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AlgebraQuestion and Answers: Page 342

Question Number 23479 Answers: 0 Comments: 0

Common solution. (d/dy)(u_x +u)+2x^2 y(u_x +u)=0.

$Common solution .$ $\frac{d}{d y} (u_{x} + u) + 2 x^{2} y (u_{x} + u) = 0 .$

Question Number 23471 Answers: 1 Comments: 0

Prove that ΣΣ_(0≤i<j≤n) ((1/(^n C_i )) + (1/(^n C_j ))) = Σ_(r=0) ^(n−1) ((n − r)/(^n C_r )) + Σ_(r=1) ^n (r/(^n C_r ))

$P r o v e t h a t$ $\underset{0 ⩽ i < j ⩽ n}{Σ Σ} (\frac{1}{^{n} C_{i}} + \frac{1}{^{n} C_{j}}) = \underset{r = 0}{\sum^{n - 1}} \frac{n - r}{^{n} C_{r}} + \underset{r = 1}{\sum^{n}} \frac{r}{^{n} C_{r}}$

Question Number 23357 Answers: 0 Comments: 0

Prove that (1/(m!))C_0 +(n/((m+1)!))C_1 +((n(n−1))/((m+2)!))C_2 +...+((n(n−1)...2.1)/((m+n)!))C_n = (((m+n+1)(m+n+2)...(m+2n))/((m+n)!)).

$P r o v e t h a t \frac{1}{m!} C_{0} + \frac{n}{(m + 1)!} C_{1} + \frac{n (n - 1)}{(m + 2)!} C_{2}$ $+ . . . + \frac{n (n - 1) . . . 2.1}{(m + n)!} C_{n} =$ $\frac{(m + n + 1) (m + n + 2) . . . (m + 2 n)}{(m + n)!} .$

Question Number 23334 Answers: 1 Comments: 3

Question Number 23323 Answers: 0 Comments: 1

Question Number 23250 Answers: 0 Comments: 0

If (1 + x)^n = C_0 + C_1 x + C_2 x^2 + C_3 x^3 + ... + C_n x^n , then prove that ΣΣ_(0≤i<j≤n) ((i/(^n C_i )) + (j/(^n C_j ))) = (n^2 /2)(Σ_(r=0) ^n (1/(^n C_r ))).

$I f {(1 + x)}^{n} = C_{0} + C_{1} x + C_{2} x^{2} + C_{3} x^{3}$ $+ . . . + C_{n} x^{n}, t h e n p r o v e t h a t$ $\underset{0 ⩽ i < j ⩽ n}{Σ Σ} (\frac{i}{^{n} C_{i}} + \frac{j}{^{n} C_{j}}) = \frac{n^{2}}{2} (\underset{r = 0}{\sum^{n}} \frac{1}{^{n} C_{r}}) .$

Question Number 23208 Answers: 1 Comments: 0

Is it possible to find how many real roots exist in the equation x^4 + ∣x∣ = 3 without find all the value of x?

$Is it possible to find how many real roots$ $exist in the equation$ $x^{4} + ∣ x ∣ = 3$ $without find all the value of x ?$

Question Number 23096 Answers: 0 Comments: 1

Solve: 3^(x ) = ((27)/x) + 18

$Solve : 3^{x} = \frac{27}{x} + 18$

Question Number 23000 Answers: 0 Comments: 0

Let n be a positive integer and p_1 , p_2 , ..., p_n be n prime numbers all larger than 5 such that 6 divides p_1 ^2 + p_2 ^2 + ... + p_n ^2 . Prove that 6 divides n.

$Let n be a positive integer and p_{1}, p_{2},$ $. . ., p_{n} be n prime numbers all larger$ $than 5 such that 6 divides p_{1}^{2} + p_{2}^{2} + . . . +$ $p_{n}^{2} . Prove that 6 divides n .$

Question Number 22945 Answers: 0 Comments: 0

Question Number 22856 Answers: 0 Comments: 1

solve: sin(x)=2, x∈C

$solve : \sin (x) = 2, x \in C$

Question Number 22768 Answers: 1 Comments: 2

Question Number 22739 Answers: 0 Comments: 0

If (1 + x)^n = C_0 + C_1 x + C_2 x^2 + C_3 x^3 + ... + C_n x^n , Prove that ΣΣ_(0≤i<j≤n) (i + j)C_i C_j = n(2^(2n−1) − (1/2)^(2n) C_n )

$I f {(1 + x)}^{n} = C_{0} + C_{1} x + C_{2} x^{2} + C_{3} x^{3}$ $+ . . . + C_{n} x^{n},$ $P r o v e t h a t \underset{0 ⩽ i < j ⩽ n}{Σ Σ} (i + j) C_{i} C_{j} =$ $n (2^{2 n - 1} - \frac{1}{2}^{2 n} C_{n})$

Question Number 22701 Answers: 2 Comments: 0

Question Number 22618 Answers: 1 Comments: 0

If (1 + x)^n = C_0 + C_1 x + C_2 x^2 + C_3 x^3 + ... + C_n x^n , prove that (2^2 /(1.2))C_0 + (2^3 /(2.3))C_1 + (2^4 /(3.4))C_2 + ... + (2^(n+2) /((n + 1)(n + 2)))C_n = ((3^(n+2) − 2n − 5)/((n + 1)(n + 2)))

$I f {(1 + x)}^{n} = C_{0} + C_{1} x + C_{2} x^{2} + C_{3} x^{3}$ $+ . . . + C_{n} x^{n}, p r o v e t h a t$ $\frac{2^{2}}{1.2} C_{0} + \frac{2^{3}}{2.3} C_{1} + \frac{2^{4}}{3.4} C_{2} + . . . +$ $\frac{2^{n + 2}}{(n + 1) (n + 2)} C_{n} = \frac{3^{n + 2} - 2 n - 5}{(n + 1) (n + 2)}$

Question Number 22640 Answers: 0 Comments: 0

With usual notation, show that (C_0 /x) − (C_1 /(x+1)) + (C_2 /(x+2)) − .... + (−1)^n (C_n /(x+n))= ((n!)/(x(x + 1)(x + 2)....(x + n)))

$W i t h u s u a l n o t a t i o n, s h o w t h a t$ $\frac{C_{0}}{x} - \frac{C_{1}}{x + 1} + \frac{C_{2}}{x + 2} - . . . . + {(- 1)}^{n} \frac{C_{n}}{x + n} =$ $\frac{n!}{x (x + 1) (x + 2) . . . . (x + n)}$

Question Number 22612 Answers: 2 Comments: 0

In the binomial expasion of (a − b)^5 , the sum of 2^(nd) and 3^(rd) term is zero, then (a/b) is

$I n t h e b i n o m i a l e x p a s i o n o f {(a - b)}^{5},$ $t h e s u m o f 2^{n d} a n d 3^{r d} t e r m i s z e r o,$ $t h e n \frac{a}{b} i s$

Question Number 22547 Answers: 1 Comments: 0

If α = (5/(2!3)) + ((5.7)/(3!3^2 )) + ((5.7.9)/(4!3^3 )) ,... then find the value of α^2 + 4α.

$If α = \frac{5}{2! 3} + \frac{5.7}{3! 3^{2}} + \frac{5.7.9}{4! 3^{3}}, . . . then find$ $the value of α^{2} + 4 α .$

Question Number 22517 Answers: 0 Comments: 0

Find the coefficient of x in the expansion of [(√(1 + x^2 )) − x]^(−1) in ascending power of x when ∣x∣ < 1.

$Find the coefficient of x in the expansion$ $of {[\sqrt{1 + x^{2}} - x]}^{- 1} in ascending power$ $of x when ∣ x ∣ < 1 .$

Question Number 22503 Answers: 1 Comments: 4

If (a + bx)^(−2) = (1/4) − 3x + ..., then (a, b) =

$If {(a + b x)}^{- 2} = \frac{1}{4} - 3 x + . . ., then (a, b) =$

Question Number 22491 Answers: 1 Comments: 0

The coefficient of x^r in the expansion of (1 − 2x)^(−1/2) is (1) (((2r)!)/((r!)^2 )) (2) (((2r)!)/(2^r (r!)^2 )) (3) (((2r)!)/((r!)^2 2^(2r) )) (4) (((2r)!)/(2^r (r + 1)!(r − 1)!))

$The coefficient of x^{r} in the expansion of$ ${(1 - 2 x)}^{- 1 / 2} is$ $(1) \frac{(2 r)!}{{(r!)}^{2}}$ $(2) \frac{(2 r)!}{2^{r} {(r!)}^{2}}$ $(3) \frac{(2 r)!}{{(r!)}^{2} 2^{2 r}}$ $(4) \frac{(2 r)!}{2^{r} (r + 1)! (r - 1)!}$

Question Number 22474 Answers: 0 Comments: 0

Let R = (5(√5) + 11)^(2n+1) and f = R − [R], then prove that Rf = 4^(2n+1) .

$Let R = {(5 \sqrt{5} + 11)}^{2 n + 1} and f = R - [R],$ $then prove that R f = 4^{2 n + 1} .$

Question Number 22472 Answers: 0 Comments: 0

If x^x ∙y^y ∙z^z = x^y ∙y^z ∙z^x = x^z ∙y^x ∙z^y such that x, y and z are positive integers greater than 1, then which of the following cannot be true for any of the possible value of x, y and z? (1) xyz = 27 (2) xyz = 1728 (3) x + y + z = 32 (4) x + y + z = 12

$If x^{x} \cdot y^{y} \cdot z^{z} = x^{y} \cdot y^{z} \cdot z^{x} = x^{z} \cdot y^{x} \cdot z^{y} such$ $that x, y and z are positive integers$ $greater than 1, then which of the$ $following cannot be true for any of the$ $possible value of x, y and z ?$ $(1) x y z = 27$ $(2) x y z = 1728$ $(3) x + y + z = 32$ $(4) x + y + z = 12$

Question Number 22468 Answers: 0 Comments: 0

If a_r is the coefficient of x^r in the expansion (1 + x + x^2 )^n , then a_1 − 2a_2 + 3a_3 − ....... 2na_(2n) =

$If a_{r} is the coefficient of x^{r} in the$ $expansion {(1 + x + x^{2})}^{n}, then$ $a_{1} - 2 a_{2} + 3 a_{3} - . . . . . . . 2 {n a}_{2 n} =$

Question Number 22463 Answers: 1 Comments: 0

Solve for real x: (1/([x])) + (1/([2x])) = (x) + (1/3), where [x] is the greatest integer less than or equal to x and (x) = x − [x], [e.g. [3.4] = 3 and (3.4) = 0.4].

$Solve for real x :$ $\frac{1}{[x]} + \frac{1}{[2 x]} = (x) + \frac{1}{3},$ $where [x] is the greatest integer less$ $than or equal to x and (x) = x - [x],$ $[e . g . [3.4] = 3 and (3.4) = 0.4] .$

Question Number 29184 Answers: 1 Comments: 0

{ (((√(x^2 −4xy))+(√(y^2 +2xy+9))=10)),((x−y=7)) :} How many real roots of the equtions system?

${\begin{cases} \sqrt{x^{2} - 4 xy} + \sqrt{y^{2} + 2 xy + 9} = 10 \\ x - y = 7 \end{cases}$ $How many real roots of$ $the equtions system ?$

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