Question Paper: Engineering Maths 1 : Question Paper Jan 2014 - First Year Engineering (C Cycle) (Semester 1) | Visveswaraya Technological University (VTU)
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## Engineering Maths 1 - Jan 2014

### First Year Engineering (C Cycle) (Semester 1)

TOTAL MARKS: 100
TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.
(2) Attempt any four from the remaining questions.
(3) Assume data wherever required.
(4) Figures to the right indicate full marks.

### Choose the correct answer for the following:-

1 (a) (i) $$\if \ y=\dfrac {x+2}{x+}, \ then ] y_n \ is \$$A) \ \dfrac {(-1)^n(n+1)!}{(x-1)^{n-1}} \$$B) \ \dfrac {(-1)^n n!}{(x+1)^{n+1}}\$$C) \ \dfrac {(-1)^n n!}{(x+1)^n} \$$D) \ \dfrac {(-1)^{n-1}n!}{(x+1)^{n+1}}$$(1 marks) 1 (a) (ii) If y=(ax+b)m with m=n, then yn is
(A) n! an
(B) 0
(C) n! bn
(C) n!
(1 marks)
1 (a) (iii) The geometrical intepretation of Lagrange's mean value theorem is
$$(A)\ f'(C)=\dfrac {f(b)-f(a)}{b-a} \$$B)\ f'(C)=\dfrac {f(b)+f(a)}{b-a} \$$C)\ \dfrac {f'(C)}{g'(C)}= \dfrac {f(b)-f(a)}{g(b)-g(a)} \$$D)\ none \ of \ these $$(1 marks) 1 (a) (iv) The Maclaurin's series expansion of e-x is$$ (A)\ 1+x+\dfrac {x^2}{2!}+\dfrac {x^3}{3!}-.....\$$B)\ 1-x+\dfrac {x^2}{2!}-\dfrac {x^3}{3!}+.....\$$C)\ x-\dfrac {x^2}{2!}+\dfrac {x^3}{3!}-.....\$$D)\ x+\dfrac {x^2}{2!}+\dfrac {x^3}{3!}+.....\\$$\lt/span\gt\ltspan class='paper-ques-marks'\gt(1 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt1 (b)\lt/b\gt If y= sin log (x\ltsup\gt2\lt/sup\gt+2x+1), prove that (x+1)\ltsup\gt2\lt/sup\gt y\ltsub\gtn+2\lt/sub\gt+(2n+1)(x+1)y\ltsub\gtn\lt/sub\gt+(n\ltsup\gt2\lt/sup\gt+4)y\ltsub\gtn\lt/sub\gt=0\lt/span\gt\ltspan class='paper-ques-marks'\gt(4 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt1 (c) \lt/b\gt If x is positive, show that $$x>\log (1+x)>x-\dfrac {1}{2}X^2$$\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt1 (d)\lt/b\gt Using Maclourin's series, expand log (1+e\ltsup\gtx\lt/sup\gt) upto the terms containing x\ltsup\gt4\lt/sup\gt.\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt -------------- \ltspan class='paper-comments'\gt ### Choose the correct answer for the following :- \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt2 (a) (i)\lt/b\gt $$\lim_{x \rightarrow \frac {\pi}{4}}\left (\dfrac {1-\tan x}{^\pi_4 -x } \right ) \ is \ equal \ to$$ \ltbr\gt (A) 2 \ltbr\gt (B) -2 \ltbr\gt (C) 1 \ltbr\gt (D) -1\lt/span\gt\ltspan class='paper-ques-marks'\gt(1 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt2 (a) (ii)\lt/b\gt If ? be the angle between the tangent and radius vector at any point on the curve r=f(?), then sin ? is equal to \ltbr\gt $$(A)\ dr/ds \$$B)\ r\dfrac{d\theta}{ds}\$$C)\ r\dfrac{d\theta}{dr}\$$D)\ ds/dr $$\lt/span\gt\ltspan class='paper-ques-marks'\gt(1 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt2 (a) (iii)\lt/b\gt The rate at which the curve is bending called \ltbr\gt (A) radius of curvature \ltbr\gt (B) curvature \ltbr\gt (C) circle of curvature \ltbr\gt (D) evolute\lt/span\gt\ltspan class='paper-ques-marks'\gt(1 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt2 (a) (iv)\lt/b\gt The radius of curvature for polar curve r=f(?) is given by \ltbr\gt$$ (A)\ \dfrac {(r^2+r^2_1)^{3/2}}{r^2+r^2_1-rr_2}\$$B)\ \dfrac {(r^2+r^2_1)^{3/2}}{r^2_1+2r^2-rr^2}\$$C)\ \dfrac {(r^2+r^2_1)^{3/2}}{r^2+2r^2_1-rr_2}\$$D)\ \dfrac {(r^2-r^2_1)^{3/2}}{r^2+2r^2_1-rr^2}$$\lt/span\gt\ltspan class='paper-ques-marks'\gt(1 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt2 (b)\lt/b\gt Find the Pedal equation of the curve r\ltsup\gtm\lt/sup\gt=a\ltsup\gtm\lt/sup\gt cos m?\lt/span\gt\ltspan class='paper-ques-marks'\gt(4 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt2 (c) \lt/b\gt Find the radius of curvature for the curve $$y^2=\dfrac {a^2(a-x)}{x}$$ where the curve meets the x-axis. \lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt2 (d)\lt/b\gt $$Evaluate \ \lim_{x\rightarrow \infty}\left ( \dfrac {ax+1}{ax-1} \right )^x$$\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt -------------- \ltspan class='paper-comments'\gt ### Choose the correct answer for the following :- \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt3 (a) (i)\lt/b\gt $$If\ u=\log (x^2+y^2+z^2),\ then \ \dfrac {\partial u}{\partial z} \ is \$$A)\ \dfrac {2x}{x^2+y^2+z^3} \$$B)\ \dfrac {2y}{x^2+y^2+z^2} \$$C)\ \dfrac {2z}{x^2+y^2+z^2} \$$D)\ \dfrac {2z}{x^2+y^2-z^2}$$\lt/span\gt\ltspan class='paper-ques-marks'\gt(1 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt3 (a) (ii)\lt/b\gt If u=f(x, y) and y is a function x, then \ltbr\gt $$(A)\ \dfrac {du}{dx}= \dfrac {\partial u}{\partial x}+ \dfrac {\partial u}{\partial y} \dfrac {dy}{dx}\$$B)\ \dfrac {\partial u}{\partial x}= \dfrac {du}{dx}+ \dfrac {\partial u}{\partial y}\dfrac {dy}{dx}\$$C)\ \dfrac {du}{dx}= \dfrac {\partial u}{\partial x}+ \dfrac {\partial u}{\partial y}\dfrac {\partial y}{\partial x}\$$D)\ \dfrac {\partial u}{\partial x}=\dfrac {du}{dx}+\dfrac {\partial u}{\partial y}\dfrac {\partial y }{\partial x}$$\lt/span\gt\ltspan class='paper-ques-marks'\gt(1 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt3 (a) (iii)\lt/b\gt$$ if \ r=\dfrac {\partial^2 f}{\partial x^2},\ S=\dfrac {\partial^2f}{\partial x \partial y} \ and \ t=\dfrac {\partial^2 f}{\partial y^2} $$then the condition for the saddle point is \ltbr\gt (A) rt-s\ltsup\gt2\lt/sup\gt<0 \ltbr\gt (B) rt-s\ltsup\gt2\lt/sup\gt=0 \ltbr\gt (C) rt-s\ltsup\gt2\lt/sup\gt>0 \ltbr\gt (D) rt-s\ltsup\gt2\lt/sup\gt ? 0\lt/span\gt\ltspan class='paper-ques-marks'\gt(1 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt3 (a) (iv)\lt/b\gt If u=x+y+z, v=y+z, z=z, then$$ J\left ( \dfrac {u,v,z}{x,y,z} \right ) $$is equal to \ltbr\gt (A) 1 \ltbr\gt (B) -1 \ltbr\gt (C) 0 \ltbr\gt (D) none of these\lt/span\gt\ltspan class='paper-ques-marks'\gt(1 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt3 (b)\lt/b\gt The focal length of a mirror is given by the formula$$ \dfrac {1}{v}-\dfrac {1}{u}=\dfrac {2}{f} $$if equal errors, 'e' are made in the determination of u and v. show that the resulting error in f is$$ e \left ( \dfrac {1}{u}+\dfrac {1}{v} \right )$$\lt/span\gt\ltspan class='paper-ques-marks'\gt(4 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt3 (c) \lt/b\gt If u=f(2x-3y, 3y-4z, 4z-2x), prove that$$ \dfrac {1}{2}\dfrac {\partial u}{\partial x}+\dfrac {1}{3}\dfrac {\partial u}{\partial y}+ \dfrac {1}{4}\dfrac {\partial u}{\partial z}=0 $$\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt3 (d)\lt/b\gt If x=u(1-v), y=uv, prove that JJ'=1\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt -------------- \ltspan class='paper-comments'\gt ### Choose the correct answer for the following :- \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt4 (a) (i)\lt/b\gt Directional derivative is maximum along \ltbr\gt (A) tangent to the surface \ltbr\gt (B) normal to the surface \ltbr\gt (C) any unit vector \ltbr\gt (D) coordinate axes\lt/span\gt\ltspan class='paper-ques-marks'\gt(1 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt4 (a) (ii)\lt/b\gt If r=|x\ltsub\gti\lt/sub\gt+y\ltsub\gtj\lt/sub\gt+2\ltsub\gtk\lt/sub\gt|, then ? r\ltsup\gtn\lt/sup\gt is \ltbr\gt (A) nr\ltsup\gtn-1\lt/sup\gt \ltbr\gt (B) r\ltsup\gtn-1\lt/sup\gt \ltbr\gt (C) ?.? r\ltsup\gtn\lt/sup\gt \ltbr\gt (D) none of these\lt/span\gt\ltspan class='paper-ques-marks'\gt(1 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt4 (a) (iii)\lt/b\gt If f=3x\ltsup\gt2\lt/sup\gt-3y\ltsup\gt2\lt/sup\gt+4z\ltsup\gt2\lt/sup\gt, then curl (grad f) is \ltbr\gt (A) 4x-6y+8z \ltbr\gt (B) 4x\ltsub\gti\lt/sub\gt-6y\ltsub\gtj\lt/sub\gt+8z k \ltbr\gt$$ \vec{0} $$\ltbr\gt (D) 3\lt/span\gt\ltspan class='paper-ques-marks'\gt(1 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt4 (a) (iv)\lt/b\gt If the base vectors e\ltsub\gt1\lt/sub\gt and e\ltsub\gt2\lt/sub\gt are orthogonal then |e\ltsub\gt1\lt/sub\gt × e\ltsub\gt2\lt/sub\gt| is \ltbr\gt (A) 0 \ltbr\gt (B) -1 \ltbr\gt (C) +1 \ltbr\gt (D) none of these\lt/span\gt\ltspan class='paper-ques-marks'\gt(1 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt4 (b)\lt/b\gt$$ If \ \vec{F}=(x+y+1)i+j-(x+y)k, \ show \ that \ \vec{F}\cdot curl \ \vec{F}=0 $$\lt/span\gt\ltspan class='paper-ques-marks'\gt(4 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt4 (c) \lt/b\gt Find constant 'a' and 'b' such that$$ \vec{F}=(axy+z^3)i+(3x^2-z)j+(bxz^2-y)k $$is irrotational. Also find a scalar function ? such that$$ \vec{F}=abla \phi $$\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt4 (d)\lt/b\gt Prove that a spherical coordinate system is orthogonal.\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt -------------- \ltspan class='paper-comments'\gt ### Choose the correct answer for the following :- \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5 (a) (i) \lt/b\gt$$ \int^\pi_0 \sin^7x \ dx \ is \ equal \ to \$$A)\ zero \$$B)\ \dfrac {32\pi}{35}\$$C)\ \dfrac {32}{35}\$$D)\ =\dfrac {35\pi}{32} $$\lt/span\gt\ltspan class='paper-ques-marks'\gt(1 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5 (a) (ii)\lt/b\gt The asymptote of (2-x)y\ltsup\gt2\lt/sup\gt=x\ltsup\gt3\lt/sup\gt is \ltbr\gt (A) x=2 \ltbr\gt (B) y-axis \ltbr\gt (C) x-axis \ltbr\gt (D) none of these\lt/span\gt\ltspan class='paper-ques-marks'\gt(1 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5 (a) (iii)\lt/b\gt The area of the cordioid r=a(1- cos ?) is \ltbr\gt$$ (A)\ \dfrac {3\pi a^2}{2}\$$B)\ \dfrac {3\pi}{2}\$$C)\ \dfrac {a^2}{2}\$$D)\ \dfrac {3a^2}{2}$$\lt/span\gt\ltspan class='paper-ques-marks'\gt(1 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5 (a) (iv)\lt/b\gt The entire length of the asteroid x\ltsup\gt2/3\lt/sup\gt+y\ltsup\gt2/3\lt/sup\gt=a\ltsup\gt2/3\lt/sup\gt is \ltbr\gt (A) 6a \ltbr\gt (B) 3a \ltbr\gt (C) 2a \ltbr\gt (D) a\lt/span\gt\ltspan class='paper-ques-marks'\gt(1 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5 (b)\lt/b\gt $$Evaluate \ \int^\pi_0 \log (1+ a \cos x)dx$$ by differentiating under the integral sign.\lt/span\gt\ltspan class='paper-ques-marks'\gt(4 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5 (c) \lt/b\gt Evaluate $$\int^{2a}_0X^2\sqrt{2ax-x^2 }dx$$ using reduction formula.\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5 (d)\lt/b\gt Find the volume of generated by the revolution of the curve r=a(1+cos ?) about the initial line.\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt -------------- \ltspan class='paper-comments'\gt ### Choose the correct answer for the following :- \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6 (a) (i)\lt/b\gt The general solution of the differential equation dy/dx=(y/x)+tan (y/x) is \ltbr\gt (A) sin (y/x)=c \ltbr\gt (B) sin (y/x)=cx \ltbr\gt (C) cos(y/x)=cx \ltbr\gt (D) cos (y/x)=c\lt/span\gt\ltspan class='paper-ques-marks'\gt(1 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6 (a) (ii)\lt/b\gt The family of straight lines passing through the origin is represented by the differential equation : \ltbr\gt (A) ydx+xdy=0 \ltbr\gt (B) xdy-ydx=0 \ltbr\gt (C) xdx+ydy=0 \ltbr\gt (D) ydy-xdx=0\lt/span\gt\ltspan class='paper-ques-marks'\gt(1 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6 (a) (iii)\lt/b\gt The homogeneous differential equation Mdx+Ndy=0 can be reduced to a differential equation, in which the variables are seperated by the substitution \ltbr\gt (A) y=vx \ltbr\gt (B) x+y=v \ltbr\gt (C) xy=v \ltbr\gt (D) x-y=v\lt/span\gt\ltspan class='paper-ques-marks'\gt(1 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6 (a) (iv)\lt/b\gt The equation y-2x=c represents the orthogonal trajectories family \ltbr\gt (A) y=ae\ltsup\gt-2x\lt/sup\gt \ltbr\gt (B) x\ltsup\gt2\lt/sup\gt+2y\ltsup\gt2\lt/sup\gt=a \ltbr\gt (C) xy=a \ltbr\gt (D) x+2y=a\lt/span\gt\ltspan class='paper-ques-marks'\gt(1 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6 (b)\lt/b\gt $$Solve \ (x+1)\dfrac {dy}{dx}-y=e^{3x}(x+1)^2$$\lt/span\gt\ltspan class='paper-ques-marks'\gt(4 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6 (c) \lt/b\gt Solve (1+xy) ydx+(1-xy)xdy=0\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6 (d) \lt/b\gt Find the orthogonal trajectory of the cordioids r=a(1- cos ?)\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt -------------- \ltspan class='paper-comments'\gt ### Choose the correct answer for the following :- \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt7 (a) (i)\lt/b\gt If every minor of order 'r' of a matrix A is zero, then rank of A is \ltbr\gt (A) greater than r \ltbr\gt (B) equal r \ltbr\gt (C) less than or equal to r \ltbr\gt (D) less\lt/span\gt\ltspan class='paper-ques-marks'\gt(1 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt7 (a) (ii)\lt/b\gt The trivial solution for the given system of equations x+2y+3z=0, 3x+4y+4z=0, 7x+10y+12z=0 is \ltbr\gt (A) (1, 1, 1) \ltbr\gt (B) (1, 0, 0) \ltbr\gt (C) (0, 1, 0) \ltbr\gt (D) (0, 0, 0)\lt/span\gt\ltspan class='paper-ques-marks'\gt(1 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt7 (a) (iii)\lt/b\gt Matrix has a value. This statement \ltbr\gt (A) is always true \ltbr\gt (B) depends upon the matrices \ltbr\gt (C) is false \ltbr\gt (D) none of these\lt/span\gt\ltspan class='paper-ques-marks'\gt(1 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt7 (a) (iv)\lt/b\gt If A is singular and ?(A)=?(A:B) then the system has \ltbr\gt (A) unique solution \ltbr\gt (B) infinitely many solution \ltbr\gt (C) trivial solution \ltbr\gt (D) no solution.\lt/span\gt\ltspan class='paper-ques-marks'\gt(1 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt7 (b)\lt/b\gt Using elementary transformations, find the rank of the matrix $$\begin {bmatrix}1&2&0&-1 \ 3&4&1&2 \ -2&3&2&5 \end{bmatrix}$$\lt/span\gt\ltspan class='paper-ques-marks'\gt(4 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt7 (c) \lt/b\gt Show that the system x+y+z=4; 2x+y-z=1; x-y+2z=2 is consistent, solve the system.\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt7 (d)\lt/b\gt Apply Gauss-Jordan method to solve the system of equation: 2x+5y+7z=52; 2x+y-z=0; x+y+z=9\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt -------------- \ltspan class='paper-comments'\gt ### Choose the correct answer for the following :- \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt8 (a) (i)\lt/b\gt A square matrix A is called orthogonal if, \ltbr\gt (A) A=A\ltsup\gtL\lt/sup\gt \ltbr\gt (B) A\ltsup\gtT\lt/sup\gt=A\ltsup\gt-1\lt/sup\gt \ltbr\gt (C) AA\ltsup\gt-1\lt/sup\gt=I \ltbr\gt (D) none of these\lt/span\gt\ltspan class='paper-ques-marks'\gt(1 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt8 (a) (ii)\lt/b\gt The eigen values of the matrix $$\begin{bmatrix}2 &\sqrt{2} \ \sqrt{2}&2 \end{bmatrix} \ are \$$A)\ 1 \pm \sqrt{6}\$$B)\ 1 \pm \sqrt{5}\$$C)\ \sqrt{5}\$$D)\ 1$$\lt/span\gt\ltspan class='paper-ques-marks'\gt(1 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt8 (a) (iii)\lt/b\gt The index and signature of the quadratic form $$x^2_1+2X^2_2-3X^2_3$$ are respectively \ltbr\gt (A) 2,1 \ltbr\gt (B) 1,2 \ltbr\gt (C) 1,1 \ltbr\gt (D) none of these\lt/span\gt\ltspan class='paper-ques-marks'\gt(1 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt8 (a) (iv)\lt/b\gt Two square matrices A and B are similar, if \ltbr\gt (A) A=B \ltbr\gt (B) B=P\ltsup\gt-1\lt/sup\gtAP \ltbr\gt (C) A\ltsup\gtT\lt/sup\gt=B\ltsup\gtT\lt/sup\gt \ltbr\gt (D) A\ltsup\gt-1\lt/sup\gt=B\ltsup\gt-1\lt/sup\gt\lt/span\gt\ltspan class='paper-ques-marks'\gt(1 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt8 (b)\lt/b\gt Reduce the quadratic form 8x\ltsup\gt2\lt/sup\gt+7y\ltsup\gt2\lt/sup\gt+3z\ltsup\gt2\lt/sup\gt-12yz+4zx-8xy to the canonical form\lt/span\gt\ltspan class='paper-ques-marks'\gt(4 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt8 (c) \lt/b\gt Determine the characteristics roots and eigen vectors of $$A=\begin{bmatrix}8 &-6 &2 \ -6&7 &-4 \ 2&- 4&3 \end{bmatrix}$$\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt8 (d)\lt/b\gt Reduce the quadratic form $$x^2_1+2x^2_2-7x^2_3-4x_1x_2+8x_2x_3$$ into sum of squares.
(6 marks)