Mathematics Questions

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

1. Q561
   Prove ​that $\left(\sqrt{3}+1\right)(3-\cot 30^{\circ })={\tan }^{3}60^{\circ }-2\sin 60^{\circ }$.
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2. Q562
   Prove ​that $1+\frac{{\cot }^2\alpha }{1+\csc \alpha }=\csc \alpha$.
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3. Q563
   Prove ​that $\tan \theta +\tan (90^{\circ }-\theta )=\sec \theta \sec (90^{\circ }-\theta )$.
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4. Q564
   Find the angle of ​elevation of ​the sun ​when the shadow ​of ​a pole $h$ metres high is $\sqrt{3}h$ ​metres long.
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5. Q565
   If $\sqrt{3}\tan \theta =1$, ​then find ​the ​value ​of ${\sin }^2\theta -{\cos }^2\theta$.
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6. Q566
   A ladder ​15 ​metres long ​just reaches the top of a vertical wall. If ​the ladder makes ​an angle of $60^{\circ }$ ​with the ​wall, find the height ​of the wall.
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7. Q567
   Simplify $(1+{\tan }^2\theta )(1-\sin \theta )(1+\sin \theta )$.
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8. Q568
   If $2{\sin }^2\theta -{\cos }^2\theta =2$, ​then ​find the ​value of $\theta$.
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9. Q569
   Show ​that $\frac{{\cos }^2(45^{\circ }+\theta )+{\cos }^2(45^{\circ }-\theta )}{\tan (60^{\circ }+\theta )\tan (30^{\circ }-\theta )}=1$.
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10. Q570
    An observer $1.5$ metres tall is $20.5$ ​metres away ​from ​a ​tower $22$ metres high. Determine the angle ​of elevation of the top of the tower ​from ​the ​eye of the observer.
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11. Q571
    Show ​that ${\tan }^4\theta +{\tan }^2\theta ={\sec }^4\theta -{\sec }^2\theta$.
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12. Q572
    If radii ​of two concentric ​circles are 4 cm and 5 ​cm, ​then ​the length ​of each chord of one circle which is tangent to ​the ​other circle is

    1. (a)$3$ cm
    2. (b)$6$ cm
    3. (c)$9$ cm
    4. (d)$1$ cm
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13. Q573
    In Fig. 9.3, ​if $\angle AOB=125^{\circ }$, ​then $\angle COD$ ​is equal to

    1. (a)$62.5^{\circ }$
    2. (b)$45^{\circ }$
    3. (c)$35^{\circ }$
    4. (d)$55^{\circ }$
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14. Q574
    In Fig. 9.4, $AB$ is a chord of the ​circle and $AOC$ is its diameter such ​that $\angle ACB=50^{\circ }$. If $AT$ is the tangent to ​the circle at ​the ​point $A$, then $\angle BAT$ is ​equal to

    1. (a)$65^{\circ }$
    2. (b)$60^{\circ }$
    3. (c)$50^{\circ }$
    4. (d)$40^{\circ }$
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15. Q575
    From a point ​P which is at a distance of ​13 cm from ​the centre ​O of a circle of radius 5 cm, the ​pair of ​tangents ​PQ and PR to the circle are drawn. ​Then the area of the quadrilateral PQOR is

    1. (a)$60$ cm${}^2$
    2. (b)$65$ cm${}^2$
    3. (c)$30$ cm${}^2$
    4. (d)$32.5$ cm${}^2$
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16. Q576
    At one end A of a diameter AB of a circle of radius ​5 cm, tangent XAY is drawn to the circle. The ​length of ​the chord CD parallel to XY and ​at a distance 8 cm from A ​is

    1. (a)4 cm
    2. (b)5 cm
    3. (c)6 cm
    4. (d)8 cm
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17. Q577
    In ​Fig. 9.5, $AT$ is ​a tangent to ​the circle with ​centre $O$ such ​that $OT=4\text{cm}$ and $\angle OTA=30^{\circ }$. ​Then $AT$ is equal to

    1. (a)4 cm
    2. (b)2 cm
    3. (c)$2\sqrt{3}$ cm
    4. (d)$4\sqrt{3}$ cm
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18. Q578
    In ​Fig. 9.6, ​if O is the centre of a ​circle, PQ ​is a chord and the tangent PR ​at P makes an ​angle of 50 ​degrees with ​PQ, then $\angle POQ$ is equal to

    1. (a)$100^{\circ }$
    2. (b)$80^{\circ }$
    3. (c)$90^{\circ }$
    4. (d)$75^{\circ }$
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19. Q579
    In Fig. 9.7, ​if PA and ​PB ​are ​tangents to ​the circle ​with centre O ​such that $\angle APB=50^{\circ }$, then $\angle OAB$ is equal to

    1. (a)$25^{\circ }$
    2. (b)$30^{\circ }$
    3. (c)$40^{\circ }$
    4. (d)$50^{\circ }$
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20. Q580
    If two tangents inclined at ​an ​angle $60^{\circ }$ ​are drawn to ​a circle of radius ​3 ​cm, then length of each tangent is ​equal ​to

    1. (a)$\frac{3}{2}\sqrt{3}$ cm
    2. (b)$6$ cm
    3. (c)$3$ cm
    4. (d)$3\sqrt{3}$ cm
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