 # IGNOU MTE-05 Solved Assignment 2023 | MTE

Solved By – Narendra Kr. Sharma – M.Sc (Mathematics Honors) – Delhi University

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Details For MTE-05 Solved Assignment

## IGNOU MTE-05 Assignment Question Paper 2023

Course Code: MTE-05

Assignment Code: MTE-05/TMA/2023

Maximum Marks: 100

1. Check whether the following statements are true or false. Justify your answer with a short explanation or a counter example.

$$(2 \times 10=20)$$

(i) The numbers $$\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{3}}, \frac{3}{4}$$ are the direction cosines of a line.

(ii) The points $$(1,2),(7,6)$$ and $$(4,4)$$ are collinear.

(iii) The conic $$12 x^{2}+12 x y+3 y^{2}+2 x+y=0$$ is degenerate.

(iv) Intersection of the ellipsoid $$\frac{x^{2}}{4}+\frac{y^{2}}{25}+\frac{z^{2}}{4}=1$$ and the plane $$y=5$$ is a circle.

(v) The conicoid $$3 x^{2}+y^{2}+2 x y+x-y-z+1=0$$ is non-central.

(vi) The line $$y=x$$ is a tangent to the parabola $$y^{2}=c x, c>0$$.

(vii) The equation $$2 x^{2}+y+z+1=0$$ represents a paraboloid.

(viii) Projection of a line segment on a line perpendicular to it is the length of the line segment.

(ix) The lines $$x=-y, z=2$$ and $$x=y, z=0$$ intersect each other.

(x) Every planar section of a cylinder is a circle.

2. (a) Trace the conic $$x^{2}-2 x y+y^{2}-3 x+2 y+3=0$$.

(b) Prove that the conic passing through the points of intersection of two rectangular hyperbolas is also a rectangular hyperbola.

(c) Show that the line $$x=y$$ touches the conic $$a x^{2}+2 h x y+b y^{2}+2 g x+2 f y+$$ $$c=0$$, if $$f+g=0$$.

3. (a) Let $$P$$ be the midpoint of the line segment joining the points $$A(a+b, b)$$ and $$B(a-b, a+b)$$. Find the slope of the line passing through $$P$$ and $$Q\left(b,-\frac{a}{2}\right)$$. Under what conditions on $$a$$ and $$b$$, this line is parallel to the $$y$$-axis?

(b) (i) Show that $$\left|\begin{array}{ccc}x & y & 1 \\ 2 & 3 & 1 \\ -4 & 7 & 1\end{array}\right|=0$$ represents the equation of a line passing through $$(2,3)$$ and $$(-4,7)$$

(ii) Prove that the equation of a line through $$\left(x_{1}, y_{1}\right)$$ and $$\left(x_{2}, y_{2}\right)$$ can be expressed in the form $$\left|\begin{array}{lll}x & y & 1 \\ x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1\end{array}\right|=0$$

(c) Find the eccentricity, foci, centre and directrices of the ellipse $$\frac{x^{2}}{16}+\frac{y^{2}}{4}=1$$. Also give a rough sketch of it.

(d) Prove that the length of the chord of a parabola which passes through the focus and which is inclined at $$30^{\circ}$$ to the axis of the parabola is four times the length of the latus rectum.

4. (a) Find the equations of the line through $$(1,3,4)$$ and parallel to the line joining the points $$(-4,5,3)$$ and $$(8,9,7)$$.

(b) Find the equation of the plane which passes through the line of intersection of the planes $$3 x+4 y-5 z=9$$ and $$2 x+6 y+6 z=7$$ and which is perpendicular to the plane $$3 x+2 y-5 z+6=0$$. (c) Find the distance of the origin from the plane which passes through $$(2,1,8),(1,0,2)$$ and $$(-3,4,6)$$.

5. (a) Show that the plane $$2 x+y+2 z=0$$ is a tangent plane to the sphere $$x^{2}+y^{2}+$$ $$z^{2}-2 x+2 y-2 z+2=0$$.

(b) Find the equation of the sphere touching the plane $$8 x+5 y+3 z+1=0$$ at $$(3,-1,-1)$$ and cutting the sphere $$x^{2}+y^{2}+z^{2}-2 x+y-z-6=0$$ orthogonally.

(c) Find the angle between the lines of intersection of the cone $$4 x^{2}+y^{2}+4 z^{2}+$$ $$4 y z+2 z x=0$$ and the plane $$x+2 y+3 z=0$$.

(d) Find the equation of the cylinder with base

$x^{2}+y^{2}+z^{2}-3 x-6 z+9=0, x-2 y+2 z-6=0$

6. (a) Show that the perpendiculars drawn from the origin to tangent planes to the cone $$x^{2}-y^{2}+5 z^{2}+4 x y=0$$ lie on the cone $$x^{2}-y^{2}+z^{2}+4 x y=0$$.

(b) Transform the equation $$x^{2}+2 y^{2}-6 z^{2}-2 x-8 y+3=0$$ by shifting the origin to $$(1,2,0)$$ without changing the directions of the coordinate axes. What object does this new equation represent? Give a rough sketch of it.

(c) Show that the conicoid $$2 x^{2}+2 y^{2}+x y-y z+z x+2 x-y+5 z+1=0$$ is central. Hence find its centre.

7. (a) Examine which of the following conicoids are central and which are non-central. Also determine which of the central conicoids have centre at the origin.

(i) $$x^{2}+y^{2}+z^{2}+4 x+3 y-z=0$$

(ii) $$2 x^{2}-y^{2}-z^{2}+x y+y z-z x=1$$

(iii) $$x^{2}+y^{2}-z^{2}-2 x y-3 y z-6 z x+x-2 y+5 z+4=0$$

(b) Find the transformation of the equation $$12 x^{2}-2 y^{2}+z^{2}=2 x y$$ if the origin is kept fixed and the axes are rotated in such a way that the direction ratios of the new axes are $$1,-3,0 ; 3,1,0 ; 0,0,1$$.

(c) Find the projection of the line segment joining the points $$(1,-1,6)$$ and $$(4,3,2)$$ on the line $$\frac{x-4}{3}=-y=\frac{z}{5}$$.

8. (a) Identify and trace the conicoid $$y^{2}+3 z^{2}=x$$. Describe its sections by the planes $$y=0$$ and $$z=0$$.

(b) Find the equation of tangent plane to the conicoid $$x^{2}+3 y^{2}=4 z$$ at $$(2,-4,13)$$. Represent the tangent plane geometrically.

$$2\:cos\:\theta \:cos\:\phi =cos\:\left(\theta +\phi \right)+cos\:\left(\theta -\phi \right)$$

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$$cot\:\theta =\frac{cos\:\theta }{sin\:\theta }$$

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