The condition \(2r^2=r+r^3\) gives (r(r-1)2=0), so the nonzero value is (1). In exams, always apply the nonzero condition.
Step 2
Why this answer is correct
The correct answer is A. (1). The condition \(2r^2=r+r^3\) gives (r(r-1)2=0), so the nonzero value is (1). In exams, always apply the nonzero condition.
Step 3
Exam Tip
शर्त \(2r^2=r+r^3\) से (r(r-1)2=0) मिलता है, इसलिए शून्येतर मान (1) है। परीक्षा में दी गई शून्येतर शर्त जरूर लगाएं।
The condition \(2k^2=k+k^3\) gives (k(k-1)2=0), and the nonzero value is (1). In exams, do not ignore conditions like nonzero.
Step 2
Why this answer is correct
The correct answer is B. (k=1). The condition \(2k^2=k+k^3\) gives (k(k-1)2=0), and the nonzero value is (1). In exams, do not ignore conditions like nonzero.
Step 3
Exam Tip
शर्त \(2k^2=k+k^3\) से (k(k-1)2=0) मिलता है, और शून्येतर मान (1) है। परीक्षा में शून्येतर जैसी शर्त न भूलें।
In a polynomial, powers of the variable are non-negative integers like \(0,1,2,\ldots\). This is the basic identification rule.
Step 2
Why this answer is correct
The correct answer is C. अऋणात्मक पूर्णांक / Non-negative integers. In a polynomial, powers of the variable are non-negative integers like \(0,1,2,\ldots\). This is the basic identification rule.
Step 3
Exam Tip
बहुपद में चर की घातें \(0,1,2,\ldots\) जैसी अऋणात्मक पूर्णांक होती हैं। यही बहुपद की मूल पहचान है।
In a polynomial, powers of the variable are non-negative integers like \(0,1,2,\ldots\). This is the key identification rule.
Step 2
Why this answer is correct
The correct answer is C. अऋणात्मक पूर्णांक / Non-negative integers. In a polynomial, powers of the variable are non-negative integers like \(0,1,2,\ldots\). This is the key identification rule.
Step 3
Exam Tip
बहुपद में चर की घातें \(0,1,2,\ldots\) जैसी अऋणात्मक पूर्णांक होती हैं। यही सबसे महत्वपूर्ण पहचान नियम है।
Since (24^{3}=\(2^{3}\cdot3\)^{3}=2^{9}\cdot3^{3}), division leaves \(2^{3}\cdot3=24\), so the correct value is not among the options.
Step 2
Why this answer is correct
The correct answer is B. (6). Since (24^{3}=\(2^{3}\cdot3\)^{3}=2^{9}\cdot3^{3}), division leaves \(2^{3}\cdot3=24\), so the correct value is not among the options.
Step 3
Exam Tip
(24^{3}=\(2^{3}\cdot3\)^{3}=2^{9}\cdot3^{3})। भाग देने पर \(2^{3}\cdot3=24\) मिलता है, इसलिए विकल्पों में सही मान नहीं है।
Since \(1024=2^{10}\), \(16^{x}=2^{4x}\) gives \(x=\frac{5}{2}\), and \(32^{y}=2^{5y}\) gives (y=2). Hence the sum is \(\frac{9}{2}\).
Step 2
Why this answer is correct
The correct answer is A. \(\frac{9}{2}\). Since \(1024=2^{10}\), \(16^{x}=2^{4x}\) gives \(x=\frac{5}{2}\), and \(32^{y}=2^{5y}\) gives (y=2). Hence the sum is \(\frac{9}{2}\).
Step 3
Exam Tip
\(1024=2^{10}\), \(16^{x}=2^{4x}\) से \(x=\frac{5}{2}\), और \(32^{y}=2^{5y}\) से (y=2)। इसलिए योग \(\frac{9}{2}\) है।
Here (125^{\frac{2}{3}}=(5)^{2}=25) and (25^{-\frac{3}{2}}=(5)^{-3}=\frac{1}{125}). The product is \(\frac{1}{5}\).
Step 2
Why this answer is correct
The correct answer is A. \(\frac{1}{5}\). Here (125^{\frac{2}{3}}=(5)^{2}=25) and (25^{-\frac{3}{2}}=(5)^{-3}=\frac{1}{125}). The product is \(\frac{1}{5}\).
Step 3
Exam Tip
(125^{\frac{2}{3}}=(5)^{2}=25) और (25^{-\frac{3}{2}}=(5)^{-3}=\frac{1}{125})। गुणनफल \(\frac{1}{5}\) है।
We get (a=4), and \(9^{b}=3^{2b}=3^{6}\) gives (b=3). Thus \(a^{b}-b^{a}=4^{3}-3^{4}=64-81=-17\), which is not among the options.
Step 2
Why this answer is correct
The correct answer is A. \(\frac{37}{8}\). We get (a=4), and \(9^{b}=3^{2b}=3^{6}\) gives (b=3). Thus \(a^{b}-b^{a}=4^{3}-3^{4}=64-81=-17\), which is not among the options.
Step 3
Exam Tip
(a=4) और \(9^{b}=3^{2b}=3^{6}\) से (b=3) है। इसलिए \(a^{b}-b^{a}=4^{3}-3^{4}=64-81=-17\), अतः विकल्पों में यह मान नहीं है।
Here (25^{\frac{3}{2}}=\(5^{2}\)^{\frac{3}{2}}=5^{3}) and (125^{-\frac{2}{3}}=\(5^{3}\)^{-\frac{2}{3}}=5^{-2}). The product is (5).
Step 2
Why this answer is correct
The correct answer is B. (5). Here (25^{\frac{3}{2}}=\(5^{2}\)^{\frac{3}{2}}=5^{3}) and (125^{-\frac{2}{3}}=\(5^{3}\)^{-\frac{2}{3}}=5^{-2}). The product is (5).
Step 3
Exam Tip
(25^{\frac{3}{2}}=\(5^{2}\)^{\frac{3}{2}}=5^{3}) और (125^{-\frac{2}{3}}=\(5^{3}\)^{-\frac{2}{3}}=5^{-2})। गुणनफल (5) है।
Here \(169^{-1}=13^{-2}\) and \(2197^{-1}=13^{-3}\), so \(\frac{13^{4}\cdot13^{-2}}{13^{-3}}=13^{5}\). In exams, division by a negative power adds the exponent.
Step 2
Why this answer is correct
The correct answer is B. \(13^{5}\). Here \(169^{-1}=13^{-2}\) and \(2197^{-1}=13^{-3}\), so \(\frac{13^{4}\cdot13^{-2}}{13^{-3}}=13^{5}\). In exams, division by a negative power adds the exponent.
Step 3
Exam Tip
\(169^{-1}=13^{-2}\) और \(2197^{-1}=13^{-3}\), इसलिए \(\frac{13^{4}\cdot13^{-2}}{13^{-3}}=13^{5}\)। परीक्षा में ऋणात्मक घात से भाग करते समय घात जुड़ती है।
Here \(6^{x+1}-6^{x}=5\cdot6^{x}=900\), so \(6^{x}=180\), which is not a listed integral power. Therefore none of the listed options is correct.
Step 2
Why this answer is correct
The correct answer is B. (3). Here \(6^{x+1}-6^{x}=5\cdot6^{x}=900\), so \(6^{x}=180\), which is not a listed integral power. Therefore none of the listed options is correct.
Step 3
Exam Tip
\(6^{x+1}-6^{x}=6\cdot6^{x}-6^{x}=5\cdot6^{x}=900\), इसलिए \(6^{x}=180\) नहीं बनता। इसलिए दिए विकल्पों में कोई भी सही नहीं है।
Since \(25^{-2}=5^{-4}\) and \(125=5^{3}\), the total exponent is (9-4+3-4=4). In exams, convert all terms to the same base.
Step 2
Why this answer is correct
The correct answer is C. \(5^{4}\). Since \(25^{-2}=5^{-4}\) and \(125=5^{3}\), the total exponent is (9-4+3-4=4). In exams, convert all terms to the same base.
Step 3
Exam Tip
\(25^{-2}=5^{-4}\) और \(125=5^{3}\), इसलिए कुल घात (9-4+3-4=4) है। परीक्षा में सभी पदों को समान आधार में बदलें।
Here \(3^{-2}+3^{-4}=\frac{1}{9}+\frac{1}{81}=\frac{10}{81}\), and \(3^{-3}=\frac{1}{27}\). Division gives \(\frac{10}{3}\).
Step 2
Why this answer is correct
The correct answer is A. \(\frac{10}{3}\). Here \(3^{-2}+3^{-4}=\frac{1}{9}+\frac{1}{81}=\frac{10}{81}\), and \(3^{-3}=\frac{1}{27}\). Division gives \(\frac{10}{3}\).
Step 3
Exam Tip
\(3^{-2}+3^{-4}=\frac{1}{9}+\frac{1}{81}=\frac{10}{81}\) और \(3^{-3}=\frac{1}{27}\)। भाग देने पर \(\frac{10}{3}\) मिलता है।
Here (49^{\frac{3}{2}}=\(7^{2}\)^{\frac{3}{2}}=7^{3}) and (343^{-\frac{2}{3}}=\(7^{3}\)^{-\frac{2}{3}}=7^{-2}). The product is \(7^{1}=7\).
Step 2
Why this answer is correct
The correct answer is A. (1). Here (49^{\frac{3}{2}}=\(7^{2}\)^{\frac{3}{2}}=7^{3}) and (343^{-\frac{2}{3}}=\(7^{3}\)^{-\frac{2}{3}}=7^{-2}). The product is \(7^{1}=7\).
Step 3
Exam Tip
(49^{\frac{3}{2}}=\(7^{2}\)^{\frac{3}{2}}=7^{3}) और (343^{-\frac{2}{3}}=\(7^{3}\)^{-\frac{2}{3}}=7^{-2})। गुणनफल \(7^{1}=7\) है।
Here \(121^{-2}=11^{-4}\) and \(1331^{-1}=11^{-3}\), so \(\frac{11^{5}\cdot11^{-4}}{11^{-3}}=11^{4}\). In exams, division by a negative power adds the exponent.
Step 2
Why this answer is correct
The correct answer is C. \(11^{4}\). Here \(121^{-2}=11^{-4}\) and \(1331^{-1}=11^{-3}\), so \(\frac{11^{5}\cdot11^{-4}}{11^{-3}}=11^{4}\). In exams, division by a negative power adds the exponent.
Step 3
Exam Tip
\(121^{-2}=11^{-4}\) और \(1331^{-1}=11^{-3}\), इसलिए \(\frac{11^{5}\cdot11^{-4}}{11^{-3}}=11^{4}\)। परीक्षा में ऋणात्मक घात से भाग करते समय घात जुड़ती है।
