Mathematics Questions

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

1. Q581
   In Fig. 9.8, if $PQR$ is ​the tangent to ​a circle at $Q$ ​whose centre is $O$, $AB$ is ​a ​chord ​parallel ​to $PR$ and $\angle BQR=70^{\circ }$, then $\angle AQB$ is ​equal to

   1. (a)$20^{\circ }$
   2. (b)$40^{\circ }$
   3. (c)$35^{\circ }$
   4. (d)$45^{\circ }$
   Open question → 🔒 Log in to view the solution
2. Q582
   State whether ​the following is true or false and ​give reasons: If ​a ​chord $AB$ subtends an angle of $60^{\circ }$ at the centre of a circle, then ​the angle between the tangents at $A$ ​and $B$ is also $60^{\circ }$.
   Open question → 🔒 Log in to view the solution
3. Q583
   The ​length of ​tangent ​from an external point on a circle is ​always greater than the ​radius of ​the ​circle.
   Open question → 🔒 Log in to view the solution
4. Q584
   Write 'True' or 'False' ​and ​justify ​your answer: The ​length of tangent from an ​external ​point P on a circle with centre O is ​always less than OP.
   Open question → 🔒 Log in to view the solution
5. Q585
   The ​angle ​between ​two ​tangents to ​a circle may ​be $0^{\circ }$.
   Open question → 🔒 Log in to view the solution
6. Q586
   If angle ​between two tangents ​drawn from a point $P$ to a circle of ​radius $a$ and centre $O$ is $90^{\circ }$, ​then $OP=a\sqrt{2}$. State whether ​this statement ​is true ​or false and justify.
   Open question → 🔒 Log in to view the solution
7. Q587
   State whether the following is true or false ​and justify: ​If the angle between ​two ​tangents drawn from a ​point P to ​a circle ​of ​radius $a$ and centre O is $60^{\circ }$, then $OP=a\sqrt{3}$.
   Open question → 🔒 Log in to view the solution
8. Q588
   The ​tangent ​to the ​circumcircle of an isosceles triangle $ABC$ at $A$, ​in which $AB=AC$, is ​parallel to $BC$.
   Open question → 🔒 Log in to view the solution
9. Q589
   If a ​number of circles touch ​a given line segment ​PQ at a point ​A, then ​their ​centres lie on the perpendicular bisector of ​PQ.
   Open question → 🔒 Log in to view the solution
10. Q590
    If a ​number ​of ​circles pass through the end points ​P and Q ​of a line segment PQ, then their centres ​lie on the ​perpendicular bisector of PQ.
    Open question → 🔒 Log in to view the solution
11. Q591
    State ​whether the following ​is true or false and justify: AB ​is a diameter ​of ​a circle and AC is its chord such that $\angle BAC=30^{\circ }$. If the tangent at ​C intersects AB extended at D, ​then $BC=BD$.
    Open question → 🔒 Log in to view the solution
12. Q592
    Out of ​the two concentric circles, the radius ​of the outer ​circle is ​5 cm and the chord $AC$ ​of length ​8 cm is a tangent to the inner circle. ​Find the ​radius of the inner circle.
    Open question → 🔒 Log in to view the solution
13. Q593
    Two tangents PQ ​and PR are drawn ​from ​an external point ​to a ​circle with centre O. Prove that QORP is a ​cyclic quadrilateral.
    Open question → 🔒 Log in to view the solution
14. Q594
    If from an external point B of a circle with centre ​O, two tangents BC ​and ​BD are ​drawn such ​that $\angle DBC=120^{\circ }$, prove ​that $BC+BD=BO$, ​i.e., $BO=2BC$.
    Open question → 🔒 Log in to view the solution
15. Q595
    Prove that ​the centre of a circle touching two ​intersecting lines lies on ​the angle ​bisector of ​the ​lines.
    Open question → 🔒 Log in to view the solution
16. Q596
    In Fig. 9.13, ​AB ​and CD ​are ​common ​tangents ​to two circles of ​unequal radii. Prove that $AB=CD$.

    Open question → 🔒 Log in to view the solution
17. Q597
    In ​Question 5 above, if ​radii of the ​two circles are equal, ​prove ​that ​AB $=$ ​CD.
    Open question → 🔒 Log in to view the solution
18. Q598
    In ​Fig. ​9.14, ​common tangents $AB$ ​and $CD$ ​to two circles intersect at $E$. ​Prove that $AB=CD$.

    Open question → 🔒 Log in to view the solution
19. Q599
    A ​chord PQ of a ​circle is ​parallel to ​the tangent drawn at a point ​R of the circle. Prove ​that R ​bisects the ​arc PRQ.
    Open question → 🔒 Log in to view the solution
20. Q600
    Prove that the ​tangents drawn at ​the ends ​of ​a chord of ​a ​circle make equal angles ​with the chord.
    Open question → 🔒 Log in to view the solution