Match the entries of List A and List B
`{:(,"List A",,"List B"),((a),"The Lines joining the points of intersection of the line 4x-y=10 with the circle "x^2+y^2+3x-6y-20=0" include an angle",1.,h=pm2/3sqrt(ab)),((b),"The two lines given "a^2x^2+2hxy+b^2y^2=0" are such that slop of one is three times of the other ,then h is equal to",2.,2),((c),"If the equation "12x^2-10xy+2y^2+11x-5y+lambda=0" represents a pair of straight lines then "lambda"=",3.,10),((d),"If "x^2-3xy+(lambda)y^2+3x-5y+2=0" represents a pair of straight lines which include an angle "theta" then "cosec^2" "theta"=",4.,pi/2):}`

Answer

To solve the problem, we will match the entries of List A with List B step by step.

### Step 1: Analyze the first entry in List A
**Entry (a)**: "The Lines joining the points of intersection of the line 4x-y=10 with the circle x²+y²+3x-6y-20=0 include an angle."

To find the angle between the lines formed by the intersection of the given line and circle, we first need to find the points of intersection. We can substitute the equation of the line into the circle's equation.

**Hint**: Substitute the line equation into the circle equation to find the points of intersection.

### Step 2: Substitute the line equation
Substituting \( y = 4x - 10 \) into the circle equation:
\[ x^2 + (4x - 10)^2 + 3x - 6(4x - 10) - 20 = 0 \]
This simplifies to a quadratic equation in \( x \).

**Hint**: Expand and simplify the equation to find the roots.

### Step 3: Find the angle between the lines
Once we have the points of intersection, we can find the angle between the lines joining the origin to these points. The angle \( \theta \) can be found using the formula:
\[ \tan(\theta) = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \]
where \( m_1 \) and \( m_2 \) are the slopes of the lines.

**Hint**: Use the slopes from the points of intersection to find the angle.

### Step 4: Analyze the second entry in List A
**Entry (b)**: "The two lines given \( a^2x^2 + 2hxy + b^2y^2 = 0 \) are such that the slope of one is three times that of the other, then h is equal to."

Using the relationship between the slopes, we have:
\[ m_1 = \frac{-h + \sqrt{h^2 - a^2b^2}}{a^2} \quad \text{and} \quad m_2 = \frac{-h - \sqrt{h^2 - a^2b^2}}{a^2} \]
Given that \( m_1 = 3m_2 \), we can set up an equation to solve for \( h \).

**Hint**: Set up the equation based on the relationship between the slopes to find \( h \).

### Step 5: Analyze the third entry in List A
**Entry (c)**: "If the equation \( 12x^2 - 10xy + 2y^2 + 11x - 5y + \lambda = 0 \) represents a pair of straight lines then \( \lambda \) ="

To find \( \lambda \), we will use the condition for the equation to represent a pair of straight lines, which requires the determinant of the coefficients to be zero.

**Hint**: Set the determinant of the coefficients to zero and solve for \( \lambda \).

### Step 6: Analyze the fourth entry in List A
**Entry (d)**: "If \( x^2 - 3xy + (\lambda)y^2 + 3x - 5y + 2 = 0 \) represents a pair of straight lines which include an angle \( \theta \), then \( \csc^2 \theta \) ="

Using the condition for the equation to represent a pair of straight lines and the relationship between the angle and the coefficients, we can find \( \lambda \).

**Hint**: Use the relationship between the angle and the coefficients to find \( \csc^2 \theta \).

### Final Matching
Now, we can match the entries from List A to List B based on our findings:

- **(a)** matches with **4** (angle \( \pi/2 \))
- **(b)** matches with **1** (h = \( \frac{2}{\sqrt{3}} \sqrt{ab} \))
- **(c)** matches with **3** (lambda = 10)
- **(d)** matches with **2** (cosec² \( \theta \) = 2)

### Summary of Matches
- (a) → 4
- (b) → 1
- (c) → 3
- (d) → 2