Appendix 1: Proofs in Mathematics — Flashcards
Madhya Pradesh Board · Class 12 · Mathematics
48 flashcards for Appendix 1: Proofs in Mathematics (Madhya Pradesh Board Class 12 Mathematics) to test yourself on key terms and facts.
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Solve by direct proof: If x^2 - 5x + 6 = 0, then find x.
Answer
Step 1: Factorise the quadratic. x^2 - 5x + 6 = x^2 - 3x - 2x + 6 = x(x - 3) - 2(x - 3) = (x - 3)(x - 2) Step 2: Use zero-product property. (x - 3)(x - 2) = 0 So, x - 3 = 0 or x - 2 = 0 Step 3: So…
प्रत्यक्ष प्रमाण द्वारा हल करें: यदि x^2 - 5x + 6 = 0, तो x ज्ञात कीजिए।
Answer
चरण 1: द्विघात का गुणनखंड कीजिए। x^2 - 5x + 6 = x^2 - 3x - 2x + 6 = x(x - 3) - 2(x - 3) = (x - 3)(x - 2) चरण 2: शून्य-गुणनफल गुणधर्म का उपयोग कीजिए। (x - 3)(x - 2) = 0 तो, x - 3 = 0 या x - 2 = 0 च…
प्रमाण में x^2 - 5x + 6 को (x - 3)(x - 2) से बदलना कैसे मदद करता है?
Answer
क्योंकि समान व्यंजकों को बिना सत्य बदले एक-दूसरे के स्थान पर रखा जा सकता है। द्विघात का गुणनखंड इस प्रकार किया जाता है x^2 - 5x + 6 = x^2 - 3x - 2x + 6 = x(x - 3) - 2(x - 3) = (x - 3)(x - 2). इसके बाद…
Why does replacing x^2 - 5x + 6 by (x - 3)(x - 2) help in a proof?
Answer
Because equal expressions can be substituted for each other without changing truth. The quadratic is factorised as x^2 - 5x + 6 = x^2 - 3x - 2x + 6 = x(x - 3) - 2(x - 3) = (x - 3)(x - 2). After this, …
Show that f(x) = 2x + 5 is one-one using the definition of one-one function.
Answer
Step 1: Start with the definition. A function is one-one if f(x1) = f(x2) implies x1 = x2. Step 2: Assume f(x1) = f(x2). 2x1 + 5 = 2x2 + 5 Step 3: Subtract 5 from both sides. 2x1 = 2x2 Step 4: Divide …
परिभाषा का उपयोग करके दिखाइए कि f(x) = 2x + 5 एक-एक है।
Answer
चरण 1: परिभाषा से शुरू कीजिए। यदि f(x1) = f(x2) होने पर x1 = x2 हो, तो फलन एक-एक होता है। चरण 2: मान लीजिए f(x1) = f(x2). 2x1 + 5 = 2x2 + 5 चरण 3: दोनों पक्षों से 5 घटाइए। 2x1 = 2x2 चरण 4: दोनों पक्षो…
एक-एक फलन परीक्षण f(x1) = f(x2) => x1 = x2 का उपयोग कब करते हैं?
Answer
इसका उपयोग तब करते हैं जब यह सिद्ध करना हो कि कोई फलन एक-एक है। इस परीक्षण का अर्थ है कि समान आउटपुट समान इनपुट से ही आते हैं। f(x) = 2x + 5 के लिए: यदि f(x1) = f(x2), तो 2x1 + 5 = 2x2 + 5. 5 घटाइए: 2…
When do you use the one-one function test f(x1) = f(x2) => x1 = x2?
Answer
Use it when proving that a function is one-one. The test means equal outputs must come from equal inputs. For f(x) = 2x + 5: If f(x1) = f(x2), then 2x1 + 5 = 2x2 + 5. Subtract 5: 2x1 = 2x2. Divide by …
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