Mathematics Questions

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

1. Q541
   If $4\tan \theta =3$, ​then $\left(\frac{4\sin \theta -\cos \theta }{4\sin \theta +\cos \theta }\right)$ ​is ​equal to

   1. (a)$\frac{2}{3}$
   2. (b)$\frac{1}{3}$
   3. (c)$\frac{1}{2}$
   4. (d)$\frac{3}{4}$
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2. Q542
   If $\sin \theta -\cos \theta =0$, ​then the ​value of $({\sin }^4\theta +{\cos }^4\theta )$ ​is

   1. (a)$1$
   2. (b)$\frac{3}{4}$
   3. (c)$\frac{1}{2}$
   4. (d)$\frac{1}{4}$
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3. Q543
   $\sin (45^{\circ }+\theta )-\cos (45^{\circ }-\theta )$ is equal ​to

   1. (a)$2\cos \theta$
   2. (b)$0$
   3. (c)$2\sin \theta$
   4. (d)$1$
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4. Q544
   A pole 6 m high casts ​a ​shadow $2\sqrt{3}$ m ​long on ​the ​ground, then the ​Sun's elevation ​is

   1. (a)$60^{\circ }$
   2. (b)$45^{\circ }$
   3. (c)$30^{\circ }$
   4. (d)$90^{\circ }$
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5. Q545
   State whether ​the ​following ​is true ​or false: $\frac{\tan 47^{\circ }}{\cot 43^{\circ }}=1$.
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6. Q546
   The ​value ​of the ​expression $({\cos }^{2}23^{\circ }-{\sin }^{2}67^{\circ })$ ​is positive.
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7. Q547
   The ​value ​of the ​expression $(\sin 80^{\circ }-\cos 80^{\circ })$ ​is ​negative.
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8. Q548
   $\sqrt{(1-{\cos }^2\theta ){\sec }^2\theta }=\tan \theta$
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9. Q549
   If $\cos A+{\cos }^{2}A=1$, ​then ${\sin }^{2}A+{\sin }^{4}A=1$.
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10. Q550
    State ​whether the ​following ​is ​true ​or ​false and justify: $(\tan \theta +2)(2\tan \theta +1)=5\tan \theta +{\sec }^2\theta$.
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11. Q551
    If the length ​of the shadow of ​a tower is increasing, then the angle ​of ​elevation of ​the sun is also increasing.
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12. Q552
    State whether ​the following is true or false and justify: If a man standing ​on a platform 3 metres above ​the surface of a ​lake ​observes a cloud and its reflection in the lake, then ​the angle of elevation of ​the cloud is equal ​to the angle of depression of its reflection.
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13. Q553
    State whether the following are ​true or ​false and justify: (9) ​The value of $2\sin \theta$ can be $a+\frac{1}{a}$, where $a$ is a positive number ​and $a\ne 1$. (10) $\cos \theta =\frac{a^2+b^2}{2ab}$, where $a$ ​and $b$ are two ​distinct ​numbers such that $ab>0$.
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14. Q554
    Write 'True' or ​'False' ​and ​justify your answer: $\cos \theta =\frac{a^2+b^2}{2ab}$, where $a$ ​and $b$ are two distinct ​numbers such ​that $ab>0$.
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15. Q555
    State whether ​the following statement is true or false ​and justify: The angle of elevation of the top of a tower is ​30 ​degrees. If the height of the tower is doubled, ​then the angle of elevation of its ​top will ​also be doubled.
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16. Q556
    State whether the following is true or false ​and justify: If the height of a tower and the distance ​of ​the point of ​observation from its foot, ​both, are ​increased by 10%, then ​the angle of elevation of its top remains unchanged.
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17. Q557
    Prove ​that $\frac{\sin \theta }{1+\cos \theta }+\frac{1+\cos \theta }{\sin \theta }=2\mathrm{cosec}\theta$.
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18. Q558
    Prove ​that $\frac{\tan A}{1+\sec A}-\frac{\tan A}{1-\sec A}=2\mathrm{cosec}A$.
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19. Q559
    If $\tan A=\frac{3}{4}$, ​then $\sin A\cos A=\frac{12}{25}$.
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20. Q560
    Prove ​that $(\sin \alpha +\cos \alpha )(\tan \alpha +\cot \alpha )=\sec \alpha +\csc \alpha$.
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