mathemerize

Find the number of common tangents to the circles \(x^2 + y^2\) = 1 and \(x^2 + y^2 – 2x – 6y + 6\) = 0.

Solution : Let \(C_1\) be the center of circle \(x^2 + y^2\) = 1 i.e.  \(C_1\) = (0, 0) And \(C_2\) be the center of circle \(x^2 + y^2 – 2x – 6y + 6\) = 0 i.e. \(C_2\) = (1, 3) Let \(r_1\) be the radius of first circle and \(r_2\) be the radius …

Find the number of common tangents to the circles \(x^2 + y^2\) = 1 and \(x^2 + y^2 – 2x – 6y + 6\) = 0. Read More »

Find the middle term in the expansion of \((3x – {x^3\over 6})^7\).

Solution : The given expression is \((3x – {x^3\over 6})^7\). Here n = 7, which is an odd number. By using middle terms in binomial expansion formula, So, \(({7 + 1\over 2}\)) th and \(({7 + 1\over 2} + 1)\) th i.e.  4th and 5th terms are two middle terms. Now, \(T_{4}\) = \(T_{3 + …

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Find the middle term in the expansion of \(({2\over 3}x^2 – {3\over 2x})^{20}\).

Solution : Here, n = 20, which is an even number. So, \({20\over 2} + 1\)th term i.e. 11th term is the middle term. Hence, the middle term = \(T_{11}\) \(T_{11}\) = \(T_{10 + 1}\) = \(^{20}C_{10}\) \(({2\over 3}x^2)^{20 – 10}\) \((-{3\over 2x})^{10}\) = \(^{20}C_{10} x^{10}\) Similar Questions Find the middle term in the expansion …

Find the middle term in the expansion of \(({2\over 3}x^2 – {3\over 2x})^{20}\). Read More »

Find the 9th term in the expansion of \(({x\over a} – {3a\over x^2})^{12}\).

Solution : We know that the (r + 1)th term or general term in the expansion of \((x + a)^n\) is given by \(T_{r + 1}\) = \(^{n}C_r x^{n – r} a^r\) In the expansion of \(({x\over a} – {3a\over x^2})^{12}\), we have \(T_{9}\) = \(T_{8 + 1}\) = \(^{12}C_8 ({x\over a})^{12 – 8} ({-3a\over …

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Find the 10th term in the binomial expansion of \((2x^2 + {1\over x})^{12}\).

Solution : We know that the (r + 1)th term or general term in the expansion of \((x + a)^n\) is given by \(T_{r + 1}\) = \(^{n}C_r x^{n – r} a^r\) In the expansion of \((2x^2 + {1\over x})^{12}\), we have \(T_{10}\) = \(T_{9 + 1}\) = \(^{12}C_9 (2x^2)^{12 – 9} ({1\over x})^9\) \(\implies\) …

Find the 10th term in the binomial expansion of \((2x^2 + {1\over x})^{12}\). Read More »

Which is larger \((1.01)^{1000000}\) or 10,000?

Solution : We have, \((1.01)^{1000000}\)  – 10000 = \((1 + 0.01)^{1000000}\) – 10000 By using binomial theorem, = \(^{1000000}C_0\) + \(^{1000000}C_1 (0.01)\)  + \(^{1000000}C_2 (0.01)^2\)  + …… + \(^{1000000}C_{1000000} (0.01)^{1000000}\) – 10000 = (1 + 1000000(0.01) + other positive terms) – 10000 = (1 + 10000 + other positive terms) – 10000 = 1 + …

Which is larger \((1.01)^{1000000}\) or 10,000? Read More »