Mathematics - MCQ Practice Questions

This question asks you to find the roots of a quadratic equation using factorization.

Step 1: Write the equation in standard form.

Step 2: Find two numbers that multiply to give $6$ (constant term) and add to give $-5$ (coefficient of $x$).

The numbers are $-2$ and $-3$, because:

Step 3: Factorize the quadratic.

Step 4: Solve for $x$ by setting each factor equal to zero.

Step 5: Verify by substituting back.

For $x=2$:

For $x=3$:

Therefore, the solutions are $\boxed{x=2,x=3}$

The answer is (A).

The factorial of a positive integer $n$ is the product of all positive integers from $1$ up to $n$.

Step 1: Understand factorial notation

The notation $7!$ means:

Step 2: Multiply step by step

Working from left to right:

Step 3: Final result

Therefore,

The slope of a line is the coefficient of $x$ in the standard linear equation form $y=mx+c$.

Step 1: Identify the standard form of a linear equation.

The equation of a line is written as:

where $m$ is the slope and $c$ is the $y$-intercept.

Step 2: Compare the given equation with standard form.

Step 3: Extract the slope.

Matching coefficients:

Therefore, the slope is $\boxed{3}$.

To convert degrees to radians, use the conversion formula that relates the two angle measurement systems.

Step 1: Recall the conversion formula from degrees to radians.

Step 2: Substitute $45°$ into the formula.

Step 3: Simplify the fraction.

Therefore, $45°=\frac{\pi }{4}$ radians.

The answer is $\boxed{\frac{\pi }{4}}$

The Highest Common Factor (HCF) is the largest number that divides both given numbers without leaving a remainder.

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Step 1: Find all factors of 24

Step 2: Find all factors of 36

Step 3: Identify the common factors

Step 4: Select the highest common factor

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Alternative Method — Prime Factorization

Taking the lowest power of each common prime factor:

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Conclusion:

Integration is the reverse process of differentiation, and we use the power rule to find antiderivatives.

Step 1: Identify the integral form

We need to find $\int 2xdx$, which means we are integrating the function $2x$ with respect to $x$.

Step 2: Apply the power rule of integration

The power rule states:

Rewrite $2x$ as $2x^1$, so the integral becomes $\int 2x^{1}dx$.

Step 3: Factor out the constant

Step 4: Apply the power rule

Step 5: Verify by differentiation

Therefore,

To find the $n$-th term of an arithmetic progression, we use the formula $a_n=a+(n-1)d$, where $a$ is the first term and $d$ is the common difference.

Step 1: Identify the first term and common difference.

From the AP: $2,5,8,11,\dots$

Step 2: Write the formula for the 10th term.

We need to find $a_{10}$ using:

**Step 3: Substitute $n=10$, $a=2$, and $d=3$.**

Therefore, the 10th term of the AP is $\boxed{29}$.

This question tests your knowledge of standard trigonometric values for common angles.

Step 1: Recall the standard angle values

The trigonometric ratios for $0°,30°,45°,60°,$ and $90°$ are fixed values that must be memorized for competitive exams.

**Step 2: Identify $\cos 60°$ from the standard table**

For $60°$, the cosine value is:

Step 3: Verify using the unit circle concept

At $60°$, if you place a point on the unit circle, the $x$-coordinate (which represents cosine) equals $\frac{1}{2}$.

Step 4: Eliminate other options

Therefore,

For a $2\times 2$ matrix, the determinant is calculated using the cross-product formula: (top-left $\times$ bottom-right) minus (top-right $\times$ bottom-left).

Step 1: Identify the matrix elements.

where $a=1$, $b=2$, $c=3$, $d=4$.

**Step 2: Apply the determinant formula for a $2\times 2$ matrix.**

Step 3: Substitute the values.

Step 4: Calculate.

Therefore, the determinant of $

$is$\boxed{-2}$.

This question asks you to find the 5th term of a geometric progression using the formula for the $n$-th term.

Step 1: Identify the first term and common ratio.

**Step 2: Write the formula for the $n$-th term of a GP.**

The $n$-th term of a GP is given by:

**Step 3: Substitute values for the 5th term ($n=5$).**

Step 4: Calculate the result.

Therefore, the 5th term is $\boxed{162}$.

The diagonals of a rhombus bisect each other at right angles. So half-diagonals are $8\text{ cm}$ and $6\text{ cm}$. Each side is:

Perimeter $=4\times 10=\boxed{40\text{ cm}}$

First, find $BC$ using the Pythagorean theorem:

A key property states that the median from the right angle to the hypotenuse equals half the hypotenuse:

When two circles touch externally, the distance between their centres equals the sum of their radii:

The arc length formula is:

By the Midpoint Theorem, the line segment joining the midpoints of two sides of a triangle is parallel to the third side and equals half its length:

Volume of a cone:

First find the other leg:

Area of right triangle:

The sum of all exterior angles of any regular polygon is $360°$. The number of sides is:

The perpendicular from the centre bisects the chord, so half the chord $=8\text{ cm}$. Using the Pythagorean theorem:

Check: $7^2+24^2=49+576=625=25^2$. So this is a right triangle with legs $7$ and $24$.