(1 - 1/2) (1 - 1/3) (1 -1/4) (1 -1/5) ... (1 - 1/m) =?

Correct Answer: 1/2

Explanation:

Let, (m - n) = a,
(n -r) = b
(r - m) = c,
Now a + b + c = (m - n) + (n -r) + (r - m)
? a + b + c = m - n + n - r + r - m
? a + b + c = 0............................. (i)
As we know the Algebra formula,
a³ + b³ + c³ ? 3abc = (a+b+c) X 1/2[(a?b)²+(b?c)²+(a?c)²]
Put the value of a + b + c from equation (i).
? a³ + b³ + c³ ? 3abc = 0 X 1/2[(a?b)²+(b?c)²+(a?c)²]
? a³ + b³ + c³ ? 3abc = 0
? a³ + b³ + c³ = 3abc
? Given expression in question is
[ (m - n)³ + (n - r)³ + (r - m)³ ]/ 6(m - n) (n - r) (r - m)
= ( a³ + b³ + c³ ) / 6abc
= 3abc/6abc
= 1/2

Correct Answer: 18

Explanation:

p/q + q/p = (p² + q²)/pq
Apply the formula of algebra
a ²+ b² = (a + b)² - 2ab

p/q + q/p = ( (p + q)² - 2pq ) / pq
By substituting the pq and p + q values in given equation.
p/q + q/p = ( (p + q)² - 2pq ) / pq
p/q + q/p = ((10)² - 2 x 5 ) / 5
p/q + q/p = (100 - 10 )/ 5 = 90/5 = 18
p/q + q/p = 90/5 = 18
p/q + q/p = 18

Correct Answer: 1

Explanation:

We know that algebraic formula,
(x + y)³ = x³ + y³ + 3xy (x + y)
put the value of x + y in given equation. [ given, x + y = 1]
1 = x³ + y³ + 3xy X 1
? x³ + y³ + 3xy = 1

Correct Answer: 180

Explanation:

Given in question,
x + y = 18
By using algebraic formula
? x² + y² = (x + y)² - 2xy
Put the value of x + y and xy as per given question,
? x² + y² = (18)² - 2 x 72
? x² + y² = 324 - 144
? x² + y² = 180

Correct Answer: 1154

Explanation:

x + 1/x = 6
On squaring both sides, we get
(x + 1/x)² = (6)²
? x² + 1/x² + 2 = 36
? x² + 1/x² = 34
On squaring both sides, we get
(x² + 1/x²)² = (34)²
? x⁴ + 1/x⁴ + 2 = 1156
? x⁴ + 1/x⁴ = 1154

Correct Answer: 2

Explanation:

Given equation is,
a + 1/a = ?3 .............................(i)
On squaring both sides
? a² + 1/a² + 2 = 3
? a² + 1/a² = 1 ..............................(ii)

Now, multiplying Equation (i) and (ii), we get
(a + 1/a) (a² + 1/a²) = ?3
? a³ + a/a² + a²/a + 1/a³ = ?3
? a³ + 1/a³ + ( a + 1/a ) = ?3
now put the value of a + 1/a in above equation,
? a³ + 1/a³ + ?3 = ?3 [from Eq. (i)]
? a³ + 1/a³ = 0
? ( a⁶ + 1 ) /a³ = 0
? a⁶ + 1 = 0 x a³
? a⁶ + 1 = 0
? a⁶ = -1
? a⁶ - 1/a⁶ + 2
put the value of a⁶
= (-1)⁶ - 1/(-1)⁶ + 2
= 1 - 1 + 2 = 2

Correct Answer: 75.5

Explanation:

(1 + 1/2) (1 + 1/3) (1 + 1/4) .... (1 + 1/150)
= 3/2 x 4/3 x 5/4 x ... 151/150
= (1/2) x 151
= 151/2 = 75.5

Correct Answer: 1

Explanation:

Given that, a + 1/b = 1
? a = (1 - 1/b) = (b - 1)/b
? 1/a = b/(b - 1) [reciprocal] ..........................(1)
and b + 1/c = 1
? 1/c = (1 - b)
? c = 1/(1 -b) .........................(2)
? c + 1/a ..........................(3)
put the value of c from equ. (2) and 1/a from equ. (1) in equ. (3)
? c + 1/a = 1/(1 -b) + b/(b -1)
? c + 1/a = 1/(1 -b) - b/(1 - b)
? c + 1/a = (1 - b)/(1 - b)
? c + 1/a = 1

Correct Answer: 63/8

Explanation:

(a² + 1/a²) = 17/4
subtracts 2 and Add 2 from above equation, we will get
? a² + 1/a² - 2 + 2 = 17/4
? a² + 1/a² - 2a x 1/a + 2 = 17/4
Now apply the formula, x² + x² - 2xy
? (a - 1/a)² + 2 = 17/4
? (a - 1/a)² = 17/4 - 2
? (a - 1/a)² = (17 - 8)/4
? (a - 1/a)² = 9/4
? (a - 1/a) = ?9/4
(a -1/a) = 3/2
After that, cubing On both sides, we get
(a -1/a)³ = (3/2)³
Apply the formula (x ? y)³ = x³ ? 3x²y + 3xy² ? c³
? (x ? y)³ = x³ ? 3xy(x - y) ? y³
? a³ - 1/a³ - 3 x a x 1/a (a - 1/a) = 27/8
? a³ - 1/a³ = 27/8 + 3 x (3/2)
? a³ - 1/a³ = 27/8 + 9/2
? (a³ - 1/a³) = 63/8

Correct Answer: 9/4

Explanation:

Simplify 2³ to 8
= 5/8 x 8/5÷4/9
By using simple algebraic rule: a ÷ b/c = a x c/b
= 5/8 x 8/5 x 9/4
= (5×8×9)/(?8×5×4)
= (5×9)/(?5×4)
= (9)/(?4)
= 9/4