Mathematics - MCQ Practice Questions

Understanding:

We must find the distance between two points in a coordinate plane.

Formula:

Step 1: Compute the differences.

Step 2: Substitute into the distance formula.

Answer:

The distance between $A$ and $B$ is $10$ units.

Quick Tip:

The pair $(-8,6)$ is a multiple of $(4,3)$ — a scaled $3$-$4$-$5$ right triangle — so $d=2\times 5=10$ can be spotted without a calculator.

Understanding:

We must find the slope of the line through two given points.

Formula:

Step 1: Substitute the coordinates.

Answer:

The slope of the line is $3$.

Quick Tip:

A common mistake is to reverse numerator and denominator, giving $\frac{1}{3}$. Always place the $y$-difference in the numerator.

Understanding:

We must write the equation of a line given its slope and $y$-intercept.

Formula:

Step 1: Substitute $m=-2$ and $c=7$.

Answer:

The equation of the line is $y=-2x+7$.

Understanding:

We must find the midpoint of the segment $PQ$.

Formula:

Step 1: Apply the formula.

Answer:

The midpoint of segment $PQ$ is $(1,1)$.

Quick Tip:

Add the coordinates first, then halve — never halve each coordinate separately before adding.

Understanding:

We must find the area of a triangle given its three vertices.

Formula:

Step 1: Substitute the coordinates.

Answer:

The area of the triangle is $12$ square units.

Quick Tip:

Note that $A$ and $C$ both have $y=2$, so $AC$ is a horizontal base of length $6$ and the height is the perpendicular distance from $B$ to that base, which is $6-2=4$. Area $=\frac{1}{2}\times 6\times 4=12$ — a faster route when two vertices share the same ordinate.

Understanding:

We must find the point $P$ that divides $AB$ internally in the ratio $m:n=1:2$.

Formula:

Step 1: Compute the $x$-coordinate.

Step 2: Compute the $y$-coordinate.

Answer:

The required point is $(4,1)$.

Quick Tip:

In the section formula, the ratio $m:n$ is applied as $m$ nearer to $B$ and $n$ nearer to $A$. A sign error on $y_1=-3$ is the most common slip here.

Understanding:

We must find the equation of a line through a given point and perpendicular to a given line.

Formula:

If the slope of a line is $m$, the slope of a perpendicular line is $-\frac{1}{m}$.

Step 1: Find the slope of the given line.

So $m=\frac{2}{3}$.

Step 2: The perpendicular slope is:

Step 3: Write the equation through $(1,-2)$.

Verification:

Check $(1,-2)$: $3(1)+2(-2)+1=3-4+1=0$. ✓

Answer:

The required equation is $3x+2y+1=0$.

Quick Tip:

For lines $ax+by+c=0$ and $bx-ay+k=0$ — swapping coefficients of $x$ and $y$ and flipping one sign always gives a perpendicular line.

Understanding:

We must find the centroid of a triangle given its three vertices.

Formula:

Step 1: Compute the $x$-coordinate.

Step 2: Compute the $y$-coordinate.

Answer:

The centroid of the triangle is $\left(\frac{10}{3},1\right)$.

Quick Tip:

The centroid divides each median in the ratio $2:1$ from the vertex. It always lies inside the triangle.

Understanding:

We must find the inclination of the given line with the positive $x$-axis.

Formula:

If the slope of the line is $m$, the inclination $\theta$ satisfies:

Step 1: Find the slope $m$ from the given equation.

So $m=\sqrt{3}$.

Step 2: Find $\theta$.

Answer:

The line makes an angle of $60^{\circ }$ with the positive $x$-axis.

Quick Tip:

Memorize: $\tan 30^{\circ }=\frac{1}{\sqrt{3}}$, $\tan 45^{\circ }=1$, $\tan 60^{\circ }=\sqrt{3}$, $\tan 120^{\circ }=-\sqrt{3}$. A positive slope greater than $1$ immediately points to $60^{\circ }$.