Kaliyuga Ekalavya

NCERT Class X Chapter 7: Coordinate Geometry Example 10 Question: If the points A(6, 1), B(8, 2), C(9, 4) and D(p, 3) are the vertices of a parallelogram, taken in order, find the value of p. Given: Points A(6, 1), B(8, 2), C(9, 4), and D(p, 3) are the vertices of a parallelogram ABCD. To Find: The value of \( p \). Formula: Midpoint formula: The midpoint of a line segment with endpoints \( (x_1, y_1) \) and \( (x_2, y_2) \) is: $$ \left( \frac{x_1 + x_2}{2},\ \frac{y_1 + y_2}{2} \right) $$ Solution: Step 1: In a parallelogram, the diagonals bisect each other. Thus, the midpoints of diagonals \( AC \) and \( BD \) must be the same. Step 2: Find the midpoint of diagonal \( AC \), where \( A(6, 1) \) and \( C(9, 4) \): $$ \text{Midpoint of } AC = \left( \frac{6 + 9}{2},\ \frac{1 + 4}{2} \right) = \left( \frac{15}{2},\ \frac{5}{2} \right) $$ Step 3: Find the midpoint of diagonal \( BD \), where \( B(8, 2) \) and \( D(p, 3) \): $$ \t...

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Example 10 Question: The sum of a two-digit number and the number obtained by reversing the digits is 66. If the digits of the number differ by 2, find the number. How many such numbers are there? Use elimination method Given: Let the two-digit number be 10a + b, where a and b are digits. The reversed number is 10b + a. Their sum is 66, and |a - b| = 2. To Find: The two-digit number and the total number of such numbers. Formula: 10a + b + 10b + a = 66 Solution: 10a + b + 10b + a = 66 ⇒ 11a + 11b = 66 ⇒ a + b = 6 Since |a - b| = 2, we have two cases: Case 1: a - b = 2. Solving a + b = 6 and a - b = 2 simultaneously, we get 2a = 8 ⇒ a = 4 and b = 2. The number is 42. Case 2: b - a = 2. Solving a + b = 6 and b - a = 2 simultaneously, we get 2b = 8 ⇒ b = 4 and a = 2. The number is 24. Result: The two-digit numbers are 42 and 24. There a...

NCERT Class X Chapter 5: Arithmetic Progression Example 10 Question: In a flower bed, there are 23 rose plants in the first row, 21 in the second, 19 in the third, and so on. There are 5 rose plants in the last row. How many rows are there in the flower bed?. Given: Number of rose plants in the first row = 23 Number of rose plants in the second row = 21 Number of rose plants in the third row = 19 Number of rose plants in the last row = 5 To Find: The number of rows in the flower bed. Formula: The number of terms (n) in an arithmetic progression can be found using the formula: l = a + (n - 1)d where: l = last term a = first term n = number of terms d = common difference Solution: This is an arithmetic progression with  first term (a) = 23,  last term (l) = 5, and  common difference (d) = -2. Using the formula l = a + (n - 1)d we have 5 = 23 + (n - 1)(-2) ⇒ 5 = 23 - 2n + 2 Group...