Tag: maths

Questions Related to maths

The curve that crosses .................. is not a simple closed curve.

maths fundamental concepts - geometry terms related to polygons curves open and closed figures

  1. other

  2. itself

  3. curves

  4. All of the above

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Correct Option: B
Explanation:

When the curve crosses itself it has end points. While a simple closed curve does not have any end points.
Therefore, B is the correct answer.

A curve which has 2 end points is called an .....................

maths fundamental concepts - geometry terms related to polygons curves open and closed figures

  1. closed curve

  2. simple curve

  3. open curve

  4. None of the above

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Correct Option: C
Explanation:

An open curve is a curve whose end points do not join. In other words it has 2 end points.
Therefore, C is the correct answer.

Which of these is an example of an open curve?

maths fundamental concepts - geometry terms related to polygons curves open and closed figures

  1. Circle

  2. Triangle

  3. A line segment

  4. Pentagon

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Correct Option: C
Explanation:

As the line segment has end points, it is an open curve.
Therefore, C is the correct answer.

A closed curve has no ....................

maths fundamental concepts - geometry terms related to polygons curves open and closed figures

  1. joints

  2. starting points

  3. end points

  4. All of the above

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Correct Option: B,C
Explanation:

The beginning point and the ending point in a closed curve is the same. There are no end points and starting points. Eg. Circle.
Therefore, B and C is the correct answer.

State whether the following statements are true of false
Circles with same radii are equal

maths fundamental concepts - geometry terms related to polygons curves open and closed figures

  1. True

  2. False

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Correct Option: A
Explanation:

The statement is true.

Dimension of circle is defined by its radius so if they have same radius the circles are equal

A circle divides a plane in which it lies including itself in :

maths fundamental concepts - geometry terms related to polygons curves open and closed figures

  1. 2 parts

  2. 3 parts

  3. 4 parts

  4. 5 parts

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Correct Option: B
Explanation:

A circle drawn on a plane divides into 3 parts:

  • Exterior- Any point outside the boundary of circle.
  • Interior- Any point inside the boundary of circle.
  • Boundary- Any point on the circle.

The straight line AB is divided at C so that $\bar{AC} = 3\bar{CB}$. Circles are described on AC and CB as diameters and a common tangent meets AB produced at D. Then $\bar{BD}$ equals.

maths fundamental concepts - geometry terms related to polygons curves open and closed figures

  1. the diameter of the smaller circle

  2. the radius of the smaller circle

  3. the radius of the larger circle

  4. $\bar{CB} \sqrt{3}$

  5. the difference of the two radii

Show answer
Correct Option: B
Explanation:

Let $ x=\overline{BD} $ and let $ r$ be the radius of the small circle. 


Draw the line from the center of each of the circles to the point of contact of the tangent of the circle. 

By similar triangles, 

$ \dfrac{x+r}{r}=\dfrac{x+5r}{3r} \implies x=r$.

$ \overline{BD} $ equals the radius of the smaller circle.

All chords of the curve $x^{2}+y^{2}-10x-4y+4=0$  which make a right angle at $(8,2)$ pass through

maths fundamental concepts - geometry terms related to polygons curves open and closed figures

  1. $(2,5)$

  2. $(-2,-5)$

  3. $(-5,-2)$

  4. $(5,2)$

Show answer
Correct Option: D

Let $m$ be the slope of tangent to the curve $e^{2y}=1+x^{2}$ then set of all values  of $m$ is :

maths fundamental concepts - geometry terms related to polygons curves open and closed figures

  1. $\left[-\dfrac{1}{2}, \dfrac{1}{2}\right]$

  2. $\left[-\infty, -\dfrac{1}{2}\right]\cup \left[\dfrac{1}{2}, 0\right]$

  3. $\left[-\dfrac{1}{2}, \dfrac{1}{2}\right]-\left{0\right}$

  4. $[-2,2]-\left{0\right}$

Show answer
Correct Option: A
Explanation:

$m$ be the slope of tangent.

Given equation of curve is

${{e}^{2y}}=1+{{x}^{2}}$


Taking log both side and we get,

$ \log {{e}^{2y}}=\log \left( 1+{{x}^{2}} \right) $

$ 2y=\log \left( 1+{{x}^{2}} \right) $


On differentiating and we get,

$ 2\dfrac{dy}{dx}=\dfrac{1}{1+{{x}^{2}}}\dfrac{d}{dx}\left( 1+{{x}^{2}} \right) $

$ \Rightarrow 2\dfrac{dy}{dx}=\dfrac{1}{1+{{x}^{2}}}\dfrac{d}{dx}\left( 1+{{x}^{2}} \right) $

$ \Rightarrow 2\dfrac{dy}{dx}=\dfrac{1}{1+{{x}^{2}}}\dfrac{d}{dx}2x $

$ \Rightarrow \dfrac{dy}{dx}=\dfrac{x}{1+{{x}^{2}}} $

$m=\dfrac{dy}{dx}=\dfrac{x}{1+{{x}^{2}}}$


On put $x=\left( -1,1 \right)$

So,

$ m=\dfrac{1}{1+1}=\dfrac{1}{2} $

$ m=\dfrac{-1}{1+1}=\dfrac{-1}{2} $

Hence, the value of $m$ is $\left[-\dfrac{1}{2},\dfrac{1}{2} \right]$


Hence, this is the answer.

The radius of the locus by the point represented by $z$, when $arg\dfrac {z-1}{z+1} =\dfrac {\pi}{4}$, is

maths fundamental concepts - geometry terms related to polygons curves open and closed figures

  1. $\sqrt {2}$

  2. $\sqrt {2}\pi$

  3. $\dfrac {\pi}{\sqrt {2}}$

  4. $none\ of\ these$

Show answer
Correct Option: A
Explanation:
Arg$\left[\cfrac{z-1}{z+1}\right]=\dfrac{\pi}{4}$

$\Rightarrow$Arg$\left[{z-1}\times \cfrac{1}{z+1}\right]=\dfrac{\pi}{4}$

$\Rightarrow$Arg$\left[{z-1}\right] \times$Arg$\left[\cfrac{1}{z+1}\right]=\dfrac{\pi}{4}$

$\Rightarrow$ Arg${\left[z-1\right]}-$Arg${\left[z+1\right]}=\dfrac{\pi}{4}$

Let $z=x+iy$ 

$\therefore$Arg$\left[\left(x-1\right)+iy\right]-$Arg$\left[\left(x+1\right)+iy\right]=\dfrac{\pi}{4}$

$\Rightarrow {\tan}^{-1}\left(\dfrac{y}{x-1}\right)-{\tan}^{-1}\left(\dfrac{y}{x+1}\right)=\dfrac{\pi}{4}$

$\Rightarrow {\tan}^{-1}{\left(\dfrac{\dfrac{y}{x-1}-\dfrac{y}{x+1}}{1+\dfrac{{y}^{2}}{{x}^{2}-1}}\right)}=\dfrac{\pi}{4}$

$\Rightarrow \dfrac{2y}{{x}^{2}-1+{y}^{2}}=1$

$\Rightarrow {x}^{2}-1+{y}^{2}=2y$

$\Rightarrow {x}^{2}-2y+{y}^{2}-1=0$

This represents a circle with center at $\left(0,1\right)$ and radius  $=\sqrt{0+1-\left(-1\right)}=\sqrt{2}$