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Byju's Answer

Standard XII

Mathematics

Sum of Product of Binomial Coefficients

If an + bn/...

Question

If $\frac{a^{n} + b^{n}}{a^{n - 1} + b^{n - 1}} = x$ , where a, b > 0 & a $\neq$ b, then
List I List II
P. If n = 1, then x is 1. $(a + b) - 2 a b (a + b)^{- 1}$
Q. If n = 1/2, then x is 2. AM of a and b
R. If n = 0, then x is 3. HM of a and b
S. If n = 2, then x is 4. GM of a and b

P - 2, Q - 4, R - 3, S - 1

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P - 1, Q - 2, R - 3, S - 4

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P - 4, Q - 2, R - 3, S - 1

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P - 3, Q - 2, R - 4, S - 1

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Solution

The correct option is B P - 2, Q - 4, R - 3, S - 1
For $n = 1,$ $x = \frac{a + b}{2}$
For $n = \frac{1}{2}, x = \frac{a^{1 / 2} + b^{1 / 2}}{a^{- 1 / 2} + b^{- 1 / 2}} = \sqrt{a b}$
For $n = 0, x = \frac{2}{a^{- 1} + b^{- 1}} = \frac{2 a b}{a + b}$
For $n = 2, x = \frac{a^{2} + b^{2}}{a + b} = \frac{{(a + b)}^{2} - 2 a b}{a + b} = (a + b) - 2 a b {(a + b)}^{- 1}$

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Similar questions

Q. Given that

n \sum x = 1

(x+1)(2x+3) =

p n^{4} + q n^{3} + r n^{2} + \frac{7}{3} n + s

, then

i) p = \frac{1}{2}, q = \frac{8}{3} i i) p = \frac{1}{3}, q = \frac{1}{4} i i i) r = \frac{9}{2}, s = 0 i v) r = \frac{5}{6}, s = 0

Q. Match the following:

$\begin{matrix} 1. Pituitary gland & P. Thyroxine 2. Thyroid & Q. Growth Hormone, ACTH, Prolactin,TSH 3. Pancreas & R. Other endocrine glands 4. Pineal gland and parathyroid & S. Insulin, glucagon \end{matrix}$

Q. Match the following (a > 0)
(1) | x|

\leq

a (P) -a

\leq

\leq

a
(2) | x|

\geq

a (Q) x

\leq

-a or x

\geq

a
(3) | x| - a = 0 (R) x = a or x = -a
(4)

| x - a |

= 0 (S) x = a
(5)

x^{2}

\leq

a^{2}

$\begin{matrix} T r i g o n o m e t r i c E q u a t i o n s & G e n e r a l S o l u t i o n s 1. t a n x = 2 & P . 2 n π \pm \frac{2 π}{3}, n \in I 2. s i n x = \frac{\sqrt{3}}{2} & Q . n π - \frac{π}{4}, n \in I 3. c o s x = \frac{- 1}{2} & R . n π + (- 1)^{n} \frac{π}{3}, n \in I 4. c o t x = - 1 & S . n π + t a n^{- 1} (2), n \in I \end{matrix}$

Match the following (|x| < 1)

(1) ${(1 + x)}^{- 1}$ (P) 1 + 2x + 3 $x^{2}$ + 4 $x^{3}$ .........

(2) ${(1 - x)}^{- 1}$ (Q) 1 - x + $x^{2}$ - $x^{3}$ .........

(3) ${(1 + x)}^{- 2}$ (R) 1 + x + $x^{2}$ + $x^{3}$ .........

(4) ${(1 - x)}^{- 2}$ (S)1 - 2x + 3 $x^{2}$ - 4 $x^{3}$ .........

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	List I		List II
P.	If n = 1, then x is	1.	$(a + b) - 2 a b (a + b)^{- 1}$
Q.	If n = 1/2, then x is	2.	AM of a and b
R.	If n = 0, then x is	3.	HM of a and b
S.	If n = 2, then x is	4.	GM of a and b

If an+bnan−1+bn−1=x, where a, b > 0 & a ≠ b, thenList IList IIP.If n = 1, then x is1.(a+b)−2ab(a+b)−1Q.If n = 1/2, then x is2.AM of a and bR.If n = 0, then x is3.HM of a and bS.If n = 2, then x is4.GM of a and b

The correct option is B P - 2, Q - 4, R - 3, S - 1For n=1, x=a+b2For n=12,x=a1/2+b1/2a−1/2+b−1/2=√abFor n=0,x=2a−1+b−1=2aba+bFor n=2,x=a2+b2a+b=(a+b)2−2aba+b=(a+b)−2ab(a+b)−1

If $\frac{a^{n} + b^{n}}{a^{n - 1} + b^{n - 1}} = x$ , where a, b > 0 & a $\neq$ b, then
List I List II
P. If n = 1, then x is 1. $(a + b) - 2 a b (a + b)^{- 1}$
Q. If n = 1/2, then x is 2. AM of a and b
R. If n = 0, then x is 3. HM of a and b
S. If n = 2, then x is 4. GM of a and b