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What are the definitions of additive identity, mantissa, 3rd quadrant, medians, horizontal line, quadratic equation, triangle inequality theorem, and binomial cube expansion?

Mathematics

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  1. The additive identity in the set of real numbers (R\mathbb{R}) is the number that, when added to any real number, leaves the number unchanged. This number is 0.

    • A) 0

  2. In a logarithm, the mantissa is the fractional part of the logarithm. The integer part is called the characteristic.

    • B) Mantissa

  3. In the Cartesian coordinate system, a point with both negative x and negative y coordinates (e.g., (−4,−3)(-4, -3)) is located in the 3rd quadrant.

    • C) 3rd quadrant

  4. The medians of a triangle are line segments joining a vertex to the midpoint of the opposite side. They intersect at the centroid, which divides each median in a 2:12:1 ratio.

    • D) Medians

  5. A horizontal line of the form y=cy = c (where cc is a constant, like y=5y=5) is always parallel to the x-axis.

    • A) x-axis

  6. A quadratic equation is a polynomial equation where the highest power of the variable is 2.

    • C) 2

  7. According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

    • B) Greater

  8. Step 1: Recognize the expression as the expansion of a binomial cube. The formula for (a−b)3(a-b)^3 is a3−3a2b+3ab2−b3a^3 - 3a^2b + 3ab^2 - b^3.

  1. Expand the given expression.

    $$
a^3 - 3ab(a-b) - b^3 = a^3 - (3a^2b - 3ab^2) - b^3
$$
$$
= a^3 - 3a^2b + 3ab^2 - b^3
$$

  2. Compare with the formula.

    The expanded form matches $(a-b)^3$.
*   **B) $(a-b)^3$**

    9. Step 1: Use the distance formula between two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2): D=(x2−x1)2+(y2−y1)2 D = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}

  3. Substitute the given points A(3,0)A(3,0) and B(0,3)B(0,3).

    $$
D = \sqrt{(0-3)^2 + (3-0)^2}
$$
$$
D = \sqrt{(-3)^2 + (3)^2}
$$
$$
D = \sqrt{9 + 9}
$$
$$
D = \sqrt{18}
$$

  4. Simplify the radical.

    $$
D = \sqrt{9 \times 2}
$$
$$
D = 3\sqrt{2}
$$
The distance is $3\sqrt{2}$ units.
*   **C) $3\sqrt{2}$ units**

    10. Step 1: Convert the logarithmic equation to exponential form. If log⁡bx=y\log_{b} x = y, then by=xb^y = x.

  5. Substitute the given values log⁡10x=0\log_{10} x = 0.

    $$
x = 10^0
$$

  6. Evaluate the exponential expression.

    Any non-zero number raised to the power of 0 is 1.
$$
x = 1
$$
*   **B) 1**

    11. Step 1: Recognize the expression as a difference of squares. The formula is (a+b)(a−b)=a2−b2(a+b)(a-b) = a^2 - b^2.

  7. Substitute a=4a=4 and b=14b=\sqrt{14}.

    $$
(4 + \sqrt{14})(4 - \sqrt{14}) = 4^2 - (\sqrt{14})^2
$$
$$
= 16 - 14
$$
$$
= 2
$$
*   **A) 2**

    12. Step 1: Multiply the coefficients and the imaginary units. (−5i)(−2i)=(−5)×(−2)×i×i (-5i)(-2i) = (-5) \times (-2) \times i \times i =10i2 = 10i^2

  8. Substitute the value of i2i^2.

    Recall that $i^2 = -1$.
$$
= 10(-1)
$$
$$
= -10
$$
*   **A) -10**

    13. The two coordinate axes (x-axis and y-axis) in a standard Cartesian system are perpendicular to each other, meaning they intersect at an angle of 90 degrees. * B) 90∘90^\circ

    1. Step 1: Factorize both expressions. x4−y4=(x2)2−(y2)2=(x2−y2)(x2+y2)x^4 - y^4 = (x^2)^2 - (y^2)^2 = (x^2 - y^2)(x^2 + y^2) x2−y2x^2 - y^2

  9. Identify the common and unique factors for the Least Common Multiple (LCM).

    The factors are $(x^2 - y^2)$ and $(x^2 + y^2)$. The LCM includes all factors raised to their highest power.
$$
\text{LCM} = (x^2 - y^2)(x^2 + y^2)
$$

  10. Simplify the LCM.

    $$
\text{LCM} = x^4 - y^4
$$
*   **C) $x^4 - y^4$**

    15. A surd is an irrational number that can be expressed as a root of an integer. 5\sqrt{5} is an irrational number because 5 is not a perfect square. Since it consists of a single term, it is a monomial surd. * A) Monomial surd

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