Mathematics Questions

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

1. Q121
   State whether ​the ​following ​are true or false. Justify your answer.

   1. (i)The value of $\tan A$ is always less ​than $1$.
   2. (ii)$\sec A=\frac{12}{5}$ for some value of angle $A$.
   3. (iii)$\cos A$ ​is the abbreviation used for the cosecant of angle $A$.
   4. (iv)$\cot A$ is the product ​of $\cot$ and $A$.
   5. (v)$\sin \theta =\frac{4}{3}$ for ​some angle $\theta$.
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2. Q122
   Evaluate ​the ​following: ​

   1. (i)$\sin 60^{\circ }\cos 30^{\circ }+\sin 30^{\circ }\cos 60^{\circ }$
   2. (ii)$2{\tan }^{2}45^{\circ }+{\cos }^{2}30^{\circ }-{\sin }^{2}60^{\circ }$ ​
   3. (iii)$\frac{\cos 45^{\circ }}{\sec 30^{\circ }+\csc 30^{\circ }}$
   4. (iv)$\frac{\sin 30^{\circ }+\tan 45^{\circ }-\csc 60^{\circ }}{\sec 30^{\circ }+\cos 60^{\circ }+\cot 45^{\circ }}$ ​
   5. (v)$\frac{5{\cos }^{2}60^{\circ }+4{\sec }^{2}30^{\circ }-{\tan }^{2}45^{\circ }}{{\sin }^{2}30^{\circ }+{\cos }^{2}30^{\circ }}$.
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3. Q123
   Choose ​the correct option and ​justify your ​choice:

   1. (i)$\frac{2\tan 30^{\circ }}{1+{\tan }^{2}30^{\circ }}=$ (A) $\sin 60^{\circ }$ (B) $\cos 60^{\circ }$ (C) $\tan 60^{\circ }$ (D) $\sin 30^{\circ }$.
   2. (ii)$\frac{1-{\tan }^{2}45^{\circ }}{1+{\tan }^{2}45^{\circ }}=$ (A) $\tan 90^{\circ }$ (B) $1$ (C) $\sin 45^{\circ }$ ​(D) $0$.
   3. (iii)$\sin 2A=2\sin A$ is ​true when $A=$ (A) $0^{\circ }$ ​(B) $30^{\circ }$ ​(C) $45^{\circ }$ (D) $60^{\circ }$.
   4. (iv)$\frac{2\tan 30^{\circ }}{1-{\tan }^{2}30^{\circ }}=$ (A) $\cos 60^{\circ }$ (B) $\sin 60^{\circ }$ (C) $\tan 60^{\circ }$ (D) $\sin 30^{\circ }$.
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4. Q124
   If $\tan (A+B)=\sqrt{3}$ ​and $\tan (A-B)=\frac{1}{\sqrt{3}}$; $0^{\circ }<A+B\le 90^{\circ }$; $A>B$, ​find $A$ ​and $B$.
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5. Q125
   State whether the following are ​true or false. Justify your ​answer.

   1. (i)$\sin (A+B)=\sin A+\sin B$.
   2. (ii)​The value of $\sin \theta$ increases as $\theta$ increases.
   3. (iii)​The ​value of $\cos \theta$ increases as $\theta$ ​increases.
   4. (iv)$\sin \theta =\cos \theta$ for all values ​of $\theta$.
   5. (v)$\cot A$ is not defined for $A=0^{\circ }$.
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6. Q126
   A ​circus artist is climbing a 20 ​m long rope, which ​is tightly stretched and ​tied from the top of a vertical pole to the ground. Find the height of ​the pole, if the angle made by ​the rope with the ground ​level is $30^{\circ }$ (see Fig. 9.11).

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7. Q127
   A tree ​breaks due to storm and the broken part bends so that the ​top of ​the tree touches the ground making ​an angle $30^{\circ }$ with it. The distance ​between the ​foot of the tree to the point where the ​top ​touches the ground is 8 m. Find the height of the tree.
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8. Q128
   A contractor plans to install two slides for the ​children to play in a park. For the children below the age of $5$ years, she prefers to have a slide ​whose top is at a height of $1.5$ m, ​and is inclined at an angle of $30^{\circ }$ to the ground, whereas for elder children, she wants to have a steep slide at a height of $3$ m, and inclined at an ​angle of $60^{\circ }$ to the ground. What should be the length of the slide in each case?

   1. (i)​For children below 5 years: a slide with its top at a height of $1.5$ m, inclined at $30^{\circ }$ to the ground. Find ​the length of this slide.
   2. (ii)For elder children: a steeper slide with its top at ​a height of $3$ m, inclined at $60^{\circ }$ to the ground. Find the length of this ​slide.
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9. Q129
   The angle of elevation of the top of a tower from a point on the ground, which ​is $30$ m away ​from the foot ​of ​the ​tower, is $30^{\circ }$. Find the height ​of the ​tower.
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10. Q130
    A kite is flying at a ​height of $60$ m above the ground. The string attached to the ​kite is temporarily tied to a point ​on the ground. ​The inclination of the string with the ​ground is $60^{\circ }$. ​Find the length of ​the string, assuming that there is no slack in the string.
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11. Q131
    A 1.5 m tall ​boy ​is standing at some distance ​from a 30 m tall building. The angle of elevation ​from his eyes to the top ​of the building ​increases from $30^{\circ }$ to $60^{\circ }$ as he walks towards ​the building. Find the distance he walked towards the building.
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12. Q132
    From ​a point on the ground, the ​angles of elevation of the bottom ​and the top ​of a transmission tower fixed at the ​top ​of a $20$ m high building are $45^{\circ }$ and $60^{\circ }$ respectively. Find the ​height of the tower.
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13. Q133
    A statue, $1.6$ m tall, stands on the top ​of a pedestal. From ​a point on the ground, the ​angle of elevation ​of the top of the statue ​is $60^{\circ }$ ​and from the same point the angle of elevation of ​the top of the pedestal is $45^{\circ }$. Find the height ​of the pedestal.
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14. Q134
    The angle of elevation of ​the top of a building from ​the foot of ​the tower is $30^{\circ }$ and the angle ​of elevation ​of the top of the tower from ​the foot ​of the building is $60^{\circ }$. If the tower is $50$ m high, find the height of the ​building.
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15. Q135
    Two ​poles ​of equal ​heights are standing opposite each other on either side of the road, ​which ​is $80$ m wide. From a point between them on ​the road, the angles of elevation of the top of the poles are $60^{\circ }$ and $30^{\circ }$, respectively. Find the ​height of the poles and the distances of the point from the poles.

    1. (i)Find the height of the poles (both are equal).
    2. (ii)Find the distances of the point from each of the two poles.
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16. Q136
    A TV tower stands vertically on a bank ​of ​a canal. ​From a point on the other bank directly ​opposite the tower, the angle of elevation of the top of the tower is $60^{\circ }$. From another point $20$ m away from this point on the line joining this point to the foot of the tower, the angle of elevation of ​the top of the tower is $30^{\circ }$ (see Fig. 9.12). Find the ​height of the tower and the width of the canal.

    1. (i)Find the height of the TV ​tower.
    2. (ii)Find the width ​of the canal.
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17. Q137
    From ​the top of a $7$ m high building, ​the angle ​of elevation of the top of a cable tower is $60^{\circ }$ and the angle of ​depression of ​its foot ​is $45^{\circ }$. Determine the height ​of the tower.
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18. Q138
    From the top of a 7 m high building, ​the ​angle of elevation of the top of a cable ​tower is ​60 degrees and the angle of ​depression of its foot ​is 45 degrees. Determine the height of the tower.
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19. Q139
    As observed from ​the top of a 75 ​m high ​lighthouse from the sea-level, the angles of depression of two ships are $30^{\circ }$ ​and $45^{\circ }$. If ​one ship is exactly ​behind the other on the same side ​of the lighthouse, find ​the distance between the two ships.
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20. Q140
    A $1.2$ m tall ​girl ​spots a balloon moving with the wind ​in a horizontal line at a height of $88.2$ m from the ground. The angle of elevation of the balloon from the ​eyes of the girl at any instant is $60^{\circ }$. After some time, the angle of elevation reduces to $30^{\circ }$ (see Fig. ​9.13). ​Find the distance travelled by ​the balloon during the interval.
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