Step 1: Recall the Key Relationship
For any two positive integers a and b: \( \text{HCF}(a, b) \times \text{LCM}(a, b)\) \(= a \times b \)

Step 2: Substitute Given HCF
Here, it's given that \( \text{HCF}(a, b) = 1 \) \(\Rightarrow 1 \times \text{LCM}(a, b) = a \times b \)

Step 3: Solve for LCM
So, \( \text{LCM}(a, b) = a \times b \)

Step 4: Final Answer
When two positive integers have HCF 1 (they are co-prime), their Least Common Multiple is simply their product: \( ab \) (option D).

Step 1: Identify the Types of Numbers
The number 3 is a rational number because it can be expressed as a fraction \(\displaystyle \frac{3}{1} \).

Step 2: Understand the Nature of \(\sqrt{2}\)
The number \( \sqrt{2} \) is known to be an irrational number because it cannot be expressed as a simple fraction.

Step 3: Sum of Rational and Irrational
The sum of a rational number \((3)\) and an irrational number \((\sqrt{2})\) is always an irrational number.

Step 4: Final Answer
Therefore, \(3 + \sqrt{2}\) is an irrational number (option B).

Step 1: Identify the quadratic equation coefficients
The given quadratic equation is \(x^2 - 3x - 2 = 0\). Here, \(a = 1\), \(b = -3\), and \(c = -2\).

Step 2: Recall the discriminant formula
The discriminant \(D\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by:
\( D = b^2 - 4ac \)

Step 3: Substitute the values
Substitute \(a = 1\), \(b = -3\), and \(c = -2\) into the formula:
\( D = (-3)^2 - 4 \times 1 \times (-2)\) \(= 9 + 8 = 17 \)

Step 4: Final Answer
The discriminant of the quadratic equation is \( 17 \) (option B).

Step 1: Given Equation
The equation is \(\displaystyle x + \frac{1}{x} = 3 \) where \( x \ne 0 \).

Step 2: Convert to Quadratic Form
Multiply both sides by \( x \) to eliminate the fraction:
\(\displaystyle x \times x + x \times \frac{1}{x} = 3 \times x \) \(\Rightarrow x^2 + 1 = 3x \)

Step 3: Rearrange to Standard Quadratic Format
Bring all terms to one side: \( x^2 - 3x + 1 = 0 \)

Step 4: Identify coefficients and calculate \( a - b + c \)
Here, \( a = 1 \), \( b = -3 \), and \( c = 1 \).
Calculate: \( a - b + c \) \(= 1 - (-3) + 1\) \(= 1 + 3 + 1 \) \(= 5 \)

Step 5: Final Answer:
The value of \( a - b + c \) is 5 (Option A).

Step 1: Understand the terms
The abscissa of a point is the x-coordinate, and the ordinate is the y-coordinate.

Step 2: Identify coordinates
Given the point is \((3, -5)\), so:
Abscissa (x) = 3
Ordinate (y) = -5

Step 3: Calculate (abscissa - ordinate)
\( 3 - (-5) = 3 + 5 = 8 \)

Step 4: Final Answer:
The value of (abscissa - ordinate) is 8 (Option D).

Step 1: Understand the concept of midpoint
The midpoint of a line segment is the point that divides the segment into two equal parts.

Step 2: Ratio of division
Since the midpoint divides the segment into two equal parts, the ratio is:
\( 1 : 1 \)

Step 3: Final Answer:
The midpoint divides the line segment in the ratio 1 : 1 (Option C).

Step 1: Understand criteria for similarity of triangles
The common criteria for triangle similarity are:

• AAA (Angle-Angle-Angle): If all corresponding angles are equal, triangles are similar.
• SSS (Side-Side-Side): If all three corresponding sides are in proportion, triangles are similar.
• SAS (Side-Angle-Side): If two sides are in proportion and the included angle is equal, triangles are similar.

Step 2: Identify the option that is not a similarity criterion
RHS (Right angle-Hypotenuse-Side) is a criterion for congruence of right-angled triangles, not for similarity.

Step 3:Final Answer:
Option D (RHS) is not a criterion for similarity of triangles.

Step 1: Understand the triangle OAP
Triangle OAP has:
• ∠OAP = 90° (right angle at point of tangency)
• ∠OPA = 30°
• OP = 10 cm (hypotenuse)

Step 2: Use the right triangle ratios
OA is opposite to angle ∠OPA (30°); AP is adjacent.
By trigonometry:
\( \sin(30^\circ) \) \(\displaystyle= \frac{\text{Opposite}}{\text{Hypotenuse}} \) \(\displaystyle= \frac{\text{OA}}{10} \)
\(\displaystyle \sin(30^\circ) = \frac{1}{2} \)
So, \( \displaystyle \frac{1}{2} = \frac{\text{OA}}{10}\)
\( \Rightarrow \text{OA} = 5 \) cm (radius).

Step 3: Find AP using cosine
\(\displaystyle \cos(30^\circ) = \frac{\text{Adjacent}}{\text{Hypotenuse}}\) \( = \frac{\text{AP}}{10} \)
\( \displaystyle \cos(30^\circ) = \frac{\sqrt{3}}{2} \)
So, \( \displaystyle\frac{\sqrt{3}}{2} = \frac{\text{AP}}{10}\)
\(\Rightarrow \text{AP}\displaystyle = 10 \times \frac{\sqrt{3}}{2} \) \(= 5\sqrt{3} \) cm.

Final Answer: The length of AP is 5\(\sqrt{3}\) cm (Option D).

Step 1: Evaluate each given trigonometric statement

• \(\tan 45^\circ = \cot 45^\circ\)
  Both \(\tan 45^\circ\) and \(\cot 45^\circ\) equal 1, so this is true.
• \(\sin 90^\circ = \tan 45^\circ\)
  \(\sin 90^\circ = 1\), \(\tan 45^\circ = 1\), so this is true.
• \(\sin 30^\circ = \cos 30^\circ\)
  \(\sin 30^\circ = \frac{1}{2}\), \(\displaystyle\cos 30^\circ = \frac{\sqrt{3}}{2} \approx 0.866\), so this is false.
• \(\sin 45^\circ = \cos 45^\circ\)
  Both equal \(\displaystyle \frac{\sqrt{2}}{2} \approx 0.707\), so this is true.

Step 2: Identify the false statement
The false statement is Option C: \(\sin 30^\circ = \cos 30^\circ\).

Final Answer: Option C is false.

Step 1: Express the given expression
The expression is: \(\displaystyle \tan^2 A - \frac{1}{\cos^2 A} \)

Step 2: Use trigonometric identities
Recall that \(\displaystyle \tan A = \frac{\sin A}{\cos A}\), so \(\displaystyle \tan^2 A = \frac{\sin^2 A}{\cos^2 A} \)
Substitute this into the expression:
\(\displaystyle \frac{\sin^2 A}{\cos^2 A} - \frac{1}{\cos^2 A}\) \( \displaystyle = \frac{\sin^2 A - 1}{\cos^2 A} \)

Step 3: Simplify numerator using Pythagorean identity
Recall that \(\sin^2 A + \cos^2 A = 1\), so \( \sin^2 A - 1 = -\cos^2 A \)
Substitute: \(\displaystyle \frac{-\cos^2 A}{\cos^2 A} = -1 \)

Final Answer: The value of the expression is \(-1\) (Option D).

Step 1: Understand the angle of depression
The angle of depression is formed between the horizontal line from the observer's eye and the line of sight down to the object.

Step 2: Locate the angle in the diagram
In the figure, the angle marked z is between the horizontal line (from observer) and the line of sight, which matches the definition of the angle of depression.

Final Answer: The angle that represents the angle of depression is z (Option C).

Step 1: Analyze the shaded region
The shaded region is a triangle within a circle, with two sides equal to the radius \(r\) and the curved part at the base (arc length) denoted by \(l\).

Step 2: Find the perimeter
The perimeter of the shaded region consists of the two radii (\(r + r = 2r\)) and the arc length (\(l\)). The segment denoted as "a" is inside the triangle, not on the boundary of the shaded region.

Final Answer: The perimeter is \(l + 2r\) (Option C).

Step 1: Find the area of a circle
The area of a circle is \( \pi r^2 \).

Step 2: Find the area of a quadrant
A quadrant is one-fourth of a circle, so its area is \( \frac{1}{4} \pi r^2 \).

Step 3: Calculate the ratio
Ratio = (area of quadrant) : (area of whole circle) = \( \frac{1}{4} \pi r^2 : \pi r^2 = 1 : 4 \)

Final Answer: The ratio is 1 : 4 (Option C).

Step 1: Definitions
The total surface area is the sum of all the areas that cover the outside of a solid.
The lateral/curved surface area is the area of only the sides, without the top and bottom faces (if any).

Step 2: Sphere uniqueness
For a sphere, there are no distinct faces or bases - its entire outer surface is curved. So, the lateral/curved surface area and total surface area are both \(4\pi r^2\).

Final Answer: The solid for which lateral/curved surface area and total surface area are the same is the sphere (Option D).

Step 1: Calculate the total frequency
Add all the frequencies given:
\( 2 + 3 + 7 + 6 + 6 + 6 = 30 \)
The total number of observations, \( n = 30 \).

Step 2: Find the median class
The median class is the class interval where the cumulative frequency is equal to or just exceeds half of the total frequency.
Half of total frequency \(= \displaystyle \frac{30}{2} = 15 \).

Calculate cumulative frequencies:
- \(10-25: 2\)
- \(25-40: 2 + 3 = 5\)
- \(40-55: 5 + 7 = 12\)
- \(55-70: 12 + 6 = 18\) → This is the first cumulative frequency greater than or equal to 15.
Therefore, the median class is \( 55 - 70 \).

Step 3: Calculate the class mark of the median class
The class mark is the midpoint of the class interval:
\( \displaystyle \text{Class mark} = \frac{55 + 70}{2} = 62.5 \)

Final Answer: The class mark of the median class is 62.5 (Option D).

Step 1: Understand what modal class means
The modal class is the class interval in a grouped frequency distribution which has the highest frequency.
It represents the interval where the data is most concentrated or common.

Step 2: Look at the frequency values
From the table, the frequencies (number of batsmen) for each class are:
- 3000 – 4000: 5
- 4000 – 5000: 10
- 5000 – 6000: 9
- 6000 – 7000: 8
The highest frequency is 10, which belongs to the class 4000 – 5000.

Step 3: Identify the lower limit of the modal class
Since the modal class is 4000 – 5000, its lower limit is 4000.

Final Answer: The lower limit of the modal class is 4000 (Option B).

Step 1: Understand the concept of a sure event
A sure event is an event that is certain to happen when an experiment is performed. Its probability is 1.

Step 2: Analyze the outcomes of throwing a die
A die has 6 faces numbered from 1 to 6.
Possible outcomes are: 1, 2, 3, 4, 5, 6.

Step 3: Evaluate each option
- Option A: Getting a number between 1 and 6 – This is true for every throw (all outcomes), so this is a sure event.
- Option B: Getting an odd number < 7 – Possible odd numbers are 1, 3, 5. Not all outcomes are odd, so this is not a sure event.
- Option C: Getting an even number < 7 – Possible even numbers are 2, 4, 6. Not all outcomes are even, so this is not a sure event.
- Option D: Getting a natural number < 7 – Natural numbers less than 7 are 1, 2, 3, 4, 5, 6. This includes all possible outcomes, so this is a sure event.

Final Answer: The sure event is Getting a natural number < 7 (Option D).

Step 1: Understand the assertion
Assertion (A) states that for any two natural numbers \(a\) and \(b\), the HCF (Highest Common Factor) of \(a\) and \(b\) is a factor of their LCM (Least Common Multiple).

Step 2: Understand the reason
Reason (R) states that the HCF of two natural numbers divides both numbers.

Step 3: Verify the assertion through example
Take \(a = 12\) and \(b = 18\) as example numbers.
- Factors of 12 are \(1, 2, 3, 4, 6, 12\).
- Factors of 18 are \(1, 2, 3, 6, 9, 18\).
- Common factors are \(1, 2, 3, 6\), so \(\text{HCF}(12, 18) = 6\).

Step 4: Calculate the LCM
- Multiples of 12 are \(12, 24, 36, 48, 60, 72, \ldots\).
- Multiples of 18 are \(18, 36, 54, 72, \ldots\).
- The smallest common multiple is \(36\), so \(\text{LCM}(12, 18) = 36\).

Step 5: Verify the relation between HCF, LCM, and the numbers
- Compute \(\text{HCF} \times \text{LCM} \) \(= 6 \times 36 = 216\).
- Compute \(a \times b = 12 \times 18 = 216\).
This confirms the formula \(a \times b \) \(= \text{HCF}(a, b) \times \text{LCM}(a, b)\) holds.

Step 6: Check if HCF divides LCM
- Calculate \(\displaystyle \frac{\text{LCM}}{\text{HCF}} = \displaystyle \frac{36}{6} = 6\) which is an integer.
- This means HCF is indeed a factor of LCM, confirming Assertion (A) is true.

Step 7: Understand how Reason (R) explains the Assertion
- Reason (R) states that the HCF divides both numbers, which is the fundamental property that allows the multiplication formula to hold.
- Because the HCF divides both \(a\) and \(b\), it also divides their LCM via the relationship.
- Therefore, Reason (R) correctly explains Assertion (A).

Final Answer: Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A) (Option A).

Step 1: Write the given system of equations

\(4x + py + 8 = 0 \quad\) \( \Rightarrow \quad 4x + py = -8\)

\(2x + 2y + 2 = 0 \quad\) \( \Rightarrow \quad 2x + 2y = -2\)

Step 2: Identify coefficients

Comparing with the general form \(a_1x + b_1y = c_1\) and \(a_2x + b_2y = c_2\):

\(a_1 = 4, \quad b_1 = p, \quad c_1 = -8\)

\(a_2 = 2, \quad b_2 = 2, \quad c_2 = -2\)

Step 3: Use condition for consistent system with infinitely many solutions

The system is consistent with infinitely many solutions if:

\(\displaystyle \frac{a_1}{a_2} = \displaystyle \frac{b_1}{b_2} = \displaystyle \frac{c_1}{c_2}\)

Step 4: Calculate the ratios

\(\displaystyle \frac{a_1}{a_2} = \displaystyle \frac{4}{2} = 2\), \(\quad \displaystyle \frac{b_1}{b_2} = \displaystyle \frac{p}{2}\), \( \displaystyle \frac{c_1}{c_2} = \displaystyle \frac{-8}{-2} = 4\)

Step 5: Equate the ratios to find \(p\)

For the system to have infinitely many solutions, ratios must be equal:

\(2 = \displaystyle \frac{p}{2} = 4\)

But \(2 \neq 4\), so the system cannot be consistent with infinitely many solutions.

Step 6: Check if the system is consistent (has some solution)

For the system to be consistent (at least one solution), check if:

\(\displaystyle \frac{a_1}{a_2} \neq \displaystyle \frac{b_1}{b_2}\) or system is consistent if:

\(4x + py = -8 \quad\) and \( \quad 2x + 2y = -2\)

Step 7: Find \(p\) for which system is consistent

Multiply second equation by 2:

\(4x + 4y = -4\) Compare with first:

\(4x + py = -8\) For consistency, these lines should not be parallel (i.e. slopes differ), so:

\(p \neq 4\)

Conclusion:

- Assertion (A) states \(p = 4\) makes system consistent — this is false because \(p \neq 4\) ensures consistency.

- Reason (R) is true as it correctly states the condition for infinite solutions.

Therefore, Option D is correct: Assertion is false, but Reason is true.