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1) In an increasing geometric series, the sum of the second and the sixth term is 25 / 2 and the product of the third and fifth term is 25. Then, the sum of 4th, 6th and 8th terms is equal to :
2. 30
3. 26
4. 35
Let a be the first term and r be the common ratio.
ar + ar5 = 25 / 2 and ar2 x ar4 = 25
a2r6 = 25
ar3 = 5
a = 5 / r3 …(1)
(5r / r3) + (5r5 / r3) = 25 / 2
(1 / r2) + r2 = 5 / 2
Put r2 = t
[t2 + 1] / t = 5 / 2
2t2 – 5t + 2 = 0
2t2 – 4t – t + 2 = 0
(2t – 1) (t – 2) = 0
t = 1 / 2, 2
r2 = 1 / 2, 2
r = √2 as the G.P. is increasing.
ar3 + ar5 + ar7
= ar3( 1 + r2 + r4)
= 5 [1 + 2 + 4]
2 ) Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.
Reason R: The de Broglie’s wavelength of the electrons emitted from an electron gun is much less than the wavelength of visible light.
In light of the above statements, choose the correct answer from the options given below:
1. A is true but R is false.
2. A is false but R is true.
3. Both A and R are true, and R is the correct explanation of A.
4. Both A and R are true, but R is NOT the correct explanation of A.
Answer: 3
Resolution limit (Δθ) = 1.22λ/d
Resolution power = 1/Resolution limit
λ ↓ Δθ↓
Δθ↓ Power ↑
3. Statement I: o-Nitrophenol is steam volatile due to intramolecular hydrogen bonding
Statement II: o-Nitrophenol has high melting point due to hydrogen bonding.
2. Statement I is true, but Statement II is false
3. Both Statement I and Statement II are true
4. Statement I is false, but Statement II is true
o-Nitrophenol is steam volatile due to intramolecular hydrogen H-bonding. but m-Nitrophenol has more melting point due to its symmetry.