\(\alpha+\beta=\frac{5}{2}\) and \(\alpha\beta=-\frac{3}{2}\). (\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\(\alpha+\beta\)2-2\alpha\beta}{\alpha\beta}=-\frac{37}{6}).
Step 2
Why this answer is correct
The correct answer is A. -\(\frac{37}{6}\). \(\alpha+\beta=\frac{5}{2}\) and \(\alpha\beta=-\frac{3}{2}\). (\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\(\alpha+\beta\)2-2\alpha\beta}{\alpha\beta}=-\frac{37}{6}).
Step 3
Exam Tip
\(\alpha+\beta=\frac{5}{2}\) और \(\alpha\beta=-\frac{3}{2}\) हैं। (\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\(\alpha+\beta\)2-2\alpha\beta}{\alpha\beta}=-\frac{37}{6})।
The new sum is \(\alpha+\beta+2=8\) and product is \(\alpha\beta+\alpha+\beta+1=15\). Thus the polynomial is \(x^2-8x+15\).
Step 2
Why this answer is correct
The correct answer is A. \(x^2-8x+15\). The new sum is \(\alpha+\beta+2=8\) and product is \(\alpha\beta+\alpha+\beta+1=15\). Thus the polynomial is \(x^2-8x+15\).
Step 3
Exam Tip
नए शून्यकों का योग \(\alpha+\beta+2=8\) और गुणनफल \(\alpha\beta+\alpha+\beta+1=15\) है। अतः बहुपद \(x^2-8x+15\) है।
(\alpha-2\beta+\alpha\beta-2=\alpha\beta\(\alpha+\beta\)). Here \(\alpha\beta=-10\) and \(\alpha+\beta=3\), so the value is (-30).
Step 2
Why this answer is correct
The correct answer is A. (-30). (\alpha-2\beta+\alpha\beta-2=\alpha\beta\(\alpha+\beta\)). Here \(\alpha\beta=-10\) and \(\alpha+\beta=3\), so the value is (-30).
Step 3
Exam Tip
(\alpha-2\beta+\alpha\beta-2=\alpha\beta\(\alpha+\beta\)) होता है। यहां \(\alpha\beta=-10\) और \(\alpha+\beta=3\), इसलिए मान (-30) है।
\(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha\beta}\), where \(\alpha+\beta=26\) and \(\alpha\beta=165\), so the value is \(\frac{676-330}{165}=\frac{346}{165}\). In exams, convert expressions into sum and product.
Step 2
Why this answer is correct
The correct answer is A. \( \frac{346}{165}\). \(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha\beta}\), where \(\alpha+\beta=26\) and \(\alpha\beta=165\), so the value is \(\frac{676-330}{165}=\frac{346}{165}\). In exams, convert expressions into sum and product.
Step 3
Exam Tip
\(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha\beta}\), जहां \(\alpha+\beta=26\) और \(\alpha\beta=165\), इसलिए मान \(\frac{676-330}{165}=\frac{346}{165}\) है। परीक्षा में अभिव्यक्ति को योग और गुणनफल में बदलें।
(\alpha-2\beta+\alpha\beta-2=\alpha\beta\(\alpha+\beta\)), where \(\alpha+\beta=12\) and \(\alpha\beta=-28\), so the value is (-336). In exams, factor the expression first.
Step 2
Why this answer is correct
The correct answer is A. (-336). (\alpha-2\beta+\alpha\beta-2=\alpha\beta\(\alpha+\beta\)), where \(\alpha+\beta=12\) and \(\alpha\beta=-28\), so the value is (-336). In exams, factor the expression first.
Step 3
Exam Tip
(\alpha-2\beta+\alpha\beta-2=\alpha\beta\(\alpha+\beta\)), जहां \(\alpha+\beta=12\) और \(\alpha\beta=-28\), इसलिए मान (-336) है। परीक्षा में अभिव्यक्ति को पहले factor करें।
\(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha\beta}\), where \(\alpha+\beta=22\) and \(\alpha\beta=117\), so the value is \(\frac{484-234}{117}=\frac{250}{117}\). In exams, convert expressions into sum and product.
Step 2
Why this answer is correct
The correct answer is A. \( \frac{250}{117}\). \(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha\beta}\), where \(\alpha+\beta=22\) and \(\alpha\beta=117\), so the value is \(\frac{484-234}{117}=\frac{250}{117}\). In exams, convert expressions into sum and product.
Step 3
Exam Tip
\(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha\beta}\), जहां \(\alpha+\beta=22\) और \(\alpha\beta=117\), इसलिए मान \(\frac{484-234}{117}=\frac{250}{117}\) है। परीक्षा में अभिव्यक्ति को योग और गुणनफल में बदलें।
(\alpha-2\beta+\alpha\beta-2=\alpha\beta\(\alpha+\beta\)), where \(\alpha+\beta=10\) and \(\alpha\beta=-24\), so the value is (-240). In exams, factor the expression first.
Step 2
Why this answer is correct
The correct answer is A. (-240). (\alpha-2\beta+\alpha\beta-2=\alpha\beta\(\alpha+\beta\)), where \(\alpha+\beta=10\) and \(\alpha\beta=-24\), so the value is (-240). In exams, factor the expression first.
Step 3
Exam Tip
(\alpha-2\beta+\alpha\beta-2=\alpha\beta\(\alpha+\beta\)), जहां \(\alpha+\beta=10\) और \(\alpha\beta=-24\), इसलिए मान (-240) है। परीक्षा में अभिव्यक्ति को पहले factor करें।
\(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha\beta}\), where \(\alpha+\beta=18\) and \(\alpha\beta=77\), so the value is \(\frac{324-154}{77}=\frac{170}{77}\). In exams, convert expressions into sum and product.
Step 2
Why this answer is correct
The correct answer is A. \( \frac{170}{77}\). \(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha\beta}\), where \(\alpha+\beta=18\) and \(\alpha\beta=77\), so the value is \(\frac{324-154}{77}=\frac{170}{77}\). In exams, convert expressions into sum and product.
Step 3
Exam Tip
\(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha\beta}\), जहां \(\alpha+\beta=18\) और \(\alpha\beta=77\), इसलिए मान \(\frac{324-154}{77}=\frac{170}{77}\) है। परीक्षा में अभिव्यक्ति को योग और गुणनफल में बदलें।
(\alpha-2\beta+\alpha\beta-2=\alpha\beta\(\alpha+\beta\)), where \(\alpha+\beta=8\) and \(\alpha\beta=-20\), so the value is (-160). In exams, factor the expression first.
Step 2
Why this answer is correct
The correct answer is A. (-160). (\alpha-2\beta+\alpha\beta-2=\alpha\beta\(\alpha+\beta\)), where \(\alpha+\beta=8\) and \(\alpha\beta=-20\), so the value is (-160). In exams, factor the expression first.
Step 3
Exam Tip
(\alpha-2\beta+\alpha\beta-2=\alpha\beta\(\alpha+\beta\)), जहां \(\alpha+\beta=8\) और \(\alpha\beta=-20\), इसलिए मान (-160) है। परीक्षा में अभिव्यक्ति को पहले factor करें।
\(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha\beta}\), where \(\alpha+\beta=14\) and \(\alpha\beta=45\), so the value is \(\frac{196-90}{45}=\frac{106}{45}\). In exams, convert expressions into sum and product.
Step 2
Why this answer is correct
The correct answer is A. \( \frac{106}{45}\). \(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha\beta}\), where \(\alpha+\beta=14\) and \(\alpha\beta=45\), so the value is \(\frac{196-90}{45}=\frac{106}{45}\). In exams, convert expressions into sum and product.
Step 3
Exam Tip
\(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha\beta}\), जहां \(\alpha+\beta=14\) और \(\alpha\beta=45\), इसलिए मान \(\frac{196-90}{45}=\frac{106}{45}\) है। परीक्षा में अभिव्यक्ति को योग और गुणनफल में बदलें।
(\alpha-2\beta+\alpha\beta-2=\alpha\beta\(\alpha+\beta\)), where \(\alpha+\beta=6\) and \(\alpha\beta=-16\), so the value is (-96). In exams, factor the expression first.
Step 2
Why this answer is correct
The correct answer is A. (-96). (\alpha-2\beta+\alpha\beta-2=\alpha\beta\(\alpha+\beta\)), where \(\alpha+\beta=6\) and \(\alpha\beta=-16\), so the value is (-96). In exams, factor the expression first.
Step 3
Exam Tip
(\alpha-2\beta+\alpha\beta-2=\alpha\beta\(\alpha+\beta\)), जहां \(\alpha+\beta=6\) और \(\alpha\beta=-16\), इसलिए मान (-96) है। परीक्षा में अभिव्यक्ति को पहले factor करें।
\(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha\beta}\), where \(\alpha+\beta=10\) and \(\alpha\beta=21\), so the value is \(\frac{100-42}{21}=\frac{58}{21}\). In exams, convert expressions into sum and product.
Step 2
Why this answer is correct
The correct answer is A. \( \frac{58}{21}\). \(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha\beta}\), where \(\alpha+\beta=10\) and \(\alpha\beta=21\), so the value is \(\frac{100-42}{21}=\frac{58}{21}\). In exams, convert expressions into sum and product.
Step 3
Exam Tip
\(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha\beta}\), जहां \(\alpha+\beta=10\) और \(\alpha\beta=21\), इसलिए मान \(\frac{100-42}{21}=\frac{58}{21}\) है। परीक्षा में अभिव्यक्ति को योग और गुणनफल में बदलें।
(\alpha-2\beta+\alpha\beta-2=\alpha\beta\(\alpha+\beta\)), where \(\alpha+\beta=4\) and \(\alpha\beta=-12\), so the value is (-48). In exams, factor the expression first.
Step 2
Why this answer is correct
The correct answer is A. (-48). (\alpha-2\beta+\alpha\beta-2=\alpha\beta\(\alpha+\beta\)), where \(\alpha+\beta=4\) and \(\alpha\beta=-12\), so the value is (-48). In exams, factor the expression first.
Step 3
Exam Tip
(\alpha-2\beta+\alpha\beta-2=\alpha\beta\(\alpha+\beta\)), जहां \(\alpha+\beta=4\) और \(\alpha\beta=-12\), इसलिए मान (-48) है। परीक्षा में अभिव्यक्ति को पहले factor करें।
\(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha\beta}\), where \(\alpha+\beta=8\) and \(\alpha\beta=15\), so the value is \(\frac{64-30}{15}=\frac{34}{15}\). In exams, convert expressions into sum and product.
Step 2
Why this answer is correct
The correct answer is A. \( \frac{34}{15}\). \(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha\beta}\), where \(\alpha+\beta=8\) and \(\alpha\beta=15\), so the value is \(\frac{64-30}{15}=\frac{34}{15}\). In exams, convert expressions into sum and product.
Step 3
Exam Tip
\(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha\beta}\), जहां \(\alpha+\beta=8\) और \(\alpha\beta=15\), इसलिए मान \(\frac{64-30}{15}=\frac{34}{15}\) है। परीक्षा में अभिव्यक्ति को योग और गुणनफल में बदलें।
(\alpha-2\beta+\alpha\beta-2=\alpha\beta\(\alpha+\beta\)), where \(\alpha+\beta=2\) and \(\alpha\beta=-8\), so the value is (-16). In exams, factor the expression first.
Step 2
Why this answer is correct
The correct answer is A. (-16). (\alpha-2\beta+\alpha\beta-2=\alpha\beta\(\alpha+\beta\)), where \(\alpha+\beta=2\) and \(\alpha\beta=-8\), so the value is (-16). In exams, factor the expression first.
Step 3
Exam Tip
(\alpha-2\beta+\alpha\beta-2=\alpha\beta\(\alpha+\beta\)), जहां \(\alpha+\beta=2\) और \(\alpha\beta=-8\), इसलिए मान (-16) है। परीक्षा में अभिव्यक्ति को पहले factor करें।
We know (\(\alpha-\beta\)2=\(\alpha+\beta\)2-4\alpha\beta). Hence (\(\alpha-\beta\)2+4\alpha\beta=\(\alpha+\beta\)2=100).
Step 2
Why this answer is correct
The correct answer is A. (100). We know (\(\alpha-\beta\)2=\(\alpha+\beta\)2-4\alpha\beta). Hence (\(\alpha-\beta\)2+4\alpha\beta=\(\alpha+\beta\)2=100).
Step 3
Exam Tip
(\(\alpha-\beta\)2=\(\alpha+\beta\)2-4\alpha\beta) होता है। इसलिए (\(\alpha-\beta\)2+4\alpha\beta=\(\alpha+\beta\)2=100)।
Here \(\alpha+\beta=7\) and \(\alpha\beta=10\). Since \(\alpha^2+\beta^2=49-20=29\), the value is (29-6(7)=-13), so none of the options is correct.
Step 2
Why this answer is correct
The correct answer is B. (-11). Here \(\alpha+\beta=7\) and \(\alpha\beta=10\). Since \(\alpha^2+\beta^2=49-20=29\), the value is (29-6(7)=-13), so none of the options is correct.
Step 3
Exam Tip
\(\alpha+\beta=7\) और \(\alpha\beta=10\) है। \(\alpha^2+\beta^2=49-20=29\), इसलिए (29-6(7)=-13), अतः विकल्पों में कोई सही नहीं है।
We use \(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha\beta}\). Here \(\alpha^2+\beta^2=169-72=97\) and \(\alpha\beta=36\), so the value is \(\frac{97}{36}\).
Step 2
Why this answer is correct
The correct answer is A. \(\frac{97}{36}\). We use \(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha\beta}\). Here \(\alpha^2+\beta^2=169-72=97\) and \(\alpha\beta=36\), so the value is \(\frac{97}{36}\).
Step 3
Exam Tip
\(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha\beta}\) है। यहाँ \(\alpha^2+\beta^2=169-72=97\) और \(\alpha\beta=36\), इसलिए मान \(\frac{97}{36}\) है।
Since \(\alpha\) is a root, \(\alpha^2=5\alpha+3\), and \(\alpha\beta=-3\). The expression becomes (5\alpha+3+5\beta-3=5\(\alpha+\beta\)=25).
Step 2
Why this answer is correct
The correct answer is C. (25). Since \(\alpha\) is a root, \(\alpha^2=5\alpha+3\), and \(\alpha\beta=-3\). The expression becomes (5\alpha+3+5\beta-3=5\(\alpha+\beta\)=25).
Step 3
Exam Tip
क्योंकि \(\alpha\) जड़ है, \(\alpha^2=5\alpha+3\) और \(\alpha\beta=-3\) है। व्यंजक (5\alpha+3+5\beta-3=5\(\alpha+\beta\)=25) बनता है।
The roots are (4) and (5). Direct substitution gives \(\frac{6}{2}+\frac{7}{3}=\frac{16}{3}\), so option (A) should be correct.
Step 2
Why this answer is correct
The correct answer is B. \(\frac{17}{3}\). The roots are (4) and (5). Direct substitution gives \(\frac{6}{2}+\frac{7}{3}=\frac{16}{3}\), so option (A) should be correct.
Step 3
Exam Tip
जड़ें (4) और (5) हैं। सीधे रखने पर \(\frac{6}{2}+\frac{7}{3}=\frac{16}{3}\) मिलता है, इसलिए विकल्प (A) सही होना चाहिए।
Here \(\alpha+\beta=5\) and \(\alpha\beta=6\). The new roots are (5) and (6), so the equation is \(x^2-11x+30=0\).
Step 2
Why this answer is correct
The correct answer is A. \(x^2-11x+30=0\). Here \(\alpha+\beta=5\) and \(\alpha\beta=6\). The new roots are (5) and (6), so the equation is \(x^2-11x+30=0\).
Step 3
Exam Tip
\(\alpha+\beta=5\) और \(\alpha\beta=6\) हैं। नई जड़ें (5) और (6) हैं, इसलिए समीकरण \(x^2-11x+30=0\) है।
We use \(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha\beta}\). Here \(\alpha^2+\beta^2=73\) and \(\alpha\beta=24\), so the value is \(\frac{73}{24}\).
Step 2
Why this answer is correct
The correct answer is C. \(\frac{73}{24}\). We use \(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha\beta}\). Here \(\alpha^2+\beta^2=73\) and \(\alpha\beta=24\), so the value is \(\frac{73}{24}\).
Step 3
Exam Tip
\(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha\beta}\) है। यहाँ \(\alpha^2+\beta^2=73\) और \(\alpha\beta=24\), इसलिए मान \(\frac{73}{24}\) है।
Since \(\alpha\) is a root, \(\alpha^2=4\alpha+2\), and \(\alpha\beta=-2\). The expression becomes (4\alpha+2+4\beta-2=4\(\alpha+\beta\)=16).
Step 2
Why this answer is correct
The correct answer is C. (16). Since \(\alpha\) is a root, \(\alpha^2=4\alpha+2\), and \(\alpha\beta=-2\). The expression becomes (4\alpha+2+4\beta-2=4\(\alpha+\beta\)=16).
Step 3
Exam Tip
क्योंकि \(\alpha\) जड़ है, \(\alpha^2=4\alpha+2\) होगा और \(\alpha\beta=-2\) है। व्यंजक (4\alpha+2+4\beta-2=4\(\alpha+\beta\)=16) बनता है।
We use \(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha\beta}\). Here \(\alpha^2+\beta^2=45\) and \(\alpha\beta=18\), so the value is \(\frac{5}{2}\).
Step 2
Why this answer is correct
The correct answer is A. \(\frac{5}{2}\). We use \(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha\beta}\). Here \(\alpha^2+\beta^2=45\) and \(\alpha\beta=18\), so the value is \(\frac{5}{2}\).
Step 3
Exam Tip
\(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha\beta}\) है। यहाँ \(\alpha^2+\beta^2=45\) और \(\alpha\beta=18\), इसलिए मान \(\frac{5}{2}\) है।
Since \(\alpha^2=3\alpha+1\), the expression becomes (3\alpha+1+3\beta-1=3\(\alpha+\beta\)=9), so no option is correct. The correct value is (9).
Step 2
Why this answer is correct
The correct answer is A. (8). Since \(\alpha^2=3\alpha+1\), the expression becomes (3\alpha+1+3\beta-1=3\(\alpha+\beta\)=9), so no option is correct. The correct value is (9).
Step 3
Exam Tip
क्योंकि \(\alpha^2=3\alpha+1\), व्यंजक (3\alpha+1+3\beta-1=3\(\alpha+\beta\)=9) बनता है, इसलिए कोई विकल्प सही नहीं है। सही मान (9) होगा।
One root can be \(\alpha=2\), which makes \(\alpha-2=0\). Therefore the expression is undefined; always check zero denominators first in exams.
Step 2
Why this answer is correct
The correct answer is A. अपरिभाषित / Undefined. One root can be \(\alpha=2\), which makes \(\alpha-2=0\). Therefore the expression is undefined; always check zero denominators first in exams.
Step 3
Exam Tip
जड़ों में से एक \(\alpha=2\) हो सकती है, जिससे \(\alpha-2=0\) बनता है। इसलिए व्यंजक अपरिभाषित है; परीक्षा में हर पहले शून्य हर देखें।
The roots are (3) and (5). Substitution gives \(\frac{5}{1}+\frac{7}{3}=\frac{22}{3}\), so none of the options is correct; the correct value should be \(\frac{22}{3}\).
Step 2
Why this answer is correct
The correct answer is A. (8). The roots are (3) and (5). Substitution gives \(\frac{5}{1}+\frac{7}{3}=\frac{22}{3}\), so none of the options is correct; the correct value should be \(\frac{22}{3}\).
Step 3
Exam Tip
जड़ें (3) और (5) हैं। सीधे रखने पर \(\frac{5}{1}+\frac{7}{3}=\frac{22}{3}\) आता है, इसलिए विकल्पों में कोई सही नहीं; सही प्रश्न के लिए उत्तर \(\frac{22}{3}\) होना चाहिए।
Use (\(\alpha-\beta\)2=\(\alpha+\beta\)2-4\alpha\beta). Since (16=\(\alpha+\beta\)2-20), we get (\(\alpha+\beta\)2=36) and \(m^2=36\).
Step 2
Why this answer is correct
The correct answer is C. (36). Use (\(\alpha-\beta\)2=\(\alpha+\beta\)2-4\alpha\beta). Since (16=\(\alpha+\beta\)2-20), we get (\(\alpha+\beta\)2=36) and \(m^2=36\).
Step 3
Exam Tip
(\(\alpha-\beta\)2=\(\alpha+\beta\)2-4\alpha\beta) लगाएँ। (16=\(\alpha+\beta\)2-20), इसलिए (\(\alpha+\beta\)2=36) और \(m^2=36\)।
(\alpha-2\beta+\alpha\beta-2=\alpha\beta\(\alpha+\beta\)). Since \(\alpha\beta=-\frac{5}{2}\) and \(\alpha+\beta=-\frac{3}{2}\), the value is \(\frac{15}{4}\).
Step 2
Why this answer is correct
The correct answer is A. \(\frac{15}{4}\). (\alpha-2\beta+\alpha\beta-2=\alpha\beta\(\alpha+\beta\)). Since \(\alpha\beta=-\frac{5}{2}\) and \(\alpha+\beta=-\frac{3}{2}\), the value is \(\frac{15}{4}\).
Step 3
Exam Tip
(\alpha-2\beta+\alpha\beta-2=\alpha\beta\(\alpha+\beta\)) होता है। \(\alpha\beta=-\frac{5}{2}\) और \(\alpha+\beta=-\frac{3}{2}\), इसलिए मान \(\frac{15}{4}\) है।
\(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha\beta}\). With \(\alpha+\beta=-4\) and \(\alpha\beta=1\), the value is (14).
Step 2
Why this answer is correct
The correct answer is C. (14). \(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha\beta}\). With \(\alpha+\beta=-4\) and \(\alpha\beta=1\), the value is (14).
Step 3
Exam Tip
\(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha\beta}\) होता है। \(\alpha+\beta=-4\) और \(\alpha\beta=1\) से मान (14) आता है।
(\alpha-2\beta+\alpha\beta-2=\alpha\beta\(\alpha+\beta\)). Here \(\alpha\beta=-21\) and \(\alpha+\beta=4\), so the value is (-84).
Step 2
Why this answer is correct
The correct answer is A. (-84). (\alpha-2\beta+\alpha\beta-2=\alpha\beta\(\alpha+\beta\)). Here \(\alpha\beta=-21\) and \(\alpha+\beta=4\), so the value is (-84).
Step 3
Exam Tip
(\alpha-2\beta+\alpha\beta-2=\alpha\beta\(\alpha+\beta\)) होता है। यहां \(\alpha\beta=-21\) और \(\alpha+\beta=4\) इसलिए मान (-84) है।
Here \(\alpha+\beta=8\) and \(\alpha\beta=15\). The value is (\frac{\(\alpha+\beta\)2-2\alpha\beta}{\alpha\beta}=\frac{34}{15}).
Step 2
Why this answer is correct
The correct answer is A. \(\frac{34}{15}\). Here \(\alpha+\beta=8\) and \(\alpha\beta=15\). The value is (\frac{\(\alpha+\beta\)2-2\alpha\beta}{\alpha\beta}=\frac{34}{15}).
Step 3
Exam Tip
यहां \(\alpha+\beta=8\) और \(\alpha\beta=15\) है। मान (\frac{\(\alpha+\beta\)2-2\alpha\beta}{\alpha\beta}=\frac{34}{15}) होगा।
(\alpha-2\beta+\alpha\beta-2=\alpha\beta\(\alpha+\beta\)). Here \(\alpha\beta=-10\) and \(\alpha+\beta=3\), so the value is (-30).
Step 2
Why this answer is correct
The correct answer is A. (-30). (\alpha-2\beta+\alpha\beta-2=\alpha\beta\(\alpha+\beta\)). Here \(\alpha\beta=-10\) and \(\alpha+\beta=3\), so the value is (-30).
Step 3
Exam Tip
(\alpha-2\beta+\alpha\beta-2=\alpha\beta\(\alpha+\beta\)) होता है। यहां \(\alpha\beta=-10\) और \(\alpha+\beta=3\) इसलिए मान (-30) है।
Here \(\alpha+\beta=7\) and \(\alpha\beta=12\). (\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\(\alpha+\beta\)2-2\alpha\beta}{\alpha\beta}=\frac{25}{12}).
Step 2
Why this answer is correct
The correct answer is A. \(\frac{25}{12}\). Here \(\alpha+\beta=7\) and \(\alpha\beta=12\). (\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\(\alpha+\beta\)2-2\alpha\beta}{\alpha\beta}=\frac{25}{12}).
Step 3
Exam Tip
यहां \(\alpha+\beta=7\) और \(\alpha\beta=12\) है। (\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\(\alpha+\beta\)2-2\alpha\beta}{\alpha\beta}=\frac{25}{12}) होगा।
Here \(\alpha+\beta=\frac{5}{4}\) and \(\alpha\beta=-\frac{3}{2}\). Therefore \(\alpha+\beta-2\alpha\beta=\frac{5}{4}+3=\frac{17}{4}\).
Step 2
Why this answer is correct
The correct answer is A. \(\frac{17}{4}\). Here \(\alpha+\beta=\frac{5}{4}\) and \(\alpha\beta=-\frac{3}{2}\). Therefore \(\alpha+\beta-2\alpha\beta=\frac{5}{4}+3=\frac{17}{4}\).
Step 3
Exam Tip
यहां \(\alpha+\beta=\frac{5}{4}\) और \(\alpha\beta=-\frac{3}{2}\) है। इसलिए \(\alpha+\beta-2\alpha\beta=\frac{5}{4}+3=\frac{17}{4}\) होगा।
Here \(\alpha+\beta=\frac{7}{2}\) and \(\alpha\beta=\frac{3}{2}\). Therefore \(\alpha+\beta-\alpha\beta=\frac{7}{2}-\frac{3}{2}=2\).
Step 2
Why this answer is correct
The correct answer is A. (2). Here \(\alpha+\beta=\frac{7}{2}\) and \(\alpha\beta=\frac{3}{2}\). Therefore \(\alpha+\beta-\alpha\beta=\frac{7}{2}-\frac{3}{2}=2\).
Step 3
Exam Tip
यहां \(\alpha+\beta=\frac{7}{2}\) और \(\alpha\beta=\frac{3}{2}\) है। इसलिए \(\alpha+\beta-\alpha\beta=\frac{7}{2}-\frac{3}{2}=2\) होगा।
We use \(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha\beta}\). Here \(\alpha+\beta=\frac{11}{3}\) and \(\alpha\beta=2\), so the value is \(\frac{85}{18}\).
Step 2
Why this answer is correct
The correct answer is A. \( \frac{85}{18} \). We use \(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha\beta}\). Here \(\alpha+\beta=\frac{11}{3}\) and \(\alpha\beta=2\), so the value is \(\frac{85}{18}\).
Step 3
Exam Tip
\(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha\beta}\) होता है। यहाँ \(\alpha+\beta=\frac{11}{3}\) और \(\alpha\beta=2\), इसलिए मान \(\frac{85}{18}\) है।
We use \(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha\beta}\). Here \(\alpha+\beta=\frac{7}{2}\) and \(\alpha\beta=\frac{3}{2}\), so the value is \(\frac{37}{6}\).
Step 2
Why this answer is correct
The correct answer is A. \( \frac{37}{6} \). We use \(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha\beta}\). Here \(\alpha+\beta=\frac{7}{2}\) and \(\alpha\beta=\frac{3}{2}\), so the value is \(\frac{37}{6}\).
Step 3
Exam Tip
\(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha\beta}\) होता है। यहाँ \(\alpha+\beta=\frac{7}{2}\) और \(\alpha\beta=\frac{3}{2}\), इसलिए मान \(\frac{37}{6}\) है।
A. \(4+\sqrt{5}\) और \(4-\sqrt{5}\)/\(4+\sqrt{5}\) and \(4-\sqrt{5}\)
Step 1
Concept
The sum of \(4+\sqrt{5}\) and \(4-\sqrt{5}\) is (8), and the product is (16-5=11). In exams check the sum and product of options.
Step 2
Why this answer is correct
The correct answer is A. \(4+\sqrt{5}\) और \(4-\sqrt{5}\) / \(4+\sqrt{5}\) and \(4-\sqrt{5}\). The sum of \(4+\sqrt{5}\) and \(4-\sqrt{5}\) is (8), and the product is (16-5=11). In exams check the sum and product of options.
Step 3
Exam Tip
\(4+\sqrt{5}\) और \(4-\sqrt{5}\) का योग (8) और गुणनफल (16-5=11) है। परीक्षा में विकल्पों का योग और गुणनफल जांचें।
Here \(\alpha+\beta=8\) and \(\alpha\beta=1\), so \(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha\beta}=\frac{64-2}{1}=62\). In such questions, first find the sum and product.
Step 2
Why this answer is correct
The correct answer is A. (62). Here \(\alpha+\beta=8\) and \(\alpha\beta=1\), so \(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha\beta}=\frac{64-2}{1}=62\). In such questions, first find the sum and product.
Step 3
Exam Tip
\(\alpha+\beta=8\) और \(\alpha\beta=1\), इसलिए \(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha\beta}=\frac{64-2}{1}=62\)। ऐसे प्रश्नों में पहले योग और गुणनफल निकालें।
\(\alpha+\beta=2\) and \(\alpha\beta=-11\), so (\alpha-2+\beta-2+\alpha\beta=\(\alpha+\beta\)2-\alpha\beta=4+11=15). Sum and product are enough for symmetric expressions.
Step 2
Why this answer is correct
The correct answer is A. (15). \(\alpha+\beta=2\) and \(\alpha\beta=-11\), so (\alpha-2+\beta-2+\alpha\beta=\(\alpha+\beta\)2-\alpha\beta=4+11=15). Sum and product are enough for symmetric expressions.
Step 3
Exam Tip
\(\alpha+\beta=2\) और \(\alpha\beta=-11\), इसलिए (\alpha-2+\beta-2+\alpha\beta=\(\alpha+\beta\)2-\alpha\beta=4+11=15)। सममित व्यंजकों में योग और गुणनफल काफी होते हैं।
Here \(\alpha+\beta=5\) and \(\alpha\beta=6\). Since \(\alpha^2+\beta^2=13\), the value is (13-4\(\alpha+\beta\)=13-20=-7).
Step 2
Why this answer is correct
The correct answer is A. (-7). Here \(\alpha+\beta=5\) and \(\alpha\beta=6\). Since \(\alpha^2+\beta^2=13\), the value is (13-4\(\alpha+\beta\)=13-20=-7).
Step 3
Exam Tip
\(\alpha+\beta=5\) और \(\alpha\beta=6\) है। \(\alpha^2+\beta^2=13\), इसलिए (13-4\(\alpha+\beta\)=13-20=-7)।
Here \(\alpha+\beta=14\) and \(\alpha\beta=45\). \(\alpha^2+\beta^2=196-90=106\), so the value is \(\frac{106}{45}\).
Step 2
Why this answer is correct
The correct answer is A. \(\frac{106}{45}\). Here \(\alpha+\beta=14\) and \(\alpha\beta=45\). \(\alpha^2+\beta^2=196-90=106\), so the value is \(\frac{106}{45}\).
Step 3
Exam Tip
यहां \(\alpha+\beta=14\) और \(\alpha\beta=45\) है। \(\alpha^2+\beta^2=196-90=106\) इसलिए मान \(\frac{106}{45}\) है।
The roots are (5) and (4), so the sum is (9) and the difference can be \(\pm1\). Hence the value is (9) or (-9).
Step 2
Why this answer is correct
The correct answer is A. (9) या (-9) / (9) or (-9). The roots are (5) and (4), so the sum is (9) and the difference can be \(\pm1\). Hence the value is (9) or (-9).
Step 3
Exam Tip
मूल (5) और (4) हैं इसलिए योग (9) और अंतर \(\pm1\) हो सकता है। इसलिए मान (9) या (-9) है।
Here \(\alpha+\beta=12\) and \(\alpha\beta=32\). \(\alpha^2+\beta^2=144-64=80\), and \(\frac{80}{32}=\frac{5}{2}\).
Step 2
Why this answer is correct
The correct answer is A. \(\frac{5}{2}\). Here \(\alpha+\beta=12\) and \(\alpha\beta=32\). \(\alpha^2+\beta^2=144-64=80\), and \(\frac{80}{32}=\frac{5}{2}\).
Step 3
Exam Tip
यहां \(\alpha+\beta=12\) और \(\alpha\beta=32\) है। \(\alpha^2+\beta^2=144-64=80\) और \(\frac{80}{32}=\frac{5}{2}\) है।
B. \(\frac{7}{3}\) या \(-\frac{7}{3}\)/\(\frac{7}{3}\) or \(-\frac{7}{3}\)
Step 1
Concept
The roots are (5) and (2), so the sum is (7) and the difference can be \(\pm3\). Hence the value is \(\frac{7}{3}\) or \(-\frac{7}{3}\).
Step 2
Why this answer is correct
The correct answer is B. \(\frac{7}{3}\) या \(-\frac{7}{3}\) / \(\frac{7}{3}\) or \(-\frac{7}{3}\). The roots are (5) and (2), so the sum is (7) and the difference can be \(\pm3\). Hence the value is \(\frac{7}{3}\) or \(-\frac{7}{3}\).
Step 3
Exam Tip
मूल (5) और (2) हैं इसलिए योग (7) और अंतर \(\pm3\) हो सकता है। इसलिए मान \(\frac{7}{3}\) या \(-\frac{7}{3}\) है।
The roots are (3) and (5), so \(\alpha-1\) and \(\beta-1\) are (2) and (4). The value is \(\frac{2}{4}+\frac{4}{2}=\frac{5}{2}\).
Step 2
Why this answer is correct
The correct answer is A. \(\frac{17}{8}\). The roots are (3) and (5), so \(\alpha-1\) and \(\beta-1\) are (2) and (4). The value is \(\frac{2}{4}+\frac{4}{2}=\frac{5}{2}\).
Step 3
Exam Tip
मूल (3) और (5) हैं इसलिए \(\alpha-1\) और \(\beta-1\) (2) और (4) होंगे। मान \(\frac{2}{4}+\frac{4}{2}=\frac{5}{2}\) है।
The roots are (5) and (6). Direct substitution gives \(\frac{6}{4}+\frac{7}{5}=\frac{29}{10}\), so none of the given options is correct.
Step 2
Why this answer is correct
The correct answer is B. \(\frac{19}{5}\). The roots are (5) and (6). Direct substitution gives \(\frac{6}{4}+\frac{7}{5}=\frac{29}{10}\), so none of the given options is correct.
Step 3
Exam Tip
जड़ें (5) और (6) हैं। सीधे रखने पर \(\frac{6}{4}+\frac{7}{5}=\frac{29}{10}\), इसलिए दिए गए विकल्पों में कोई सही नहीं है।
For three AP terms, \(2\beta=\alpha+\gamma\), so \(\beta=7\). In exams, treat the middle term as the average of the extremes.
Step 2
Why this answer is correct
The correct answer is C. (7). For three AP terms, \(2\beta=\alpha+\gamma\), so \(\beta=7\). In exams, treat the middle term as the average of the extremes.
Step 3
Exam Tip
तीन पदों में \(2\beta=\alpha+\gamma\), इसलिए \(\beta=7\)। परीक्षा में मध्य पद को सिरों का औसत मानें।
For infinitely many solutions, \(\frac{9}{27}=\frac{\beta+4}{18}=\frac{15}{45}\) must hold. This gives \(\beta=2\).
Step 2
Why this answer is correct
The correct answer is B. \(\beta=2\). For infinitely many solutions, \(\frac{9}{27}=\frac{\beta+4}{18}=\frac{15}{45}\) must hold. This gives \(\beta=2\).
Step 3
Exam Tip
अनंत हलों में \(\frac{9}{27}=\frac{\beta+4}{18}=\frac{15}{45}\) होना चाहिए। इससे \(\beta=2\) मिलता है।
Equating coefficient ratios gives \(\beta=3\). The constant ratio \(\frac{5}{9}\) is different, so there is no solution.
Step 2
Why this answer is correct
The correct answer is C. \(\beta=3\). Equating coefficient ratios gives \(\beta=3\). The constant ratio \(\frac{5}{9}\) is different, so there is no solution.
Step 3
Exam Tip
गुणांक अनुपात समान करने पर \(\beta=3\) मिलता है। स्थिर अनुपात \(\frac{5}{9}\) अलग है, इसलिए कोई हल नहीं।
\(\alpha+\beta=12\) and \(\alpha\beta=35\). \(\frac{1}{\alpha-1}+\frac{1}{\beta-1}=\frac{\alpha+\beta-2}{\alpha\beta-\alpha-\beta+1}=\frac{10}{24}=\frac{5}{12}\).
Step 2
Why this answer is correct
The correct answer is D. \(\frac{11}{24}\). \(\alpha+\beta=12\) and \(\alpha\beta=35\). \(\frac{1}{\alpha-1}+\frac{1}{\beta-1}=\frac{\alpha+\beta-2}{\alpha\beta-\alpha-\beta+1}=\frac{10}{24}=\frac{5}{12}\).
Step 3
Exam Tip
\(\alpha+\beta=12\) और \(\alpha\beta=35\) हैं। \(\frac{1}{\alpha-1}+\frac{1}{\beta-1}=\frac{\alpha+\beta-2}{\alpha\beta-\alpha-\beta+1}=\frac{10}{24}=\frac{5}{12}\)।
Here \(\alpha+\beta=\frac{10}{3}\) and \(\alpha\beta=\frac{7}{3}\). Hence (\alpha-2+\beta-2=\(\alpha+\beta\)2-2\alpha\beta=\frac{100}{9}-\frac{14}{3}=\frac{58}{9}).
Step 2
Why this answer is correct
The correct answer is A. \(\frac{58}{9}\). Here \(\alpha+\beta=\frac{10}{3}\) and \(\alpha\beta=\frac{7}{3}\). Hence (\alpha-2+\beta-2=\(\alpha+\beta\)2-2\alpha\beta=\frac{100}{9}-\frac{14}{3}=\frac{58}{9}).
Step 3
Exam Tip
यहाँ \(\alpha+\beta=\frac{10}{3}\) और \(\alpha\beta=\frac{7}{3}\) हैं। इसलिए (\alpha-2+\beta-2=\(\alpha+\beta\)2-2\alpha\beta=\frac{100}{9}-\frac{14}{3}=\frac{58}{9})।
\(\alpha+\beta=\frac{9}{2}\) and \(\alpha\beta=2\). Hence \(\frac{1}{\alpha}+\frac{1}{\beta}=\frac{\alpha+\beta}{\alpha\beta}=\frac{9}{4}\).
Step 2
Why this answer is correct
The correct answer is A. \(\frac{9}{4}\). \(\alpha+\beta=\frac{9}{2}\) and \(\alpha\beta=2\). Hence \(\frac{1}{\alpha}+\frac{1}{\beta}=\frac{\alpha+\beta}{\alpha\beta}=\frac{9}{4}\).
Step 3
Exam Tip
\(\alpha+\beta=\frac{9}{2}\) और \(\alpha\beta=2\) हैं। इसलिए \(\frac{1}{\alpha}+\frac{1}{\beta}=\frac{\alpha+\beta}{\alpha\beta}=\frac{9}{4}\)।
