If “x + ” = 1 and y + . then z + = ?
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Solution
Let, x = –1
then -1+ = 1
⇒ y =
∵ y+
⇒ +
⇒ ⇒ z = 2
So z + = 2+ = 1
At which condition that one of the roots of equation ax2 + bx + c = 0 will be double of another root ?
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Solution
Equation = ax2 + bx + c = 0
Let α and β are two roots.
So α + β = -----------(1)
αβ = ------------(2)
since α = 2β substituting
3β =
⇒ 9β2 = , 2β2 =
comparing 2b2 = 9ac
If x +, then what will value of ?
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Solution
⇒ x2 + a = bx .......(i)
⇒
=
Term “712 – 412” will always be divisible by which number ?
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Solution
Since 'n' is even so 'xn – yn' will be divided by (x + y) and (x – y)
So 712 – 412 will be divisible by (7+4) and (7-4)
So. 11 × 3 = 33 will always be a factor.
If a2 + b2 + c2 = 2(a – b – c) – 3 then what will be value of (2a – 3b + 4c) ?
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Solution
∵ a2 + b2 + c2 = 2(a – b – c) – 3
⇒ a2 – 2a + 1 + b2 + 2b + 1 + c2 + 2c + 1 = 0
⇒ (a-1)2 + (b+1)2 + (c+1)2 = 0
⇒ a = 1, b = – 1, c = – 1
So,
2a – 3b + 4c = 2×1 – 3×(–1) + 4 (–1)
= 1
If x = 2 + , y = 2– then what will be value of ?
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Solution
x = 2 + y = 2 –
⇒ x + y = 4
& xy = 4 – 3 = 1
So अत: =
=
=
If = 98 (a > 0) then what will be value of ?
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Solution
a2 + = 98 ........(i)
⇒ a2 + = 98 + 2 = 100
⇒ ........(ii)
Multiplying,
= 98 × 10
= = 980
⇒ = 980 – 10 = 970
If ‘a’ and ‘b’ are two distinct odd integers then largest integer by which term ‘a4 –b4‘ will always be divisible by is ?
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Solution
Since a, b are odd integers so all operations on them will result into even numbers.
So Breaking. a4 – b4 = (a2 + b2) (a2 – b2)
= (a2 + b2) (a – b) (a + b)
So 2 is the smallest factor of even number.
Required largest number = 2 × 2 × 2 = 8
The roots of the equation “12x2 + mx + 5 = 0″ are in ratio 5 : 4. Then what will be the value of ‘m’ ?
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Solution
Equation → 12x2 + mx + 5 = 0
Let two roots are α and β
So α + β = ----------(1)
and αβ = -----------(2)
and
⇒
⇒ β = , so, α =
from equation (1)
⇒ m = – 9