JEE Mathematics - MCQ Practice Questions

Understanding:

We need to find $p^2-2q$ given that the roots of $x^2-px+q=0$ are $\sin \theta$ and $\cos \theta$.

Formula:

For a quadratic $x^2-px+q=0$ with roots $\alpha$ and $\beta$:

Also, the Pythagorean identity:

Step 1: Express $p$ and $q$ using Vieta's formulas

Step 2: Compute $p^2$

Step 3: Compute $p^2-2q$

Answer:

The value of $p^2-2q$ is $1$, independent of $\theta$.

Quick Tip:

The identity $(\sin \theta +\cos \theta {)}^2=1+2\sin \theta \cos \theta$ is key — it connects $p^2$ directly to $q$, making the result constant.

Understanding:

We must evaluate ${\int }_0^{1}x{e}^{x}dx$.

Formula:

Integration by parts:

Step 1: Choose $u$ and $dv$

Step 2: Apply the formula

Step 3: Substitute limits

Answer:

The value of the integral is $1$.

Quick Tip:

When applying integration by parts, always choose $u$ as the polynomial part — differentiation reduces it, making the second integral simpler.

Understanding:

We must count 4-digit numbers formed from digits $\{1,2,3,4,5\}$ without repetition that are divisible by $4$.

Formula:

A number is divisible by $4$ if and only if its last two digits form a number divisible by $4$:

Step 1: List all valid last-two-digit pairs (from $\{1,2,3,4,5\}$, no repetition, forming a 2-digit number divisible by 4)

Checking all ordered pairs:

Also checking: $44$ (repetition), $20,40$ (0 not available).

Valid pairs (last two digits): $12,24,32,52$.

Step 2: Count arrangements for each valid ending

For each valid pair, the remaining $3$ digits are chosen from the remaining $3$ available digits and arranged in $3!=6$ ways for the first two positions.

Step 3: Total count

Answer:

The number of valid 4-digit numbers is $24$.

Quick Tip:

For divisibility by $4$, only the last two digits matter. Systematically list all valid 2-digit endings first, then multiply by the permutations of remaining digits.

Understanding:

We must find the determinant $|3AB|$ given $|A|=2$, $|B|=-3$, and the order of each matrix is $3$.

Formula:

For an $n\times n$ matrix:

Step 1: Compute $|AB|$

Step 2: Compute $|3AB|$

Answer:

The value of $|3AB|$ is $-162$.

Quick Tip:

A common mistake is forgetting to raise the scalar to the power of the matrix order. For a $3\times 3$ matrix, $|kM|=k^3|M|$, not $k|M|$.

Understanding:

We must find the equation of the tangent to $y=x^2-2x+3$ at $x=2$.

Formula:

Equation of tangent at a point $(x_0,y_0)$:

Step 1: Find the $y$-coordinate at $x=2$

So the point of tangency is $(2,3)$.

Step 2: Find the slope

Step 3: Write the tangent equation

Answer:

The equation of the tangent at $x=2$ is $y=2x-1$.

Quick Tip:

Always verify the point lies on the curve before writing the tangent equation — a small arithmetic slip in $y_0$ shifts the entire line.

Understanding:

We must evaluate $(1+\omega -{\omega }^2{)}^3$ where $\omega$ is a non-real cube root of unity.

Formula:

Properties of cube roots of unity:

Step 1: Simplify the expression inside the brackets

Using $1+\omega =-{\omega }^2$:

Step 2: Raise to the power $3$

Step 3: Simplify using ${\omega }^3=1$

Answer:

The value of $(1+\omega -{\omega }^2{)}^3$ is $-8$.

Quick Tip:

The identity $1+\omega +{\omega }^2=0$ is the single most useful fact for cube-root-of-unity problems — memorise it and use it immediately to collapse sums.

Understanding:

We must solve the inequality $\frac{x-1}{x+2}>1$.

Formula:

To solve $\frac{f(x)}{g(x)}>1$, rewrite as $\frac{f(x)-g(x)}{g(x)}>0$ to avoid multiplying by an expression of unknown sign:

Step 1: Rearrange

Step 2: Solve the simplified inequality

Since $-3<0$, the fraction $\frac{-3}{x+2}$ is positive only when $x+2<0$:

Step 3: Check the boundary

At $x=-2$, the expression is undefined, so $x=-2$ is excluded.

Answer:

The solution set is $x<-2$.

Quick Tip:

Never cross-multiply an inequality by $(x+2)$ without knowing its sign. Always bring everything to one side and form a single fraction — this approach is always safe.

Understanding:

We must find the general solution of $\frac{dy}{dx}=e^{x+y}$.

Formula:

This is a separable ODE. Separate variables and integrate:

Step 1: Separate variables

Step 2: Integrate both sides

Step 3: Rearrange

Writing $-C_1=C$ (an arbitrary constant):

Answer:

The general solution is $e^{-y}-e^x=C$.

Quick Tip:

After integrating $e^{-y}$ with respect to $y$, the result is $-e^{-y}$ — the negative sign is easy to drop accidentally, which changes the form of the answer significantly.

Understanding:

We must find side $c$ in $△ABC$ using the cosine rule.

Formula:

The Law of Cosines:

Step 1: Substitute values

Step 2: Find $c$

Answer:

The length of side $c$ is $\sqrt{39}$.

Quick Tip:

$\cos 60^{\circ }=\frac{1}{2}$ is a standard value. Memorising it (and $\cos 30^{\circ },\cos 45^{\circ }$) avoids look-up time under exam pressure.

Understanding:

We must find the term independent of $x$ (i.e., the constant term) in ${\left(x^2+\frac{1}{x}\right)}^9$.

Formula:

The general term in the binomial expansion of $(a+b{)}^n$:

Step 1: Write the general term

Step 2: Set the power of $x$ to zero

Step 3: Compute the term

Answer:

The term independent of $x$ is $84$.

Quick Tip:

Always simplify $\left(\genfrac{}{}{0ex}{}{9}{6}\right)$ as $\left(\genfrac{}{}{0ex}{}{9}{3}\right)$ using the symmetry $\left(\genfrac{}{}{0ex}{}{n}{r}\right)=\left(\genfrac{}{}{0ex}{}{n}{n-r}\right)$ — it requires fewer multiplications.