Mathematics - MCQ Practice Questions

Step 1: Set up the arithmetic progression.

When three numbers are in arithmetic progression, they can be written as

where $a$ is the middle term and $d$ is the common difference.

Step 2: Apply the sum condition.

Step 3: State the result.

Therefore, the middle term is

Step 1: Apply Vieta's Formulas

For the quadratic $x^2-5x+6=0$, Vieta's formulas give:

Step 2: Simplify the Expression

Combine the fractions algebraically:

Step 3: Substitute and Evaluate

This approach avoids finding the individual roots and uses the relationship between them directly.

Final Answer:

Step 1: State the Geometric Mean Altitude Theorem

When an altitude is drawn from the right angle to the hypotenuse of a right triangle, the altitude is the geometric mean of the two segments into which it divides the hypotenuse. That is, if the two segments have lengths $p$ and $q$, then

Step 2: Identify the Known Values

The altitude divides the hypotenuse into two segments of lengths

Step 3: Apply the Theorem

Substituting into the formula:

Result

This is a fundamental property of right triangles, frequently applied in competitive examinations. The length of the altitude drawn to the hypotenuse is

Step 1: State the Angle Bisector Theorem

By the Angle Bisector Theorem, the angle bisector from a vertex divides the opposite side in the ratio of the two adjacent sides:

Step 2: Substitute the Known Values

Given $AB=6$ cm, $AC=9$ cm, and $BD=4$ cm, substituting into the ratio yields:

**Step 3: Solve for $DC$

Result

This theorem is essential for solving numerous geometry problems in competitive exams.

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

Step 1: Recall the standard trigonometric ratios table for common angles ($0°,30°,45°,60°,90°$).

Step 2: Locate the value of $\sin 30°$ from the standard table:

Step 3: Verify using the unit circle or right triangle approach — in a $30°$-$60°$-$90°$ triangle, the side opposite to $30°$ is half the hypotenuse, giving us:

Step 4: Convert the fraction to decimal form:

Therefore, $\sin 30°=0.5$.

This is a linear equation problem where we need to isolate the variable $x$ on one side.

Step 1: Start with the equation

Step 2: Subtract $5$ from both sides to remove the constant term

Step 3: Divide both sides by $2$ to isolate $x$

Step 4: Verify by substituting $x=5$ back into the original equation

Therefore, $x=5$.

To find the area of a circle, we use the formula $\text{Area}=\pi r^2$, where $r$ is the radius.

Step 1: Write the formula for the area of a circle.

Step 2: Substitute the given values: radius $r=7\text{cm}$ and $\pi =\frac{22}{7}$.

Step 3: Calculate $7^2$.

Step 4: Simplify by canceling $7$.

Therefore, the area of the circle is $\boxed{154{\text{cm}}^2}$.

The sum of all three interior angles of any triangle is always constant, regardless of the triangle's shape or size.

Step 1: Understand the Fundamental Property

Every triangle has exactly $3$ interior angles. These angles are formed where two sides of the triangle meet.

Step 2: Apply the Angle Sum Theorem

The angle sum property of a triangle states:

Step 3: Verify with Examples

This property holds for all triangles without exception. It is a foundational result in Euclidean geometry.

Therefore, the sum of the interior angles of a triangle is:

The logarithm is the inverse operation of exponentiation; ${\log }_b(x)=y$ means $b^y=x$.

Step 1: Understand what we need to find.

We need to find the value of ${\log }_{10}(1000)$. This asks: "10 raised to what power equals 1000?"

Step 2: Express 1000 as a power of 10.

Step 3: Apply the logarithm definition.

Since $10^3=1000$, we have:

Therefore, ${\log }_{10}(1000)=\boxed{3}$

To find $f(2)$, substitute $x=2$ into the given function and calculate the result.

Step 1: Write the function.

Step 2: Substitute $x=2$ into the function.

Step 3: Calculate the square term.

Step 4: Calculate the multiplication term.

Step 5: Add all terms.

Therefore, $f(2)=\boxed{12}$

The square root is the inverse operation of squaring, and asks: what number multiplied by itself equals $144$?

**Step 1: Understand what $\sqrt{144}$ means**

We need to find a number that, when multiplied by itself, gives $144$:

**Step 2: Check which number squared equals $144$

We test candidates by squaring them:

Step 3: Verify the answer

Since $12\times 12=144$, we conclude:

Therefore, the answer is $\boxed{12}$.

When flipping a fair coin, the probability of any single outcome equals the number of favorable outcomes divided by the total number of possible outcomes.

Step 1: Identify the total possible outcomes when tossing a coin.

A coin has exactly $2$ possible outcomes: Head or Tail.

Step 2: Identify the favorable outcomes for getting a Head.

Step 3: Apply the probability formula.

Step 4: Calculate.

Therefore, the probability of getting a Head in a coin toss is $\frac{1}{2}$, or $0.5$, or $50\%$.

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 $\boxed{3}$.

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:

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,