If x2 + 1/x2 = 79, find the value of x + 1/x
Question: If $x^{2}+1 / x^{2}=79$, find the value of $x+1 / x$ Solution: We have, $(x+1 / x)^{2}=x^{2}+(1 / x)^{2}+2 * x * 1 / x$ $\Rightarrow(x+1 / x)^{2}=x^{2}+1 / x^{2}+2$ $\Rightarrow(x+1 / x)^{2}=79+2 \quad\left[\therefore x^{2}+1 / x^{2}=79\right]$ $\Rightarrow(x+1 / x)^{2}=81$ $\Rightarrow(x+1 / x)^{2}=(\pm 9)^{2}$ $\Rightarrow x+1 / x=\pm 9$...
Read More →If α, β are the zeros of a polynomial such that α + β = −6 and αβ = −4,
Question: If $\alpha, \beta$ are the zeros of a polynomial such that $\alpha+\beta=-6$ and $\alpha \beta=-4$, then write the polynomial. Solution: Let S and P denotes respectively the sum and product of the zeros of a polynomial We are given $S=-6$ and $P=-4$. Then The required polynomial $g(x)$ is given by $g(x)=x^{2}-S x+P$ $g(x)=x^{2}-(-6) x+(-4)$ $=x^{2}+6 x-4$ Hence, the polynomial is $x^{2}+6 x-4$...
Read More →If x2 + 1/x2 = 66, find the value of x − 1/x
Question: If $x^{2}+1 / x^{2}=66$, find the value of $x-1 / x$ Solution: We have, $(x-1 / x)^{2}=x^{2}+(1 / x)^{2}-2 * x * 1 / x$ $\Rightarrow(x-1 / x)^{2}=x^{2}+1 / x^{2}-2$ $\Rightarrow(x-1 / x)^{2}=66-2\left[\therefore x^{2}+1 / x^{2}=66\right]$ $\Rightarrow(x-1 / x)^{2}=64$ $\Rightarrow(x-1 / x)^{2}=(\pm 8)^{2}$ $\Rightarrow x-1 / x=\pm 8$...
Read More →If 1 is a zero of the polynomial p(x) = ax2 − 3(a − 1) x − 1,
Question: If 1 is a zero of the polynomial $p(x)=a x^{2}-3(a-1) x-1$, then find the value of $a$. Solution: We know that if $x=\alpha$ is a zero of polynomial then $x-\alpha$ is a factor of $p(x)$ Since 1 is zero of $p(x)$ Therefore, $x-1$ is a factor of $p(x)$ Now, we divide $p(x)=a x^{2}-3(a-1) x-1$ by $x-1$. Now, Remainder $=0$ $-2 a+2=0$ $-2 a=-2$ $a=\frac{-2}{-2}$ $a=1$ Hence, the value of a is 1...
Read More →For what value of k, −4 is a zero of the polynomial x2 − x − (2k + 2)?
Question: For what value of $k,-4$ is a zero of the polynomial $x^{2}-x-(2 k+2) ?$ Solution: We know that if $x=\alpha$ is zero polynomial then $x-2$ is a factor of $f(x)$ Since $-4$ is zero of $f(x)$ Therefore $x+4$ is a factor of $f(x)$ Now, we divide $f(x)=x^{2}-x-(2 k+2)$ by $g(x)=x+4$ to find the value of $k$ Now, Remainder $=0$ $-2 k+18=0$ $-2 k=-18$ $k=\frac{-18}{-2}$ $k=9$ Hence, the value of $k$ is 9...
Read More →Solve this
Question: If $x+\frac{1}{x}=\sqrt{5}$, find the value of $x^{2}+\frac{1}{x^{2}}$ and $x^{4}+\frac{1}{x^{4}}$ Solution: We have, $(x+1 / x)^{2}=x^{2}+(1 / x)^{2}+2 * x * 1 / x$ $\Rightarrow(x+1 / x)^{2}=x^{2}+1 / x^{2}+2$ $\Rightarrow(\sqrt{5})^{2}=x^{2}+\frac{1}{x^{2}}+2\left[\therefore x+\frac{1}{x}=\sqrt{5}\right]$ $\Rightarrow 5=x^{2}+1 / x^{2}+2$ $\Rightarrow x^{2}+1 / x^{2}=3 \ldots(1)$ Now, $\left(x^{2}+1 / x^{2}\right)^{2}=x^{4}+1 / x^{4}+2 * x^{2} * 1 / x^{2}$ $\Rightarrow\left(x^{2}+1 \...
Read More →If (x + a) is a factor of 2x2 + 2ax + 5x + 10, find a.
Question: If $(x+a)$ is a factor of $2 x^{2}+2 a x+5 x+10$, find $a .$ Solution: Given $(x+a)$ is a factor of $f(x)=2 x^{2}+2 a x+5 x+10$. Let us now divide $f(x)$ by $x+a$. We have, Now, remainder $=0$ $10-5 a=0$ $10=5 a$ $\frac{10}{5}=a$ $2=a$ Hence, the value of $a$ is 2...
Read More →If x − 1/x = −1, find the value of x2 + 1/x2
Question: If $x-1 / x=-1$, find the value of $x^{2}+1 / x^{2}$ Solution: We have, $x-1 / x=-1$ Now, $(x-1 / x)^{2}=x^{2}+(1 / x)^{2}-2 * x * 1 / x$ $\Rightarrow(x-1 / x)^{2}=x^{2}+1 / x^{2}-2$ $\Rightarrow(-1)^{2}=x^{2}+1 / x^{2}-2[\therefore x-1 / x=-1]$ $\Rightarrow 2+1=x^{2}+1 / x^{2}$ $\Rightarrow x^{2}+1 / x^{2}=3$...
Read More →Write the zeros of the polynomial x2 − x − 6.
Question: Write the zeros of the polynomial $x^{2}-x-6$ Solution: We have to find the zeros of the polynomial $x^{2}-x-6$ $f(x)=x^{2}-x-6$ $f(x)=x^{2}-3 x+2 x-6$ $f(x)=x(x-3)+2(x-3)$ $f(x)=(x+2)(x-3)$ We know that if $(x-\alpha)$ is a factor of $f(x)$ then $x=\alpha$ is a zero of polynomial Therefore we have $x+2=0$ $x=-2$ Also $x-3=0$ $x=3$ Hence, the zeros of polynomial $x^{2}-x-6$ is $3,-2$...
Read More →If x + 1/x = 11, find the value of x2 + 1/x2
Question: If $x+1 / x=11$, find the value of $x^{2}+1 / x^{2}$ Solution: We have, $x+1 / x=11$ Now, $(x+1 / x)^{2}=x^{2}+(1 / x)^{2}+2 * x * 1 / x$ $\Rightarrow(x+1 / x)^{2}=x / 2+1 / x^{2}+2$ $\Rightarrow(11)^{2}=x^{2}+1 / x^{2}+2[? x+1 / x=11]$ $\Rightarrow 121=x^{2}+1 / x^{2}+2$ $\Rightarrow x^{2}+1 / x^{2}=119$...
Read More →Write the coefficient of the polynomial
Question: Write the coefficient of the polynomial $p(z)=z^{5}-2 z^{2}+4$ Solution: We have to find the co-efficient of the polynomial $p(z)=z^{5}-2 z^{2}+4$ Co-efficient of $z^{5}=1$ Co-efficient of $z^{4}=0$ Co-efficient of $z^{3}=0$ Co-efficient of $z^{2}=-2$ Co-efficient of $z=0$ Constant term $=4$ Hence, the co-efficient of $z^{5}, z^{4}, z^{3}, z^{2}, z$ and constant term is $1,0,0,-2,0,4$...
Read More →Simplify each of the following:
Question: Simplify each of the following: (i) $175 \times 175+2 \times 175 \times 25+25 \times 25$ (ii) $322 \times 322-2 \times 322 \times 22+22 \times 22$ (iii) $0.76 \times 0.76+2 \times 0.76 \times 0.24+0.24 \times 0.24$ (iv) $\frac{7.83 * 7.83-1.17 * 1.17}{6.66}$ Solution: (i) We have (ii) We have, $322 \times 322-2 \times 322 \times 22+22 \times 22$ $=(322-22)^{2} \quad\left[a^{2}+b^{2}-2 a b=(a-b)^{2}\right]$ $=(300)^{2} \quad[$ Where $a=322$ and $b=22]$ $=90000$ Therefore, 322 322 - 2 32...
Read More →If a − b, a and b are zeros of the polynomial
Question: If $a-b, a$ and $b$ are zeros of the polynomial $f(x)=2 x^{3}-6 x^{2}+5 x-7$, write the value of $a$. Solution: Let $a-b, a$ and $a+b$ be the zeros of the polynomial $f(x)=2 x^{3}-6 x^{2}+5 x-7$ then Sum of the zeros $=\frac{-\text { Coefficient of } x^{2}}{\text { Coefficient of } x^{3}}$ $(a-d)+a+(a+d)=-\left(\frac{-6}{2}\right)$ $a+a+a-\mu+\mu=\frac{6}{2}$ $3 a=3$ $a=\frac{3}{3}$ $a=1$ Hence, the value of $a$ is 1 ....
Read More →If f(x) = x3 + x2 − ax + b is divisible
Question: If $l(x)=x^{3}+x^{2}-a x+b$ is divisible by $x^{2}-x$ write the value of $a$ and $b$. Solution: We are given $f(x)=x^{3}+x^{2}-a x+b$ is exactly divisible by $x^{2}-x$ then the remainder should be zero Therefore Quotient $=x+2$ and Remainder $=x(2-a)+b$ Now, Remainder $=0$ $x(2-a)+b=0$ $x(2-a)+b=0 x+0$ Equating coefficient of $\mathrm{x}$, we get $2-a=0$ $2=a$ Equating constant term $b=0$ Hence, the value of $a$ and $b$ are $a=2, b=0$...
Read More →If fourth degree polynomial is divided by a quadratic polynomial,
Question: If fourth degree polynomial is divided by a quadratic polynomial, write the degree of the remainder. Solution: Here $f(x)$ represent dividend and $g(x)$ represent divisor. $g(x)=$ quadratic polynomial $g(x)=a x^{2}+b x+c$ Therefore degree of $(f(x))=4$ Degree of $(g(x))=2$ The quotient $\mathrm{q}(\mathrm{x})$ is of degree $2(=4-2)$ The remainder $r(x)$ is of degree 1 or less. Hence, the degree of the remainder is equal to 1 or less than 1...
Read More →Evaluate each of the following using identities:
Question: Evaluate each of the following using identities: (i) $(399)^{2}$ (ii) $(0.98)^{2}$ (iii) $991 \times 1009$ (iv) $117 \times 83$ Solution: (i) We have, $399^{2}=(400-1)^{2}$ $=(400)^{2}+(1)^{2}-2 \times 400 \times 1 \quad\left[(a-b)^{2}=a^{2}+b^{2}-2 a b\right]$ Where, a = 400 and b = 1 = 160000 + 1 - 8000 = 159201 Therefore, $(399)^{2}=159201$. (ii) We have, $(0.98)^{2}=(1-0.02)^{2}$ $=(1)^{2}+(0.02)^{2}-2 \times 1 \times 0.02$ = 1 + 0.0004 - 0.04 [Where, a = 1 and b = 0.02] = 1.0004 -...
Read More →Write a quadratic polynomial,
Question: Write a quadratic polynomial, sum of whose zeros is $2 \sqrt{3}$ and their product is 2 . Solution: Let $S$ and $P$ denotes respectively the sum and product of the zeros of a polynomial are $2 \sqrt{3}$ and 2 . The required polynomial $g(x)$ is given by $g(x)=k\left(x^{2}-S x+P\right)$ $g(x)=k\left(x^{2}-2 \sqrt{3} x+2\right)$ Hence, the quadratic polynomial is $g(x)=k\left(x^{2}-2 \sqrt{3} x+2\right)$ where $k$ is any non-zeros real number....
Read More →Give an example of polynomials f(x), g(x),
Question: Give an example of polynomialsf(x),g(x),q(x) andr(x) satisfyingf(x) =g(x),q(x) +r(x), where degreer(x) = 0. Solution: Using division algorithm, we have $f(x)=g(x) \times q(x)+r(x)$ $x^{5}-4 x^{3}+x^{2}+3 x+1=\left(x^{3}-3 x+1\right)\left(x^{2}-1\right)+2$ $x^{5}-4 x^{3}+x^{2}+3 x+1=x^{5}-3 x^{3}+x^{2}-x^{3}+3 x-1+2$ $x^{5}-4 x^{3}+x^{2}+3 x+1=x^{5}-3 x^{3}-x^{3}+x^{2}+3 x-1+2$ $x^{5}-4 x^{3}+x^{2}+3 x+1=x^{5}-4 x^{3}+x^{2}+3 x+1$ Hence an example for polynomial $f(x), g(x), q(x)$ and $...
Read More →Evaluate each of the following using identities:
Question: Evaluate each of the following using identities: (i) $(2 x-1 / x)^{2}$ (ii) $(2 x+y)(2 x-y)$ (iii) $\left(a^{2} b-a b^{2}\right)^{2}$ (iv) $(a-0.1)(a+0.1)$ (v) $\left(1.5 x^{2}-0.3 y^{2}\right)\left(1.5 x^{2}+0.3 y^{2}\right)$ Solution: (i) Given, $(2 x-1 / x)^{2}=(2 x)^{2}+(1 / x)^{2}-2 * 2 x * 1 / x$ $(2 x-1 / x)^{2}=4 x^{2}+1 / x^{2}-4 \quad\left[\therefore(a-b)^{2}=a^{2}+b^{2}-2 a b\right]$ Where, $a=2 x, b=1 / x$ $\therefore(2 x-1 / x)^{2}=4 x^{2}+1 / x^{2}-4$ (ii) Given, $(2 x+y)...
Read More →State division algorithm for polynomials.
Question: State division algorithm for polynomials. Solution: If $f(x)$ and $g(x)$ are any two polynomials with $g(x) \neq 0$, then we can always find polynomials $q(x)$ and $r(x)$ such that $f(x)=q(x) g(x)+r(x)$, where $r(x)=0$ or degree $r(x)$ degree $g(x)$...
Read More →Iffind the value of
Question: If $x=\frac{\sqrt{3}+1}{2}$, find the value of $4 x^{3}+2 x^{2}-8 x+7$ Solution: Given, $x=\frac{\sqrt{3}+1}{2}$, and given to find the value of $4 x^{3}+2 x^{2}-8 x+7$ $2 x=\sqrt{3}+1$ $2 x-1=\sqrt{3}$ Now, squaring on both the sides, we get, $(2 x-1)^{2}=3$ $4 x^{2}-4 x+1=3$ $4 x^{2}-4 x+1-3=0$ $4 x^{2}-4 x-2=0$ $2 x^{2}-2 x-1=0$ Now taking $4 x^{3}+2 x^{2}-8 x+7$ $2 x\left(2 x^{2}-2 x-1\right)+4 x^{2}+2 x+2 x^{2}-8 x+7$ $2 x\left(2 x^{2}-2 x-1\right)+6 x^{2}-6 x+7$ As, $2 x^{2}-2 x-...
Read More →If x = 1 is a zero of the polynomial f(x) = x3 − 2x2 + 4x + k,
Question: If $x=1$ is a zero of the polynomial $f(x)=x^{3}-2 x^{2}+4 x+k$, write the value of $k$. Solution: We have to find the value of $K$ if $x=1$ is a zero of the polynomial $f(x)=x^{3}-2 x^{2}+4 x+k$. $f(x)=x^{3}-2 x^{2}+4 x+k$ $f(1)=1^{3}-2(1)^{2}+4 \times 1+k$ $0=1-2+4+k$ $0=5-2+k$ $0=3+k$ $-3=k$ Hence, the value of $k$ is $k=-3$...
Read More →The graph of a polynomial f(x) is as shown in Fig. 2.21.
Question: The graph of a polynomialf(x) is as shown in Fig. 2.21. Write the number of real zeros off(x). Solution: The graph of a polynomial $f(x)$ touches $x$-axis at two points We know that if a curve touches the $x$-axis at two points then it has two common zeros of $f(x)$. Hence the number of zeros of $f(x)$, in this case is 2 ....
Read More →Find the values of each of the following correct to three places of decimals, it being given that
Question: Find the values of each of the following correct to three places of decimals, it being given that $\sqrt{2}=1.414, \sqrt{3}=1.732, \sqrt{5}=2.236, \sqrt{6}=2.4495, \sqrt{10}=3.162$ (i) $\frac{3-\sqrt{5}}{3+2 \sqrt{5}}$ (ii) $\frac{1+\sqrt{2}}{3-2 \sqrt{2}}$ Solution: (i) $\frac{3-\sqrt{5}}{3+2 \sqrt{5}}$ Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor $3-2 \sqrt{5}$ $=\frac{(3-\sqrt{5})(3-2 \sqrt{5})}{(3+2 \sqrt{5})(3+2 \sqrt{5}...
Read More →In Q. No. 15, write the sign of c.
Question: In Q. No. 15, write the sign ofc. Solution: The parabola $y=a x^{2}+b x+c$ cuts $y$-axis at P which lies on OY. Putting $x=0$ in $y=a x^{2}+b x+c$, we get $y=c$. So the coordinates of $\mathrm{P}$ are $(0, c)$. Clearly, $\mathrm{P}$ lies on $O Y^{\prime}$. Therefore $c0$...
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