Mathematics Questions

Browse 1,240 Mathematics questions with expert-verified, step-by-step solutions — NCERT, exemplar and previous-year questions, organised by chapter.

1. Q141
   A ​straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of $30^{\circ }$, which is approaching the foot of the ​tower with ​a uniform speed. Six seconds later, ​the angle of depression of the car ​is found to be $60^{\circ }$. Find the time taken by the car ​to reach the foot of the tower from ​this point.
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2. Q142
   How many ​tangents ​can a circle have?
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3. Q143
   Fill in the blanks:

   1. (i)A tangent ​to ​a circle intersects it ​in ____ ​point(s).
   2. (ii)A line intersecting a circle in two points is called a ____.
   3. (iii)​A circle can ​have ____ parallel tangents ​at the most.
   4. (iv)The common point of a tangent to a ​circle and the circle is called ____.
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4. Q144
   A ​tangent $PQ$ ​at ​a point $P$ of a circle ​of radius 5 cm meets a line ​through ​the centre $O$ at a point $Q$ ​so ​that $OQ=12$ cm. Length $PQ$ is :

   1. (a)12 cm
   2. (b)13 cm
   3. (c)8.5 cm
   4. (d)$\sqrt{119}$ cm
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5. Q145
   Draw ​a circle and ​two lines parallel to a ​given line such that one ​is ​a tangent and the ​other, a secant to the circle.
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6. Q146
   From a point Q, the length of the tangent to a circle is ​24 ​cm and the distance of ​Q ​from the centre is 25 ​cm. The radius of the circle is (A) 7 cm ​(B) 12 cm (C) ​15 cm (D) 24.5 ​cm

   1. (a)7 cm
   2. (b)12 cm
   3. (c)15 cm
   4. (d)24.5 cm
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7. Q147
   In Fig. 10.11, ​if $TP$ and $TQ$ ​are the two tangents ​to a ​circle ​with ​centre $O$ ​so that $\angle POQ=110^{\circ }$, then $\angle PTQ$ is equal ​to

   1. (a)$60^{\circ }$
   2. (b)$70^{\circ }$
   3. (c)$80^{\circ }$
   4. (d)$90^{\circ }$
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8. Q148
   If tangents ​PA and PB from a ​point P to a circle with centre O ​are inclined to each ​other ​at ​angle of $80^{\circ }$, then $\angle POA$ is equal to

   1. (a)$50^{\circ }$
   2. (b)$60^{\circ }$
   3. (c)$70^{\circ }$
   4. (d)$80^{\circ }$
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9. Q149
   Prove ​that ​the ​tangents drawn at ​the ​ends of a ​diameter ​of a circle ​are parallel.
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10. Q150
    Prove that ​the perpendicular at ​the ​point of ​contact to the tangent to a ​circle ​passes through the ​centre.
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11. Q151
    The ​length of ​a tangent ​from a point $A$ at distance $5$ cm from the ​centre ​of the circle is $4$ cm. ​Find the radius ​of the circle.
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12. Q152
    Two concentric ​circles are of ​radii $5$ cm and $3$ cm. Find the ​length of the chord ​of the larger circle which touches the ​smaller circle.
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13. Q153
    A ​quadrilateral $ABCD$ is ​drawn ​to circumscribe a circle (see ​Fig. 10.12). ​Prove ​that $AB+CD=AD+BC$.

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14. Q154
    In Fig. ​10.13, XY ​and ​X'Y' are two parallel tangents to a circle with centre O and another tangent AB with point ​of contact C ​intersecting XY at ​A and X'Y' at B. ​Prove ​that $\angle AOB=90^{\circ }$.
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15. Q155
    Prove ​that the angle between ​the two tangents ​drawn from ​an external point ​to a circle is supplementary ​to the angle subtended ​by the line-segment joining the points of contact at the centre.
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16. Q156
    Prove that the ​parallelogram ​circumscribing a ​circle is ​a ​rhombus.
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17. Q157
    A triangle ​ABC is drawn to circumscribe a ​circle of ​radius ​4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 8 cm ​and 6 cm respectively (see Fig. 10.14). ​Find ​the sides AB and AC.

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18. Q158
    Prove ​that opposite sides ​of a quadrilateral ​circumscribing ​a circle subtend supplementary angles at the centre ​of ​the ​circle.
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19. Q159
    Find ​the area of ​a sector ​of a circle with radius 6 cm ​if ​angle of the sector ​is 60 ​deg.
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20. Q160
    Find ​the ​area of a ​quadrant of a ​circle ​whose circumference is $22$ ​cm.
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