Solve the following quadratic equations by factorization:
Question: Solve the following quadratic equations by factorization: $6 x^{2}+11 x+3=0$ Solution: We have been given $6 x^{2}+11 x+3=0$ $6 x^{2}+9 x+2 x+3=0$ $3 x(2 x+3)+1(2 x+3)=0$ $(2 x+3)(3 x+1)=0$ $2 x+3=0$ $x=\frac{-3}{2}$ or, $3 x+1=0$ $x=\frac{-1}{3}$ Hence $x=\frac{-3}{2}$ or $x=\frac{-1}{3}$....
Read More →Zero of the zero polynomial is
Question: Zero of the zero polynomial is(a) 0(b) 1(c) every real number(d) not defined Solution: (d) not definedZero of the zero polynomial isnot defined....
Read More →Solve the following quadratic equations by factorization:
Question: Solve the following quadratic equations by factorization: $6 x^{2}-x-2=0$ Solution: We have been given $6 x^{2}-x-2=0$ Now we solve the above quadratic equation. Therefore, $6 x^{2}-4 x+3 x-2=0$ $2 x(3 x-2)+1(3 x-2)=0$ $(2 x+1)(3 x-2)=0$ Now, one of the products must be equal to zero for the whole product to be zero. Hence we equate both the products to zero in order to find the value ofx. Therefore, $2 x+1=0$ $2 x=-1$ $x=\frac{-1}{2}$ Or $3 x-2=0$ $3 x=2$ $x=\frac{2}{3}$ Hence, $x=\fr...
Read More →Degree of the zero polynomial is
Question: Degree of the zero polynomial is(a) 1(b) 0(c) not defined(d) none of these Solution: (c) not definedDegree of the zero polynomial isnot defined....
Read More →Show that the function f : R − {3} → R − {2} given by
Question: Show that the function $f: R-\{3\} \rightarrow R-\{2\}$ given by $f(x)=\frac{x-2}{x-3}$ is a bijection. Solution: $f: R-\{3\} \rightarrow R-\{2\}$ given by $f(x)=\frac{x-2}{x-3}$ Injectivity:Letxandybe any two elements in the domain (R {3}), such thatf(x) = f(y). $f(x)=f(y)$ $\Rightarrow \frac{x-2}{x-3}=\frac{y-2}{y-3}$ $\Rightarrow(x-2)(y-3)=(y-2)(x-3)$ $\Rightarrow x y-3 x-2 y+6=x y-3 y-2 x+6$ $\Rightarrow x=y$ So,fis one-one Surjectivity:Letybe any element in the co-domain (R {2}), ...
Read More →If the complex number z=x+iy satisfies the condition
Question: If the complex number $z=x+i y$ satisfies the condition $|z+1|=1$, then $z$ lies on (a)xaxis (b) circle with centre (1, 0) and radius 1 (c)yaxis (d) none of these Solution: $|z+1|=1$ $\Rightarrow|z+1|^{2}=1^{2}$ $\Rightarrow(z+1) \overline{(z+1)}=1$ $\Rightarrow(z+1)(\bar{z}+1)=1$ $\Rightarrow z \bar{z}+z+\bar{z}+1=1$ $\Rightarrow z \bar{z}+z+\bar{z}=0$ Since, $z=x+i y$ $\therefore z \bar{z}+z+\bar{z}=0$ $\Rightarrow(x+i y)(x-i y)+x+i y+x-i y=0$ $\Rightarrow x^{2}+y^{2}+2 x=0$ $\Righta...
Read More →Solve the following quadratic equations by factorization:
Question: Solve the following quadratic equations by factorization: $9 x^{2}-3 x-2=0$ Solution: We have been given, $9 x^{2}-3 x-2=0$ $9 x^{2}-6 x+3 x-2=0$ $3 x(3 x-2)+1(3 x-2)=0$ $(3 x+1)(3 x-2)=0$ Therefore, $3 x+1=0$ $3 x=-1$ $x=\frac{-1}{3}$ or, $3 x-2=0$ $3 x=2$ $x=\frac{2}{3}$ Hence, $x=\frac{-1}{3}$ or $x=\frac{2}{3}$....
Read More →Solve this
Question: $\sqrt{3}$ is a polynomial of degree (a) $\frac{1}{2}$ (b) 2 (c) 1 (d) 0 Solution: (d) 0 $\sqrt{3}$ is a constant term, so it is a polynomial of degree 0 ....
Read More →Which of the following is correct for any two complex numbers
Question: Which of the following is correct for any two complex numbersz1andz2? (a) $\left|z_{1} z_{2}\right|=\left|z_{1}\right|\left|z_{2}\right|$ (b) $\arg \left(z_{1} z_{2}\right)=\arg \left(z_{1}\right) \arg \left(z_{2}\right)$ (c) $\left|z_{1}+z_{2}\right|=\left|z_{1}\right|+\left|z_{2}\right|$ (d) $\left|z_{1}+z_{2}\right| \geq\left|z_{1}\right|+\left|z_{2}\right|$ Solution: Since, it is known that $\left|z_{1} z_{2}\right|=\left|z_{1}\right|\left|z_{2}\right|$ $\arg \left(z_{1} z_{2}\righ...
Read More →Solve the following quadratic equations by factorization:
Question: Solve the following quadratic equations by factorization: $4 x^{2}+5 x=0$ Solution: We have been given, $4 x^{2}+5 x=0$ Therefore we have, $x(4 x+5)=0$ Now, one of the products must be equal to zero for the whole product to be zero. Hence we equate both the products to zero in order to find the value ofx. Therefore, $x=0$ Or $4 x+5=0$ $4 x=-5$ $x=\frac{-5}{4}$ Hence, $x=0$ or $x=\frac{-5}{4}$....
Read More →Which of the following is correct for any two complex numbers
Question: Which of the following is correct for any two complex numbersz1andz2? (a) $\left|z_{1} z_{2}\right|=\left|z_{1}\right|\left|z_{2}\right|$ (b) $\arg \left(z_{1} z_{2}\right)=\arg \left(z_{1}\right) \arg \left(z_{2}\right)$ (c) $\left|z_{1}+z_{2}\right|=\left|z_{1}\right|+\left|z_{2}\right|$ (d) $\left|z_{1}+z_{2}\right| \geq\left|z_{1}\right|+\left|z_{2}\right|$ Solution: Since, it is known that $\left|z_{1} z_{2}\right|=\left|z_{1}\right|\left|z_{2}\right|$ $\arg \left(z_{1} z_{2}\righ...
Read More →If z is a complex number, then
Question: Ifzis a complex number,then (a) $|z|^{2}|z|^{2}$ (b) $|z|^{2}=|z|^{2}$ (c) $|z|^{2}|z|^{2}$ (d) $|z|^{2} \geq|z|^{2}$ Solution: It is obvious that, for any complex numberz, $|z|^{2}=|z|^{2}$ Hence, the correct option is (b)....
Read More →If z is a complex number, then
Question: Ifzis a complex number,then (a) $|z|^{2}|z|^{2}$ (b) $|z|^{2}=|z|^{2}$ (c) $|z|^{2}|z|^{2}$ (d) $|z|^{2} \geq|z|^{2}$ Solution: It is obvious that, for any complex numberz, $|z|^{2}=|z|^{2}$ Hence, the correct option is (b)....
Read More →Solve the following quadratic equations by factorization:
Question: Solve the following quadratic equations by factorization: (2x+ 3)(3x 7) = 0 Solution: We have been given, $(2 x+3)(3 x-7)=0$ Therefore, $(2 x+3)=0$ $2 x=-3$ $x=\frac{-3}{2}$ or $(3 x-7)=0$ $3 x=7$ $x=\frac{7}{3}$ Therefore, $x=\frac{-3}{2}$ or $x=\frac{7}{3}$....
Read More →Which of the following is a binomial?
Question: Which of the following is a binomial? (a) $x^{2}+x+3$ (b) $x^{2}+4$ (c) $2 x^{2}$ (d) $x+3+\frac{1}{x}$ Solution: (b) $x^{2}+4$ Clearly, $x^{2}+4$ is an expression having two non-zero terms. So, it is a binomial....
Read More →If f : A → B is an injection, such that range of f = {a}, determine the number of elements in A.
Question: Iff:ABis an injection, such that range off= {a}, determine the number of elements inA. Solution: Range off= {a}So, the number of images off= 1Since,fis an injection, there will be exactly one image for each element off.So, number of elements inA= 1....
Read More →The complex number z which satisfies the condition
Question: The complex number $z$ which satisfies the condition $\left|\frac{i+z}{i-z}\right|=1$ lies on (a) circlex2+y2= 1 (b) thexaxis (c)theyaxis (d) the linex+y= 1 Solution: $\left|\frac{i+z}{i-z}\right|=1$ $\Rightarrow\left|\frac{i+z}{i-z}\right|^{2}=1^{2}$ $\Rightarrow\left(\frac{i+z}{i-z}\right) \overline{\left(\frac{i+z}{i-z}\right)}=1$ $\Rightarrow\left(\frac{i+z}{i-z}\right)\left(\frac{-i+\bar{z}}{-i-\bar{z}}\right)=1$ $\Rightarrow\left(\frac{-i^{2}-z i+\bar{z} i+z \bar{z}}{-i^{2}+z i-\...
Read More →Which of the following is a linear polynomial?
Question: Which of the following is a linear polynomial? (a) $x+x^{2}$ (b) $x+1$ (c) $5 x^{2}-x+3$ (d) $x+\frac{1}{x}$ Solution: (b) $x+1$ Clearly, $x+1$ is a polynomial of degree 1 . So, it is a linear polynomial....
Read More →Solve the following quadratic equations by factorization:
Question: Solve the following quadratic equations by factorization: (x 4) (x+ 2) = 0 Solution: We have been given, $(x-4)(x+2)=0$ Therefore, $(x-4)=0$ $x=4$ or $(x+2)=0$ $x=-2$ Therefore, $x=4$ or $x=-2$....
Read More →A real value of x satisfies the equation
Question: A real value of $x$ satisfies the equation $\frac{3-4 i x}{3+4 i x}=a-i b(a, b \in \mathbb{R})$, if $a^{2}+b^{2}=$ (a) 1 (b) 1 (c) 2 (d) 2 Solution: $a-i b=\frac{3-4 i x}{3+4 i x}$ $=\frac{3-4 i x}{3+4 i x} \times \frac{3-4 i x}{3-4 i x}$ $=\frac{9+16 x^{2} i^{2}-24 x i}{9-16 x^{2} i^{2}}$ $=\frac{\left(9-16 x^{2}\right)-i(24 x)}{9+16 x^{2}}$ $\Rightarrow|a-i b|^{2}=\left|\frac{\left(9-16 x^{2}\right)-i(24 x)}{9+16 x^{2}}\right|^{2}$ $\Rightarrow a^{2}+b^{2}=\frac{\left(9-16 x^{2}\righ...
Read More →Which of the following is quadratic polynomial?
Question: Which of the following is quadratic polynomial? (a) $x+4$ (b) $x^{3}+x$ (c) $x^{3}+2 x+6$ (d) $x^{2}+5 x+4$ Solution: (d) $x^{2}+5 x+4$ $x^{2}+5 x+4$ is a polynomial of degree 2 . So, it is a quadratic polynomial....
Read More →Which of the following is a polynomial?
Question: Which of the following is a polynomial? (a) $x^{-2}+x^{-1}+3$ (b) $x+x^{-1}+2$ (c) $x^{-1}$ (d) 0 Solution: (d) 00 is a polynomial whose degree is not defined....
Read More →Solve the following
Question: If $f(z)=\frac{7-z}{1-z^{2}}$, where $z=1+2 i$, then $|f(z)|$ is (a) $\frac{|z|}{2}$ (b) $|z|$ (c) $2|z|$ (d) none of these Solution: $f(z)=\frac{7-z}{1-z^{2}}$ $=\frac{7-(1+2 i)}{1-(1+2 i)^{2}}$ $=\frac{7-1-2 i}{1-\left(1^{2}+2^{2} i^{2}+4 i\right)}$ $=\frac{6-2 i}{1-1+4-4 i}$ $=\frac{6-2 i}{4-4 i}$ $=\frac{6-2 i}{4-4 i} \times \frac{4+4 i}{4+4 i}$ $=\frac{24+24 i-8 i-8 i^{2}}{4^{2}-4^{2} i^{2}}$ $=\frac{24+16 i+8}{16+16}$ $=\frac{32+16 i}{32}$ $=1+\frac{1}{2} i$ Since $z=1+2 i$ $\the...
Read More →Classify the following functions as injection, surjection or bijection :
Question: Classify the following functions as injection, surjection or bijection : (i) $f: \mathbf{N} \rightarrow \mathbf{N}$ given by $f(x)=x^{2}$ (ii) $f: \mathbf{Z} \rightarrow \mathbf{Z}$ given by $f(x)=x^{2}$ (iii) $f: \mathbf{N} \rightarrow \mathbf{N}$ given by $f(x)=x^{3}$ (iv) $f: \mathbf{Z} \rightarrow \mathbf{Z}$ given by $f(x)=x^{3}$ (v) $f: \mathbf{R} \rightarrow \mathbf{R}$, defined by $f(x)=|x|$ (vi) $f: \boldsymbol{Z} \rightarrow \boldsymbol{Z}$, defined by $f(x)=x^{2}+x$ (vii) $f...
Read More →Which of the following is a polynimial?
Question: Which of the following is a polynimial? (a) $x-\frac{1}{x}+2$ (b) $\frac{1}{x}+5$ (c) $\sqrt{x}+3$ (d) $-4$ Solution: (d) 4 -4 is a constant polynomial of degree zero....
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