Excercise 10B
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: 
$\sqrt{3} x^{2}+10 x+7 \sqrt{3}=0$ $\Rightarrow 3 x^{2}+10 \sqrt{3} x+21=0 \quad$ (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: 
$\sqrt{2} x^{2}-3 x-2 \sqrt{2}=0$ $\Rightarrow 2 x^{2}-3 \sqrt{2} x-4=0 \quad$ (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: 
$\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: 
$\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: 
$\frac{2}{x^{2}}-\frac{5}{x}+2=0$ $\Rightarrow \frac{2-5 x+2 x^{2}}{x^{2}}=0$ $\Rightarrow 2 x^{2}-5 x+2=0$ $\Rightarrow 4 x^{2}-10 x+4=0 \quad$ (Multiplying both sides by 2) $\Rightarrow 4 x^{2}-10...
Find the roots of the given equation: 
$\begin{array}{l} 5 x^{2}-6 x-2=0 \\ \Rightarrow 25 x^{2}-30 x-10=0 \\ \Rightarrow 25 x^{2}-30 x=10 \end{array}$ $\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: 
$3 x^{2}-2 x-1=0$ $\Rightarrow 9 x^{2}-6 x-3=0 \quad$ (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: 
$\begin{array}{l} 7 x^{2}+3 x-4=0 \\ \Rightarrow 49 x^{2}+21 x-28=0 \end{array}$ $\Rightarrow 49 x^{2}+21 x=28$ $\Rightarrow(7 x)^{2}+2 \times 7 x \times...
Find the roots of the given equation: 
$8 x^{2}-14 x-15=0$ $\Rightarrow 16 x^{2}-28 x-30=0 \quad$ (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: 
$\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: 
$2 x^{2}+5 x-3=0$ $\Rightarrow 4 x^{2}+10 x-6=0 \quad$ (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: 
$\begin{array}{l} 4 x^{2}+4 \sqrt{3} x+3=0 \\ \Rightarrow 4 x^{2}+4 \sqrt{3} x=-3 \\ \Rightarrow(2 x)^{2}+2 \times 2 x \times \sqrt{3}+(\sqrt{3})^{2}=-3+(\sqrt{3})^{2} \end{array}$ $\begin{array}{l}...
Find the roots of the given equation: 
$\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: 
$x^{2}-4 x+1=0$ $\Rightarrow x^{2}-4 x=-1$ $\Rightarrow x^{2}-2 \times x \times 2+2^{2}=-1+2^{2} \quad$ (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: 
$\begin{array}{l} x^{2}-6 x+3=0 \\ \Rightarrow x^{2}-6 x=-3 \\ \Rightarrow x^{2}-2 \times x \times 3+3^{2}=-3+3^{2} \\ \Rightarrow(x-3)^{2}=-3+9=6 \\ \Rightarrow x-3=\pm \sqrt{6} \end{array}$...