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: 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: 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:

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:

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 $6x$.

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,

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 $\boxed{\frac{\pi }{4}}$

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 $\boxed{29}$.

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: