Plot the points A(1, -1) and B(4, 5): i. Draw a line segment joining these points. Write the coordinates of a point on this line segment between the points A and B. ii. Extend this line segment and write the coordinates of a point on this line which lies outside the line segment AB.

In point A(1, -1), x-coordinate is positive and y-coordinate is negative, so it lies in IV quadrant. In point B(4, 5), both coordinates are positive, so it lies in I quadrant. On plotting these point, we get the following graph. i. On joining the points A and B, we get the line segment AB. Now, to find the coordinates of a point on this line segment between A and B draw a perpendicular to X-axis from x = 2 and 3. [since, x = 2 and 3 lies between A and B] say it intersect line segment AB at P and p ’. Now, draw a perpendicular to Y-axis from P and p ’, they intersect Y axis at y = 1 and 3, respectively. Thus, we get points (2,1) and (3, 3) which lie between line segment AB. ii. Extent the line segment AB. Now, draw a perpendicular to X-axis from x = 5, say it intersects extended line segment at Q. Again, draw a perpendicular to Y-axis from Q and it intersects Y-axis at y = 7. Thus, we get the point Q(5,7) which lies outside the line segment AB.

Plot the points A(1, -1) and B(4, 5): i. Draw a line segment join…

Following table gives the distribution of students of sections $A$ and $B$ of a class according to the marks obtained by them.

Section A

Section B

Marks

Frequency

Marks

Frequency

$0-15$

$15-30$

$30-45$

$45-60$

$60-75$

$75-90$

$5$

$12$

$28$

$30$

$35$

$13$

$0-15$

$15-30$

$30-45$

$45-60$

$60-75$

$75-90$

$3$

$16$

$25$

$27$

$40$

$10$

Represent the marks of the students of both the sections on the same graph by two frequency polygons.What do you observe?

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If $a=\frac{\sqrt{2}+1}{\sqrt{2}-1}$ and $b=\frac{\sqrt{2}-1}{\sqrt{2}+1}$, then find the value of $a ^2+ b ^2-4 ab$.

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The following observed values of $x$ and $y$ are thought to satisfy a linear equation. Write the linear equation

$x$	$6$	$-6$
$y$	$-2$	$6$

Draw the graph, using the values of $x, y$ as given in the above table. At what points the graph of the linear equation
$i.$ Cuts the $X-$axis$?$
$ii.$ Cuts the $Y- $axis$?$

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A chord of length $16\ cm$ is drawn in a circle of radius $10\ cm$. Find the distance of the chord from the centre of the circle.

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Two lines $AB$ and $CD$ intersect at a point $O,$ such that $\angle\text{BOC}+\angle\text{AOD}=280^\circ,$ as shown in the figure. Find all the four angles.

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In a $\triangle\text{ABC},\angle\text{ABC}=\angle\text{ACB}$ and the bisectors of $\angle\text{ABC}$ and $\angle\text{ACB}$ intersect at O such that $\angle\text{BOC}=120^\circ.$ Show that $\angle\text{A}=\angle\text{B}=\angle\text{C}=60^\circ.$

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Students of a school staged a rally for cleanliness campaign. They walked through the lanes in two groups. One group walked through the lanes $A B, B C$ and $C A$; while other through $A C, C D$ and $D A$ (See in a given figure). Then they cleaned the area enclosed within their lanes. If $A B=9 m, B C=40 m, C D=15 m, DA =28 m$, and $\angle B=$ $90^{\circ}$, which group cleaned more area and by how much? Find the total area cleaned by the students (neglecting the width of the lanes).

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In the given figure, $O P, O Q, O R$ and $O S$ are four rays. Prove that $\angle POQ +\angle ROQ +\angle SOR +\angle POS =360^{\circ}$.

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In the given figure, $AB\ ||\ CD$. Find the value of $x$.

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The heights of $75$ students in a school are given below:

Height (in cm)	$130-136$	$136-142$	$142-148$	$148-154$	$154-160$	$160-166$
Number of students	$9$	$12$	$18$	$23$	$10$	$3$

Draw a histogram to represent the above data.

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