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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here are the solutions to the questions: (iv) If the diagonals of a parallelogram are equal and perpendicular to each other, then quadrilateral will be. If the diagonals of a parallelogram are equal, it is a rectangle. If they are perpendicular, it is a rhombus. A quadrilateral that is both a rectangle and a rhombus is a square. (v) Write the name of longest chord of a circle. The longest chord of a circle is its diameter. (vi) Write the Heron's formula to find the area of a triangle. Heron's formula for the area of a triangle with side lengths a, b, c is: Area = sqrt(s(s-a)(s-b)(s-c)) where s is the semi-perimeter, given by: s = (a+b+c)/(2) प्र.6 सरल कीजिए - (Simplify -) 7^(1)/(2) · 8^(1)/(2) Step 1: Use the exponent rule a^m · b^m = (ab)^m. 7^(1)/(2) · 8^(1)/(2) = (7 · 8)^(1)/(2) Step 2: Multiply the bases. (7 · 8)^(1)/(2) = 56^(1)/(2) Step 3: Convert the fractional exponent to a square root and simplify. 56^(1)/(2) = sqrt(56) = sqrt(4 · 14) = 2sqrt(14) The simplified expression is 2sqrt(14). अथवा / OR Multiply 6sqrt(5) and 2sqrt(5). Step 1: Group the integer parts and the radical parts. (6sqrt(5)) · (2sqrt(5)) = (6 · 2) · (sqrt(5) · sqrt(5)) Step 2: Perform the multiplications. (6 · 2) · (sqrt(5) · sqrt(5)) = 12 · 5 Step 3: Calculate the final product. 12 · 5 = 60 The product is 60. प्र.7 (1)/(sqrt(7)) का हर का परिमेयकरण कीजिए। (Rationalise the denominator of (1)/(sqrt(7)).) Step 1: To rationalise the denominator, multiply both the numerator and the denominator by sqrt(7). (1)/(sqrt(7)) = (1)/(sqrt(7)) × sqrt(7)sqrt(7) Step 2: Perform the multiplication. 1 × sqrt(7)sqrt(7) × sqrt(7) = sqrt(7)7 The rationalised expression is sqrt(7)7. अथवा / OR (3)/(5) और (4)/(5) के बीच दो परिमेय संख्या ज्ञात कीजिए। (Find two rational numbers between (3)/(5) and (4)/(5).) Step 1: To find rational numbers between two fractions, express them with a larger common denominator. Multiply both fractions by (10)/(10). (3)/(5) = (3 × 10)/(5 × 10) = (30)/(50) (4)/(5) = (4 × 10)/(5 × 10) = (40)/(50) Step 2: Identify two rational numbers between (30)/(50) and (40)/(50). Two such numbers could be (31)/(50) and (32)/(50). Two rational numbers between (3)/(5) and (4)/(5) are (31)/(50), (32)/(50). (Other valid answers exist, e.g., (33)/(50), (34)/(50), etc.) प्र.8 गुणनखंड ज्ञात कीजिए - (Factorise -) 12x^2 - 7x + 1 Step 1: Find two numbers that multiply to 12 × 1 = 12 and add up to -7. These numbers are -3 and -4. Step 2: Rewrite the middle term using these numbers. 12x^2 - 3x - 4x + 1 Step 3: Group the terms and factor out common factors from each group. (12x^2 - 3x) - (4x - 1) 3x(4x - 1) - 1(4x - 1) Step 4: Factor out the common binomial factor (4x - 1). (4x - 1)(3x - 1) The factorised expression is (4x - 1)(3x - 1). अथवा / OR सीधे गुणा किये बिना 104 × 96 का गुणनफल ज्ञात कीजिए। (Evaluate the product of 104 × 96 without multiplying directly.) Step 1: Express the numbers as a sum and difference from a common base (100). 104 = 100 + 4 96 = 100 - 4 Step 2: Use the algebraic identity (a+b)(a-b) = a^2 - b^2. Here, a=100 and b=4. 104 × 96 = (100 + 4)(100 - 4) = 100^2 - 4^2 Step 3: Calculate the squares and subtract. 100^2 - 4^2 = 10000 - 16 = 9984 The product is 9984. प्र.9 SSS सर्वांगसमता नियम लिखिए। (Write SSS Congruence rule.) SSS (Side-Side-Side) Congruence Rule states that if three sides of one triangle are equal to the three corresponding sides of another triangle, then the two triangles are congruent. Drop the next question.