What are the solutions to the following questions: 1) The product of (3+sqrt(5))(3-sqrt(5)) is? 2) The base of common logarithm is? 3) If A = , then P(A) is? 4) If n(A U B) = 50, n(A) = 30 and n(B) =

Here, $a=3$ and $b=\sqrt{5}$.

The product is 4. This is not among the given options (Prime number, Odd number, Rational number, Irrational number). However, 4 is a rational number.
The correct option is D) Rational number.

2) The base of common logarithm is:
The common logarithm, also known as the decimal logarithm, is the logarithm with base 10.
The correct option is C) 10.

3) If $A = \{\}$, then $P(A)$ is:

If $A = \{\}$ (the empty set), its only subset is the empty set itself.

The correct option is B) $\{\{\}\}$.

4) If $n(A \cup B) = 50$, $n(A) = 30$ and $n(B) = 35$, then $n(A \cap B)$ is equal to:

$n(A \cup B) = n(A) + n(B) - n(A \cap B)$.
Substitute the given values into the formula.

The correct option is B) 15.

5) The L.C.M. of $16x^2$, $4x$ and $30xy$ is:

16, 4, and 30.
Prime factorization:
$16 = 2^4$
$4 = 2^2$
$30 = 2 \times 3 \times 5$
LCM of (16, 4, 30) = $2^4 \times 3 \times 5 = 16 \times 3 \times 5 = 240$.

$x^2$, $x$, and $xy$.
For $x$: the highest power is $x^2$.
For $y$: the highest power is $y$.

LCM = $240x^2y$.
The correct option is A) $240x^2y$.

6) The factorization of $12x + 36$ is:

The GCF of 12 and 36 is 12.

The correct option is D) $12(x+3)$.

7) Solution of $5x - 10 = 10$ is:

The correct option is B) 4.

8) $\sin 60^\circ =$
The value of $\sin 60^\circ$ is $\frac{\sqrt{3}}{2}$.
The correct option is A) $\frac{\sqrt{3}}{2}$.

9) $\sec^2 \theta - \tan^2 \theta =$
This is a fundamental trigonometric identity.

The correct option is C) 1.

10) Distance between two points $P(1,2)$ and $Q(4,6)$ is:

$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$.
Here, $(x_1, y_1) = (1,2)$ and $(x_2, y_2) = (4,6)$.

The correct option is D) 5.

11) If the volume of two similar solids is $125 \text{ cm}^3$ and $27 \text{ cm}^3$, the ratio of their corresponding heights is:

Let $V_1$ and $V_2$ be the volumes, and $h_1$ and $h_2$ be the corresponding heights.

The ratio of their corresponding heights is $5:3$.
The correct option is B) $5:3$.

12) Slope of the line $y = 5x + 3$ is:

Compare $y = 5x + 3$ with $y = mx + c$.
The slope $m$ is 5.
The correct option is B) 5.

13) An equilateral triangle:
An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal to $60^\circ$.

• can be isosceles: An equilateral triangle is a special case of an isosceles triangle (where at least two sides are equal).
• can be right angled: A right-angled triangle has one angle of $90^\circ$. An equilateral triangle has all angles $60^\circ$, so it cannot be right-angled.
• can be obtuse angled: An obtuse-angled triangle has one angle greater than $90^\circ$. An equilateral triangle has all angles $60^\circ$, so it cannot be obtuse-angled.
• has each angle equal to $50^\circ$: This is incorrect; each angle is $60^\circ$. The most accurate description among the choices is that an equilateral triangle can be isosceles. The correct option is A) can be isosceles.

14) Which data takes only some specific values?

• Continuous data: Can take any value within a given range (e.g., height, temperature).
• Discrete data: Can only take specific, distinct values (e.g., number of students, shoe size).
• Grouped data: Data organized into groups or intervals.
• Ungrouped data: Raw data not yet organized. Data that takes only some specific values is discrete data. The correct option is B) Discrete data.

15) While rolling a pair of dice, what will be the probability of double 2?

There is only one outcome for this: (2, 2).

The correct option is A) $\frac{1}{36}$.