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AlgebraQuestion and Answers: Page 230

Question Number 118298 Answers: 0 Comments: 1

b, x, y, c are consecutive terms of a G.P. ∴ x = br, y = br^2 , c = br^3 A.M. between b and c is a ∴ ((b+c)/2) = a ⇒ 2a = b+c 2abc = (b+c)bc = b^2 c + bc^2 = b^2 (br^3 ) + b(br^3 )^2 = b^3 r^3 + b^3 r^6 = (br)^3 + (br^2 )^3 = x^3 + y^3 ∴ x^3 + y^3 = 2abc ←

$b, x, y, c are consecutive terms of a G . P .$ $∴ x = br, y = {br}^{2}, c = {br}^{3}$ $A . M . between b and c is a$ $∴ \frac{b + c}{2} = a \Rightarrow 2 a = b + c$ $2 a bc = (b + c) bc$ $= b^{2} c + b c^{2}$ $= b^{2} ({br}^{3}) + b {({br}^{3})}^{2}$ $= b^{3} r^{3} + b^{3} r^{6}$ $= {(br)}^{3} + {({br}^{2})}^{3}$ $= x^{3} + y^{3}$ $∴ x^{3} + y^{3} = 2 abc \leftarrow$

Question Number 118286 Answers: 2 Comments: 0

What condition should be satisfied by the vectors a and b for the following relations to hold true :(a)∣a+b∣=∣a−b∣ ;(b)∣ a+b∣>∣ a−b∣;(c)∣a+ b∣< ∣a−b∣

$What condition should be satisfied by$ $the vectors a and b for the following$ $relations to hold true : (a) ∣ a + b ∣=∣ a - b ∣$ $; (b) ∣ a + b ∣>∣ a - b ∣; (c) ∣ a + b ∣< ∣ a - b ∣$

Question Number 118280 Answers: 0 Comments: 1

(26) OA^(→) = a^→ = (((4.8)),((3.6)) ) , OB^(→) = b^→ = ((( 8)),((15)) ) a^→ . b^→ = (4.8)(8) + (3.6)(15) = 92.4^(→ ) a = (√(4.8^2 +3.6^2 )) = (√(23.04+12.96)) = (√(36)) = 6 b = (√(8^2 +15^2 )) = (√(64+225)) = (√(289)) = 17 a.b = 6 . 17 = 102 cos AOB = ((a^→ .b^→ )/(a.b)) = ((92.4)/(102)) = 0.9059 = cos 25.06° ∴ ∠AOB = 25.06°

$(26) \vec{OA} = \vec{a} = (\begin{matrix} 4.8 \\ 3.6 \end{matrix}), \vec{OB} = \vec{b} = (\begin{matrix} 8 \\ 15 \end{matrix})$ $\vec{a} . \vec{b} = (4.8) (8) + (3.6) (15) = 92 . \vec{4}$ $a = \sqrt{4 . 8^{2} + 3 . 6^{2}} = \sqrt{23.04 + 12.96} = \sqrt{36} = 6$ $b = \sqrt{8^{2} + 15^{2}} = \sqrt{64 + 225} = \sqrt{289} = 17$ $a . b = 6 . 17 = 102$ $\cos AOB = \frac{\vec{a} . \vec{b}}{a . b} = \frac{92.4}{102} = 0.9059 = \cos 25.06 °$ $∴ ∠ AOB = 25.06 °$

Question Number 118277 Answers: 0 Comments: 0

(23) Let a^→ = ((( 7)),((24)) ) and b^→ = ((( 3)),((−4)) ) a^→ .b^→ = (7)(3) + (24)(−4) = 21−96 = −75 a = (√(7^2 +24^2 )) = (√(49+576)) = (√(625)) = 25 b = (√(3^2 +(−4)^2 )) = (√(9+16)) = (√(25)) = 5 a.b = 25 . 5 = 125 Let θ be the angle between a^→ and b^→ . cosθ = ((a^→ .b^→ )/(a.b)) = ((−75)/(125)) = −0.6 = cos126.87° ∴ θ = 126.87°

$(23) Let \vec{a} = (\begin{matrix} 7 \\ 24 \end{matrix}) and \vec{b} = (\begin{matrix} 3 \\ - 4 \end{matrix})$ $\vec{a} . \vec{b} = (7) (3) + (24) (- 4) = 21 - 96 = - 75$ $a = \sqrt{7^{2} + 24^{2}} = \sqrt{49 + 576} = \sqrt{625} = 25$ $b = \sqrt{3^{2} + {(- 4)}^{2}} = \sqrt{9 + 16} = \sqrt{25} = 5$ $a . b = 25 . 5 = 125$ $Let θ be the angle between \vec{a} and \vec{b} .$ $\cos θ = \frac{\vec{a} . \vec{b}}{a . b} = \frac{- 75}{125} = - 0.6 = \cos 126 . 87 °$ $∴ θ = 126.87 °$

Question Number 118275 Answers: 1 Comments: 1

Question Number 118263 Answers: 0 Comments: 0

Question Number 118227 Answers: 1 Comments: 0

If x = (√(42−(√(42−(√(42−...)))))) y = (√(x+(√(x+(√(x+...)))))) z=(√(y.(√(y.(√(y.(√(y...)))))))) . Find x+y+z .

$I f x = \sqrt{42 - \sqrt{42 - \sqrt{42 - . . .}}}$ $y = \sqrt{x + \sqrt{x + \sqrt{x + . . .}}}$ $z = \sqrt{y . \sqrt{y . \sqrt{y . \sqrt{y . . .}}}} . F i n d x + y + z .$

Question Number 118196 Answers: 1 Comments: 0

solve in N: b^3 (2b^2 +2b+1)=18360

$s o l v e i n N :$ $b^{3} (2 b^{2} + 2 b + 1) = 18360$

Question Number 118193 Answers: 1 Comments: 0

Given A=n^2 −2n+2 , B=n^2 +2n+2 n ∈ N^∗ −{1}. Show that ∀ divisor of A which divise n can also divise 2. Show that all common divisor of A and B can divise 4n.

$G i v e n A = n^{2} - 2 n + 2, B = n^{2} + 2 n + 2$ $n \in N^{*} - {1} .$ $S h o w t h a t \forall d i v i s o r o f A w h i c h d i v i s e$ $n c a n a l s o d i v i s e 2 .$ $S h o w t h a t a l l c o m m o n d i v i s o r o f$ $A a n d B c a n d i v i s e 4 n .$

Question Number 118184 Answers: 2 Comments: 1

factorise x^4 +4

$f a c t o r i s e x^{4} + 4$

Question Number 118181 Answers: 1 Comments: 1

find all numbers >1 from N which their cube are <18360

$f i n d a l l n u m b e r s > 1 f r o m N w h i c h$ $t h e i r c u b e a r e < 18360$

Question Number 118180 Answers: 1 Comments: 0

show that if n is odd , n(n^2 +3) is even.

$s h o w t h a t i f n i s o d d, n (n^{2} + 3) i s e v e n .$

Question Number 118172 Answers: 1 Comments: 0

Find the area of a rhombus with side 8 cm

$Find the area of a rhombus with side 8 cm$

Question Number 118023 Answers: 1 Comments: 0

..calculus.. x,y,z ∈R^+ and x^2 +y^2 +z^2 =1 find min_(x,y,z∈R^(+ ) ) ((((yz)/x)+((xz)/y)+((xy)/z)) )=? m.n.1970..

$. . c a l c u l u s . .$ $x, y, z \in R^{+} a n d x^{2} + y^{2} + z^{2} = 1$ $f i n d$ ${m i n}_{x, y, z \in R^{+}} ((\frac{y z}{x} + \frac{x z}{y} + \frac{x y}{z})) = ?$ $m . n . 1970 . .$

Question Number 118011 Answers: 4 Comments: 0

Question Number 118002 Answers: 1 Comments: 0

Question Number 117984 Answers: 1 Comments: 0

If f(x) is a polynomial function satisfying the relation f(x)+f((1/x))=f(x)f((1/x)) for all 0≠x∈R and if f(2)=9, then f(6) is (A) 216 (B) 217 (C) 126 (D) 127

$If f (x) is a polynomial function satisfying the relation$ $f (x) + f (\frac{1}{x}) = f (x) f (\frac{1}{x})$ $for all 0 \neq x \in R and if f (2) = 9, then f (6) is$ $(A) 216 (B) 217 (C) 126 (D) 127$

Question Number 117972 Answers: 2 Comments: 0

The number of surjections of {1,2,3,4} onto {x,y} is (A) 16 (B) 8 (C) 14 (D) 6

$The number of surjections of {1, 2, 3, 4} onto {x, y} is$ $(A) 16 (B) 8 (C) 14 (D) 6$

Question Number 117934 Answers: 1 Comments: 0

Let f : [1,∞)→[2,∞) be the function defined by f(x)=x+(1/x) If g : [2,∞)→[1,∞), is a function such that (g○f)(x)=x for all x≥1. Show that g(t)=((t+(√(t^2 −4)))/2)

$Let f : [1, \infty) \to [2, \infty) be the function defined by$ $f (x) = x + \frac{1}{x}$ $If g : [2, \infty) \to [1, \infty), is a function such that (g \circ f) (x) = x$ $for all x ⩾ 1 . Show that g (t) = \frac{t + \sqrt{t^{2} - 4}}{2}$

Question Number 117828 Answers: 1 Comments: 0

1)((√3)−1)((√3)+1)=(√3)×(√3)−(√3)−1 =3−(√3)−1 =2−(√3) 2)(2x+(√3))(2x−(√3))=(2x)^2 −2x(√3)+2x(√3)−3 =4x^2 −3

$1) (\sqrt{3} - 1) (\sqrt{3} + 1) = \sqrt{3} \times \sqrt{3} - \sqrt{3} - 1$ $= 3 - \sqrt{3} - 1$ $= 2 - \sqrt{3}$ $2) (2 x + \sqrt{3}) (2 x - \sqrt{3}) = {(2 x)}^{2} - 2 x \sqrt{3} + 2 x \sqrt{3} - 3$ $= 4 x^{2} - 3$

Question Number 117827 Answers: 0 Comments: 5

Question Number 117781 Answers: 1 Comments: 0

Log (cosβ) = p ⇒ cos β = 10^p ∴ secβ = (1/(cosβ)) = (1/(10^p )) = 10^(−p) ∴ Log (secβ) = Log 10^(−p) = −p Log 10 = −p

$Log (\cos β) = p \Rightarrow \cos β = 10^{p}$ $∴ \sec β = \frac{1}{\cos β} = \frac{1}{10^{p}} = 10^{- p}$ $∴ Log (\sec β) = Log 10^{- p}$ $= - p Log 10$ $= - p$

Question Number 117767 Answers: 1 Comments: 1

x^2 +y_ ^2 =a^2 (√(2 )) x^2 +y^2 =a^2 what is intersection Angle=?

$x^{2} + y_{}^{2} = a^{2} \sqrt{2}$ $x^{2} + y^{2} = a^{2}$ $w h a t i s i n t e r s e c t i o n A n g l e = ?$

Question Number 117666 Answers: 0 Comments: 3

Question Number 117649 Answers: 1 Comments: 0

Let P(x) be a polynomial function of degree n such that P(k)=(k/(k+1)) for k=0,1,2,...,n. Then P(n+1) is equal to (A) −1 if n is even (B) 1 if n is odd (C) (n/(n+2)) if n is even (D) (n/(n+2)) if n is odd Which among the four proposals is/are correct ?

$Let P (x) be a polynomial function of degree n such that$ $P (k) = \frac{k}{k + 1} for k = 0, 1, 2, . . ., n . Then P (n + 1) is equal to$ $(A) - 1 if n is even (B) 1 if n is odd$ $(C) \frac{n}{n + 2} if n is even (D) \frac{n}{n + 2} if n is odd$ $Which among the four proposals is / are correct ?$

Question Number 117603 Answers: 0 Comments: 0

Let f : R→R be a function satisfying the following : (a) f(−x)=−f(x) (b) f(x+1)=f(x)+1 (c) f((1/x))=((f(x))/x^2 ) for all x≠0 Show that (i)f(x)=x for all x,y∈R (ii) f(x+y)=f(x)+f(y) for all x,y∈R (iii) f(xy)=f(x)f(y) for all x,y∈R (iv) f((x/y))=((f(x))/(f(y))) for all x,y∈R with y≠0

$Let f : R \to R be a function satisfying the following :$ $(a) f (- x) = - f (x)$ $(b) f (x + 1) = f (x) + 1$ $(c) f (\frac{1}{x}) = \frac{f (x)}{x^{2}} for all x \neq 0$ $Show that$ $(i) f (x) = x for all x, y \in R$ $(ii) f (x + y) = f (x) + f (y) for all x, y \in R$ $(iii) f (x y) = f (x) f (y) for all x, y \in R$ $(iv) f (\frac{x}{y}) = \frac{f (x)}{f (y)} for all x, y \in R with y \neq 0$

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