Let <math><msub><mrow><mrow><mi>a</mi></mrow></mrow><mrow><mrow><mn>1</mn></mrow></mrow></msub><mo>,</mo><msub><mrow><mrow><mi>a</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msub><mo>,</mo><mo>…</mo><msub><mrow><mrow><mi>a</mi></mrow></mrow><mrow><mrow><mi>n</mi></mrow></mrow></msub></math> be a given A.P. whose common difference is an integer and <math><msub><mrow><mrow><mi>S</mi></mrow></mrow><mrow><mrow><mi>n</mi></mrow></mrow></msub><mo>=</mo><msub><mrow><mrow><mi>a</mi></mrow></mrow><mrow><mrow><mn>1</mn></mrow></mrow></msub><mo>+</mo><msub><mrow><mrow><mi>a</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msub></math> <math><mo>+</mo><mo>…</mo><mo>+</mo><msub><mrow><mrow><mi>a</mi></mrow></mrow><mrow><mrow><mi>n</mi></mrow></mrow></msub></math> . If <math><msub><mrow><mrow><mi>a</mi></mrow></mrow><mrow><mrow><mn>1</mn></mrow></mrow></msub><mo>=</mo><mn>1</mn><mo>,</mo><msub><mrow><mrow><mi>a</mi></mrow></mrow><mrow><mrow><mn>1</mn></mrow></mrow></msub><mo>=</mo><mn>300</mn></math> and <math><mn>15</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mn>50</mn></math> , then the ordered pair <math><mfenced separators="|"><mrow><mrow><msub><mrow><mrow><mi>S</mi></mrow></mrow><mrow><mrow><mi>n</mi><mo>-</mo><mn>4</mn></mrow></mrow></msub><mo>,</mo><msub><mrow><mrow><mi>a</mi></mrow></mrow><mrow><mrow><mi>n</mi><mo>-</mo><mn>4</mn></mrow></mrow></msub></mrow></mrow></mfenced></math> is equal to:

xdydx-y=x2(xcos?x+sin?x),x>0dydx-yx=x(xcos?x+sin?x)⇒dydx+Py=QSo, I.F. =e∫-1xdx1|x|=1x(x>0)Thus, yx=∫1x(x(xcos?x+sin?x))dx⇒yx=xsin?x+C?y(π)=π⇒C=1So, y=x2sin?x+x⇒(y)π/2=π24+π2Also, dydx=x2cos?x+2xsin?x+1⇒d2ydx2=-x2sin?x+4xcos?x+2sin?x

Let&nbsp;<math><msub><mrow><mrow><mi>a</mi></mrow></mrow><mrow><mrow><mn>1</mn></mrow></mrow></msub><mo>,</mo><msub><mrow><mrow><mi>a</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msub><mo>,</mo><mo>…</mo><msub><mrow><mrow><mi>a</mi></mrow></mrow><mrow><mrow><mi>n</mi></mrow></mrow></msub></math>&nbsp;be a given A.P. whose common difference is an integer and&nbsp;<math><msub><mrow><mrow><mi>S</mi></mrow></mrow><mrow><mrow><mi>n</mi></mrow></mrow></msub><mo>=</mo><msub><mrow><mrow><mi>a</mi></mrow></mrow><mrow><mrow><mn>1</mn></mrow></mrow></msub><mo>+</mo><msub><mrow><mrow><mi>a</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msub></math>&nbsp;<math><mo>+</mo><mo>…</mo><mo>+</mo><msub><mrow><mrow><mi>a</mi></mrow></mrow><mrow><mrow><mi>n</mi></mrow></mrow></msub></math>&nbsp;. If&nbsp;<math><msub><mrow><mrow><mi>a</mi></mrow></mrow><mrow><mrow><mn>1</mn></mrow></mrow></msub><mo>=</mo><mn>1</mn><mo>,</mo><msub><mrow><mrow><mi>a</mi></mrow></mrow><mrow><mrow><mn>1</mn></mrow></mrow></msub><mo>=</mo><mn>300</mn></math>&nbsp;and&nbsp;<math><mn>15</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mn>50</mn></math>&nbsp;, then the ordered pair&nbsp;<math><mfenced separators="|"><mrow><mrow><msub><mrow><mrow><mi>S</mi></mrow></mrow><mrow><mrow><mi>n</mi><mo>-</mo><mn>4</mn></mrow></mrow></msub><mo>,</mo><msub><mrow><mrow><mi>a</mi></mrow></mrow><mrow><mrow><mi>n</mi><mo>-</mo><mn>4</mn></mrow></mrow></msub></mrow></mrow></mfenced></math>&nbsp;is equal to:

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Maths NCERT Exemplar Solutions Class 12th Chapter Twelve Maths Class 12th Sequences and Series Maths NCERT Exemplar Solutions Class 12th Chapter Nine

Let $a_{1}, a_{2}, \dots a_{n}$ be a given A.P. whose common difference is an integer and $S_{n} = a_{1} + a_{2}$ $+ \dots + a_{n}$ . If $a_{1} = 1, a_{1} = 300$ and $15 \leq n \leq 50$ , then the ordered pair $(S_{n - 4}, a_{n - 4})$ is equal to:

Option 1 -

$(2490,248)$

Option 2 -

$(2480,248)$

Option 3 -

$(2490,249)$

Option 4 -

$(2480,249)$

4 Views | Posted 2 months ago

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1 Answer

Answered by

alok kumar singh | Contributor-Level 10

2 months ago

Correct Option - 3

Detailed Solution:

$x \frac{d y}{d x} - y = x^{2} (x c o s ? x + s i n ? x), x > 0$

$\frac{d y}{d x} - \frac{y}{x} = x (x c o s ? x + s i n ? x) \Rightarrow \frac{d y}{d x} + P y = Q$

So, I.F. $= e^{\int - \frac{1}{x} d x} \frac{1}{| x |} = \frac{1}{x} (x > 0)$

Thus, $\frac{y}{x} = \int \frac{1}{x} (x (x c o s ? x + s i n ? x)) d x$

$\Rightarrow \frac{y}{x} = x s i n ? x + C$

$? y (π) = π \Rightarrow C = 1$

So, $y = x^{2} s i n ? x + x \Rightarrow (y)_{π / 2} = \frac{π^{2}}{4} + \frac{π}{2}$

Also, $\frac{d y}{d x} = x^{2} c o s ? x + 2 x s i n ? x + 1$

$\Rightarrow \frac{d^{2} y}{d x^{2}} = - x^{2} s i n ? x + 4 x c o s ? x + 2 s i n ? x$

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Let $a_{1}, a_{2}, \dots a_{n}$ be a given A.P. whose common difference is an integer and $S_{n} = a_{1} + a_{2}$ $+ \dots + a_{n}$ . If $a_{1} = 1, a_{1} = 300$ and $15 \leq n \leq 50$ , then the ordered pair $(S_{n - 4}, a_{n - 4})$ is equal to:

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Let a1,a2,…an be a given A.P. whose common difference is an integer and Sn=a1+a2 +…+an . If a1=1,a1=300 and 15≤n≤50 , then the ordered pair Sn-4,an-4 is equal to:

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Let $a_{1}, a_{2}, \dots a_{n}$ be a given A.P. whose common difference is an integer and $S_{n} = a_{1} + a_{2}$ $+ \dots + a_{n}$ . If $a_{1} = 1, a_{1} = 300$ and $15 \leq n \leq 50$ , then the ordered pair $(S_{n - 4}, a_{n - 4})$ is equal to: