Paper I PYQs-2016

1.(a)(i) Using elementary row operations, find the inverse of \(A= \begin{bmatrix}{1} & {2} & {1} \\ {1} & {3} & {2} \\ {1} & {0} & {1}\end{bmatrix}\).

[6M]

1.(a)(ii) If \(A=\begin{bmatrix}{1} & {1} & {3} \\ {5} & {2} & {6} \\ {-2} & {-1} & {-3}\end{bmatrix}\) then find \(A^{14}+3 A-2 I\).

[4M]

1.(b)(i) Using elementary row operation find the condition that the linear equations have a solution: \(\quad x-2 y+z=a\)
\(\quad 2 x+7 y-3 z=b\)
\(\quad 3 x+5 y-2 z=c\)

[7M]

1.(b)(ii) If \(w_{1}=\{(x, y, z) \vert x+y-z=0\}\), \(w_{2}=\{(x, y, z) \vert 3 x+y-2 z=0\}\), \(w_{3}=\{(x, y, z) \vert x-7 y+3 z=0\}\), then find \(\operatorname{dim}\left(w_{1} \cap w_{2} \cap w_{3}\right)\) and \(\operatorname{dim}\left(w_{1}+w_{2}\right)\).

[3M]

1.(c) Evaluate \(I=\int_{0}^{1} \sqrt[3]{x \log \left(\dfrac{1}{x}\right)} d x\)

[10M]

1.(d) Find the equation of the sphere which passes though the circle \(x^{2}+y^{2}=4\); \(z=0\) and is cut by the plane \(x+2 y+2 z=0\) in a circle of radius 3.

[10M]

1.(e) Find the shortest distance between the lines \(\dfrac{x-1}{2}=\dfrac{y-2}{4}=z-3\) and \(y-m x=z=0\). For what value of \(m\) will the two lines intersect?

[10M]

2.(a)(i) If \(M_{2}(R)\) is space of real matrices of order \(2 \times 2\) and \(P_{2}(x)\) is the space of real polynomials of degree at most 2, then find the matrix representation of \(T: M_{2}(R) \rightarrow P_{2}(x)\) such that \(T\left(\begin{bmatrix}{a} & {b} \\ {c} & {d}\end{bmatrix}\right)=a+c+(a-d) x+(b+c) x^{2}\), with respect to the standard bases of \(M_{2} (R)\) and \(P_{2}(x)\), further find null space of \(T\).

[10M]

2.(a)(ii) If \(T : P_{2}(x) \rightarrow P_{3}(x)\) is such that \(T(f(x))=f(x)+5 \int_{0}^{x} f(t) d t\), then choosing \(\left\{1,1+x, 1-x^{2}\right\}\) and \(\left\{1, x, x^{2}, x^{3}\right\}\) as bases of \(P_{2}(x)$ and $P_{3}(x)\) respectively, find the matrix of \(T\).

[6M]

2.(b)(i) If \(A= \begin{bmatrix}{1} & {-1} & {2} \\ {-2} & {1} & {-1} \\ {1} & {2} & {3}\end{bmatrix}\) is the matrix representation of a linear transformation \(T : P_{2}(x) \rightarrow P_{2}(x)\) with respect to the bases \(\{1-x, x(1-x), x(1+x)\}\) and \(\left\{1,1+x, 1+x^{2}\right\}\) then find \(T\).

[6M]

2.(b)(ii) Prove that eigen values of a Hermitian matrix are all real.

[8M]

2.(c) If \(A= \begin{bmatrix}{1} & {1} & {0} \\ {1} & {1} & {0} \\ {0} & {0} & {1}\end{bmatrix}\), then find the eigen values and eigenvectors of \(A\).

[6M]

3.(a) Find the maximum and minimum values of \(x^{2}+y^{2}+z^{2}\) subject to the conditions \(\dfrac{x^{2}}{4}+\dfrac{y^{2}}{5}+\dfrac{z^{2}}{25}=1\) and \(x+y-z=0\).

[20M]

3.(b) Let \(f(x, y)=\left\{\begin{array}{ll}{\dfrac{2 x^{4}y-5 x^{3s} y^{2}+y^{5}}{\left(x^{2}+y^{2}\right)^{2}},} & {(x, y ) \neq(0,0)} \\ {0} & {, (x, y )=(0,0)}\end{array}\right.\)
Find a \(\delta>0\) such that \(\vert f(x, y)-f(0,0) \vert< 0.01\) whenever \(\sqrt{x^{2}+y^{2}}<\delta\).

[15M]

Remarks: Original question has printing mistake

3.(c) Find the surface area of the plane \(x+2 y+2 z=12\) cut off by \(x=0\), \(y=0\) and \(x^{2}+y^{2}=16\).

[15M]

4.(a) Find the surface generated by a line which intersects the line \(y=a=z, x+3 z=a=y+z\) and parallel to the plane \(x+y=0\).

[10M]

4.(b) Show that the cone \(3 y z-2 z x-2 x y=0\) has an infinite set of three mutually perpendicular generators. If \(\dfrac{x}{1}=\dfrac{y}{1}=\dfrac{z}{z}\) is a generator belonging to one such set, find the other two.

[10M]

4.(c) Evaluate \(\iint_{R} f(x, y) d x d y\) over the rectangle \(R=[0,1 ; 0,1]\), where \(f(x, y)=\left\{\begin{array}{c}{x+y, \text { if } x^{2}<y<2 x^{2}} \\ {0,}\end{array}\right.\)

[15M]

4.(d) Find the locus of the point of intersection of three mutually perpendicular tangent planes to the conicoid \(a x^{2}+b y^{2}+c z^{2}=1\).

[15M]

5.(a) Find a particular integral of \(\dfrac{d^{2} y}{d x^{2}}+y=e^{x / 2} \sin \dfrac{x \sqrt{3}}{2}\).

[10M]

5.(b) Prove that the vectors \(\vec{a}= \hat{i} \hat{j} \hat{k}\), \(\vec{b}= \hat{i} \hat{j} \hat{k}\), \(\vec{c}= \hat{i} \hat{j} \hat{k}\) can form the sides of a triangle. Find the lengths of the medians of the triangle.

[10M]

5.(c) Solve:

[10M]

5.(d) Show that the family of parabolas \(y^{2}=4 c x+4 c^{2}\) is self orthogonal.

[10M]

5.(e) A particle moves with a central acceleration which varies inversely as the cube of the distance. If it is projected from an apse at a distance \(a\) from the origin with a velocity which is \(\sqrt{2}\) times the velocity for a circle of radius \(a\), then find the equation to the path.

[10M]

6.(a) Solve \(\left\{y(1-x \tan x)+x^{2} \cos x\right\} d x-x d y=0\).

[10M]

6.(b) Using the method of variation of parameters, solve the differential equation \(\left(D^{2}+2 D+1\right) y=e^{-x} \log (x),\left[D \equiv \dfrac{d}{d x}\right]\).

[15M]

6.(c) Find the general solution of the equation \(x^{2} \dfrac{d^{3} y}{d x^{3}}-4 x \dfrac{d^{2} y}{d x^{2}}+6 \dfrac{d y}{d x}=4\).

[15M]

6.(d) Using Laplace transformation, solve the following: \(y^{\prime \prime}-2 y^{\prime}-8 y=0\), \(y(0)=3\), \(y^{\prime}(0)=6\).

[10M]

7.(a) A uniform rod \(AB\) of length 2 a movable about a hinge at \(A\) rests with other end against a smooth vertical wall. If \(\alpha\) is inclination of the rod to the vertical, prove that the magnitude of reaction of the hinge is \(\dfrac{1}{2} W \sqrt{4+\tan ^{2} \alpha}\), where \(W\) is the weight of the rod.

[15M]

7.(b) Two weights \(P\) and \(Q\) are suspended from a fixed point \(O\) by strings \(O A\), \(O B\) and are kept apart by a light rod \(AB\). If the strings \(OA\) and \(OB\) make angles \(\alpha\) and \(\beta\) with the rod \(AB\), show that the angle \(\theta\) which the rod makes with the vertical is given by \(\tan \theta=\dfrac{P+Q}{P \cot \alpha-Q \cot \beta}\).

[15M]

7.(c) A square \(ABCD\), the length of whose sides is \(a\), is fixed in a vertical plane with two of its sides horizontal. An endless string of length \(l(>4 a)\) passes over four pegs at the angles of the board and through a ring of weight \(W\) which is hanging vertically. Show that the tension of the string is \(\dfrac{W(l-3 a)}{2 \sqrt{l^{2}-6 l a+8 a^{2}}}\).

[20M]

8.(a) Find \(f(r)\) such that \(\nabla f=\dfrac{\vec{r}}{r^{5}}\) and \(f(1)=0\).

[10M]

8.(b) Prove that the vector \(\vec{a}=3 \vec{i}+\hat{j}-2 \hat{k}\), \(\vec{b}=-\hat{i}+3 \hat{j}+4 \hat{k}\), \(\vec{c}=4 \hat{i}-2 \hat{j}-6 \hat{k}\) can from the sides of a triangle. Find the length of the medians of the triangle.

[10M]

8.(c) A particle moves in a straight line. Its acceleration is directed towards a fixed point \(O\) in the line and is always equal to \(\mu\left(\dfrac{a^{5}}{x^{2}}\right)^{1 / 3}\) when it is at a distance \(x\) from \(O\). If it starts from rest at a distance \(a\) from \(O\), then find the time the particle will arrive at \(O\).

[15M]

8.(d) For the cardioid \(r=a(1+\cos \theta)\), show that the square of the radius of curvature at any point \((r, \theta)\) is proportion to \(r\). Also find the radius of curvature if \(\theta=0\), \(\dfrac{\pi}{4}\), \(\dfrac{\pi}{2}\).

[15M]