Kaliyuga Ekalavya

NCERT Class X Chapter 7: Coordinate Geometry Example 7 Question: In what ratio does the point (–4, 6) divide the line segment joining the points A(–6, 10) and B(3, –8)? Given: A = (–6, 10), B = (3, –8), and P = (–4, 6) divides AB. To Find: The ratio in which the point P divides the line segment AB. Formula: Section formula: If a point \( P(x, y) \) divides the line segment joining \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio \( m:n \), then \[ x = \frac{mx_2 + nx_1}{m + n}, \quad y = \frac{my_2 + ny_1}{m + n} \] Solution: Step 1: Let the required ratio be \( m:n \). Using the section formula for the x-coordinate: \[ -4 = \frac{m \times 3 + n \times (-6)}{m + n} \] Step 2: Cross multiply and simplify the equation for x-coordinate. \[ -4(m + n) = 3m - 6n \] \[ -4m - 4n = 3m - 6n \] \[ -4m - 4n - 3m + 6n = 0 \] \[ -7m + 2n = 0 \] \[ 2n = 7m \] Step 3: Express the ratio \( \frac{m}{n} \) from the above equation. \[ 2n = 7m \implies \fr...

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Example 7 Question: Two rails are represented by the equations x + 2y – 4 = 0 and 2x + 4y – 12 = 0. Will the rails cross each other? Given: Equation of rail 1: x + 2y – 4 = 0 Equation of rail 2: 2x + 4y – 12 = 0 To Find: Whether the two rails cross each other. Formula: If two lines are represented by the equations a 1 x + b 1 y + c 1 = 0 and a 2 x + b 2 y + c 2 = 0, then the lines are parallel if a 1 a 2 = b 1 b 2 . Solution: Let the given equations be x + 2y – 4 = 0 .....(1) 2x + 4y – 12 = 0 .....(2) From equation (1), we have x = 4 - 2y Substituting this in equation (2): 2(4 - 2y) + 4y – 12 = 0 ⇒ 8 – 4y + 4y – 12 = 0 ⇒ –4 = 0 This is a contradiction. Alternatively, we can check if the lines are parallel: For equation (1): a 1 = 1, b 1 = 2 For equation (2): a 2 = 2, b 2 = 4 a 1 a 2 = 1 2 = 1 2 b 1 b 2 = 2 4 = 1 2 Since a 1 a 2 = b 1 b 2 , the lines are parallel...

NCERT Class X Chapter 9: Some Applications of Trigonometry Example 7 Question: From a point on a bridge across a river, the angles of depression of the banks on opposite sides of the river are 30° and 45°, respectively. If the bridge is at a height of 3 m from the banks, find the width of the river. Given: Angles of depression to the two banks: \(30^\circ\) and \(45^\circ\) Height of the bridge above the banks: \(3\,\text{m}\) To Find: The width of the river. Formula: \[ \tan \theta = \frac{\text{Opposite side}}{\text{Adjacent side}} \] Solution: Step 1: Let the points where the lines of sight from the bridge meet the two banks be \(A\) and \(B\). Let \(P\) be the point on the bridge directly above the river at height \(3\,\text{m}\). Let the distances from \(P\) vertically down to \(A\) and \(B\) along the river banks be \(x\) and \(y\) respectively. The total width of the river is \(x + y\). Step 2: For the bank with angle of depression \(45^\circ\)...

NCERT Class X Chapter 1: Real Numbers Example 7 Question: Show that \(3\sqrt{2}\) is irrational. Given: The number \(3\sqrt{2}\). To Find: To prove that \(3\sqrt{2}\) is irrational. Formula: We use proof by contradiction and the property that the product of a non-zero rational number and an irrational number is irrational. Solution: Step 1: Assume, to the contrary, that \(3\sqrt{2}\) is rational. $$ 3\sqrt{2} = \frac{a}{b} $$ where \(a\) and \(b\) are coprime integers, \(b \neq 0\). Step 2: Divide both sides by 3 to isolate \(\sqrt{2}\). $$ \sqrt{2} = \frac{a}{3b} $$ Step 3: Note that \(\frac{a}{3b}\) is a rational number, since \(a\) and \(b\) are integers and \(3b \neq 0\). Step 4: But it is a well-known fact that \(\sqrt{2}\) is irrational. Step 5: This leads to a contradiction, since \(\sqrt{2}\) cannot be rational. Step 6: Therefore, our assumption i...

NCERT Class X Chapter 14: Probability Example 7 Question: There are 40 students in Class X of a school of whom 25 are girls and 15 are boys. The class teacher has to select one student as a class representative. She writes the name of each student on a separate card, the cards being identical. Then she puts cards in a bag and stirs them thoroughly. She then draws one card from the bag. What is the probability that the name written on the card is the name of (i) a girl? (ii) a boy? Given: Total number of students = 40 Number of girls = 25 Number of boys = 15 To Find: (i) Probability of selecting a girl (ii) Probability of selecting a boy Formula: Probability = Number of favorable outcomes Total number of possible outcomes Solution: (i) Probability of selecting a girl: Number of favorable outcomes (girls) = 25 Total number of possible outcomes (students) = 40 Probability (girl) = 25 40 = 5 ...

NCERT Class X Chapter 5: Arithmetic Progression Example 7 Question: How many two-digit numbers are divisible by 3?. Given: Two-digit numbers. To Find: The number of two-digit numbers divisible by 3. Formula: The number of multiples of 3 between a and b is given by \( \frac{m}{3} - \frac{n}{3} + 1 \). Where, m and n are numbers between a and b which are divisible by 3.  \( m \leq b \) and \( n \geq a \) Solution: The smallest two-digit number is 10. The largest two-digit number is 99. So, a = 10 and b = 99 We need to find the number of multiples of 3 between 10 and 99. The smallest multiple of 3 greater than or equal to 10 is 12 (3 x 4).  The largest multiple of 3 less than or equal to 99 is 99 (3 x 33). Therefore, m = 99 and n = 12 The number of multiples is \( \frac{99}{3} - \frac{12}{3} + 1 \) = 33 - 4 + 1 = 30. Result: There are 30 two-digit numbers divisible by 3. Next question solution: NCERT Class X Chapter 5...

NCERT Class X Chapter 4: Quadratic Equation Example 7 Question: Find the discriminant of the quadratic equation 2x 2 – 4x + 3 = 0, and hence find the nature of its roots. Given: The quadratic equation: 2x 2 – 4x + 3 = 0 To Find: 1. The discriminant of the equation. 2. The nature of its roots. 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). Solution: The given quadratic equation is 2x 2 – 4x + 3 = 0. Comparing it with the standard form ax 2 + bx + c = 0, we have: a = 2 b = -4 c = 3 Now, calculate the discriminant (Δ): ⇒ Δ = b 2 - 4ac ⇒ Δ = (-4)...