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- Question
- [ (m - n) 3 + (n - r) 3 + (r - m) 3 ] / 6[(m - n) (n - r) (r - m)] =?
Options- A. 1/2
- B. 1/3
- C. 1/5
- D. 1/6
- Correct Answer
- 1/2
ExplanationLet, (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 - 1. If p + q = 10 and pq = 5, then the numerical value if p/q + q/p will be?
Options- A. 22
- B. 18
- C. 16
- D. 20 Discuss
- 2. If x + y = 1, then the value of (x3 + y3 + 3xy) is?
Options- A. -2
- B. 1
- C. 2
- D. 3 Discuss
- 3. If x + y = 18 and xy = 72, what is the value of x2 + y2?
Options- A. 120
- B. 90
- C. 180
- D. Cannot be determined Discuss
- 4. If x + 1/x = 6, then x 4 + 1/x 4 is?
Options- A. 1152
- B. 1154
- C. 1148
- D. 1150 Discuss
- 5. If x + 1/x = 2, then what is value of x - 1/x ?
Options- A. 0
- B. 1
- C. 2
- D. -2 Discuss
- 6. (1 - 1/2) (1 - 1/3) (1 -1/4) (1 -1/5) ... (1 - 1/m) =?
Options- A. 1/m
- B. m
- C. m + 1
- D. 1/(m - 1) Discuss
- 7. If a + 1/a = ? 3, then the value of a6 - 1/a6 + 2 will be
Options- A. 3?3
- B. 5
- C. 1
- D. 2 Discuss
- 8. The value of (1 + 1/2) (1 + 1/3) (1 + 1/4) ..... (1 + 1/150) is?
Options- A. 65.5
- B. 50.5
- C. 105
- D. 75.5 Discuss
- 9. If a + 1/b = 1 and b + 1/c = 1, then c + 1/a =?
Options- A. 4
- B. 24
- C. 3
- D. 1 Discuss
- 10. If (a2 + 1/a2) = 17/4, then what is (a3 - 1/a3) equal to?
Options- A. 75/16
- B. 63/8
- C. 95/8
- D. None of these Discuss
Simplification problems
Search Results
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: 0
Explanation:
Given that, x + 1/x = 2 ...................(i)
On squaring both sides, we get
(x + 1/x)2 = 4
? x2 + 1/x2 + 2 = 4
? x2 + 1/x2 = 2 ..................................(ii)
Now, we have
(x - 1/x)2 = (x2 + 1/x2) - 2x X 1/x
now put the value of x2 + 1/x2
(x - 1/x)2 = 2 - 2 = 0 [from Eq. (ii)]
? x - 1/x = 0
Correct Answer: 1/m
Explanation:
Given expression
= (1 - 1/2) x (1 - 1/3) x (1 - 1/4) x (1 - 1/5) ...(1 - 1/m - 1) x (1 - 1/m)
= ((2 - 1)/2) x ((3 - 1)/3) ((4 - 1)/4) ((5 - 1)/5) ...((m-1 - 1)/m - 1) x ((m - 1)/m)
= (1/2) x (2/3) x (3/4) x (4/5) x ... x (m -2)/(m-1) x ((m - 1)/m
use the multiplication rule of Algebra,
= 1 x 1/m
= 1/m
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
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