In Q. No. 14, write the sign of c.
Question: In Q. No. 14, write the sign ofc. Solution: The parabola $y=a x^{2}+b x+c$ cuts $y$-axis at point P which lies on $\mathrm{y}$-axis. Putting $x=0$ in $y=a x^{2}+b x+c$, we get $y=c$. So the coordinates of P are $(0, c)$. Clearly, P lies on OY. Therefore $c0$...
Read More →The graph of the polynomial f(x) = ax2 + bx + c is as shown
Question: The graph of the polynomial $f(x)=a x^{2}+b x+c$ is as shown in Fig. 2.20. Write the value of $b^{2}-4 a c$ and the number of real zeros of $f(x)$. Solution: The graph of the polynomial $f(x)=a x^{2}+b x+c$ or the curve touches $x$-axis at point $\left(\frac{-b}{2 a}, 0\right)$. The $x$-coordinate of this point gives two equal zeros of the polynomial and $b^{2}-4 a c=0$....
Read More →The graph of the polynomial f(x) = ax2 + bx + c is as shown below
Question: The graph of the polynomial $f(x)=a x^{2}+b x+c$ is as shown below (Fig. 2.19). Write the signs of 'a' and $b^{2}-4 a c$. Solution: Clearly, $f(x)=a x^{2}+b x+c$ represent a parabola opening upwards. Therefore, $a0$ Since the parabola cutsx-axis at two points, this means that the polynomial will have two real solutions Hence $b^{2}-4 a c0$ Hence $a0$ and $b^{2}-4 a c0$...
Read More →Find the value of
Question: Find the value of $\frac{6}{\sqrt{5}-\sqrt{3}}$, it being given that $\sqrt{3}=1.732$ and $\sqrt{5}=2.236$. Solution: Given, $\frac{6}{\sqrt{5}-\sqrt{3}}$ Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor $\sqrt{5}+\sqrt{3}$ for $\frac{1}{\sqrt{5}-\sqrt{3}}$ $=\frac{6(\sqrt{5}+\sqrt{3})}{(\sqrt{5}-\sqrt{3})(\sqrt{5}+\sqrt{3})}$ Since, $(a+b)(a-b)=\left(a^{2}-b^{2}\right)$ $=\frac{6 \sqrt{5}+6 \sqrt{3}}{5-3}$ $=\frac{6 \sqrt{5}+6 \...
Read More →The graph of a polynomial y = f(x), shown in Fig. 2.18.
Question: The graph of a polynomialy=f(x), shown in Fig. 2.18. Find the number of real zeros off(x). Solution: A real number $\alpha$ is a zero of polynomial $f(x)$, if $f(\alpha)=0$ In the above figure the curve intersectsx-axis at one point and touches at one point When a curve touchesx-axis at one point, it means it has two common zeros at that point Hence the number of real zeroes is 3...
Read More →If x = 3+√8, find the value of
Question: If $x=3+\sqrt{8}$, find the value of $\left(x^{2}+\frac{1}{x^{2}}\right)$ Solution: Given, $x=3+\sqrt{8}$ $\left(x^{2}+\frac{1}{x^{2}}\right)$ We have, $x=3+\sqrt{8}$, $\frac{1}{x}=\frac{1}{3+\sqrt{8}}$ Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor $3-\sqrt{8}$ for $\frac{1}{3+\sqrt{8}}$ $\frac{1}{x}=\frac{3-\sqrt{8}}{(3+\sqrt{8})(3-\sqrt{8})}$ Since, $(a+b)(a-b)=\left(a^{2}-b^{2}\right)$ $\frac{1}{x}=\frac{3-\sqrt{8}}{9-8}$ $...
Read More →In Fig. 2.17, the graph of a polynomial p(x) is given. Find the zeros of the polynomial.
Question: In Fig. 2.17, the graph of a polynomialp(x) is given. Find the zeros of the polynomial. Solution: Just see the point of intersection of the curve andx-axis and find out thex-coordinate of these points. Thesex-coordinates will be the zeros of the polynomial Since the intersection points are $(-3,0)$ and $(-1,0)$...
Read More →If the sum of the zeros of the quadratic polynomial f(x) = kx2 − 3x + 5 is 1,
Question: If the sum of the zeros of the quadratic polynomial $1(x)=k x^{2}-3 x+5$ is 1 , write the value of $k$. Solution: We have to find the value of $k$, if the sum of the zeros of the quadratic polynomial $f(x)=k x^{2}-3 x+5$ is Given Sum of the polynomial $=1$ $\frac{-\text { Coefficient of } x}{\text { Coefficient of } x^{2}}=1$ $-\left(\frac{-3}{k}\right)=1$ $3=1 \times k$ $3=k$ Hence, the value of $k$ is 3...
Read More →If the product of zeros of the quadratic polynomial f(x) = x2 − 4x + k is 3,
Question: If the product of zeros of the quadratic polynomial $f(x)=x^{2}-4 x+k$ is 3, find the value of $k$. Solution: We have to find the value ofk. Given, The product of the zeros of the quadratic polynomial $f(x)=x^{2}-4 x+k$. is 3 Product of the polynomial $=3$ $\frac{\text { Constant term }}{\text { Coefficient of } x^{3}}=3$ $\frac{k}{1}=3$ $k=3 \times 1$ $k=3$ Hence, the value of $k$ is $k=3$....
Read More →Write the family of quadratic polynomials having
Question: Write the family of quadratic polynomials having $-\frac{1}{4}$ and 1 as its zeros. Solution: We know that, if $x=\alpha$ is a zero of a polynomial then $x-2$ is a factor of quadratic polynomials. Since $\frac{-1}{4}$ and 1 are zeros of polynomial. Therefore $\left(x+\frac{1}{4}\right)(x-1)$ $=x^{2}+\frac{1}{4} x-x-\frac{1}{4}$ $=x^{2}+\frac{1}{4} x-\frac{1 \times 4}{1 \times 4} x-\frac{1}{4}$ $=x^{2}+\frac{1-4}{4} x-\frac{1}{4}$ $=x^{2}-\frac{3}{4} x-\frac{1}{4}$ Hence, the family of ...
Read More →If x = 2+√3, find the value of
Question: If $x=2+\sqrt{3}$, find the value of $x^{3}+\frac{1}{x^{3}}$ Solution: Given, $x=2+\sqrt{3}$ To find the value of $x^{3}+\frac{1}{x^{3}}$ We have, $x=2+\sqrt{3}$, $\frac{1}{x}=\frac{1}{2+\sqrt{3}}$ Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor $2-\sqrt{3}$ for $\frac{1}{2+\sqrt{3}}$ $=\frac{2-\sqrt{3}}{(2+\sqrt{3})(2-\sqrt{3})}$ Since, $(a+b)(a-b)=\left(a^{2}-b^{2}\right)$ $\frac{1}{x}=\frac{2-\sqrt{3}}{4-3}$ $x+\frac{1}{x}=2+...
Read More →The sum and product of the zeros of a quadratic polynomial are
Question: The sum and product of the zeros of a quadratic polynomial are $-\frac{1}{2}$ and $-3$ respectively. What is the quadratic polynomial. Solution: Let sum of quadratic polynomial is $\alpha+\beta=\frac{-1}{2}$ Product of the quadratic polynomial is $\alpha \beta=-3$ Let $S$ and $P$ denote the sum and product of the zeros of a polynomial as $\frac{-1}{2}$ and $-3$. Then The required polynomial $g(x)$ is given by $g(x)=k\left(x^{2}-S x+P\right)$ $=k\left[x^{2}-\left(\frac{-1}{2}\right) x+(...
Read More →Define the zero of a polynomial.
Question: Define the zero of a polynomial. Solution: The zero of a polynomial $f(x)$ is defined as any real number $\alpha$ such that $f(\alpha)=0$...
Read More →Define value of polynomial at a point.
Question: Define value of polynomial at a point. Solution: If $f(x)$ is a polynomial and $\alpha$ is any real number, then the real number obtained by replacing $x$ by $\alpha$ in $f(x)$, is called the value of $f(x)$ at $x=\alpha$ and is denoted by $f(\alpha)$...
Read More →Write the standard form of a cubic polynomial with real coefficients.
Question: Write the standard form of a cubic polynomial with real coefficients. Solution: The most general form of a cubic polynomial with coefficients as real numbers is of the form $f(x)=a x^{3}+b x^{2}+c x+d$, where $a, b, c, d$ are real number and $a \neq 0$...
Read More →Write the standard form of a quadratic polynomial with real coefficients.
Question: Write the standard form of a quadratic polynomial with real coefficients. Solution: Any quadratic polynomial in variable $x$ with real coefficients is of the form $f(x)=a x^{2}+b x+c$, where $a, b, c$ are real numbers and $a \neq 0$...
Read More →Write the standard form of a linear polynomial with real coefficients.
Question: Write the standard form of a linear polynomial with real coefficients. Solution: Any linear polynomial in variable $x$ with real coefficients is of the form $f(x)=a x+b$, where $a, b$ are real numbers and $a \neq 0$...
Read More →In each of the following determine rational numbers a and b:
Question: In each of the following determine rational numbers a and b: (i) $\frac{\sqrt{3}-1}{\sqrt{3}+1}=a-b \sqrt{3}$ (ii) $\frac{4+\sqrt{2}}{2+\sqrt{2}}=a-\sqrt{b}$ (iii) $\frac{3+\sqrt{2}}{3-\sqrt{2}}=a+b \sqrt{2}$ (iv) $\frac{5+3 \sqrt{3}}{7+4 \sqrt{3}}=a+b \sqrt{3}$ (v) $\frac{\sqrt{11}-\sqrt{7}}{\sqrt{11}+\sqrt{7}}=a-b \sqrt{77}$ (vi) $\frac{4+3 \sqrt{5}}{4-3 \sqrt{5}}=a+b \sqrt{5}$ Solution: Given, (i) $\frac{\sqrt{3}-1}{\sqrt{3}+1}=a-b \sqrt{3}$ Rationalizing the denominator by multiply...
Read More →Define degree of a polynomial.
Question: Define degree of a polynomial. Solution: The exponent of the highest degree term in a polynomial is known as its degree. In other words, the highest power of $x$ in a polynomial $f(x)$ is called the degree of the polynomial $f(x)$. For Example: $g(x)=2 x^{2}+3 x+4$ is a polynomial in the variable $x$ of degree 2 ....
Read More →Define a polynomial with real coefficients.
Question: Define a polynomial with real coefficients. Solution: In the polynomial $f(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}$, $a_{n} x^{n}, a_{n-1} x^{n-1}, \cdots, a_{1} x$, and $a_{0}$ are known as the terms of the polynomial and $a_{n}, a_{n-1}, \cdots, a_{1}$ and $a_{0}$ are their real coefficients. For example, $p(x)=3 x-2$ is a polynomial and 3 is a real coefficient...
Read More →Find all the zeros of the polynomial x3 + 3x2 − 2x − 6,
Question: Find all the zeros of the polynomial $x^{3}+3 x^{2}-2 x-6$, if two of its zeros are $-\sqrt{2}$ and $\sqrt{2}$. Solution: We know that if $x=\alpha$ is a zero of a polynomial, and then $x-\alpha$ is a factor of $f(x)$. Since $\sqrt{2}$ and $-\sqrt{2}$ are zeros of $f(x)$. Therefore $(x+\sqrt{2})(x-\sqrt{2})=x^{2}-(\sqrt{2})^{2}$ $=x^{2}-2$ $x^{2}-2$ is a factor of $f(x)$.Now, we divide $x^{3}+3 x^{2}-2 x-6$ by $g(x)=x^{2}-2$ to find the other zeros of $f(x)$. By using division algorith...
Read More →Find all the zeros of the polynomial 2x3 + x2 − 6x − 3,
Question: Find all the zeros of the polynomial $2 x^{3}+x^{2}-6 x-3$, if two of its zeros are $-\sqrt{3}$ and $\sqrt{3}$. Solution: We know that if $x=\alpha$ is a zero of a polynomial, and then $x-\alpha$ is a factor of $f(x)$. Since $\sqrt{3}$ and $-\sqrt{3}$ are zeros of $f(x)$. Therefore $(x+\sqrt{3})(x-\sqrt{3})=x^{2}-3$ $=x^{2}-3$ $x^{2}-3$ is a factor of $f(x)$.Now, we divide $2 x^{3}+x^{2}-6 x-3$ by $g(x)=x^{2}-3$ to find the other zeros of $f(x)$. By using division algorithm we have $f(...
Read More →Find all zeros of the polynomial 2x4 + 7x3 − 19x2 − 14x + 30,
Question: Find all zeros of the polynomial $2 x^{4}+7 x^{3}-19 x^{2}-14 x+30$, if two of its zeros are $\sqrt{2}$ and $-\sqrt{2}$. Solution: We know that if $x=\alpha$ is a zero of a polynomial, and then $x-\alpha$ is a factor of $f(x)$. Since $\sqrt{2}$ and $-\sqrt{2}$ are zeros of $f(x)$. Therefore $(x+\sqrt{2})(x-\sqrt{2})=x^{2}-(\sqrt{2})^{2}$ $=x^{2}-2$ $x^{2}-2$ is a factor of $f(x)$. Now, we divide $2 x^{4}+7 x^{3}-19 x^{2}-14 x+30$ by $g(x)=x^{2}-2$ to find the zero of $f(x)$. By using d...
Read More →Simplify:
Question: Simplify: (i) $\frac{3 \sqrt{2}-2 \sqrt{3}}{3 \sqrt{2}+2 \sqrt{3}}+\frac{\sqrt{12}}{\sqrt{3}-\sqrt{2}}$ (ii) $\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}+\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}$ (iii) $\frac{7+3 \sqrt{5}}{3+\sqrt{5}}-\frac{7-3 \sqrt{5}}{3-\sqrt{5}}$ (iv) $\frac{1}{2+\sqrt{3}}+\frac{2}{\sqrt{5}-\sqrt{3}}+\frac{1}{2-\sqrt{5}}$ (v) $\frac{2}{\sqrt{5}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{2}}-\frac{3}{\sqrt{5}+\sqrt{2}}$ Solution: (i) $\frac{3 \sqrt{2}-2 \sqrt{3}}{3 \sqrt{...
Read More →Find all the zeros of the polynomial x4 + x3 − 34x2 − 4x + 120,
Question: Find all the zeros of the polynomial $x^{4}+x^{3}-34 x^{2}-4 x+120$, if two of its zeros are 2 and $-2$. Solution: We know that if $x=\alpha$ is a zero of a polynomial, then $x-\alpha$ is a factor of $f(x)$. Since, 2 and $-2$ are zeros of $f(x)$. Therefore $(x+2)(x-2)=x^{2}-2^{2}$ $=x^{2}-4$ $x^{2}-4$ is a factor of $f(x)$.Now, we divide $x^{4}+x^{3}-34 x^{2}-4 x+120$ by $g(x)=x^{2}-4$ to find the other zeros of $f(x)$. By using division algorithm we have $f(x)=g(x) \times q(x)-r(x)$ $...
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