Question

The number of integer value(s) of $k$ for which the expression $ {{x}^{2}}-2(4k-1)x+15{{k}^{2}} $ $ -2k-7>0 $ for every real number $x$ is/are

• A. None
• B. one
• C. finitely many, but greater than 1
• D. infinitely many

Answer 1

The Correct Option is B

Given, expression is $ {{x}^{2}}-2(4k-1)+x+15{{k}^{2}}-2k-7>0 $
Its discriminant, $ D={{b}^{2}}-4ac $
$={{\{-2(4k-1)\}}^{2}}-4\times 1\times (15{{k}^{2}}-2k-7) $
$=4{{(4k-1)}^{2}}-4(15{{k}^{2}}-2k-7) $
$=4[{{(4k-1)}^{2}}-(15{{k}^{2}}-2k-7)] $
$=4[16{{k}^{2}}-8k+1-15{{k}^{2}}+2k+7] $
$=4[{{k}^{2}}-6k+8] $
$=4[{{k}^{2}}-4k-2k+8|=4|(k-4)(k-2)] $ Now, for real values of x, $ D<0 $
$ \Rightarrow $ $ (k-4)(k-2)<0 $
$ \Rightarrow $ $ k<4 $ or $ k>2 $
$ \therefore $ Integer value of k is 3. Hence, number of integer value of k is noe.

Answer 2

Ans. All of the numbers in mathematics, excluding fractions, are integers. Numbers containing both negative and positive signs, as well as zero, are considered integers. The integers do not include fractions. Integers are capable of performing all arithmetic operations.

Examples include 1, -17, 0, etc.

On a number line, all integers can be shown. Positive numbers are to the right of zero, whereas negative numbers are to the left of zero.

Quadratic Equations are defined as polynomial equations with degree 2 in one variable. The general form of a quadratic equation is given as: 

ax² + bx + c = 0

Where 

x denotes the variable.

a and b are the numerical coefficients and a ≠ 0.

c is the constant/absolute term.

The values that fulfill a given quadratic equation are called roots of the quadratic equation and each equation has a maximum of 2 roots. The roots may be real or imaginary. When a quadratic polynomial is equivalent to zero it will become a quadratic equation.

Quadratic Formula is one of the easiest ways to find the roots of a quadratic equation. The roots or zeroes of a quadratic equation ax2 + bx + c = 0 are given by the quadratic formula which is as follows: 

(α, β) = [-b ± √(b² – 4ac)]/2a

The (+) and the (-) signs can be alternatively used to obtain the two distinct roots of the quadratic equation.