13. If the deepest point in the sea is 11,600 m below sea level and the highest mountain top is 8,846 mm
(a) 8 - 15
(e-8 - 12
(f) 9 - (-9)
(g) 89 - (-11)
(h) -129-6-1
(d) 2,952 m
(d) - 12
sea level, then the difference in their elevation is
(a) 2,754 m
(b) 20,446 m
(c) 21,406 m
14. The sum of two integers is 50. If one of them is -38, the other is
(a) 88
(b) 12
(c) -88
15. If p and q are two integers such that p is the predecessor of q, then p -q is equal to
(a) 1
(b) o
(c) 2
16. Which sum is not negative?
(a) -40 + (-30)
(b) - 70 + 65
(c) - 63 + 82
(d) -1
(d) -49 +0
PROPERTIES OF ADDITION
Just as with whole numbers, addition of integers has the following properties:
1. Closure Property. The sum of two integers is always an integer.
For example, 7 + (-10 = -3, which is an integer.
2. Commutative Property. The order of the addends in an addition does not matter.
If a and b are two integers, then a + b = b + a.
For example, – 8 + 10 = 10 + (- 8) = 2, -7 +(-6) = (-6) + (-7) = - 13
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Answers
Step-by-step explanation:
Given : -
If α and β are the zeroes of the polynomial x² + 12x + 4 = 0 .
Required to find : -
find the value of ( 1/α + 1/β ) ?
Solution : -
Quadratic equation : x² + 12x + 4 = 0
α and β are the zeroes of the polynomial .
Here we can solve this question using 2 methods !
The standard form of the Quadratic equation is
ax² + bx + c = 0
On comparing the standard form of a quadratic equation with the given polynomial
Here,
a = 1
b = 12
c = 4
1st method
We know that ;
There is a relationship between the zeroes of the Quadratic equation with respective to coefficients of the quadratic equation .
So,
The relation between the sum of the zeroes and the coefficients is ;
α + β = - coefficient of x/coefficient of x²
α + β = - b/a
α + β = -(12)/1
α + β = - 12
Similarly,
The relation between the product of the zeroes and the coefficients is ;
α.β = constant term/coefficient of x²
α.β = c/a
α.β = 4/1
α.β = 4
Now,
Let's find the value of ( 1/α + 1/β )
1/α + 1/β =
α + β/α.β
substituting the values ;
- 12/4
-3
2nd method
The Quadratic formula is ;
\boxed{ \sf{x = \dfrac{ - b \pm \sqrt{ {b}^{2} - 4ac } }{2a} }}x=2a−b±b2−4ac
Here,
a = 1
b = 12
c = 4
Substituting these in the formula ;
\begin{gathered}\sf x = \dfrac{ - ( 12 ) \pm \sqrt{ ( 12 )^2 - 4 ( 1 ) ( 4)} }{ 2 ( 1 )} \\ \\ \\ \sf x = \dfrac{ - 12 \pm \sqrt{ ( 12 )^2 - 4 ( 1 ) ( 4)} }{ 2 } \\ \\ \\ \sf x = \dfrac{ - 12 \pm \sqrt{ 144 - 16} }{ 2 } \\ \\ \\ \sf x = \dfrac{ - 12 + \sqrt{128}}{2 } \quad ( and ) \quad x = \dfrac{ - 12 - \sqrt{128}}{2} \\ \\ \\ \implies \sf \alpha = \dfrac{ - 12 + \sqrt{128}}{2 } \quad ( and ) \quad \beta = \dfrac{ - 12 - \sqrt{128}}{2}\end{gathered}x=2(1)−(12)±(12)2−4(1)(4)x=2−12±(12)2−4(1)(4)x=2−12±144−16x=2−12+128(and)x=2−12−128⟹α=2−12+128(and)β=2−12−128
Now,
Let's find the value of 1/α + 1/β !
\begin{gathered}\sf \to \dfrac{1}{ \alpha} + \dfrac{1}{ \beta} \\ \\ \to \sf \dfrac{ \alpha + \beta }{ \alpha \beta } \\ \\ \to \sf \dfrac{ \dfrac{-12 + \sqrt{128}}{2} + \dfrac{- 12 - \sqrt{128}}{2} }{ \dfrac{- 12 + \sqrt{128}}{2} \times \dfrac{- 12 - \sqrt{ 128}}{2} } \\ \\ \\ \to \sf \dfrac{ \dfrac{- 12 + \sqrt{128} - 12 - \sqrt{128}}{2} }{\dfrac{( - 12 )^2 - ( \sqrt{128} )^2}{4} } \\ \\ \\ \to \sf \dfrac{ \dfrac{ - 12 - 12}{2 } }{ \dfrac{144 - 128}{4} } \\ \\ \\ \to \sf \dfrac{ \dfrac{- 24}{2} }{ \dfrac{16}{4} } \\ \\ \to \sf \frac{ - 24 }{ \dfrac{16}{2} } \\ \\ \to \sf \dfrac{ - 48 }{16} \\ \\ \implies \sf - 3\end{gathered}→α1+β1→αβα+β→2−12+128×2−12−1282−12+128+2−12−128→4(−12)2−(128)