\sin 3x)$ (d) none of these.

1 M

1 (b) 5 If y = ln√(2x) the derivative of the function y with respect to x is ____
(a) 0
(b)-1/2x (
(c) 1/2x
(d)1/x

1 M

1 (b) 6 f ( x ) = x²+ 4 x + 5 has the minimum value ____
(a) 0
(b)1
(c)-1
(d)2

1 M

1 (b) 7 A curve which passes through the origin and has the slope -1/3 is given by
(a) x + 3 y ? 1 = 0 (b) x + 3 y = 0
(c)x ? 3 y = 0
(d)none of these

1 M

Answer the given MCQ.

1 (c) 1 F(x) is strictly increasing function on R then______
A) f(x) = 0 for all x
B) f(x) > 0 for all x
C) f(x) < 0 for all x
D) none of these.

1 M

1 (c) 2 F(x) is strictly decreasing function on R then______
A) f(x) = 0 for all x
B) f(x) > 0 for all x
C) f(x) < 0 for all x
D) none of these.

1 M

1 (c) 3 dy/dx = ky, k > 0 is the deferential equation for_______
A) Population Model
C) Cooling Model
B) Mixing problem model
D) none of these.

1 M

1 (c) 4 dy/dx = x then y = _______
(a) y=x²

$(b)y=\dfrac{\infty^{2} }{0}+c$ (c) y=x (d) none of these.

1 M

1 (c) 5 Curve of y = x² +3 is______
A) Symmetric with respect to x axis.
B) Symmetric with respect to y axis.
C) Symmetric with respect to origin.
D) none of these.

1 M

1 (c) 6 r = a cosθ is_____
(a) Line (b) laminiscate (c) circle (d) None of these.

1 M

1 (c) 7

$\int_{0}^{1}x^{2}dx$ is_____
(a) area under a line (b) area under circle (c) area under parabola (d) none of these.

1 M

Answer the given MCQ.

1 (d) 1

$\int_{a}^{-a} f(x)dx=0$ if_________
(a) f is an odd function (b) f is neither odd nor even function (c) f is an even function (d) none of these.

1 M

1 (d) 2 z = x² + y² is ________
A) Cone
B) Paraboloid
C) Sphere
D) None of there.

1 M

1 (d) 3

$\lim_{(x,y)\to(0,0)}\dfrac{x^{2}-yx}{x+y}=$ _____
A)2
B) 1
C) 0
D) -1

1 M

1 (d) 4

$If z=x^{2}-y^{2}then \dfrac{\partial z }{\partial x}=$ _______
A)2y
B) 0
C) 2z
D) none of these.

1 M

1 (d) 5 Equation of tangent plane of z = x at (2,0,2) is
A)z=x
B) x+y+z=2
C) x+z=0
D) none of these.

1 M

1 (d) 6 If z = x² +y² + 3 minimum value z is ______
A) 3
B) ∞
C) 0
D) none of these.

1 M

1 (d) 7 Y = sin2x is increasing in interval _________
A) (0,π)

$B) (0,\dfrac{\pi}{4})\ (c)\ (0,\dfrac{\pi}{2})$
D) none of these.

1 M

2 (a) Test the convergence of

$\dfrac{1}{1.2.3}+\dfrac{3}{2.3.4}+\dfrac{5}{3.4.5}+.........$ .

3 M

2 (b) Prove that

$1+\dfrac{2}{3}+\dfrac{4}{9}+\dfrac{8}{27}+\dfrac{16}{81}+......$ converges and find its sum.

4 M

2 (c) State Taylor's series for one variable and hence find √(36.12).

7 M

3 (a) Find

$\dfrac{\partial w }{\partial r},\dfrac{\partial w }{\partial s}$ in terns of r and s if w =2y+z² where

$x=\dfrac{r}{s},y=r^{2}+ln s, z=2r$ .

3 M

3 (c)

$If u=\cos ec^{-1}\left [ \dfrac{x^{1/2}+y^{1/2}}{x^{1/3}+y^{1/3}} \right ]^{1/2} show that x^{2}\dfrac{\partial^2 u}{\partial x^2}+2xy\dfrac{\partial^2 u}{\partial x \partial y}+y^{2}\dfrac{\partial^2 u}{\partial y^2}=\dfrac{\tan u}{144}(13+\tan ^{2} u)$ . Also state Euler's modified theorem.

7 M

4 (a) Evaluate

$\lim_{x\to 0}\left [ \dfrac{1}{x^{2}}-\dfrac{1}{\sin^{2}x} \right ]$ .

3 M

4 (b) Expand e^x sin y in powers of x and y upto third degree.

4 M

4 (c) A rectangular box, open at the top, is to have a volume 32 c.c. Find the dimensions of the box requiring least material for its construction.

7 M

5 (a) Find the volume of the tetrahedron bounded by the plane x + y + z = 2 and the planes x = 0 , y = 0 and z = 0 .

3 M

5 (b) Trace the curve r = a ( 1 + cos θ ).

4 M

5 (c) Evaluate

$\int_{0}^{4a}\int_{x^{2}/4a}^{2\sqrt{ax}} \limits dydx$ by changing the order of integration.

7 M

6 (a) Evaluate

$\int_{0}^{\infty }\dfrac{dx}{x^{2}+1}$ .

3 M

6 (b) Evaluate the integral

$\int_{0}^{1}\int_{0}^{1-x}\limits e^{\dfrac{y}{e^{x+y}}}$ dydx by changing the variables x + y = u , y = uv.

4 M

6 (c) Evaluate

$\iint_{R}\limits x^{2}$ dA where R is the region in the first quadrant bounded by the hyperbola xy = 16 and the lines y = x , y = 0 and x = 8 .

7 M

7 (a) Test the convergence of

$\sum_{n=1}^{\infty }\dfrac{2tan^{-1}n}{1+n^{2}}$ .

3 M

7 (b) Find the equation of tangent plane and normal line to the surface xyz = 6 at ( 1 , 2 , 3 )

4 M

7 (c) Evaluate

$\int_{0}^{1}\limits \int_{0}^{\sqrt{1-x^{2}}}\limits\int_{0}^{\sqrt{1-x^{2}-y^{2}}}\limits$ xyz dzdydx.

7 M

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