Kaliyuga Ekalavya

NCERT Class X Chapter 7: Coordinate Geometry Example 9 Question: Find the ratio in which the y-axis divides the line segment joining the points (5, –6) and (–1, –4). Also find the point of intersection. Given: Points \( A(5, -6) \) and \( B(-1, -4) \). To Find: The ratio in which the y-axis divides the line segment AB. The coordinates of the point of intersection. Formula: Section formula: The coordinates of the point dividing the line segment joining \( (x_1, y_1) \) and \( (x_2, y_2) \) in the ratio \( m:n \) are: $$ \left( \frac{mx_2 + nx_1}{m+n},\ \frac{my_2 + ny_1}{m+n} \right) $$ Solution: Step 1: Let the y-axis divide the line segment AB at point P in the ratio \( k:1 \). Step 2: The coordinates of P using the section formula are: $$ P = \left( \frac{k \cdot (-1) + 1 \cdot 5}{k+1},\ \frac{k \cdot (-4) + 1 \cdot (-6)}{k+1} \right) $$ Step 3: Since P lies on the y-axis, its x-coordinate is 0. $$ \frac{k \cdot (-1) + 1 \cdot 5}{k+1}...

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Example 9 Question: Use elimination method to find all possible solutions of the following pair of linear equations: \( 2x + 3y = 8 \) \( 4x + 6y = 7 \) Given: Equation (1): \( 2x + 3y = 8 \) Equation (2): \( 4x + 6y = 7 \) To Find: All possible solutions of the given pair of linear equations using elimination method. Formula: The elimination method involves manipulating the equations to eliminate one variable, allowing us to solve for the other variable. Solution: Step 1: Write the given equations. $$ \begin{aligned} &\text{Equation (1):} \quad 2x + 3y = 8 \\ &\text{Equation (2):} \quad 4x + 6y = 7 \end{aligned} $$ Step 2: Multiply equation (1) by 2 to make the coefficients of \(x\) the same in both equations. $$ 2 \times (2x + 3y) = 2 \times 8 \\ \Rightarrow 4x + 6y = 16 \quad \text{(Equation 3)} $$ Step 3: Subtract equation (2) from equation (3) to eliminate \(x\) ...

NCERT Class X Chapter 14: Probability Example 9 Question: Harpreet tosses two different coins simultaneously (say, one is of Rs 1 and other of Rs 2). What is the probability that she gets at least one head? Given: Two different coins are tossed simultaneously. To Find: The probability of getting at least one head. Formula: Probability = Favorable Outcomes Total Outcomes Solution: Possible outcomes when tossing two coins are: HH, HT, TH, TT Total number of outcomes = 4 Favorable outcomes (at least one head) are: HH, HT, TH Number of favorable outcomes = 3 Probability (at least one head) = 3 4 Result: The probability that Harpreet gets at least one head is 3 4 . Next question solution: NCERT Class X Chapter 14: Probability Example 10 Explore more in Probability: Click this link to explore more NCERT Class X Chapter 14 Probability solutions Explore more: Click this link to explore more NCERT Class X chapter solutions ...

NCERT Class X Chapter 5: Arithmetic Progression Example 9 Question: A sum of Rs 1000 is invested at 8% simple interest per year. Calculate the interest at the end of each year. Do these interests form an AP? If so, find the interest at the end of 30 years making use of this fact. Given: Principal (P) = Rs 1000 Rate of interest (R) = 8% per year To Find: Interest at the end of each year.  Whether the interests form an AP.  Interest at the end of 30 years. Formula: Simple Interest (SI) = P × R × T 100  where, P is principal,  R is rate, and  T is time in years. Solution: Interest at the end of 1st year (T=1) = 1000 × 8 × 1 100 = Rs 80 Interest at the end of 2nd year (T=2) = 1000 × 8 × 2 100 = Rs 160 Interest at the end of 3rd year (T=3) = 1000 × 8 × 3 100 = Rs 240  The interest after at the end of each year is 80, 160, 240, .... Let a 1 = 80, a 2 = 160, a 3 = 240 d = a 2 - a 1...

NCERT Class X Chapter 4: Quadratic Equation Example 9 Question: Find the discriminant of the equation 3x 2 – 2x + (1/3) = 0 and hence find the nature of its roots. Find them, if they are real.. Given: The quadratic equation: 3x 2 – 2x + 1 3 = 0 To Find: 1. The discriminant of the equation. 2. The nature of its roots. 3. The roots, if they are real. Formula: WKT, for a quadratic equation of the form ax 2 + bx + c = 0, the discriminant (Δ) is given by: Δ = b 2 - 4ac WKT, the nature of roots is determined by the value of the discriminant (Δ): • If Δ > 0, there are two distinct real roots. • If Δ = 0, there are two equal real roots. • If Δ < 0, there are no real roots (or two distinct complex roots). WKT, if the roots are real, they can be found using the quadratic formula: x = -b ± √Δ 2a Solution: The given quadratic equation is 3x 2 – 2x ...