Maths Class X Question Bank

Question 1 of 38

1. Question
1 point(s)
Will it be true to say that the perimeter of a square circumscribing a circle of radius a cm is 8a cm? Give reasons for your answer.
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Hint

Radius, $r=a\ cm$
Diameter, $d=2a\ cm$
Side of square $=2a\ cm$
Perimeter $=4\times 2a=8a\ cm$
Question 2 of 38

2. Question
1 point(s)
Is the following statement true? Give reasons for your answer. Area of a segment of a circle = area of the corresponding sector – area of the corresponding triangle
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Hint

False
It is true only for a minor segment. In the case of a major segment area of the triangle will have to be added to the corresponding area of the sector.
Question 3 of 38

3. Question
1 point(s)
Is the area of the circle inscribed in a square of side a $cm,\ \pi a^2\ cm^2 $ ? Give reason for your answer
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Hint

False
Let a be the side of square
Diameter of circle = side of square = a
Radius of the circle $=\dfrac{a}{2}$
Area of the circle $=\pi r^2$
$=\pi\left(\dfrac{a}{2}\right)^2\\ \\ =\dfrac{\pi a^2}{4}$
Area of the circle is $\dfrac{\pi a^2}{4}\ cm^2$
Question 4 of 38

4. Question
1 point(s)
In the figure, a circle is inscribed in a square of side 5 cm and another circle is circumscribing the square. Is it true to say that area of the outer circle is two times the area of the inner circle? Give reasons for your answer.
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Hint

True
Area of circle inner circle $=\pi \left(\dfrac{a}{2}\right)^2$
$=\dfrac{\pi a^2}{4}$
area of outer circle $=\pi\left(\dfrac{\sqrt2a}{2}\right)^2$
$=\dfrac{1}{2}\left(\pi a^2\right)$
Ratio $=1:2$
Question 5 of 38

5. Question
1 point(s)
In the figure, a square is inscribed in a circle of diameter d and another square is circumscribing the circle. Is the area of the outer square four times the area of the inner square? Give reasons for your answer.
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Hint

False
Diagonal of inner square = Diameter of circle = d
Let side of inner square be $x$
$d^2=x^2+x^2=2x\\ \\ \Rightarrow x^2=\dfrac{d^2}{2}$
But area of outer square $d^2$
Question 6 of 38

6. Question
1 point(s)
Is it true to say that area of a segment of a circle is less than the area of its corresponding sector ? Why ?
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Hint

False
It is true only in the case of mirror segment but in case of minor segment area is always greater than the area of sector.
Question 7 of 38

7. Question
1 point(s)
In covering a distance s metres, circular wheel of radius r metres makes $\dfrac{s}{2\pi r} $ revolutions. Is this statement true ? Why ?
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Hint

True
The distance covered in one revolution is $2\pi r$
Question 8 of 38

8. Question
1 point(s)
The numerical value of the area of a circle is greater than the numberical value of its circumference. Is this statement true ? Why ?
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Hint

If $O<r<2$ then numerical value of circumference is greater than numerical value of area of circle and if $r>2$ , area is greater than circumference.
Question 9 of 38

9. Question
1 point(s)
If the length of an arc of a circle r is equal to that of an arc of a circle of radius 2r, then the angle of the corresponding sector of the first circle is double the angle of the corresponding sector of the other circle. Is this statement false ? Why ?
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Hint

False
Let two circles C and C’ of radius r and 2r with centres O and O’ respectively
$\widehat{AB}=\widehat{CD}=l\\ \\ \widehat{AB}=l=\dfrac{\theta_1}{360}\times2\pi r$ …(1)
$\widehat{CD}=l=\dfrac{\theta_2}{360}\times4\pi r$ …(2)
From eqn (1) and (2) $\theta_1=2\theta_2$
Question 10 of 38

10. Question
1 point(s)
The areas of two sectors of two different circles with equals corresponding arc lengths are equal. Is this statement true ? Why ?
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Hint

False
It is true for arcs of the same circle. But in defferent circle it is not possible.
Question 11 of 38

11. Question
1 point(s)
The areas of two sectors of two different circles are equal. Is it necessary that their corresponding arc lengths are equal ? Why ?
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Hint

False
It is true for arcs of the same circle. But in defferent circle it is not possible.
Question 12 of 38

12. Question
1 point(s)
Is the area of the largest circle that can be drawn inside a rectangle of length a cm and breadth b cm (a > b) is $ \pi b^2\ cm^2$ ? Why ?
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Hint

The area of the largest circle than can be drawn inside a rectangle is $\pi\left(\dfrac{b}{2}\right)^2\ cm$ where $\pi\ \dfrac{b}{2}$ is the radius of the circle and it is possible when rectangle becomes a square.
Question 13 of 38

13. Question
1 point(s)
Circumferences of two circles are equal. Is it necessary that their areas be equal ? Why ?
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Hint

True
If circumference of two circles are equal then their corresponding radii are equal so there areas will be equal.
Question 14 of 38

14. Question
1 point(s)
Areas of two circles are equal. Is it necessary that their circumferences are equal ? Why ?
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Hint

True
If area of two circle are equal then their corresponding radii are equal so their circumference will be equal.
Question 15 of 38

15. Question
1 point(s)
Is it true to say that area of a square inscribed in a circle of diameter $p\ cm $ is $p^2\ cm^2 $ ? Why ?
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Hint

False
When the square is inscribed in the circle, the diameter of a circle is equal to the diagonal of a square but not the side of the square.
$P^2=a^2+a^2\Rightarrow a^2=\dfrac{P^2}{2}$
Question 16 of 38

16. Question
1 point(s)
A bucket is raised from a well by means of a rope which is wound round a wheel of diameter 77 cm. Given that the bucket ascends in 1 minute 28 seconds with a uniform speed of 1.1 m/s. Calculate the number of complete revolutions the wheel makes in raising the bucket.
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Hint

Distance $=s\times t$
$=(1.1\times 88)\ m=96.8\ m$
circumference $=2\pi r$
$=2\times \dfrac{22}{7}\times \dfrac{77}{2}=242\ cm=2.42\ m$
No. of complete revolution
$=\dfrac{96.8}{2.42}=40$
Question 17 of 38

17. Question
1 point(s)
Diameters of three concentric circles are in the ratio 1 : 2 : 3. The sum of the circumferences of these circles is 264 cm. Find the area enclosed between second and third circles.
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Hint

$d_1=x,d_2=2y,d_3=3x$
Sum of circumference $=\pi (x+2x+3x)\\ \\ \Rightarrow 264=\pi(6x)\Rightarrow x=14\\ \\ A_2=\pi r^2_2=\dfrac{\pi}{4}d^2_2=\dfrac{\pi}{4}\times 4x^2=196\ cm^2\\ \\ A_3=\dfrac{\pi d^2_3}{4}=\dfrac{\pi}{4}\times 9x^2=441\ \pi\ cm^2$
Question 18 of 38

18. Question
1 point(s)
A chord of a circle of radius 10 cm subtends a right angle at the centre. Find the area of the corresponding (i) minor segment (ii) minor sector.
[Use latex]\pi =3.14 [/latex]]
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Hint

(i) Area of minor sector $=\dfrac{\theta}{360}\times \pi r^2$
$=\dfrac{90}{360}\times3.14\times(10)^2=3.14\times25=78.5\ cm^2$
(ii) Area of major sector $=\pi r^2-78.5$
$=3.14\times(10)^2-78.5\\ \\ =235.5\ cm^2$
Question 19 of 38

19. Question
1 point(s)
Find the area of both the segments of a circle of radius 15 cm, one of which makes an angle of 60° at the centre of the circle.
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Hint

Area of sector $=\pi r^2\ \dfrac{\theta}{360}$
$=\dfrac{22}{7}\times15\times15\times\dfrac{60}{360}=117.85$
Area of triangle $=\dfrac{1}{2}r^2\sin60=97.312$
Minor segment = Area of sector – Area of triangle
$=20.53$
Area of circle $=\pi r^2$
$=\dfrac{22}{7}\times15\times15\\ \\ =707.14$
Major segment $=707.14-20.5=686\ cm^2$ (approx)
Question 20 of 38

20. Question
1 point(s)
A chord 10 cm long is drawn in a circle whose radius is $5\sqrt3\ cm $. Find the area of minor segment.
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Hint

Area of major segment = area of circle – area of minor segment
Area of minor segment
$=\dfrac{\theta}{360}\times\pi r^2-\dfrac{1}{2}\times5\sqrt2\times5\sqrt2\\ \\ =\dfrac{275}{7}-25=\dfrac{100}{7}\ cm^2$
Area of major segment
$=\dfrac{22}{7}\times(5\sqrt2)^2-\dfrac{100}{7}\\ \\ =\dfrac{1000}{7}\ cm^2$
Question 21 of 38

21. Question
1 point(s)
A chord of a circle of radius 28 cm subtends an angle 45° at the centre of the circle. Find the area of the minor segment.
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Hint

Area $=\dfrac{45}{360}\times\dfrac{22}{7}\times28\times28$
$=\dfrac{2648}{8}=308\ cm^2$
Question 22 of 38

22. Question
1 point(s)
The perimeter of a sector of a circle with central angle 90° is 25 cm. Find the area of the minor segment of the circle.
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Hint

$25=\dfrac{2\pi r}{4}\Rightarrow r=\dfrac{50}{\pi }$
Area of minor segment
$=\dfrac{1}{4}\times\pi\left(\dfrac{50}{\pi}\right)^2=199.04\ cm^2$
Question 23 of 38

23. Question
1 point(s)
Find the area of the shaded portions of the following figure with given measurements :
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Hint

Area of sector $=\dfrac{\theta}{360}\times \pi r^2$
$=\dfrac{60}{360}\times\dfrac{22}{7}\times15\times15\\ \\ =\dfrac{297000}{2520}=117.86\ cm^2$
Question 24 of 38

24. Question
1 point(s)
The diameter of a circular pond is 17.5 m. It is surrounded by a path of width 3.5 m. Find the area of the path.
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Hint

Radius $=\dfrac{17.5}{2}\ m$
Radius of the park with path $=\dfrac{17.5}{2}+3.5=12.25$
Area of path $=\pi\ (R^2-r^2)$
$=\dfrac{22}{7}\ \left[(12.25)^2-\left(\dfrac{17.5}{2}\right)^2\right]\\ \\ =\dfrac{22}{7}\times\dfrac{7}{2}\times\dfrac{42}{2}=231\ m^2$
Question 25 of 38

25. Question
1 point(s)
A sector is cut from a circle of radius 21 cm. The angle of the sector is 150°. Find the area of the sector.
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Hint

Length of the arc $=\dfrac{\theta}{360}$
$=\dfrac{15}{360}\times2\times\dfrac{22}{7}\times21=55\ cm$
Area $=\dfrac{lr}{2}=\dfrac{55\times21}{2}=577.5\ cm^2$
Question 26 of 38

26. Question
1 point(s)
The diameter of the wheel of a bus is 140 cm. How many revolutions per minute must the wheel make in order to keep a speed of 66 km/hr ?
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Hint

Distance covered in 1 revolution
$=2\times\dfrac{22}{7}\times70=440\ cm$
Speed $=66\ km/hr$
$=\dfrac{66\times1000\times100}{60}=110000\ cm/min$
No. of revolutions $=\dfrac{110000}{440}=250$
Question 27 of 38

27. Question
1 point(s)
A playground is in the form of a rectangle having semicircles on the shorter sides. Find its area when the length of the rectangular portion is 80 m and the breadth is 42 m.
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Hint

Area = Area of rectangle + Area of semicircle
$=80+42+\dfrac{22}{7}\times21\times21\\ \\ =3360+1386\\ \\ =4746\ cm^2$
Question 28 of 38

28. Question
1 point(s)
A bicycle wheel makes 500 revolutions in moving 22 km. Find the diameter of the wheel.
(take $\pi=\dfrac{22}{7} $)
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Hint

Number of revolutions $=\dfrac{2200000}{2\pi r}=500$
$\Rightarrow 22000=10\pi r\\ \\ \Rightarrow r=700\ cm$
Diameter $=2\times700=1400\ cm$
$=14\ m$
Question 29 of 38

29. Question
1 point(s)
A plot is in the form of a rectangle ABCD having semicircle on BC as shown in the figure. The semicircle portion is grassy while the remaining plot is without grass. Find the area of the plot without grass where AB = 60 m and BC = 28 m.
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Hint

Area of plot = Area of rectangle + Area of semicircle
$=60\times28+\dfrac{22}{7}\times\dfrac{14\times14}{2}\\ \\ =1680+308=1988\ m^2$
Question 30 of 38

30. Question
1 point(s)
Find the area of a right angled triangle if the radius of its circumcircle is 2.5 cm and the long altitude drawn to the hypotenuse is 2 cm long.
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Hint

Hypotenuse $=2\times2.5=5\ cm$
Altitude drawn to the hypotenuse $=2\ cm$
Area of right triangle $=\dfrac{1}{2}\times5\times2=5\ cm^2$
Question 31 of 38

31. Question
1 point(s)
PQRS is a diameter of a circle of radius 6 cm. The lengths PQ, QR and RS are equal. Semi-circles are drawn on PQ and QS as diameters. Find the perimeter of the shaded region.
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Hint

$QR+RS=4+4=8\ cm$
Perimeter of the shaded part $=\pi\times6+\pi\times4+\pi\times2$
$=12\pi=12\times\dfrac{22}{7}=37.71\ cm$
Area $=\dfrac{1}{2}\times\dfrac{22}{7}(6^2+2^2-4^2)$
$=\dfrac{1}{2}\times\dfrac{22}{7}\times24=37.71\ cm^2$
Question 32 of 38

32. Question
1 point(s)
OABC is a rhombus whose three vertices A, B and C lie on a circle with centre O. If the radius of the circle is 10 cm, find the area of the rhombus.
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Hint

$AB^2=AD^2+DB^2\\ \\ AD^2=10^2-5^2=75\\ \\ AD=5\sqrt3\ cm\\ \\ AC=2\times5\sqrt3=10\sqrt3\ cm$
Area $=\dfrac{1}{2}\times10\sqrt3\times10=50\sqrt3\ cm^2$
Question 33 of 38

33. Question
1 point(s)
The difference between circumference and diameter of a circle is 135 cm. Find the radius of the circle. $ \left[Take\ \pi=\dfrac{22}{7}\right]$
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Hint

$2\pi r-2r=135\\ \\ \Rightarrow 2r\left(\dfrac{22}{7}-1\right)=135\\ \\ \Rightarrow r=31.5\ cm$
Question 34 of 38

34. Question
1 point(s)
The length of the minute hand of a clock is 7 cm. Find the area swept by the minute hand from 6.00 pm to 6.10 pm.
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Hint

Area $=\dfrac{\theta}{360}\times \pi r^2$
$=\dfrac{60}{360}\times\dfrac{22}{7}\times7\times7\\ \\ =25.67\ cm^2$
Question 35 of 38

35. Question
1 point(s)
The sum of circumferences of two circles is 132 cm. If the radius of one circle is 14 cm, find the radius of the second circle.
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Hint

$2\pi\ (R+r)=132\\ \\ \Rightarrow 2\pi(14+r)=132\\ \\ (14+r)\times2\times\dfrac{22}{7}=132\\ \\ 14+r=\dfrac{132\times7}{44}\\ \\ 14+r=21\\ \\ r=7\ cm$
Question 36 of 38

36. Question
1 point(s)
What will be the increase in area of a circle, if its radius is increased by 40%?
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Hint

New radius $=140\%$ of $r$
$=\dfrac{7}{5}r$
New area $=\pi\left(\dfrac{7}{5}r\right)^2=\dfrac{49}{25}r^2$
$\%$ increase $=\dfrac{\dfrac{24}{25}\pi r^2}{\pi r^2}\times100=96\%$
Question 37 of 38

37. Question
1 point(s)
The radius of the wheels of a bus is 70 cm. How many revolutions per minute must a wheel make in order to move at a speed of 66 km/h?
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Hint

Circumference of the wheel $=2\pi\times\dfrac{70}{100}m$
Speed of the wheel $=66\ km/hr$
$=66\times16.67\ m/min$
Total resulution $=\dfrac{66\times16.67}{2\pi\times\dfrac{70}{100}}$
$=250$ resulution per minute
Question 38 of 38

38. Question
1 point(s)
What will be the ratio of perimeters a square and a circle if their areas are equal?
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Hint

$a^2=\pi r^2\\ \\ \Rightarrow \dfrac{a}{r}=\sqrt{\pi}\\ \\ \dfrac{4a}{2\pi r}=\dfrac{2a}{\pi r}=\dfrac{2}{\pi }\sqrt{\pi}\\ \\ =\dfrac{2}{\sqrt{\pi}}$