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MensurationQuestion and Answers: Page 9

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{ ((u_0 = 3 : u_1 = 4)),((u_(n+1) = u_n + 6u_(n−1) )) :} Express u_n in terms of n

${\begin{cases} u_{0} = 3 : u_{1} = 4 \\ u_{n + 1} = u_{n} + 6 u_{n - 1} \end{cases}$ $E x p r e s s u_{n} i n t e r m s o f n$

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Given that f(x) = ∫_x ^(2x) (1/( (√(1+t^4 ))))dt (a) state its domain (b) is f(x) even or odd?

$Given that f (x) = \int_{x}^{2 x} \frac{1}{\sqrt{1 + t^{4}}} d t$ $(a) state its domain$ $(b) is f (x) even or odd ?$

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Q 158528 P=Π_(n=1) ^∞ ((((n+1)^3 −1)/((n+1)^3 +1))) ⇒ P = Π_(n=1) ^∞ ((((n+1)^3 −1^3 )/((n+1)^3 +1^3 ))) ⇒ P = Π_(n=1) ^∞ {(((n+1−1)(n^2 +2n+1+n+1+1))/((n+1+1)(n^2 +2n+1−n−1+1)))} ⇒ P = Π_(n=1) ^∞ {(n/(n+2))}•Π_(n=1) ^∞ {((n^2 +3n+3)/(n^2 +n+1))} = lim_(n→∞) Π_(k=1) ^n {(k/(k+2))}•lim_(n→∞) Π_(k=1) ^n {((k^2 +3k+3)/(k^2 +k+1))} = lim_(n→∞) {(1/3)•(2/4)•(3/5)•...•(n/(n+2))}×lim_(n→∞) {(7/3)•((13)/7)•...•((n^2 +3n+3)/(n^2 +n+1))} =2lim_(n→∞) {(1/((n+1)(n+2)))}×(1/3)lim_(n→∞) {n^2 +3n+3} =(2/3)lim_(n→∞) {((n^2 +3n+3)/(n^2 +3n+2))} = (2/3)lim_(n→∞) {((1+(3/n)+(3/n^2 ))/(1+(3/n)+(2/n^2 )))} = (2/3). P = Π_(n=1) ^∞ ((((n+1)^3 −1)/((n+1)^3 +1))) = (2/3).. ...............Le puissant...............

$Q 158528$ $P = \underset{n = 1}{\prod^{\infty}} (\frac{{(n + 1)}^{3} - 1}{{(n + 1)}^{3} + 1})$ $\Rightarrow P = \underset{n = 1}{\prod^{\infty}} (\frac{{(n + 1)}^{3} - 1^{3}}{{(n + 1)}^{3} + 1^{3}})$ $\Rightarrow P = \underset{n = 1}{\prod^{\infty}} {\frac{(n + 1 - 1) (n^{2} + 2 n + 1 + n + 1 + 1)}{(n + 1 + 1) (n^{2} + 2 n + 1 - n - 1 + 1)}}$ $\Rightarrow P = \underset{n = 1}{\prod^{\infty}} {\frac{n}{n + 2}} ∙ \underset{n = 1}{\prod^{\infty}} {\frac{n^{2} + 3 n + 3}{n^{2} + n + 1}}$ $= \lim_{n \to \infty} \underset{k = 1}{\prod^{n}} {\frac{k}{k + 2}} ∙ \lim_{n \to \infty} \underset{k = 1}{\prod^{n}} {\frac{k^{2} + 3 k + 3}{k^{2} + k + 1}}$ $= \lim_{n \to \infty} {\frac{1}{3} ∙ \frac{2}{4} ∙ \frac{3}{5} ∙ . . . ∙ \frac{n}{n + 2}} \times \lim_{n \to \infty} {\frac{7}{3} ∙ \frac{13}{7} ∙ . . . ∙ \frac{n^{2} + 3 n + 3}{n^{2} + n + 1}}$ $= 2 \lim_{n \to \infty} {\frac{1}{(n + 1) (n + 2)}} \times \frac{1}{3} \lim_{n \to \infty} {n^{2} + 3 n + 3}$ $= \frac{2}{3} \lim_{n \to \infty} {\frac{n^{2} + 3 n + 3}{n^{2} + 3 n + 2}} = \frac{2}{3} \lim_{n \to \infty} {\frac{1 + \frac{3}{n} + \frac{3}{n^{2}}}{1 + \frac{3}{n} + \frac{2}{n^{2}}}} = \frac{2}{3} .$ $P = \underset{n = 1}{\prod^{\infty}} (\frac{{(n + 1)}^{3} - 1}{{(n + 1)}^{3} + 1}) = \frac{2}{3} . .$ $. . . . . . . . . . . . . . . L e p u i s s a n t . . . . . . . . . . . . . . .$

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8. A number can be expressed as a terminating decimal,if the denominator has factors : (a) 2,3 or 5 (b) 2 or 3 (c) 3 or 5 (d) 2 or 5 9. Given that : HCF of 2520 and 6600= 120, LCM of 2520 and 6600= 252×k, then the value of k is : (a) 165 (b) 1625 (c) 550 (d) 600 10. The decimal expansion of the rational number ((47)/(2^4 ×5^(3 ) )) will terminate after : (a) 3 places (b) 4 places (c) 5 places (d) 1 place 11. The perimeter of two similar triangles ABC and LMN are 60 cm and 48 cm respectively . If LM = 8cm,then lenght of AB is : (a) 10 cm (b) 8 cm (c) 5 cm (d) 6 cm 12. Ratio in which the line segment joining (1,−7) and (6,4) are divided by x-axis is given as: (a) 4 :7 (b) 2 : 5 (c) 7 : 4 (d) 5 : 2 13. 119^2 − 111^2 is : (a) Prime number (b) Composite number ( c) An odd composite number (d)An odd prime number 14. Side of square , whose diagonal is 16 cm is given by: (a) 6(√(2 )) cm (b) 4(√2) cm (c) 7(√(2 ))cm (d) 8(√(2 ))cm

$8 . A number can be expressed as a terminating decimal, if the denominator has factors :$ $(a) 2, 3 or 5$ $(b) 2 or 3$ $(c) 3 or 5$ $(d) 2 or 5$ $9 . Given that : HCF of 2520 and 6600 = 120, LCM of 2520 and 6600 = 252 \times k, then the value of k is :$ $(a) 165$ $(b) 1625$ $(c) 550$ $(d) 600$ $10 . The decimal expansion of the rational number \frac{47}{2^{4} \times 5^{3}} will terminate after :$ $(a) 3 places$ $(b) 4 places$ $(c) 5 places$ $(d) 1 place$ $11 . The perimeter of two similar triangles ABC and LMN are 60 cm and 48 cm respectively . If LM = 8 cm, then lenght of AB is :$ $(a) 10 cm$ $(b) 8 cm$ $(c) 5 cm$ $(d) 6 cm$ $12 . Ratio in which the line segment joining (1, - 7) and (6, 4) are divided by x - axis is given as :$ $(a) 4 : 7$ $(b) 2 : 5$ $(c) 7 : 4$ $(d) 5 : 2$ $13 . 119^{2} - 111^{2} is :$ $(a) Prime number$ $(b) Composite number$ $(c) An odd composite number$ $(d) An odd prime number$ $14 . Side of square, whose diagonal is 16 cm is given by :$ $(a) 6 \sqrt{2} cm$ $(b) 4 \sqrt{2} cm$ $(c) 7 \sqrt{2} cm$ $(d) 8 \sqrt{2} cm$

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lim_(x→0) (((tanx)/x))^(1/x^2 ) = ?

$\lim_{x \to 0} {(\frac{t a n x}{x})}^{\frac{1}{x^{2}}} = ?$

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The mean and standard deviation of 20 observation are found to be 10 and 2 respectively .On rechecking it was found that an observation 8 was incorrect.Calculate the incorrect mean and standard deviation (a)If the wrong iterm was ommited (b) If it is replaced by 12

$The mean and standard deviation$ $of 20 observation are found to be$ $10 and 2 respectively . On rechecking$ $it was found that an observation$ $8 was incorrect . Calculate the incorrect$ $mean and standard deviation$ $(a) I f the wrong iterm was ommited$ $(b) If it is replaced by 12$

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