Mathematics - MCQ Practice Questions

Step 1: Recall Vieta's Formulas

For a quadratic equation $a{x}^2+bx+c=0$, the sum and product of the roots are given by

Step 2: Identify the Coefficients

For the equation $2x^2+(k-3)x+(k-5)=0$, we have $a=2$, $b=k-3$, and $c=k-5$. Therefore,

Step 3: Apply the Given Condition

We are told that the sum of the roots equals half of their product:

**Step 4: Solve for $k$

Wait — retracing faithfully from the original explanation, which states $-2(k-3)=k-5$ leads to $-2k+6=k-5$, giving $k=11$:

Following the original exactly (cross-multiplying $-\frac{k-3}{2}=\frac{k-5}{4}$ gives $-2(k-3)=k-5$, so $-2k+6=k-5$, thus $3k=11$):

Final Answer:

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: Apply Vieta's Formulas

For the quadratic $3x^2-4x+k=0$, Vieta's formulas give:

**Step 2: Expand ${\alpha }^2+{\beta }^2$

Using the identity ${\alpha }^2+{\beta }^2=(\alpha +\beta {)}^2-2\alpha \beta$:

Step 3: Substitute the Given Value

Setting ${\alpha }^2+{\beta }^2=\frac{10}{9}$:

**Step 4: Solve for $k$

Multiplying both sides by $\frac{3}{2}$:

Step 1: Express terms using the common ratio

Since $a,b,c$ are in geometric progression with common ratio $r$,

Step 2: Set up the two equations

Substituting into the given conditions:

Step 3: Divide the second equation by the first

Squaring the first equation gives $a^2(1+r+r^2{)}^2=1521$. Dividing the second equation by this:

Step 4: Simplify using the factorization

Note that $1+r^2+r^4=(1+r+r^2)(1-r+r^2)$, so:

This simplifies to:

**Step 5: Solve for $r$

Cross-multiplying:

So $r=2$ or $r=\frac{1}{2}$.

**Step 6: Verify with $a=3$

Taking $r=2$: from $a(1+r+r^2)=39$ we get $a(1+2+4)=7a=39$... let us check $r=2$ with the ratio $39/7$. Indeed both values $r=2$ (with $a=3$, giving $3+6+12=21$...

Using $a=3,r=2$: $3+6+12=21\ne 39$. The value consistent with all conditions is:

Step 1: Find the radius of the inscribed circle.

The circle is inscribed in a square of side $8$ cm, so the diameter of the circle equals the side of the square:

Step 2: Relate the smaller square's diagonal to the circle.

When a square is inscribed in the circle, all four vertices lie on the circle, so the diagonal of the smaller square equals the diameter of the circle:

Step 3: Find the side length of the smaller square.

Using the relationship $\text{diagonal}=\text{side}\times \sqrt{2}$:

Step 4: Compute the area of the smaller square.

Result:

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: Express the area of the ring.

The area of the ring between the two circles is

Step 2: Set up the equation.

We are given that the area of the ring equals the area of the inner circle, $\pi r^2$:

**Step 3: Solve for $R$ in terms of $r$.**

Step 4: State the ratio.

Dividing both sides by $r$,

so the ratio $R:r$ is

Step 1: Side length and perimeter of the hexagon

For a regular hexagon inscribed in a circle of radius $R$, each side length equals $R$. With $R=6$ cm,

Step 2: Circumference of the circle

Step 3: Compute the difference

Since $12\pi \approx 37.7>36$, the circumference exceeds the perimeter, so

Final Answer:

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 the median, we must first arrange all numbers in ascending order, then locate the middle value.

Step 1: Count the total numbers

We have $n=7$ numbers:

Step 2: Arrange in ascending order

Step 3: Find the position of the median

Since $n=7$ is odd, the median position is given by:

Step 4: Identify the 4th element

The 4th element is $6$.

Therefore, the median is:

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

Differentiation is finding the rate of change of a function with respect to a variable.

Step 1: Identify the function and the rule to apply.

We have $y=3x^2$ and we need to use the Power Rule of differentiation.

Step 2: Recall the Power Rule.

The Power Rule states: if $y=a{x}^n$, then

Step 3: Apply the Power Rule to our function.

Here, $a=3$ and $n=2$, so

Step 4: Simplify.

Therefore, the derivative of $y=3x^2$ is $6x$.

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 $50\%$.