This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Fresh day BILIJITIN, let's solve. Q.9 Step 1: Write down the given system of linear equations. Equation (1): 2x + 3y = 12 Equation (2): 3x - 2y = 4 Step 2: Choose a variable to eliminate and multiply the equations to make their coefficients equal. To eliminate y, multiply Equation (1) by 2 and Equation (2) by 3: (2x + 3y = 12) × 2 4x + 6y = 24 (3) (3x - 2y = 4) × 3 9x - 6y = 12 (4) Step 3: Add the new equations to eliminate the chosen variable. Add Equation (3) and Equation (4): (4x + 6y) + (9x - 6y) = 24 + 12 4x + 9x + 6y - 6y = 36 13x = 36 Step 4: Solve for the remaining variable. x = (36)/(13) Step 5: Substitute the value of x into one of the original equations to find y. Using Equation (1): 2x + 3y = 12 2((36)/(13)) + 3y = 12 (72)/(13) + 3y = 12 3y = 12 - (72)/(13) 3y = (12 × 13 - 72)/(13) 3y = (156 - 72)/(13) 3y = (84)/(13) y = (84)/(13 × 3) y = (28)/(13) The final answer is x=(36)/(13), y=(28)/(13). Q.10 Step 1: Find the zeroes of the polynomial p(x) = x^2 - 5x + 6. Set p(x) = 0: x^2 - 5x + 6 = 0 Factor the quadratic equation: (x-2)(x-3) = 0 The zeroes are x=2 and x=3. Let = 2 and = 3. Step 2: Verify the relation between zeroes and coefficients. For a quadratic polynomial ax^2 + bx + c, the sum of zeroes is -(b)/(a) and the product of zeroes is (c)/(a). For p(x) = x^2 - 5x + 6, we have a=1, b=-5, c=6. Sum of zeroes: + = 2 + 3 = 5 -(b)/(a) = -((-5))/(1) = 5 The sum of zeroes matches. Product of zeroes: = 2 × 3 = 6 (c)/(a) = (6)/(1) = 6 The product of zeroes matches. The final answer is Zeroes are 2, 3. Verified relations: sum 5 = -((-5))/(1), product 6 = (6)/(1). Q.11 Step 1: Rewrite the linear equation 2x - y + 1 = 0 in slope-intercept form (y = mx + c). y = 2x + 1 Step 2: Find two points that satisfy the equation. If x=0: y = 2(0) + 1 = 1. So, point A is (0, 1). If x=1: y = 2(1) + 1 = 3. So, point B is (1, 3). If x=-1: y = 2(-1) + 1 = -1. So, point C is (-1, -1). Step 3: Explain how to draw the graph. Plot the points (0, 1) and (1, 3) (or any other two points you found) on a coordinate plane. Draw a straight line passing through these two points. This line represents the graph of the equation 2x - y + 1 = 0. The final answer is The graph is a line passing through points like (0,1) and (1,3). Q.12 Step 1: Identify the given information for PQR. The triangle is right-angled at Q. Hypotenuse PR = 13 cm. Leg PQ = 5 cm. Step 2: Find the length of the other leg, QR, using the Pythagorean theorem. PR^2 = PQ^2 + QR^2 (13 cm)^2 = (5 cm)^2 + QR^2 169 cm^2 = 25 cm^2 + QR^2 QR^2 = 169 cm^2 - 25 cm^2 QR^2 = 144 cm^2 QR = sqrt(144 cm)^2 QR = 12 cm Step 3: Find P. In a right-angled triangle, (angle) = Opposite sideHypotenuse. For angle P, the opposite side is QR and the hypotenuse is PR. P = (QR)/(PR) P = 12 cm13 cm P = (12)/(13) The final answer is QR=12 cm, P=(12)/(13). Q.13 Step 1: Assume, for the sake of contradiction, that sqrt(3) is a rational number. If sqrt(3) is rational, it can be expressed as a fraction (a)/(b), where a and b are integers, b 0, and a and b are coprime (their greatest common divisor is 1). sqrt(3) = (a)/(b) Step 2: Square both sides of the equation. (sqrt(3))^2 = ((a)/(b))^2 3 = (a^2)/(b^2) 3b^2 = a^2 (*) Step 3: Deduce that a^2 is divisible by 3, which implies a is divisible by 3. From equation (*), a^2 is a multiple of 3. If a^2 is divisible by 3, then a must also be divisible by 3 (a property of prime numbers). So, we can write a = 3c for some integer c. Step 4: Substitute a=3c back into equation (*). 3b^2 = (3c)^2 3b^2 = 9c^2 Divide both sides by 3: b^2 = 3c^2 Step 5: Deduce that b^2 is divisible by 3, which implies b is divisible by 3. From this equation, b^2 is a multiple of 3. Therefore, b must also be divisible by 3. Step 6: Conclude the contradiction. We have shown that both a and b are divisible by 3. This contradicts our initial assumption that a and b are coprime. Since our initial assumption leads to a contradiction, it must be false. Therefore, sqrt(3) is an irrational number. The final answer is Proof by contradiction: Assume sqrt(3) is rational, leading to a and b both being divisible by 3, contradicting their coprimality. Thus, sqrt(3) is irrational.. Q.14 Step 1: Create a table to calculate class marks (x_i), frequencies (f_i), and the product f_i x_i. | Class Interval | Frequency (f_i) | Class Mark (x_i) | f_i x_i | |:--------------:|:-------------------:|:------------------:|:---------:| | 0 - 10 | 5 | 5 | 25 | | 10 - 20 | 8 | 15 | 120 | | 20 - 30 | 12 | 25 | 300 | | 30 - 40 | 10 | 35 | 350 | | 40 - 50 | 5 | 45 | 225 | Step 2: Calculate the sum of frequencies ( f_i) and the sum of f_i x_i ( f_i x_i). f_i = 5 + 8 + 12 + 10 + 5 = 40 f_i x_i = 25 + 120 + 300 + 350 + 225 = 1020 Step 3: Calculate the mean using the formula Mean = ( f_i x_i)/( f_i). Mean = (1020)/(40) Mean = 25.5 The final answer is Mean=25.5. Q.15 Step 1: Label the given points. Let A = (-1, 8), B = (0, 5), and C = (2, -1). Step 2: Calculate the slope of the line segment AB. The slope m_AB between two points (x_1, y_1) and (x_2, y_2) is given by m = (y_2 - y_1)/(x_2 - x_1). m_AB = (5 - 8)/(0 - (-1)) = (-3)/(0 + 1) = (-3)/(1) = -3 Step 3: Calculate the slope of the line segment BC. m_BC = (-1 - 5)/(2 - 0) = (-6)/(2) = -3 Step 4: Compare the slopes. Since m_AB = m_BC = -3, the slopes of the line segments AB and BC are equal. This means that points A, B, and C lie on the same straight line, i.e., they are collinear. The final answer is The slope of AB is -3 and the slope of BC is -3. Since the slopes are equal, the points are collinear.. Got more? Send 'em