MORE IN Engineering Maths 1
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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) Then nthderivation of cos2x is
$(A) \ 2^{n-1}\cos \left(2x+\dfrac {n\pi}{2} \right )\\(B) \ 2^{n}\cos \left(2x+\dfrac {n\pi}{2} \right )\\(C) \ 2^{n-1}\cos \left(2x+n\pi \right )\\(D) \ 2^{n-1}\cos \left(\dfrac {n\pi}{2} \right )\\$
1 M
1 (a) (ii) The value of C of the Cauchy mean value theorem for f(x) = ex and g(x) = e-x in [4, 5] is
$(A) \ \dfrac {5}{2}\\(B) \ \dfrac {3}{2}\\(C) \ \dfrac {9}{2}\\(D) \ \dfrac {1}{2}\\$
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1 (a) (iii) Find the nth derivative of y=xn-1 log x is
$(A) \ y_n=\dfrac {(n+1)!}{x}\\(B) \ y_n=\dfrac {n!}{x}\\(C) \ y_n=\dfrac {(n-1)!}{x}\\(D) \ y_n=\dfrac {n!}{x^2}\\$
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1 (a) (iv) Maclaurin's series expansion of log (1+x) is
$(A) \ x+\dfrac {x^2}{2}+\dfrac {x^3}{5}+...\\(B) \ x-\dfrac {x^2}{3}+\dfrac {x^4}{5}-...\\(C) \ x-\dfrac {x^2}{2}+\dfrac {x^3}{3}-\dfrac {x^4}{5}+...\\(D) \ x+\dfrac {x^2}{3}+\dfrac {x^3}{16}+...\\$
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1 (b) By information in two different ways the nth derivation of x2n. Prove that
$1+\dfrac {n^2}{1^2}+\dfrac {n^2(n-1)^2}{1^2 \times 2^2}+\dfrac {n^2 (n-1)^2(n-2)^2}{1^2\times 2^2 \times 3^2}+..... =\dfrac {(2n)!}{(n!)^2}$
6 M
1 (c) Verify Rolle's theorem for the function $f(x)=\dfrac {\sin 2x}{e^{2x}}\ in \ \left [ 0, \dfrac {\pi}{2} \right ]$
4 M
1 (d) Using Maclaurin's series expand log sec x upto the term containing x6.
6 M

2 (a) (i) $The \ value \ of \ \lim_{x\rightarrow \infty} \dfrac {\log x}{x} \ is$
(A) 0
(B) 1
(C) 2
(D) -2
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2 (a) (ii) If s is the arc length of the curve x=f(y) then $\dfrac {ds}{dy} \ is \$
$(A) \ \sqrt{1+y^2_1}\\(B) \ \sqrt{1+y_1} \\(C) \ \sqrt{1+ \left ( \dfrac {dx}{dy} \right )^2}\\(D) \ \sqrt{\left ( \dfrac {dy}{dx} \right )^2 + \left (\dfrac {dx}{dy} \right)^2}\\$
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2 (a) (iii) Pedal equation of the curve $\dfrac {2a}{r}=1 - \cos \theta \ is \$
(A) P=ar2
(B) P2=a2r
(C) P2=a2r2
(D) P2=ar
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2 (a) (iv) The angle between the two curves r=ae? and re?=b is
$(A)\ \dfrac {\pi}{2}\\(B)\ \dfrac {\pi}{4}\\(C)\ 0\\(D)\ \pi$
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2 (b) For the curve $y=\dfrac {ax}{a+x},$ if ? is the radius of curvature at any point (x, y) show that; $\left (\dfrac {2\rho}{2} \right )^{2/3}= \left ( \dfrac {y}{x} \right )^2 + \left ( \dfrac {x}{y} \right )^2$
6 M
2 (c) $Evaluate \ \lim_{x\rightarrow 0}\left (\dfrac {\sin x}{x} \right )^{1/x^2}$
4 M
2 (d) Find the angle between the curves $r= \dfrac {a}{1+\cos \theta}; \ r=\dfrac {b}{1-\cos \theta}$
6 M

3 (a) (i) $When \ u=y^2 \ \log\left ( \dfrac {x}{y} \right ) \ then \ x\dfrac {\partial u}{\partial x}+ y\dfrac {\partial u}{\partial y} \ is$
(A) u
(B) u2
(C) 2u
(D) 3u
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3 (a) (ii) The Taylor's series of f(x, y)=xy at (1, 1) is
(A) 1+[(x-1)+(y-1)]
(B) 1+[(x-1)+(y-1)]+[(x-1)(y-1)]
(C) [(x-1)(y-1)]
(D) None of these
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3 (a) (iii) The Jacobian of transformation from the Cartesian to polar coordinates system is
(A) r3
(B) r
(C) r2
(D) r sin ?
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3 (a) (iv) The rectangle solid of maximum volume which can be inscribed in a sphere is
(A) parallerlogram
(B) rectangle
(C) cube
(D) triangle
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3 (b) Examine the function sin x + sin y + sin (x+y) for extreme values.
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3 (c) Find the possible error in percent in computing the parallel resistance 'r' of two resistances r1 and r2 from the formula $\dfrac {1}{r}= \dfrac {1}{r_1} + \dfrac {1}{r_2}$ are both in error by 2%.
4 M
3 (d) $If \ z(x+y)=x^2 + y^2 \ show \ that \ \left [ \dfrac {\partial z}{\partial x}- \dfrac {\partial z}{\partial y} \right ]^2 = 4 \left [ 1- \dfrac {\partial z}{\partial x}- \dfrac {\partial z}{\partial y} \right ]$
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4 (a) (i) A gradient of the scalar point function ? that is ? ? is
(A) vector function
(B) scalar function
(C) zero
(D) ?
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4 (a) (ii) The directional derivative of f(x, y, z)=x2yz+4xz2 at the (1, -2, -1) in the direction PQ where P=(1, 2, -1), Q=(-1, 2, 3) is
$(A) \ \dfrac {28}{\sqrt{5}}\\(B) \ \dfrac {30}{\sqrt{4}}\\(C) \ \dfrac {-28}{\sqrt{5}}\\(D) \ \dfrac {20}{\sqrt{6}}$
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4 (a) (iii) If R is the position vector of any point P(x, y, z) then ? . R is
(A) 3
(B) -3
(C) 2
(D) 0
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4 (a) (iv) $If \ \bar{r}=x\widehat{i}+y\widehat{j}+z\widehat{k} \ then \ Curl \ \bar{r}=.....$
(A) 0
(B) 1
(C) -1
(D) ?
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4 (b) Find the constant a and b such that F=(axy+z3)i+(3x2-z)j+(bxz2-y)k is irrotational and find scalar potential function ? such that F=??
6 M
4 (c) $Prove \ that \ abla x \left [ \dfrac {ax\bar{r}}{r^n} \right ]=\dfrac {-\bar{a}}{r^3}+\dfrac {3(a.\bar{r})\bar{r}}{r^5}$
4 M
4 (d) Prove that the cylindrical coordinates system is orthogonal.
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5 (a) (i) $The \ value \ of \ \int^1_0 x^2 (1-x^2)^{1/2}dx \ is \\(A) \ \dfrac {\pi}{23}\\(B) \ \dfrac {1}{32}\\(C) \ \dfrac {\pi}{32}\\(D) \ \dfrac {\pi}{16}$
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5 (a) (ii) The tangent to the curve y2=4ax at origin is
(A) y-axis
(B) x-axis
(C) both x-axis and y-axis
(D) does not exist
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5 (a) (iii) $The \ value \ of \ \int^{\pi}_0 \sin^4 \left ( \dfrac {x}{2} \right )dx \ is \\(A) \ \dfrac {3\pi}{18}\\(B) \ \dfrac {3\pi}{8}\\(C) \ \dfrac {3\pi}{16}\\(D) \ \dfrac {3\pi^2}{8}$
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5 (a) (iv) The surface area of the sphere of radius 'a' is
(A) 4?a2
(B) 4?2a
(C) 4?a
(D) 2?a2
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5 (b) Obtain the reduction formula for ? sinn x cosn x dx.
6 M
5 (c) Evaluate $\int^\pi_0 \dfrac {\tan^{-1}(ax)}{x(1+x^2)}dx$ using the method of differentiation under integral sign.
4 M
5 (d) Find the area of the loop of the curve ay2=x2(a-x).
6 M

6 (a) (i) The solution of the differential equation $\dfrac {dy}{dx}=e^{x+y} \ is$
(A) ex+e-y=c
(B) e-x + e-y=c
(C) ex+ ey=c
(D) ex+y =c
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6 (a) (ii) If the homogeneous differential equation $\dfrac {dy}{dx}=\dfrac {f_1(x, y)}{f_2(x,y)}$ the degree of the homogeneous functions f1(x,y) and f2(x,y) are
(A) different
(B) same
(C) relatively prime
(D) degree of f1(x,y) > degree of f2(x,y)
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6 (a) (iii) The integrating factor of the differential equation $(1+x^2)\dfrac {dy}{dx}+xy = \sin h^{-1}x\ is \\(A) \ \dfrac {1}{\sqrt{1+x^2}}\\(B) \ \sqrt{1-x^2}\\(C) \ \sqrt{1+x^2}\\(D) \ \dfrac {x}{\sqrt{1+x^2}}$
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6 (a) (iv) If replacing $\dfrac {dy}{dx}\ by \ -\dfrac {dx}{dy}$ in the differential equation $f \left ( x,y \dfrac {dy}{dx} \right )=0$
(A) polar trajectory
(B) orthogonal trajectrory
(C) parametric trajectory
(D) parallel trajectory
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6 (b) $Solve \ (1+xy^2)\dfrac {dy}{dx}=1$
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6 (c) $Solve \ \dfrac {dy}{dx}= \dfrac {x(2\log x+)}{\sin y+ y \cos y}$
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6 (d) Find the orthogonal trajectory of rn=an sin n?
6 M

7 (a) (i) In a system of linear equations if the rank of the co-efficient matrix=rank of the augmented matrix=n number of unknown then the system has
(A) no solution
(B) unique solution
(C) infinite number of solution
(D) trivial solutions
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7 (a) (ii) The rank of matrix $\begin{bmatrix}2 &-1 &3 &1 \\ 1&4 &-2 &1 \\ 5&2 &4 &3 \end{bmatrix} \ is$
(A) 3
(B) 4
(C) 2
(D) 5
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7 (a) (iii) A square matrix in which aij=aji for all I and j then it is called a
(A) unique matrix
(B) symmetric matrix
(C) skew symmetric
(D) triangular matrix
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7 (a) (iv) The inverse of the square matrix A is
$(A) \ |A| \\(B) \ \dfrac {adj A}{|A|}\\(C) \ adjA \\(D) \ \dfrac {|A|}{adjA}$
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7 (b) Investigate for what value of ? and ? the simultaneous equation x+y+=6, x+2y+3z=10, x+2y+?z=? have
(A) no solutions
(B) unique solutions
(C) infinite number of solutions
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7 (c) Apply Gauss-elimination method to solve the following equations:
2x-y+3z=1, -3x+4y-5z=0, x+3y-6z=0
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7 (d) Find the rank of $\begin{bmatrix}2 &3 &-1 &-1 \\ 1&-1 &-2 &-4 \\ 3&1 &3 &-2 \\ 6&3 &0 &-7 \end{bmatrix}$
6 M

8 (a) (i) The eigen values of the matrix $\begin{bmatrix}6 &-2 &2 \\ -2&3 &-1 \\ 2&-1 &3 \end{bmatrix} \ are$
(A) 2,3,8
(B) 2,2,8
(C) 8,4,3
(D) 2,-2,8
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8 (a) (ii) A homogeneous expression of the second degree in any number of variables is called a
(B) diagonal form
(C) symmetric form
(D) spectral form
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8 (a) (iii) A square matrix A of order 3 has 3 linearly independent eigen vectors then a matrix P can be found such that P-1 AP is a
(A) digonal matrix
(B) symmetric matrix
(C) unit matrix
(D) singular matrix
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8 (a) (iv) if the eigen vector is (1, 1, 1) then its normalized form is
$(A) \ \left (\dfrac {1}{\sqrt{3}}, \dfrac {1}{\sqrt{3}}, \dfrac {1}{\sqrt{3}} \right ) \\(B) \ \left (\dfrac {1}{\sqrt{2}}, 0,-\dfrac {1}{\sqrt{2}} \right )\\(C) \ \left (-\dfrac {1}{\sqrt{3}},\dfrac {1}{\sqrt{2}},-\dfrac {1}{\sqrt{2}} \right )\\(D) \ \left (-\dfrac {1}{\sqrt{3}}, - \dfrac {1}{\sqrt{3}}, \dfrac {1}{\sqrt{3}} \right )$
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8 (b) Reduce 6x2 + 3y2 - 4xy - 2yz + 4zx ito canonical form.
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8 (c) Find all the eigen values for the matrix, $A=\begin{bmatrix}7 &-2 &0 \\ -2&6 &-2 \\ 0&-2 &5 \end{bmatrix}$
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8 (d) Reduce the matrix, $A=\begin{bmatrix}11 &-4 &7 \\ 7&-2 &-5 \\ 10&-4 &-6 \end{bmatrix}$ into a digonal matrix.
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