STUPIDSID
1 (a)
Examine for consistency of the system of equations:
2x-3y+5z=1
3x+y-z=2
x+4y-6z=1
if consistent solve it.
2x-3y+5z=1
3x+y-z=2
x+4y-6z=1
if consistent solve it.
4 M
1 (b)
Find the eigenvalues of matrix: \[ \begin{bmatrix}
1 & 0 &-4 \\0
&5 &4 \\-4
&4 &3
\end{bmatrix} \] hence find eigenvector corresponding to highest eigenvalue.
4 M
1 (c)
Find the complex number z if amp\((z+2i) = \dfrac {\pi}{4} \text { and amp}(z-2i) = \dfrac {3\pi }{4} . \)
4 M
2 (a)
Examine for the linear dependence or independence. If dependent, find the relation among the following vectors (1, 1, 1), (1, 2, 3), (2, 3, 8).
4 M
2 (b)
Find all values of (1+i)1/4.
4 M
2 (c)
If sin (α+iβ)=x+iy, then prove that: \[ i) \ \dfrac {x^2}{\cosh^2 \beta} + \dfrac {y^2}{\sinh^2 \beta} = 1, \\ ii) \ \dfrac {x^2}{\sin^2 \alpha} - \dfrac {y^2}{\cos^2 \alpha} = 1 \]
4 M
Test convergence of the series (any one):
3 (a) (i)
\[ \dfrac {1}{2+3} + \dfrac {2}{2+3^2} + \dfrac {3}{2+3^3}+ \cdots \ \cdots \]
4 M
3 (a) (ii)
\[ \sum^{\infty}_{n=1} \dfrac {\sqrt{n+1} + \sqrt{n}}{n^3}\]
4 M
3 (b)
Prove that: \[ e^{x\cos x} = 1 + x+ \dfrac {x^2}{2} - \dfrac {x^3}{3} - \cdots \ \cdots \]
4 M
3 (c)
Find nth derivative of \[ \dfrac {x+1}{(x-1)(x+2)(x-3)} \]
4 M
Solve any one:
4 (a) (i)
\[ \lim_{x\to 0} (\cos x)^{1/x^2}\]
4 M
4 (a) (ii)
\[ \lim_{x\to 0 } \dfrac {e^{ax} - e^{-ax}} {\log (1+bx)} \]
4 M
4 (b)
Using Taylor's theorem, expand x4-5x3+5x2+x+2 in powers of x-2.
4 M
4 (c)
if y=cos(m log x), then prove that: \[ x^2 y_{n+2} + (2n+1) xy_{n+1} + (m^2 + n^2) y_n = 0 \]
4 M
Solve any two:
5 (a)
If u=tan(y+ax)+(y-ax)3/2, where a is a constant, then show that: \[ \dfrac {\partial^2 u}{\partial x^2} = a^2 \dfrac {\partial ^2 u}{\partial y^2} \]
6 M
5 (b)
If \[ \displaystyle u=x^3 f\left ( \dfrac {y}{x} \right )+ \dfrac {1}{y^3} \phi \left ( \dfrac {x}{y} \right ), \] prove that: \[ x^2 \dfrac {\partial ^2 u }{\partial x^2} + 2xy \dfrac {\partial^2 u}{\partial x \partial y} + y^2 \dfrac {\partial ^2 u }{\partial y^2}+ x \dfrac {\partial u}{\partial y} + y \dfrac {\partial u}{\partial y} = 9u. \]
7 M
5 (c)
If u=f(x-y, y-z, z-x), then find the value of: \[ \dfrac {\partial u}{\partial x} + \dfrac {\partial u}{\partial y} + \dfrac {\partial u}{\partial z} \]
6 M
Solve any two:
6 (a)
If u=mx+ny, v=nx-my, where m, n are constants, then find the value of: \[ \left ( \dfrac {\partial u}{\partial x} \right )_y \cdot \left ( \dfrac {\partial y}{\partial v} \right )_x \cdot \left ( \dfrac {\partial x}{\partial u} \right )_v \cdot \left ( \dfrac {\partial v}{\partial y} \right )_u \]
6 M
6 (b)
\[ \text{If } u=\tan^{-1} \left [ \dfrac {x^3 + y^3}{x+y} \right ], \] then prove that: \[ x^2 \dfrac {\partial ^2 u }{\partial x^2} + 2xy \dfrac {\partial ^2 u}{\partial x \partial y} + y^2 \dfrac {\partial ^2 u }{\partial y^2} = \sin 2u \left [ 1-4 \sin ^2 u \right ]. \]
7 M
6 (c)
\[ \text{If } z=f(x,y), \ x=\dfrac {\cos u}{v}, \ y = \dfrac {\sin u}{v},\], then prove that: \[ v \dfrac {\partial z}{\partial v} - \dfrac {\partial z}{\partial u} = (y-x) \dfrac {\partial z}{\partial x} - ( y +x) \dfrac {\partial z}{\partial y}. \]
6 M
7 (a)
\[ \text{If } u= \dfrac {y-x}{1+xy}, \ v=\tan^{-1} y -\tan^{-1}x \\ \text{find } \dfrac {\partial (u,v)}{\partial( x, y)} \]
4 M
7 (b)
Prove that:
u=y+z, v=x+2x2, w=x-4yz-2y2,
are functionally dependent and find relation.
u=y+z, v=x+2x2, w=x-4yz-2y2,
are functionally dependent and find relation.
5 M
7 (c)
As dimensions of a triangle ABC are varied, shown that the maximum value of cos A cos B cos C is obtained when the triangle is equilateral.
4 M
8 (a)
If u+v2=x, v+w2=y, w+u2=z find \(\dfrac {\partial u}{\partial x}\).
4 M
8 (b)
In estimating the cost of a pile of bricks measured 2m×15m×1.2 m, the top of the pile is streched 1% beyond the standard length. If the count is 450 bricks in 1 cubic meter and bricks cost Rs. 450 per thousand, find the approximate error in cost.
5 M
8 (c)
Find the minimum value of x2+y2, subject to the condition ax+by=c.
4 M
More question papers from Engineering Mathematics-1