According to question , we can say that
The equation x² ? px + q = 0, p, q ? R has no real root
On comparing with quadratic eq. Ax² + Bx + C = 0 , we get
? A =1, B= - p, C = q
Now ,we have B² < 4AC
? ( - P )² < 4 x 1 x q
? p² < 4q.
Hence , option B is correct answer .

The given quadratic equation 3x² + (k ? 1)x + 9 = 0
Putting x = 3 in above given eq. , we get
27 + 3(k ? 1) + 9 = 0
? 27 + 3k - 3 + 9 = 0
? 3k = ?33 ? k = ?11.
Hence , the value of k is ?11.

The given expression x⁴ + 7x² + 16 = ( x⁴ + 8x² + 16 ) - x²
= ( x⁴ + 2 * 4 * x² + 4² ) - x²
= ( x² + 4 )² - x²
= ( x² + 4 + x ) ( x² + 4 - x )
Hence , x⁴ + 7x² + 16 = ( x² + x + 4 ) ( x² - x + 4 ).
Option C is correct answer .

As per the given above question , we can say that
The given quadratic equations are
x² - 7x + 10 = 0 ? (x ? 5)(x ? 2) = 0 ? x = 5, 2
And x² - 10x + 16 = 0 ? (x ? 8)(x ? 2) = 0 ? x = 8, 2
? Common root is 2.

Let the smaller part be x. Then the larger part = 16 ? x.
According to question , we have
2( larger part )² - ( smaller part )² = 164
Now , 2( 16 - x )² - x² = 164
? 2( 256 + x² - 32x ) - x² = 164
? x² - 64x + 348 = 0
? x² - 6x - 58x + 348 = 0
? x(x - 6) - 58(x - 6) = 0
? (x - 6) (x - 58)
x = 6 or x = 58
But m = 58 is not possible, since sum of the two parts is 16
? smaller part ( x ) = 6,
? other part = 16 - x = 16 - 6 = 10.
Hence , required answer will be option A .

Given equation is
2x - x² = x - 2 ? x² - x - 2 = 0
? ( x + 1 ) ( x - 2 ) = 0
? x = 2 or = - 1.
Therefore , option B is correct answer .

Given that : - log₁₀ ( x² - 6x + 45 ) = 2
? x² - 6x + 45 = 10² = 100
? x² - 6x - 55 = 0
? ( x - 11 ) ( x + 5 ) = 0
? x = 11 or x = - 5.
Hence , the values of x are 11 and - 5.