Mathematics

CBSE Class 10

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The 14^{th} term of an AP is twice its 8^{th} term. If its 6^{th} terms is -8, then find the sum of its first 20 terms.

Let a and d be the first term and the common difference of the AP, respectively.

∴ n^{th} term of the AP, a_{n} = a + (n-1)d

So,

a_{14} = a+ (14-1)d = a + 13d

a_{8} = a + (8 - 1)d = a + 7d

a_{6} = a+ (6 - 1)d = a + 5d

According to the question,

a_{14} = 2a_{8}

⇒ a + 13d = 2 (a + 7d)

⇒ a + d = 0 .... (i)

Also,

a_{6} = a + 5d = - 8 ... (ii)

solving (i) and (ii), we get

a = 2 and d = -2

∴ S_{20} = 20/2 [2 x 2 + (20 - 1)(-2)]

= - 340

Hence, the sum of the first 20 terms is -340.

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13.
The angle of elevation of an aeroplane from a point A on the ground 60°. After a flight of 15 seconds, the angle of elevation changes to 30°. If the aeroplane is flying at a constant height of , find the speed of the plane in km/hr.

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14.

If the coordinates of points A and B are (-2, -2) and (2, -4) respectively, find the coordinates of P such that AP = 3/7 AB, where P lies on the line segment AB.

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15.
The probability of selecting a red ball at random from a jar that contains only red, blue and orange balls is 1/4. The probability of selecting a blue ball at random from the same jar is 1/3. If the jar contains 10 orange balls, find the total number of balls in the jar.

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16.

Find the area of the minor segment of a circle of radius 14 cm, when its central angle is 60°. Also find the area of the corresponding major segment. [ use π = 22/7 ]

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17.

Due to sudden floods, some welfare associations jointly requested the government to get 100 tents fixed immediately and offered to contribute 50% of the cost. If the lower part of each tent is of the form of a cylinder of diameter 4.2 m and height 4 m with the conical upper part of same diameter but of height 2.8 m, and the canvas to be used costs Rs. 100 per sq. m, find the amount, the associations will have to pay. What values are shown by these associations? [ use π = 22/7]

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18.
A hemispherical bowl of internal diameter 36 cm contains liquid. This liquid is filled into 72 cylindrical bottles of diameter 6 cm. Find the height of the each bottle, if 10% liquid is wasted in this transfer.

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19.
A cubical block of side 10 cm is surmounted by a hemisphere. What is the largest diameter that the hemisphere can have ? Find the cost of painting the total surface area of the solid so formed, at the rate of Rs 5 per 100 sq. cm. [Use π =3.14]

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20.
504 cones, each of diameter 3.5 cm and height 3 cm, are melted and recast into a metallic sphere. Find the diameter of the sphere and hence find its surface area. [Use π =22/7]

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