GTU Calculus - June 2015 Exam Question Paper

GTU First Year Engineering (Semester 1)
Calculus
June 2015

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

Attempt the following.

1 (a) 1 Maclaurin series of sin x is

$a) \sum_{n=0}^{\infty }\dfrac{x^{2n+1}}{(2n+1)!}\ b)\sum_{n=0}^{\infty }\dfrac{x^{2n+1}}{(2n+1)!}\ c)\sum_{n=0}^{\infty }\dfrac{x^{2n}}{(2n)!}\ d)\sum_{n=0}^{\infty }(-1)^{n}\dfrac{x^{2n}}{(2n)!}$ .

1 M

1 (a) 2 The total area enclosed between y=sin x and X-axis in [0,π] is
a) 0
b) 4
c) 2
d) 1.

1 M

1 (a) 3 The value of

$\lim_{x\to 2}{x^{1/x}}$
a) ∞
(b) -∞
(c) 1
(d) 0,

1 M

1 (a) 4 How many leaves r=sin5 θ has?
a) 6
(b) 2
(c) 3
(d) 5.

1 M

1 (a) 5 The sum of the series \\sum_{n=1}^{\infty }\dfrac{1}{2^{n}}\] is
a) 0
(b) 1
(c) -1
(d) 1/2.

1 M

1 (a) 6 The values of x at which the curves y=x& y=4x-x² intersects each other are
a) 0 & 4
b) -4 & 4
c) 0 & 3
d) -3 & 3.

1 M

1 (a) 7 The area enclosed between the curve r=f(θ) and two rays θ=α & θ=β is

$a) \int_{\alpha }^{\beta }\limits r^{2}d\theta \ b)\int_{\alpha }^{\beta }\limits r^{3}d\theta \ c)\dfrac{4\pi}{3} \int_{\alpha }^{\beta }\limits r^{3}d\theta \ d) \dfrac{1}{2} \int_{\alpha }^{\beta }\limits r^{2}d\theta$ .

1 M

Attempt the following.

1 (b) 1 If

$u=\sin^{-1}\dfrac{x}{y}+\tan^{-1}\dfrac{y}{x}$ the the value of xu_x+yu_y is
a) u
(b) -u
(c) 0
(d) 1.

1 M

1 (b) 2 For an implicit function f(x,y)=c the value of

$\dfrac{dy}{dx}$ is

$a) \dfrac{f_{x}}{f_{y}} \ b)\dfrac{f_{y}}{f_{x}} \ c)-\dfrac{f_{x}}{f_{y}} \ d)-\dfrac{f_{y}}{f_{x}}$ .

1 M

1 (b) 3 For the function u=(x²+y²)^1/3 the value of x²u_xx+2xyu_xy+y²u_yy is

$a) -2u\ (b)-\dfrac{2u}{3} \ (c)\dfrac{u}{9} \ (d)-\dfrac{2u}{9}$ .

1 M

1 (b) 4

$If\ x=r \cos \theta \ and\ y=r \sin \theta \ then\ J=\dfrac{\partial (r,\theta)}{\partial (x,y)}$ is

$a) r\ b) -r\ (c) \dfrac{1}{r}\ (d)-\dfrac{1}{r}$ .

1 M

1 (b) 5 The fundamental period of sin 2x is
a) π
(b) 2π
(c) 4π
(d) π/2.

1 M

1 (b) 6 The focus pf parabola y²4ax is (ae,0) if
a) e<1
(b) e>1
(c) e=1
(d) e≠1.

1 M

1 (b) 7 The function f(x)=2x³+3x²-12x+7 is decreasing in
a) [-2,1]
(b) R-[-2,1]
(c) [0,2]
(d) [1,3].

1 M

Objective Question (MCQ)

1 (c) 1

$\lim_{x\to 2}\dfrac{x^{2}-4}{x^{2}+4}=$
a) 1
(b) 0
(c) -1/2
(d) None of these.

1 M

1 (c) 10 Use matrices to solve the following simultaneous equations x+2 y= 3
2x+3 y=5
(a) x= 2 ; y=1
(b)x=1 ; y=1 (
(c) x=2 ; y=2
(d)None of these.

1 M

1 (c) 11 Determine the area bounded by the x axis, the curve y=2x²+x-6, and the line x=4 and x=6
a) 298
(b) 126

$c)\ 99\dfrac{1}{3}\ (d)\ 26\dfrac{2}{3}$ .

1 M

1 (c) 12 Given the function f ( x )=x² which value of c satisfies the conclusion of mean value theorem on the interval [-4, 5]?
(a) 0
(b)1
(c)1/2
(d)None of these.

1 M

1 (c) 13 Eliminate the parameter in the equations x=t²; y=t⁴
a) y=x² for x≥0
(b) y=√(x) for x≥0
(c) y=2x² for x≥0
(d) None of these.

1 M

1 (c) 14 At how many places does the curve x=cos 3 t ; y=sin t cross the x-axis?
a) 2
(b) 1
(c) 3
(d) None of these.

1 M

1 (c) 2

$f(x)\dfrac{x^{2}-x}{2x};x\ne 0 \\ f(0)=k$
If
and if f is continuous on x=0 then k=
a) -1
(b) -1/2
(c) 0
(d) None of these.

1 M

1 (c) 3 What is the slope of the tangent line to the curve x+y=xy at point (2, 2)
a) -1
(b) -2
(c) -3
(d) -4

1 M

1 (c) 4 Determine the second derivative of the function f(x)=x². 1n 2x
a) 21n2x+3
(b)

$21n2x+\dfrac{3}{2}$
(c) 0
(d) None of these.

1 M

1 (c) 5 At a minimum, the second differential function of the form y=axⁿ+bx^n-1+...... is
a) Positive
(b)Negative
(c)Zero
(d)Infinite.

1 M

1 (c) 6

$If y=\dfrac{3}{x^{4}}\ then \ \int \ y\ dx\\ a) \dfrac{15}{x^{5}}+c\ b) -\dfrac{1}{x^{3}}+c\ c)\dfrac{12}{x^{3}}+c\ d) None \ of \ these$ .

1 M

1 (c) 7 Evaluate

$y=\int_{0}^{\dfrac{\pi}{2}}\limits \ \sin(4x) \sin(6x)dx$

$a) y=\dfrac{\pi}{2}\ b) y=\dfrac{\pi}{4}\ c) y=\dfrac{3\pi}{4}\ d y=0$ .

1 M

1 (c) 8 Use matrices to solve the following simultaneous equations
x+2z=9 4x+2y+z=14 x+3y+4 z=26
(a) x=3; y=-1/2;z=3
(b)x=-1;y=1;z=6 (
(c) x=1;y=3;z=4
(d)x=1;y=4;z=2.

1 M

1 (c) 9 Determine the area bounded by The x axis, the curve y=sin 2 x ,and the line

$x=\dfrac{\pi}{4} \ and\ x=\pi/2$
a) 0.5
(b) 1
(c) 0.25
(d) 1.5.

1 M

2 (a) Show that the P-series

$\sum_{n=1}^{\infty }\dfrac{1}{(n)^{p}}$ (P- a real constant ) converges if P>1 and diverges if p≤1.

3 M

2 (b) Define the Geometric series and find the sum of the following series

$\sum_{n=1}^{\infty }\dfrac{3^{n-1}-1}{6^{n-1}}$ .

4 M

2 (c) 1)Investigate the convergence of the series

$\sum_{n=1}^{\infty }\dfrac{2^{n}+5}{3^{n}}$
2) Is the series

$\sum_{n=1}^{\infty }\dfrac{2n+1}{(n+1)^{2}}$ converges or diverges?

7 M

3 (a) Define Taylor?s series for the function of one variable and using it show that

$\tan ^{-1}(x+h)= \tan^{-1}x+(h\sin \alpha)\dfrac{\sin \alpha}{1}-(h \sin \alpha)^{2}\dfrac{\sin 2 \alpha}{2}+(h \sin \alpha)^{3} \dfrac{\sin 3 \alpha}{3}+----\ where\ \alpha=\cot ^{-1}x$ .

3 M

3 (b) Trace the curve 9ay²=x(x-3a)².

4 M

3 (c) Attempt the following.
1)

$\lim_{x\to 2}\left [ \dfrac{a^{x}+b^{x}+c^{x}}{3} \right ]^{1/3x}$
2)Define volume of solid of revolution by Washer's method and use it to find the volume of solid generated when the region between the graphs

$f(x)=\dfrac{1}{2}+x^{2}$ and g(x)=x over the interval [0,2] is revolved about x-axis.

7 M

4 (a) Define Improper integral of both the kinds. Check the convergence of

$\int_{0}^{3}\limits \dfrac{dx}{\sqrt{9-x^{2}}}$ .

3 M

4 (b) Trace the curve r²=a² cos 2θ.

4 M

Attempt the following.

4 (c) 1) Evaluate

$\lim_{x\to 2}\left [ \dfrac{1}{x^{2}}-\dfrac{1}{\sin ^{2}x} \right ]$
2)Define the volume of solid of revolution by disk and use it to find the volume of the solid that is obtained when the region under the curve y=√(x) over the Interval [1,4]is revolved about X- axis.

7 M

5 (a) Define Homogeneous function of two variables x and y of degree n . Also prove the following Euler's theorem for this homogeneous function of degree n .

$x\dfrac{\partial u}{\partial x}+ y\dfrac{\partial u}{\partial y}=nu$

3 M

5 (b) If

$u=\tan ^{-1}\left [ \dfrac{x^{3}+y^{3}}{x-y} \right ]$ then prove that

$x^{2}{\dfrac{\partial^2 u}{\partial x^{2}}}+2xy{\dfrac{\partial^2 u}{\partial x \partial y}}+y^{2}{\dfrac{\partial^2 u}{\partial y^{2}}}=2 \cos3u\sin u$ .

4 M

Attempt the following.

5 (c) 1) If x=ρsinφcosφ, y=ρsinφsinφand z=ρcosφ then obtain the Jacobean

$J=\dfrac{\partial (x,y,z) }{\partial (\rho, \phi, \theta)}$
2) In a plane triangle ABC find the extreme values of cos A cos B cos C .

7 M

6 (a) If u=sin^-1(x-y),x=3t.y=4t³ then show that

$\dfrac{du}{dt}=\dfrac{3}{\sqrt{1-t^{2}}}$ .

3 M

6 (b) If u=f(r) and r²=x²+y²+z² then show that

$\dfrac{\partial^2u }{\partial x^2}+\dfrac{\partial^2u }{\partial y^2}+\dfrac{\partial^2u }{\partial z^2}=f(r)+\dfrac{2}{r}f(r)$ .

4 M

Attempt the following.

6 (c) 1) Find the equations of the tangent plane and normal line to the surface
2xz²-3xy-4x=7 at (1,-1,2).
2) Find the minimum value of x²+y²+z², given that ax+by+cz=p.

7 M

7 (a) Evaluate

$\iint_{R}\limits \left ( 2x-y^{2} \right )dA$ over the triangular region R enclosed between the lines y=-x+1,y=x+1& y=3.

3 M

7 (b) Evaluate the integral

$\int_{0}^{2}\limits \int_{y/2}^{1}\limits e^{x^{2}}$ dxdy by changing the order of integration.

4 M

Attempt the following.

7 (c) 1) Use double integral in polar integral to find the area enclosed by three petalled rose r= sin 3θ.
2).Use triple integral to find the volume of the solid within the cylinder x²+ y²=9 between the planes z= 1 and x+z=1.

7 M

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