Q.Find the sum of 20 terms of the series: 12-22+32-42+52...
If T1 , T2 , T3 .....Tn are consecutive terms of an AP then prove that 1/T1T2 + 1/T2 T3 +........ + (n-1)/T1 Tn
Let a,b,c,d be positive rationals such that a + √b = c + √d, then show that either (a = c) and (b = d) or b and d are squares of rationals.
Q. In the given figure, a semicircle is drawn in AB as diameter and P is a point on AC produced. If BP meets the semicircle on AC produced. If BP meets the semicircle in D, Prove that AB2 = AC x AP + BD x BP
In Triangle ABC, <ACB = 900 Þ <BCP = 900
So, In Triangle BPC: BC2 = BP2 – CP2
Now, In Triangle ABC,
AB2 = AC2 + BC2 Þ AB2 = AC2 + BP2 – CP2
AB2 = AC2 – CP2 + BP2
AB2 = (AC - CP) (AC + CP) + BP2
AB2 = (AC - CP) (AP) + BP2
AB2 = (AC x AP - CP x AP) + BP2
AB2 = AC x AP + BP2 - CP x AP
But, AP x CP = BP x PD (D APB ~ D CPB)
AB2 = AC x AP + BP2 - BP x PD
AB2 = AC x AP + BP(BP – PD)
AB2 = AC x AP + BP x BD
D is the midpoint of side BC of a triangle ABC.AD is bisected at a point E and BE produced cuts AC at the point X. Prove that BE : EX = 3:1
Robbers in a car travelling at 20m/s pass a policeman on a motorcycle at rest. The policeman fastly starts chasing the robbers. The policeman accelerates at 3m/s2 for 12s and thereafter travels at a constant velocity. Calculate the distance covered by th
The distance covered by the policeman in 12 sec
s = ut + 1/2 at2
= 0 x12 + 0.5 x3 x 12x12 = 216m
v = u + at = 0 + 12x3 = 36m/s
The distance covered by the robber in 12 sec
s = 20 x 12 = 240m
Suppose he over takes in 12 + t sec, in which he travels = 216 + 36 * t
r robbers will travel in the same time 20 * ( 12+t) m
216 + 36 t = 240 + 20 t
t = 24 / 16 = 1.5 s
Distance travelled by policeman in 12 + 1.5 = 13.5 s
= 216 + 36 x 1.5 = 270 m
1. Ostrich has the largest eggs.
2. Amoeba and Hydra do not reproduce sexually because they do not have distinct male and female sexes. They do not produce sex cells or gametes. Hence, they reproduce asexually to grow and multiply or continue their life form.
3. In frogs, fertilisation takes place outside the body of the female. Female frogs lay eggs in large number because they are exposed to the external environment that may destroy many eggs. Hence, laying eggs in large number not only promotes fertilisation but also increases the chances of survival of at least few of the fertilised eggs.
4. No, the transformation of puppies into adult dogs is not called metamorphosis. Metamorphosis is the transformation of a larva into an adult. Dogs give birth to puppies, which grow into adults slowly.
5. If semen is not formed in male reproductive organs, then the male cannot fertilise the egg or ovum, as sperms will not reach the egg
6. Hermaphrodites are the organisms in which both the male and female sex organs are located. Earthworm and Hydra are the examples of hermaphrodites.
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Basic science Class 8 chapter-conservation By Jsunil
150 workers were engaged to finish a job in a certain number of days. 4 workers dropped out on second day, 4 more workers dropped out on the third day and so on. It took 8 more days to finish the work. Find the number of days in which the work was comple
Q. 150 workers were engaged to finish a job in a certain number of days. 4 workers dropped out on second day, 4 more workers dropped out on the third day and so on. It took 8 more days to finish the work. Find the number of days in which the work was completed.
Sol: Suppose the work is completed in n days.
Since 4 workers went away on every day except the first day.
∴ Total number of worker who worked all the n days is the sum of n terms of A.P. with first term 150 and common difference – 4.
Total number of worker who worked all the n days = n/2[2 x 150 + (n-1) x -4 ] = n (152 – 2n)
If the workers would not have went away, then the work would have finished in (n – 8) days with 150 workers working on every day.
∴ Total number of workers who would have worked all n days = 150 (n – 8)
∴ n (152 – 2n) = 150 (n – 8) ⇒ 152n – 2n2 = 150n – 1200 ⇒ 2n2 – 2n – 1200 = 0
⇒ n2 – n – 600 = 0 ⇒ n2 – 25n + 24n – 600 = 0 ⇒ n(n – 25) + 24 (n + 25) = 0
⇒ (n – 25) (n + 24) = 0 ⇒ n – 25 = 0 or n + 24 = 0 ⇒ n = 25 or n = – 24
⇒ n = 25 ( Number of days cannot be negative)
Thus, the work is completed in 25 days.
Q. Divide 32 into 4 parts which are in A.P. such that the product the product of means is 7:15.
Let four numbers in A.P. be a – 3d, a – d, a + d and a + 3d.
(a – 2d) + (a – d) + (a + d) + (a + 2d) = 32
⇒ 4a = 32
⇒ a = 8
Product of first and fourth term(extremes) = (a – 3d) (a + 3d) = (8 – 3d) (8 +32d) = 64 – 9d2
Product of second and third term (means) = (a – d) (a + d) = (8 – d) (8 + d) = 64 – d2
Product of extremes: product of means = 7:15 (Given)
( 64 – 9d2) / (64 – d2) = 7 : 15
Magnetic needle placed at distance r from straight conductor of length 8r carrying current I shows a deflection of Ф degree. what may be the deflection on passing the same current when conductor is bent in form of square and magnetic needle is placed
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