Mathematics - MCQ Practice Questions

Understanding:

The system is:

For infinitely many solutions, the two equations must be proportional.

Formula:

For $a_{1}x+b_{1}y=c_1$ and $a_{2}x+b_{2}y=c_2$, infinitely many solutions require:

Step 1: Set up the proportionality conditions.

Step 2: Solve $\frac{m}{12}=\frac{3}{m}$.

Step 3: Check with the third ratio using $m=6$.

All three ratios are equal. ✓

Step 4: Check $m=-6$.

The ratios are not equal, so $m=-6$ does not give infinitely many solutions. ✗

Answer:

The value of $m$ for infinitely many solutions is $m=6$.

Quick Tip:

Always check ALL three ratios. Getting two ratios equal is necessary but not sufficient — the constant ratio must also match.

Understanding:

We find the sum $S_n=\sum _{r=1}^n\frac{1}{1+2+3+\cdots +r}$.

Formula:

The sum of the first $r$ natural numbers is:

Step 1: Express the general term.

Step 2: Apply partial fractions.

Step 3: Sum using telescoping.

Verification with $n=1$:

$S_1=1$. Formula: $\frac{2(1)}{2}=1$. ✓

With $n=2$: $S_2=1+\frac{1}{3}=\frac{4}{3}$. Formula: $\frac{4}{3}$. ✓

Answer:

The sum of the series up to $n$ terms is $\frac{2n}{n+1}$.

Quick Tip:

Whenever the denominator of a general term is a product of two consecutive integers, partial fractions with a telescoping sum is the fastest approach.

Understanding:

We need the scalar projection of $\vec{b}$ onto $\vec{a}$.

Formula:

Step 1: Compute the dot product.

Step 2: Compute $|\vec{a}|$.

Step 3: Find the projection.

Answer:

The projection of $\vec{b}$ onto $\vec{a}$ is $1$.

Quick Tip:

Projection of $\vec{b}$ on $\vec{a}$ and projection of $\vec{a}$ on $\vec{b}$ are generally different — don't confuse the two.

Understanding:

Three unit vectors sum to zero; find the sum of their pairwise dot products.

Formula:

Square both sides of the vector equation:

Expanding:

Step 1: Substitute the magnitudes.

Step 2: Solve.

Answer:

The value of $\vec{a}\cdot \vec{b}+\vec{b}\cdot \vec{c}+\vec{c}\cdot \vec{a}$ is $-\frac{3}{2}$.

Quick Tip:

This identity — squaring a sum of vectors — is one of the most reused tricks in vector theory. Memorise the expansion $|\vec{a}+\vec{b}+\vec{c}{|}^2=|\vec{a}{|}^2+|\vec{b}{|}^2+|\vec{c}{|}^2+2(\vec{a}\cdot \vec{b}+\vec{b}\cdot \vec{c}+\vec{c}\cdot \vec{a})$.

Understanding:

Find the area of a triangle given three vertices in 3D.

Formula:

Step 1: Find the edge vectors.

Step 2: Compute the cross product.

Step 3: Find the magnitude.

Step 4: Compute the area.

Answer:

The area of the triangle is $\frac{3\sqrt{3}}{2}$ square units.

Quick Tip:

If all three components of $\overrightarrow{AB}\times \overrightarrow{AC}$ are equal, the triangle is equilateral — a useful symmetry check.

Understanding:

Find the perpendicular distance between two parallel planes.

Formula:

First, write both planes in the form $ax+by+cz+d=0$ with the same normal direction, then use:

Step 1: Make the coefficients of Plane 2 match Plane 1 by dividing by 2.

So $d_1=4$ and $d_2=\frac{7}{2}$.

Step 2: Compute the denominator.

Step 3: Compute the distance.

Answer:

The distance between the two parallel planes is $\frac{1}{2\sqrt{14}}$.

Quick Tip:

Always reduce both planes to the same coefficient scale before applying the distance formula — the most common mistake is using $|d_1-d_2|=|4-7|=3$ without halving, giving a wrong answer.

Understanding:

Find $\lambda$ such that $\vec{a}\perp \vec{b}$.

Formula:

Two vectors are perpendicular if and only if their dot product is zero:

Step 1: Expand the dot product.

Step 2: Set equal to zero and solve.

Answer:

The value of $\lambda$ is $-\frac{1}{3}$.

Understanding:

Find the foot of perpendicular from $P(1,3,4)$ to the plane $2x-y+z+3=0$.

Formula:

The foot $Q$ lies on the line through $P$ in the direction of the normal $\vec{n}=(2,-1,1)$:

Substitute $Q$ into the plane equation to find $t$.

Step 1: Substitute $Q$ into $2x-y+z+3=0$.

Step 2: Find the coordinates of the foot.

Verification:

Check $(-1,4,3)$ lies on the plane:

Answer:

The foot of the perpendicular is $(-1,4,3)$.

Quick Tip:

Always verify the foot by substituting back into the plane equation — it catches sign errors instantly.

Understanding:

Find the volume of the parallelepiped with edge vectors:

Formula:

Step 1: Expand along the first row.

Step 2: Evaluate the $2\times 2$ determinants.

Step 3: Combine.

Answer:

The volume of the parallelepiped is $10$ cubic units.

Quick Tip:

If the scalar triple product equals zero, the three vectors are coplanar — a key fact tested separately in many exams.

Understanding:

Find the shortest distance between two skew lines.

Formula:

Step 1: $\overrightarrow{a_2}-\overrightarrow{a_1}$.

Step 2: $\overrightarrow{d_1}\times \overrightarrow{d_2}$.

Step 3: Dot product of $\overrightarrow{a_2}-\overrightarrow{a_1}$ and $\overrightarrow{d_1}\times \overrightarrow{d_2}$.

Step 4: $|\overrightarrow{d_1}\times \overrightarrow{d_2}|$.

Step 5: Shortest distance.

Answer:

The shortest distance between the two lines is $\frac{3\sqrt{2}}{2}$.

Quick Tip:

If $\overrightarrow{d_1}\times \overrightarrow{d_2}=\vec{0}$, the lines are parallel and a different formula applies. Always check the cross product first.

Understanding:

Find the unit vector in the direction of $\vec{a}+\vec{b}+\vec{c}$.

Formula:

Step 1: Compute the sum.

Step 2: Find the magnitude.

Step 3: Compute the unit vector.

Answer:

The unit vector in the direction of $\vec{a}+\vec{b}+\vec{c}$ is $\frac{1}{\sqrt{3}}(\hat{i}+\hat{j}+\hat{k})$.

Quick Tip:

The unit vector along $\hat{i}+\hat{j}+\hat{k}$ has each component $\frac{1}{\sqrt{3}}$ — this appears frequently in problems involving body diagonals of a cube.

Understanding:

Find the plane through $(1,-1,2)$ perpendicular to the planes $2x+3y-2z=5$ and $x+2y-3z=8$.

Formula:

The normal to the required plane is perpendicular to the normals of both given planes, so it equals their cross product:

where $\overrightarrow{n_1}=(2,3,-2)$ and $\overrightarrow{n_2}=(1,2,-3)$.

Step 1: Compute $\overrightarrow{n_1}\times \overrightarrow{n_2}$.

Step 2: The required plane has normal $(-5,4,1)$, equivalently $(5,-4,-1)$. Write the plane equation using point $(1,-1,2)$:

Verification:

Check that $(1,-1,2)$ satisfies $5(1)-4(-1)-(2)=5+4-2=7$ ✓

Answer:

The equation of the required plane is $5x-4y-z=7$.

Quick Tip:

The cross product of two normals gives the normal to the plane perpendicular to both — this is the standard construction for such problems.

Understanding:

Find the position vector of $P$ dividing $AB$ internally in ratio $2:3$.

Formula:

Section formula for internal division in ratio $m:n$:

Step 1: Substitute $m=2$, $n=3$.

Verification:

When $m=n$ (i.e., midpoint), this gives $\frac{\vec{a}+\vec{b}}{2}$, which is the standard midpoint formula — a useful sanity check.

Answer:

The position vector of $P$ is $\frac{2\vec{b}+3\vec{a}}{5}$.

Quick Tip:

A very common trap is writing $m\vec{a}+n\vec{b}$ instead of $m\vec{b}+n\vec{a}$. Remember: the ratio $m:n$ means $m$ multiplies the far end $B$ and $n$ multiplies the near end $A$.

Understanding:

We need the probability that both balls drawn (without replacement) are of the same colour.

Formula:

Step 1: Compute the total ways to choose 2 balls from 12.

Step 2: Compute favourable outcomes (both red, both blue, or both green).

Step 3: Add favourable outcomes.

Step 4: Compute the probability.

Answer:

The probability that both balls are of the same colour is $\frac{19}{66}$.

Quick Tip:

For "same colour" problems without replacement, always use combinations $\left(\genfrac{}{}{0ex}{}{n}{2}\right)$ per colour — never multiply probabilities sequentially unless you adjust the denominator at each draw.

Understanding:

We are told the mean of 10 observations is 15, and one value is replaced.

Formula:

Step 1: Find the original sum.

Step 2: Find the new sum after replacement.

Step 3: Compute the new mean.

Answer:

The new mean after replacing 20 with 30 is 16.

Quick Tip:

Whenever a single value is replaced, the mean changes by $\frac{\text{(new value)}-\text{(old value)}}{n}$. Here: $\frac{30-20}{10}=1$, so new mean $=15+1=16$.

Understanding:

Two fair dice are thrown. We need the probability that their sum is prime.

Formula:

Step 1: Identify prime sums in range $[2,12]$: $2,3,5,7,11$.

Step 2: Count outcomes for each prime sum.

Step 3: Total favourable outcomes.

Step 4: Compute the probability.

Answer:

The probability that the sum is a prime number is $\frac{5}{12}$.

Quick Tip:

Note that $4$ is not prime ($4=2\times 2$), and neither are $6,8,9,10,12$. A common error is to include $4$ or miss $11$.

Understanding:

We need to find the variance of the data set $\{2,4,6,8,10\}$.

Formula:

Step 1: Compute the mean.

Step 2: Compute each squared deviation.

Step 3: Sum the squared deviations.

Step 4: Divide by $n$.

Answer:

The variance of the given data set is $8$.

Quick Tip:

For an arithmetic progression $a,a+d,a+2d,\dots$ with $n$ terms, variance $=\frac{(n^2-1)d^2}{12}$. Here $d=2$, $n=5$: $\frac{(25-1)\times 4}{12}=\frac{96}{12}=8$. ✓

Understanding:

We need to find the conditional probability $P(A\mid B)$.

Formula:

Step 1: Find $P(A\cap B)$.

Step 2: Compute $P(A\mid B)$.

Answer:

The conditional probability $P(A\mid B)=0.4$.

Quick Tip:

A quick sanity check: since $P(A\mid B)=0.4=P(A)$, the events $A$ and $B$ are independent. This is consistent — independent events satisfy $P(A\cap B)=P(A)\cdot P(B)=0.4\times 0.5=0.2$. ✓

Understanding:

Using inclusion–exclusion to find students who passed in at least one subject, then subtracting from the total.

Formula:

Step 1: Find students who passed in at least one subject.

Step 2: Find students who failed in both.

Answer:

The number of students who failed in both subjects is $15$.

Understanding:

We need to compute the expected value (mean) of a discrete random variable $X$.

Formula:

Step 1: Multiply each value by its probability.

Step 2: Sum the products.

Verification:

Check probabilities sum to 1: $0.1+0.3+0.4+0.2=1.0$ ✓

Answer:

The expected value $E(X)=2.7$.

Quick Tip:

A common mistake is to simply average the $x$-values as $\frac{1+2+3+4}{4}=2.5$, ignoring the non-uniform probabilities.