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Question Number 7323 Answers: 1 Comments: 4

find the value of Δ′Λ b and (B′U A) 𝚲 B if U={1,2,3,4,5,6,7,8,9,10} A={2,4,5,6} B={1,5,8,9} Λ is a unoin

$f i n d t h e v a l u e o f$ $Δ^{'} Λ b a n d (B^{'} U A) Λ B$ $i f U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}$ $A = {2, 4, 5, 6}$ $B = {1, 5, 8, 9}$ $Λ i s a u n o i n$

Question Number 7300 Answers: 1 Comments: 0

∫_0 ^(Π/2) ((sinx)/(sinx + cosx)) dx

$\int_{0}^{\frac{Π}{2}} \frac{s i n x}{s i n x + c o s x} d x$

Question Number 7299 Answers: 0 Comments: 6

Question Number 7294 Answers: 0 Comments: 3

If a polynomial p(x) satisfies the identity x^(2015) + x^(2014) + x^(2013) + 1 = (x^3 + x^2 + x + 1)p(x) Find p(−1)

$I f a p o l y n o m i a l p (x) s a t i s f i e s t h e i d e n t i t y$ $x^{2015} + x^{2014} + x^{2013} + 1 = (x^{3} + x^{2} + x + 1) p (x)$ $F i n d p (- 1)$

Question Number 7289 Answers: 0 Comments: 3

For any number x>1:x∈Z x can be expressed as a combination of numbers multiplied together. e.g. 10=5×2 20=5×4=5×2×2 100=10×10=5×2×5×2 ∴ x = p_1 ^e_1 p_2 ^e_2 ...p_n ^e_n where p_n is the nth prime factor e_n is the exponent e.g. x=810=8×10×11=4×2×5×2×11 x=2^4 ×5^1 ×11 ⇒ p_1 =2, e_1 =4, etc. let ω(x) be the number of discrete prime factors ∴ ω(x)=n x=Π_(t=1) ^(ω(x)) p_t ^e_t Therefore the number of prime factors is given by: ℧(x)=Σ_(t=0) ^(ω(x)) e_t Question How many combinations are there to represent x as the product of k numbers? (exluding 1. e.g. 10=10×1×1×1×1) e.g. 3 number ⇒ 30=3×10=3×5×2 ℧(30)=3!=6 ways My working so far: x=a×b×...×k combinations of k numbers x=(a_1 ×a_2 ×...×a_(ω(a)) )...(k_1 ×...×k_(ω(k)) ) a_1 , a_2 , ..., k_1 , etc are the prime factors of a, b, ...., k ∴ any arramgement with the bracets work so long as k brackets are present let C_k (x) be the combination of ways to express x as the product of k integers (not equal to 1). x=(Π_(t=1) ^(ω(a)) a_t ^e_t )(Π_(t=1) ^(ω(b)) b_t ^e_t )...(Π_(t=1) ^(ω(k)) k_1 ^e_t ) C_k (x)=???

$For any number x > 1 : x \in Z$ $x can be expressed as a combination$ $of numbers multiplied together .$ $e . g .$ $10 = 5 \times 2$ $20 = 5 \times 4 = 5 \times 2 \times 2$ $100 = 10 \times 10 = 5 \times 2 \times 5 \times 2$ $∴ x = p_{1}^{e_{1}} p_{2}^{e_{2}} . . . p_{n}^{e_{n}} where p_{n} is the n th$ $prime factor$ $e_{n} is the exponent$ $e . g .$ $x = 810 = 8 \times 10 \times 11 = 4 \times 2 \times 5 \times 2 \times 11$ $x = 2^{4} \times 5^{1} \times 11 \Rightarrow p_{1} = 2, e_{1} = 4, e t c .$ $let ω (x) be the number of discrete prime factors$ $∴ ω (x) = n$ $x = \underset{t = 1}{\prod^{ω (x)}} p_{t}^{e_{t}}$ $Therefore the number of prime factors$ $is given by :$ $℧ (x) = \underset{t = 0}{\sum^{ω (x)}} e_{t}$ $Question$ $How many combinations are there to$ $represent x as the product of k numbers ?$ $(exluding 1 . e . g . 10 = 10 \times 1 \times 1 \times 1 \times 1)$ $e . g . 3 number \Rightarrow 30 = 3 \times 10 = 3 \times 5 \times 2$ $℧ (30) = 3! = 6 ways$ $My working so far :$ $x = a \times b \times . . . \times k combinations of k numbers$ $x = (a_{1} \times a_{2} \times . . . \times a_{ω (a)}) . . . (k_{1} \times . . . \times k_{ω (k)})$ $a_{1}, a_{2}, . . ., k_{1}, e t c are the prime factors$ $of a, b, . . . ., k$ $∴ any arramgement with the bracets work$ $so long as k brackets are present$ $let C_{k} (x) be the combination of ways$ $to express x as the product of k$ $integers (not equal to 1) .$ $x = (\underset{t = 1}{\prod^{ω (a)}} a_{t}^{e_{t}}) (\underset{t = 1}{\prod^{ω (b)}} b_{t}^{e_{t}}) . . . (\underset{t = 1}{\prod^{ω (k)}} k_{1}^{e_{t}})$ $C_{k} (x) = ? ? ?$

Question Number 7314 Answers: 1 Comments: 1

Question Number 7287 Answers: 0 Comments: 2

7a_1 +7^2 a_2 +∙∙∙+7^n a_n =3^n −1 Σ_(n=1) ^∞ (a_n /3^(n−1) )=?

$7 a_{1} + 7^{2} a_{2} + \cdot \cdot \cdot + 7^{n} a_{n} = 3^{n} - 1$ $\underset{n = 1}{\sum^{\infty}} \frac{a_{n}}{3^{n - 1}} = ?$

Question Number 7317 Answers: 0 Comments: 4

Prove that [((n+1)/2)]+[((n+2)/4)]+[((n+4)/8)]+[((n+8)/( 16))]+........=n. Such that [.] denotes greatest integer function and n∈N.

$P r o v e t h a t [\frac{n + 1}{2}] + [\frac{n + 2}{4}] + [\frac{n + 4}{8}] + [\frac{n + 8}{16}] + . . . . . . . . = n .$ $S u c h t h a t [.] d e n o t e s g r e a t e s t i n t e g e r f u n c t i o n a n d n \in N .$

Question Number 7316 Answers: 0 Comments: 1

x−y and y−x x={−2,−1,0,2,5} y={0,5,7,8,10}

$x - y a n d y - x$ $x = {- 2, - 1, 0, 2, 5}$ $y = {0, 5, 7, 8, 10}$

Question Number 7269 Answers: 0 Comments: 0

Question Number 7265 Answers: 0 Comments: 1

Q1. A balloon in the shape of cone surrmo surmounted by hemispherical top . the daimeter of balloon is equal to hieght of cone . find the rate of change of volume of the balloone with respect to its total height h .

$Q 1 . A b a l l o o n i n t h e s h a p e o f c o n e s u r r m o$ $s u r m o u n t e d b y h e m i s p h e r i c a l$ $t o p . t h e d a i m e t e r o f b a l l o o n i s$ $e q u a l t o h i e g h t o f c o n e . f i n d t h e$ $r a t e o f c h a n g e o f v o l u m e o f t h e$ $b a l l o o n e w i t h r e s p e c t t o i t s t o t a l$ $h e i g h t h .$