Mathematics

Coordinate Geometry and Graphs

77 Questions

Coordinate geometry involves plotting linear equations, analyzing graphical data, and understanding planar graphs using vertices, edges, and faces. These concepts are essential for calculating intersection points and interpreting graphical information. Such questions frequently appear in engineering, state, and civil services examinations.

Euler graph formulaLinear equation graphsGraph intersection pointsBar graphsLinear functionsTangent equations

Coordinate Geometry and Graphs Questions

Multiple choice maths lines graphs of linear equations equations of lines parallel to the x-axis and y-axis graph of linear equations in two variables

The graph of $x =8 $ represents:

  1. line parallel to $y$-axis and at a distance of $8$ units.
  2. line parallel to $x$-axis and at a distance of $8$ units.
  3. line parallel to $y$-axis and at a distance of $0$ units.

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The line $x=8$ or a point $(8,0)$ lies on the $x$-axis whwich is parallel to $y$-axis and at a distance of $8$ units.

Multiple choice maths lines graphs of linear equations equations of lines parallel to the x-axis and y-axis graph of linear equations in two variables

The graph of $|x| + |y|$ consists of

  1. one straight line

  2. a pair of straight lines

  3. the sides of a square

  4. a circle

  5. a point
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given $|x|+|y|$

  1. For $x>0$ and $y>0$ , we have $x+y$
  2. For $x>0$ and $y<0$ , we have $x-y$
  3. For $x<0$ and $y>0$ , we have $-x+y$
  4. For $x<0$ and $y<0$ , we have $-x-y$
  5. observe that the four equations will form a square

Multiple choice mathematics and statistics set language de morgan's law for set theory complement of sets different sets de morgan's law

In order to draw a graph of $f(x) = ax^{2} + bx + c$, a table of values was constructed. These values of the function for a set of equally spaced increasing values of $x$ were $3844, 4096, 4227, 4356, 4489, 4624$, and $4761$. The one which is incorrect is

  1. $4096$
  2. $4356$
  3. $4489$
  4. $4761$

  5. None of these
Reveal answer Fill a bubble to check yourself
E Correct answer
Explanation

We are told that the values of $f(x)$ listed correspond to
7$f(x), f(x + h), f(x + 2h), ...., f(x + 7h)$.
Observe that the difference between successive values is given by
$f(x + h) - f(x) = a(x + h)^{2} + b(x + h) + c - (ax^{2} + bx + c)$
$= 2ahx + ah^{2} + bh$.
Since the difference is a linear function of $x$, it must change by the same amount whenever $x$ is increased by $h$. But the successive differences of the listed values
$3844\ 3969\ 4096\ 4227\ 4356\ 4489\ 4624\ 4761$
are $125\ 127\ 131\ 129\ 133\ 135\ 137$
so that, if only one value is incorrect; $4227$ is the value.

Multiple choice business maths sets, relations and functions graphs of the form y=ax^2+bx+c some more types of functions functions and their graphs

The graph of the function $\cos x\cos x(x+2)-\cos^{2}(x+1)$ is  

  1. A straight line through $(0, -\sin^{2}1)$ with slope $2$.
  2. A straight line through $(0, 0)$
  3. A parabola with vertex $(1, -\sin^{2}1)$
  4. A straight line through $\left(\dfrac{\pi}{2},-\sin^{2}1\right)$ and parallel to the $x-axis$.
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Expanding and simplifying the given expression using trigonometric identities reveals that the function is a constant with respect to x or simplifies to a line parallel to the x-axis passing through the specified point.

Multiple choice combining transformations transformations vectors and transformations maths

The area of triangle formed by the lines $x+y-3=0$, $x-3y+9=0$ and $3x-2y+1=0$ is:

  1. $\cfrac { 16 }{ 7 } $ sq. units
  2. $\cfrac { 10 }{ 7 } $ sq. units
  3. $4$ sq. units
  4. $9$ sq. units
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$L _1:x+y-3=0$

$L _2:x-3y+9=0$
$L _3:3x-2y+1=0$

$A$ is intersection point of $L _1$ and $L _2$ which is $(0,3)$ on solving $L _1$ and $L _2$.

$B$ is intersection point of $L _2$ and $L _3$ which is $(1,2)$ on solving $L _2$ and $L _3$.

$C$ is intersection point of $L _3$ and $L _1$ which is $(\dfrac{15}{7},\dfrac{26}{7})$ on solving $L _3$ and $L _1$.

Now, area of $\Delta ABC=\dfrac{1}{2}|(x _1y _2-x _2y _1)+(x _2y _3-x _3y _2)+(x _3y _1-x _1y _3)|$

$=\dfrac{1}{2}|(0-3)+(\dfrac{26}{7}-\dfrac{30}{7})+(\dfrac{45}{7}-0)|$

$=\dfrac{1}{2}|-3-\dfrac{4}{7}+\dfrac{45}{7}|=\dfrac{1}{2}\times \dfrac{20}{7}$

$=\dfrac{10}{7} sq. units$

Multiple choice mathematics and statistics inverse of a matrix and linear equations application of determinants area of triangle and collinearity of three points applications of determinants

The co-ordinates of the vertices A, B, C of a triangle are $ \displaystyle \left ( 6,3 \right ),\left ( -3,5 \right ),\left ( 4,-2 \right ) $ respectively and P is any point $ \displaystyle \left ( x,y \right ), $ then the ratio of areas of triangles PBC and ABC is

  1. $ \displaystyle \begin{vmatrix}x-y-2\end{vmatrix}:7 $
  2. $ \displaystyle \begin{vmatrix}x+y+2\end{vmatrix}:7 $
  3. $ \displaystyle \begin{vmatrix}x+y-2\end{vmatrix}:7 $

  4. None of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
Let  $ P=(x,y)$

We have area of $\displaystyle \triangle PBC=\left| \frac { 1 }{ 2 } \begin{vmatrix} x\quad  & y\quad  & 1 \\ -3 & 5 & 1 \\ 4 & -2 & 1 \end{vmatrix} \right| $

$\displaystyle =\frac { 1 }{ 2 } \left| \left[ x\left( 5+2 \right) -3\left( -2-y \right) +4\left( y-5 \right)  \right]  \right| $

$\displaystyle =\frac { 1 }{ 2 } \left| 7x+7y-14 \right| =\frac { 7 }{ 2 } \left| x+y-2 \right| $

Area of $\displaystyle \triangle ABC=\left| \frac { 1 }{ 2 } \begin{vmatrix} 6\quad  & 3\quad  & 1 \\ -3 & 5 & 1 \\ 4 & -2 & 1 \end{vmatrix} \right| $

$\displaystyle =\frac { 1 }{ 2 } \left| \left[ 6\left( 5+2 \right) -3\left( -2-3 \right) +4\left( 3-5 \right)  \right]  \right| $

$\displaystyle \\ =\dfrac { 1 }{ 2 } \left| 42+15-8 \right| =\dfrac { 49 }{ 2 } $

$\displaystyle \therefore \frac { area\triangle PBC }{ area\triangle ABC } =\dfrac { \left| x+y-2 \right|  }{ 7 } $
Multiple choice maths measurement (length) conversion of length using decimals in units of length define weight and units of weight how much does it weigh?

The graphs of $2x+3y-6=0$, $4x-3y-6=0$, $x=2$ and $\displaystyle y=\frac{2}{3}$ intersect in __________.

  1. Four points

  2. One point

  3. In no points

  4. In infinite number of points
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

On solving  2x+3y−6=02x+3y−6=0 and 4x−3y−6=04x−3y−6=0, we get

x=2x=2 and y=23y=23

Hence, from all the four given equations, the!li value of x is 2 and y is 2323

Therefore, the graphs of all the four lines intersect in one point.

Multiple choice maths drawing of different geometrical figures constructing a perpendicular bisector construction of a perpendicular bisector construction of penpendicual bisector

A vertex of square is $(3,4)$ and diagonal's equation is given by $x+2y=1$,then the second diagonal which passes through given vertex will be 

  1. $2x-y+2=0$
  2. $x+2y=11$
  3. $2x-y=2$

  4. None of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
Diagonals of Square are perpendicular bisector of each other.

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

Comparing with $y=mx+c$ we have $slope=m _1=\dfrac{-1}{2}$

As diagonals are perpendicular, so

 $m _1\times m _2=-1$

$\dfrac{-1}{2}\times m _2=-1$

$m _2=2$

equation of other diagonal is $y=2x+c$

as this passes through $(3,4)$ it satifies the equation

$\Rightarrow 4=2(3)+c \Rightarrow c=-2$

Thus equation of diagonal $\equiv 2x-y-2=0$

Hence, the answer is option $C$
Multiple choice maths lines equations of lines parallel to the x-axis and y-axis graphs of linear equations graph of linear equations in two variables

Read the following statements carefully and select the correct option.
Statement-I : The graph of the linear equation x + 2y = 6 passes through (8, -1).
Statement II : Every point which satisfies the linear equation is a solution of the equation.

  1. Both Statement-I and Statement-II are true.

  2. Only Statement-I is true.

  3. Only Statement-II is true.

  4. Neither Statement-I nor Statement-II are true.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Statement I: Substituting (8, -1) into x + 2y = 6 gives 8 + 2(-1) = 6, which is true. Statement II is a standard definition of a solution to an equation.