D

We know that the product of the roots of a quadratic equation a[latex]{x}^{2}[/latex] + bx + c = 0 is [latex]\frac{c}{a}[/latex]
In the given equation, [latex]{x}^{2}[/latex] + mx + 24 = 0, the product of the roots = [latex]\frac{24}{1}[/latex] = 24.
The question states that one of the roots of this equation = 1.5.
If [latex]{x}_{1}[/latex] and [latex]{x}_{2}[/latex] are the roots of the given quadratic equation and let [latex]{x}_{1}[/latex] = 1.5.
Therefore, [latex]{x}_{2}[/latex] = [latex]\frac{24}{15}[/latex]= 16.
In the given equation, m is the co-efficient of the x term.
We know that the sum of the roots of the quadratic equation a[latex]{x}^{2}[/latex] + bx + c = 0 is
[latex]\frac{-b}{a}[/latex] = [latex]\frac{-m}{1}[/latex] = -m
Sum of the roots = 16 + 1. 5 = 17 = -17.5.
Therefore, the value of m = -17.5

A

By remainder theorem: If a polynomial in one variable 'x' is divided by (x - a), where 'a' is any real number, then the remainder is the value of the polynomial at x = a.
Therefore, remainder = [latex]{(2)}^{4}[/latex] - 3[latex]{(2)}^{2}[/latex] + 7(2) - 10 = 8

C

Substituting x = 3 in the given equation we get
2([latex]{(3)}^{2}[/latex]) - 7 (3) + q = 0. Therefore, q = 3
Now, the given equation becomes 2[latex]{x}^{2}[/latex] - 7x + 3 = 0
By solving this we get (x - 3) (2x - 1) = 0
i.e., x = 3 & [latex]\frac{1}{2}[/latex]. The second root is [latex]\frac{1}{2}[/latex].

C

[latex]{(x + 2)}^{2}[/latex] = 9
[latex]\rightarrow[/latex] x + 2 = ± 3
[latex]\rightarrow[/latex] x = 1 or -5
and [latex]{(y + 3)}^{2}[/latex] = 25
[latex]\rightarrow[/latex] y + 3 = ± 5
[latex]\rightarrow[/latex] y = 2 or -8
Therefore, maximum value of [latex]\frac{x}{y}[/latex] = [latex]\frac{-5}{-8}[/latex] = [latex]\frac{5}{8}[/latex]

C

The question states that the roots of the equation are real and equal.
Note:
The roots of a quadratic equation of the form a[latex]{x}^{2}[/latex] + bx + c = 0 will be real and equal if its discriminant D = [latex]{b}^{2}[/latex] - 4ac = 0
In this case, b = m, a = 4 and c = 4.
Therefore, [latex]{m}^{2}[/latex] - 4 * 4 * 1 = 0
Or [latex]{m}^{2}[/latex] = 16
Or m = +4 or m = -4.