In the binomial expansion of (a + b)n, the coefficients of the 4th and 13thterms are equal to each other. Find the value of n.
Find the coefficient of xn in the expansion of (1 + x) (1 – x)n.
Write the 4th term from the end in the expansion of
If the coefficients of (r – 5)th and (2r – 1)th terms in the expansion of (1 + x)34 are equal, find the value of r.
Write the coefficient of x7y2 in the expansion of (x + 2y)9
Write the coefficient of the middle term in the expansion of (1 + x)2n.
Write the coefficient of the middle term in the expansion of (1 + x)2n.
Which term is independent of x in the expansion of ?
Write the number of terms in the expansion of
Show that the coefficient of x4 in the expansion of (1 + 2x + x2)5 is 212.
Prove that there is no term involving x6 in the expansion of .
By using the method of completing the square, show that the equation
has no real roots.
$2 x^{2}+x+4=0$ $\Rightarrow 4 x^{2}+2 x+8=0 \quad$ (Multiplying both sides by 2) $\Rightarrow 4 x^{2}+2 x=-8$ $\Rightarrow(2 x)^{2}+2 \times 2 x \times...
Find the roots of the given equation: ![Rendered by QuickLaTeX.com \sqrt{3} x^{2}+10 x+7 \sqrt{3}=0](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-cb3f8986c378b726ec06f568ef06fe4f_l3.png)
$\sqrt{3} x^{2}+10 x+7 \sqrt{3}=0$ $\Rightarrow 3 x^{2}+10 \sqrt{3} x+21=0 \quad$ (Multiplying both sides by $\left.\sqrt{3}\right)$ $\Rightarrow 3 x^{2}+10 \sqrt{3} x=-21$ $\Rightarrow(\sqrt{3}...
Find the roots of the given equation: ![Rendered by QuickLaTeX.com \sqrt{2} x^{2}-3 x-2 \sqrt{2}=0](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-5041576279f092b72ce4775f9b4cd839_l3.png)
$\sqrt{2} x^{2}-3 x-2 \sqrt{2}=0$ $\Rightarrow 2 x^{2}-3 \sqrt{2} x-4=0 \quad$ (Multiplying both sides by $\sqrt{2}$ ) $\Rightarrow 2 x^{2}-3 \sqrt{2} x=4$ $\Rightarrow(\sqrt{2} x)^{2}-2 \times...
Find the roots of the given equation: ![Rendered by QuickLaTeX.com \quad x^{2}-(\sqrt{2}+1) x+\sqrt{2}=0](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-2785ffe7671b5299f18efb4d4791fde8_l3.png)
$\begin{array}{l} x^{2}-(\sqrt{2}+1) x+\sqrt{2}=0 \\ \Rightarrow x^{2}-(\sqrt{2}+1) x=-\sqrt{2} \\ \Rightarrow x^{2}-2 \times x...
Find the roots of the given equation: ![Rendered by QuickLaTeX.com 4 x^{2}+4 b x-\left(a^{2}-b^{2}\right)=0](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-80a20f7766a931af43a643fa4dbf2cd7_l3.png)
$\begin{array}{l} 4 x^{2}+4 b x-\left(a^{2}-b^{2}\right)=0 \\ \Rightarrow 4 x^{2}+4 b x=a^{2}-b^{2} \\ \Rightarrow(2 x)^{2}+2 \times 2 x \times b+b^{2}=a^{2}-b^{2}+b^{2} \text { (Adding } b^{2}...
Find the roots of the given equation: ![Rendered by QuickLaTeX.com \frac{2}{x^{2}}-\frac{5}{x}+2=0](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-673440138a253aaeabdbe2167f064569_l3.png)
$\frac{2}{x^{2}}-\frac{5}{x}+2=0$ $\Rightarrow \frac{2-5 x+2 x^{2}}{x^{2}}=0$ $\Rightarrow 2 x^{2}-5 x+2=0$ $\Rightarrow 4 x^{2}-10 x+4=0 \quad$ (Multiplying both sides by 2) $\Rightarrow 4 x^{2}-10...
Find the roots of the given equation: ![Rendered by QuickLaTeX.com 5 x^{2}-6 x-2=0](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-70a2ac7121803ff2fd9f47f6ec12a166_l3.png)
$\begin{array}{l} 5 x^{2}-6 x-2=0 \\ \Rightarrow 25 x^{2}-30 x-10=0 \\ \Rightarrow 25 x^{2}-30 x=10 \end{array}$ $\begin{array}{l} \Rightarrow(5 x)^{2}-2 \times 5 x \times 3+3^{2}=10+3^{2} \\...
Find the roots of the given equation: ![Rendered by QuickLaTeX.com 3 x^{2}-2 x-1=0](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-841f519e00ff63e1f05a0268128f66f9_l3.png)
$3 x^{2}-2 x-1=0$ $\Rightarrow 9 x^{2}-6 x-3=0 \quad$ (Multiplying both sides by 3) $\Rightarrow 9 x^{2}-6 x=3$ $\Rightarrow(3 x)^{2}-2 \times 3 x \times 1+1^{2}=3+1^{2} \quad$ [Adding $1^{2}$ on...
Find the roots of the given equation: ![Rendered by QuickLaTeX.com 7 x^{2}+3 x-4=0](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-74716bc2e80af4c97c159de3346b91d8_l3.png)
$\begin{array}{l} 7 x^{2}+3 x-4=0 \\ \Rightarrow 49 x^{2}+21 x-28=0 \end{array}$ $\Rightarrow 49 x^{2}+21 x=28$ $\Rightarrow(7 x)^{2}+2 \times 7 x \times...
Find the roots of the given equation: ![Rendered by QuickLaTeX.com 8 x^{2}-14 x-15=0](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-88b0a9deae586e4acab5c3404562592f_l3.png)
$8 x^{2}-14 x-15=0$ $\Rightarrow 16 x^{2}-28 x-30=0 \quad$ (Multiplying both sides by 2) $\Rightarrow 16 x^{2}-28 x=30$ $\Rightarrow(4 x)^{2}-2 \times 4 x \times...
Find the roots of the given equation: ![Rendered by QuickLaTeX.com 3 x^{2}-x-2=0](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-8fe90b3069d2e387f293c3cab9abb96f_l3.png)
$\begin{array}{l} 3 x^{2}-x-2=0 \\ \Rightarrow 9 x^{2}-3 x-6=0 \quad \text { (Multiplying both sides by 3) } \\ \Rightarrow 9 x^{2}-3 x=6 \\ \Rightarrow(3 x)^{2}-2 \times 3 x \times...
Find the roots of the given equation: ![Rendered by QuickLaTeX.com 2 x^{2}+5 x-3=0](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-ae2736ffec509a0caff07794a4c41a56_l3.png)
$2 x^{2}+5 x-3=0$ $\Rightarrow 4 x^{2}+10 x-6=0 \quad$ (Multiplying both sides by 2) $\Rightarrow 4 x^{2}+10 x=6$ $\Rightarrow(2 x)^{2}+2 \times 2 x \times...
Find the roots of the given equation: ![Rendered by QuickLaTeX.com 4 x^{2}+4 \sqrt{3} x+3=0](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-9419ab0c08a6fd9e519dcc210dcfd114_l3.png)
$\begin{array}{l} 4 x^{2}+4 \sqrt{3} x+3=0 \\ \Rightarrow 4 x^{2}+4 \sqrt{3} x=-3 \\ \Rightarrow(2 x)^{2}+2 \times 2 x \times \sqrt{3}+(\sqrt{3})^{2}=-3+(\sqrt{3})^{2} \end{array}$ $\begin{array}{l}...
Find the roots of the given equation: ![Rendered by QuickLaTeX.com x^{2}+8 x-2=0](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-e78622d4fdd7ada3eb12936657d32d3a_l3.png)
$\begin{array}{l} x^{2}+8 x-2=0 \\ \Rightarrow x^{2}+8 x=2 \\ \Rightarrow x^{2}+2 \times x \times 4+4^{2}=2+4^{2} \\ \Rightarrow(x+4)^{2}=2+16=18 \\ \Rightarrow x+4=\pm \sqrt{18}=\pm 3 \sqrt{2} \\...
Find the roots of the given equation: ![Rendered by QuickLaTeX.com x^{2}-4 x+1=0](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-59ff267449d6dce33daeee624e91cfed_l3.png)
$x^{2}-4 x+1=0$ $\Rightarrow x^{2}-4 x=-1$ $\Rightarrow x^{2}-2 \times x \times 2+2^{2}=-1+2^{2} \quad$ (Adding $2^{2}$ on both sides) $\Rightarrow(x-2)^{2}=-1+4=3$ $\Rightarrow x-2=\pm \sqrt{3}...
Find the roots of the given equation: ![Rendered by QuickLaTeX.com x^{2}-6 x+3=0](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-15042ec71ef4ef811ccf99ad92e5d275_l3.png)
$\begin{array}{l} x^{2}-6 x+3=0 \\ \Rightarrow x^{2}-6 x=-3 \\ \Rightarrow x^{2}-2 \times x \times 3+3^{2}=-3+3^{2} \\ \Rightarrow(x-3)^{2}=-3+9=6 \\ \Rightarrow x-3=\pm \sqrt{6} \end{array}$...