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BITSAT Mathematics Mastery Hub: The Industry Foundation Prac

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Q1Domain Verified
Consider the system of linear equations represented by the matrix equation $AX = B$, where $A = \begin{pmatrix} 2 & -1 & 3 \\ 1 & 2 & -1 \\ 3 & 1 & 2 \end{pmatrix}$, $X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}$, and $B = \begin{pmatrix} 5 \\ 1 \\ 6 \end{pmatrix}$. If the determinant of $A$ is non-zero, which of the following statements accurately describes the nature of the solution set?
The system has a unique solution, but it can only be found using Gaussian elimination.
C) The system has a unique solution, obtainable via Cramer's Rule or matrix inversion, as $\det(A) \neq 0$.
The system has no solution because the matrix $A$ is not symmetri
The system has infinitely many solutions due to the presence of three variables and three equations.
Q2Domain Verified
A parabola has its focus at $(1, 2)$ and its directrix is the line $x = -3$. If the vertex of this parabola is denoted by $V$, which of the following is the correct coordinate of $V$?
$(-1, -1)$
$(-1, 2)$
$(2, 2)$
$(1, -3)$
Q3Domain Verified
Given two points $P(3, -2)$ and $Q(-1, 4)$ on a coordinate plane, consider a point $R$ that divides the line segment $PQ$ internally in the ratio $2:1$. If $S$ is the midpoint of $PR$, what are the coordinates of $S$?
$(\frac{13}{3}, 0)$
$(\frac{1}{3}, \frac{2}{3})$
$(\frac{5}{3}, \frac{2}{3})$
$(\frac{7}{3}, \frac{4}{3})$

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