Mathematics 1 - May 2014

First Year Engineering (Semester 1)

TOTAL MARKS:
TOTAL TIME: HOURS

1 Find the Eigen values of the inverse of the matrix

$$A=\begin{bmatrix} 2&1 &0 \\0 &3 &4 \\0 &0 &4 \end{bmatrix}$$ (2 marks) 10 $$Evaluate \ \int^{\frac {\pi}{2}}_0\int^{\sin \theta}_0 rd\theta dr$$ (2 marks)

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Answer any one question from Q11 (a) & Q11 (b)

11.(a) (i) Verify Cayley Hamilton theorem for the matrix

$$\begin{bmatrix} 1&0 &3 \\2 &1 &-1 \\1 &-1 &1 \end{bmatrix}$$ hence find it?s A^-1(8 marks) 11.(a) (ii) Find the Eigen values and Eigen vectors of $$\begin{bmatrix} -2&2 &-3 \\2 &1 &-6 \\-1 &-2 &0 \end{bmatrix}$$ (8 marks) 11.(b) (i) Reduce the quadratic form $$x^2_1+5x^2_2+x^2_3+2x_1x_2+2x_2x_3+6x_3x_1$$ to the canonical form through orthogonal transformation and find its nature.(10 marks) 11.(b) (ii) Prove that the Eigen values of a real symmetric matrix are real.(6 marks)

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Answer any one question from Q12 (a) & Q12 (b)

12.(a) (i) Prove that the harmonic series is divergent(8 marks) 12.(a) (ii) Test the convergence of the series

$$\dfrac {1}{4.7.10}+\dfrac {4}{7.10.13}+\dfrac {9}{10.13.16}+.....$$ (8 marks) 12.(b) (i) Find the nature of the series $$\sum^\infty_{n=2}\dfrac {1}{n(\log n)^p}$$ by Cauchy's integral test.(8 marks) 12.(b) (ii) Test the convergence of the series $$1+\dfrac {2^p}{2!}+\dfrac {3^p}{3!}+\dfrac {4^p}{4!}+.....$$ by D'Alembert's ratio test.(8 marks)

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Answer any one question from Q13 (a) & Q13 (b)

13.(a) (i) Find the envelope of

$$\dfrac {x}{a}+\dfrac {y}{b}=1$$ subject to aⁿ+bⁿ=cⁿ, where c is constant.(8 marks) 13.(a) (ii) Find the Evolute of $$\sqrt{x}+\sqrt{y}=\sqrt{a}$$ (8 marks) 13.(b) (i) Find the equation of the circle of curvature of $$\dfrac {x^2}{4}+\dfrac {y^2}{9}=2 \ at \ (2,3)$$ (8 marks) 13.(b) (ii) Find the radius of curvature at any point on x=e^t cos t, y=e^t sin t.(8 marks)

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Answer any one question from Q14 (a) & Q14 (b)

14.(a) (i) Find the extreme value of x²+y²+z² subject to the condition x+y+z=3a(8 marks) 14.(a) (ii)

$$If \ u=(x-y)f\left (\dfrac {y}{x} \right ),\ then \ find \ x^2\dfrac {\partial^2u}{\partial x^2}+2xy\dfrac {\partial^2 u}{\partial x \partial y}+ y^2\dfrac {\partial^2 u}{\partial y^2}$$ (8 marks) 14.(b) (i) $$If \ u=\dfrac {yz}{x}, \ v=\dfrac {zx}{y}, \ w=\dfrac {xy}{z} \ the \ find \ \dfrac {\partial (u,v,w)}{\partial (x,y,z)}$$ (8 marks) 14.(b) (ii) Expand e^x cos y at $$\left ( 0, \dfrac {\pi}{2} \right )$$ upto the third terms using Taylor's series.(8 marks)

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Answer any one question from Q15 (a) & Q15 (b)

15.(a) (i) Find the volume of region bounded by the paraboloid z=x²+y² and the plane z=4(8 marks) 15.(a) (i) Find the surface area of the section of the cylinder x²+y²=a² made by the plane x+y+z=a(8 marks) 15.(b) (i) Change the order of Integration

$$\int^a_0\int^{2a-x}_{\frac {x^2}{a}} xy \ dxdy$$ and hence evaluate it.(10 marks) 15.(b) (ii) Find the area of the cardioid r=a(1+cos ?)(6 marks) 2 If 2, -1, -3 are the Eigen values of the matrix A, then find the Eigen values of the matrix A²-2I(2 marks) 3 Find the nature of the series 1+2+3+....+n+.(2 marks) 4 Define Cauchy's integral test.(2 marks) 5 Find the centre of curvature of y=x² at the origin.(2 marks) 6 Define Involutes and Evolutes.(2 marks) 7 $$Evaluate: \ \lim_ {x\to \infty,y\to 2} \dfrac {xy+5}{x^2+2y^2}$$ (2 marks) 8 $$If \ x^y+y^x=c, \ then \ find \ \dfrac {dy}{dx}$$ (2 marks) 9 $$Evaluate \ \int^{5}_0\int^2_0x^2+y^2 \ dxdy$$ (2 marks)