Write down the decimal expansions of the following rational numbers by writing their denominators in the form
Question: Write down the decimal expansions of the following rational numbers by writing their denominators in the form 2m 5n, where,m,nare non-negative integers. (i) $\frac{3}{8}$ (ii) $\frac{13}{125}$ (iii) $\frac{7}{80}$ (iv) $\frac{14588}{625}$ (v) $\frac{129}{2^{2} \times 5^{7}}$ [NCERT] Solution: (i) The given number is $\frac{3}{8}$. Clearly, $8=2^{3}$ is of the form $2^{m} \times 5^{n}$, where $m=3$ and $n=0$. So, the given number has terminating decimal expansion. $\therefore \frac{3}{8...
Read More →Discuss with your teacher and find out how to distinguish between
Question: Discuss with your teacher and find out how to distinguish between (a)Plasmid DNA and chromosomal DNA (b)RNA and DNA (c)Exonuclease and Endonuclease Solution: (a)Plasmid DNA and chromosomal DNA (b)RNA and DNA (c)Exonuclease and Endonuclease...
Read More →Find the equation of the circle passing through the points (4, 1)
Question: Find the equation of the circle passing through the points (4, 1) and (6, 5) and whose centre is on the line 4x+y= 16. Solution: Let the equation of the required circle be $(x-h)^{2}+(y-k)^{2}=r^{2}$. Since the circle passes through points (4, 1) and (6, 5), $(4-h)^{2}+(1-k)^{2}=r^{2} .$ $(6-h)^{2}+(5-k)^{2}=r^{2} .$ Since the centre (h, k) of the circle lies on line 4x+y= 16, $4 h+k=16$ (3) From equations (1) and (2), we obtain $(4-h)^{2}+(1-k)^{2}=(6-h)^{2}+(5-k)^{2}$ $\Rightarrow 16...
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Question: Prove that $2 \sqrt{3}-1$ is an irrational number. Solution: Let us assume that $2 \sqrt{3}-1$ is rational .Then, there exist positive co primes $a$ and $b$ such that $2 \sqrt{3}-1=\frac{a}{b}$ $2 \sqrt{3}=\frac{a}{b}+1$ $\sqrt{3}=\frac{\frac{a}{b}+1}{2}$ $\sqrt{3}=\frac{a+b}{2 b}$ This contradicts the fact that $\sqrt{3}$ is an irrational Hence $2 \sqrt{3}-1$ is irrational...
Read More →Explain briefly
Question: Explain briefly (a)PCR (b)Restriction enzymes and DNA (c)Chitinase Solution: (a)PCR: - Polymerase chain reaction (PCR) is a technique in molecular biology to amplify a gene or a piece of DNA to obtain its several copies. It is extensively used in the process of gene manipulation. The process involvesin-vitrosynthesis of sequences using a primer, a template strand, and a thermostable DNA polymerase enzyme obtained from a bacterium, calledThermus aquaticus. The enzyme utilizes building b...
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Question: Prove that $4-5 \sqrt{2}$ is an irrational number. Solution: Let us assume that $4-5 \sqrt{2}$ is rational . Then, there exist positive co primes $a$ and $b$ such that $4-5 \sqrt{2}=\frac{a}{b}$ $5 \sqrt{2}=\frac{a}{b}-4$ $\sqrt{2}=\frac{\frac{a}{b}-4}{5}$ $\sqrt{2}=\frac{a-4 b}{5 b}$ This contradicts the fact that $\sqrt{2}$ is an irrational Hence $4-5 \sqrt{2}$ is irrational...
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Question: Prove that $2-3 \sqrt{5}$ is an irrational number. Solution: Let us assume that $2-3 \sqrt{5}$ is rational .Then, there exist positive co primes $a$ and $b$ such that $2-3 \sqrt{5}=\frac{a}{b}$ $3 \sqrt{5}=\frac{a}{b}-2$ $3 \sqrt{5}=\frac{\frac{a}{b}-2}{3}$ $\sqrt{5}=\frac{a-2 b}{3 b}$ This contradicts the fact that $\sqrt{5}$ is an irrational number Hence $2-3 \sqrt{5}$ is irrational...
Read More →Find the centre and radius of the circle 2x2 + 2y2 – x = 0
Question: Find the centre and radius of the circle $2 x^{2}+2 y^{2}-x=0$ Solution: The equation of the given circle is $2 x^{2}+2 y^{2}-x=0$. $2 x^{2}+2 y^{2}-x=0$ $\Rightarrow\left(2 x^{2}-x\right)+2 y^{2}=0$ $\Rightarrow 2\left[\left(x^{2}-\frac{x}{2}\right)+y^{2}\right]=0$ $\Rightarrow\left\{x^{2}-2 x\left(\frac{1}{4}\right)+\left(\frac{1}{4}\right)^{2}\right\}+y^{2}-\left(\frac{1}{4}\right)^{2}=0$ $\Rightarrow\left(x-\frac{1}{4}\right)^{2}+(y-0)^{2}=\left(\frac{1}{4}\right)^{2}$which is of t...
Read More →Describe briefly the followings:
Question: Describe briefly the followings: (a)Origin of replication (b)Bioreactors (c)Downstream processing Solution: (a)Origin of replication Origin of replication is defined as the DNA sequence in a genome from where replication initiates. The initiation of replication can be either uni-directional or bi-directional. A protein complex recognizes the on site, unwinds the two strands, and initiates the copying of the DNA. (b)Bioreactors Bioreactors are large vessels used for the large-scale prod...
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Question: Show that $5-2 \sqrt{3}$ is an irrational number. Solution: Let us assume that $5-2 \sqrt{3}$ is rational .Then, there exist positive co primes $a$ and $b$ such that $5-2 \sqrt{3}=\frac{a}{b}$ $2 \sqrt{3}=\frac{a}{b}-5$ $\sqrt{3}=\frac{\frac{a}{b}-5}{2}$ $\sqrt{3}=\frac{a-5 b}{2 b}$ This contradicts the fact that $\sqrt{3}$ is an irrational number Hence $5-2 \sqrt{3}$ is irrational...
Read More →Can you think and answer how a reporter enzyme can be used to monitor transformation
Question: Can you think and answer how a reporter enzyme can be used to monitor transformation of host cells by foreign DNA in addition to a selectable marker? Solution: A reporter gene can be used to monitor the transformation of host cells by foreign DNA. They act as a selectable marker to determine whether the host cell has taken up the foreign DNA or the foreign gene gets expressed in the cell. The researchers place the reporter gene and the foreign gene in the same DNA construct. Then, this...
Read More →Differentiate the function with respect to x.
Question: Differentiate the function with respect tox. $(\log x)^{x}+x^{\log x}$ Solution: Let $y=(\log x)^{x}+x^{\log x}$ Also, let $u=(\log x)^{x}$ and $v=x^{\log x}$ $\therefore y=u+v$ $\Rightarrow \frac{d y}{d x}=\frac{d u}{d x}+\frac{d v}{d x}$ ...(1) $u=(\log x)^{x}$ $\Rightarrow \log u=\log \left[(\log x)^{x}\right]$ $\Rightarrow \log u=x \log (\log x)$ Differentiating both sides with respect tox, we obtain $\frac{1}{u} \frac{d u}{d x}=\frac{d}{d x}(x) \times \log (\log x)+x \cdot \frac{d...
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Question: Show that $3+\sqrt{2}$ is an irrational number. Solution: Let us assume that $3+\sqrt{2}$ is rational . Then, there exist positive co primes $a$ and $b$ such that $3+\sqrt{2}=\frac{a}{b}$ $\sqrt{2}=\frac{a}{b}-3$ $\sqrt{2}=\frac{a-3 b}{b}$ This implies, $\sqrt{2}$ is a rational number which is a contradication. Hence $3+\sqrt{2}$ is irrational...
Read More →Can you recall meiosis and indicate at what stage a recombinant DNA is made?
Question: Can you recall meiosis and indicate at what stage a recombinant DNA is made? Solution: Meiosis is a process that involves the reduction in the amount of genetic material. It is two types, namely meiosis I and meiosis II. During the pachytene stage of prophase I, crossing over of chromosomes takes place where the exchange of segments between non-sister chromatids of homlogous chromosomes takes place. This results in the formation of recombinant DNA....
Read More →Find the centre and radius of the circle x2 + y2 – 8x + 10y – 12 = 0
Question: Find the centre and radius of the circle $x^{2}+y^{2}-8 x+10 y-12=0$ Solution: The equation of the given circle is $x^{2}+y^{2}-8 x+10 y-12=0$. $x^{2}+y^{2}-8 x+10 y-12=0$ $\Rightarrow\left(x^{2}-8 x\right)+\left(y^{2}+10 y\right)=12$ $\Rightarrow\left\{x^{2}-2(x)(4)+4^{2}\right\}+\left\{y^{2}+2(y)(5)+5^{2}\right\}-16-25=12$ $\Rightarrow(x-4)^{2}+(y+5)^{2}=53$ $\Rightarrow(x-4)^{2}+\{y-(-5)\}^{2}=(\sqrt{53})^{2}$, which is of the form $(x-h)^{2}+(y-k)^{2}=r^{2}$, where $h=4, k=-5$, and...
Read More →Collect 5 examples of palindromic DNA sequences by consulting your teacher.
Question: Collect 5 examples of palindromic DNA sequences by consulting your teacher. Better try to create a palindromic sequence by following base-pair rules. Solution: The palindromic sequence is a certain sequence of the DNA that reads the same whether read from 5 3direction or from 3 5direction. They are the site for the action of restriction enzymes. Most restriction enzymes are palindromic sequences. Five examples of palindromic sequences are:...
Read More →Find the centre and radius of the circle x2 + y2 – 4x – 8y – 45 = 0
Question: Find the centre and radius of the circle $x^{2}+y^{2}-4 x-8 y-45=0$ Solution: The equation of the given circle is $x^{2}+y^{2}-4 x-8 y-45=0$. $x^{2}+y^{2}-4 x-8 y-45=0$ $\Rightarrow\left(x^{2}-4 x\right)+\left(y^{2}-8 y\right)=45$ $\Rightarrow\left\{x^{2}-2(x)(2)+2^{2}\right\}+\left\{y^{2}-2(y)(4)+4^{2}\right\}-4-16=45$ $\Rightarrow(x-2)^{2}+(y-4)^{2}=65$ $\Rightarrow(x-2)^{2}+(y-4)^{2}=(\sqrt{65})^{2}$, which is of the form $(x-h)^{2}+(y-k)^{2}=r^{2}$, where $h=2, k=4$, and $r=\sqrt{6...
Read More →Besides better aeration and mixing properties,
Question: Besides better aeration and mixing properties, what other advantages do stirred tank bioreactors have over shake flasks? Solution: The shake flask method is used for a small-scale production of biotechnological products in a laboratory. On the other hand, stirred tank bioreactors are used for a large-scale production of biotechnology products. Stirred tank bioreactors have several advantages over shake flasks: (1)Small volumes of culture can be taken out from the reactor for sampling o...
Read More →Find the centre and radius of the circle (x + 5)2 + (y – 3)2 = 36
Question: Find the centre and radius of the circle $(x+5)^{2}+(y-3)^{2}=36$ Solution: The equation of the given circle is $(x+5)^{2}+(y-3)^{2}=36$. $(x+5)^{2}+(y-3)^{2}=36$ $\Rightarrow\{x-(-5)\}^{2}+(y-3)^{2}=6^{2}$, which is of the form $(x-h)^{2}+(y-k)^{2}=r^{2}$, where $h=-5, k=3$, and $r=6$. Thus, the centre of the given circle is (5, 3), while its radius is 6....
Read More →Do eukaryotic cells have restriction endonucleases?
Question: Do eukaryotic cells have restriction endonucleases? Justify your answer. Solution: No, eukaryotic cells do not have restriction endonucleases. This is because the DNA of eukaryotes is highly methylated by a modification enzyme, called methylase. Methylation protects the DNA from the activity of restriction enzymes .These enzymes are present in prokaryotic cells where they help prevent the invasion of DNA by virus....
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Question: Show that $2-\sqrt{3}$ is an irrational number. Solution: Let us assume that $2-\sqrt{3}$ is rational .Then, there exist positive co primes $a$ and $b$ such that $2-\sqrt{3}=\frac{a}{b}$ $\sqrt{3}=2-\frac{a}{b}$ This implies, $\sqrt{3}$ is a rational number, which is a contradiction. Hence, $2-\sqrt{3}$ is irrational number....
Read More →What would be the molar concentration of human DNA in a human cell?
Question: What would be the molar concentration of human DNA in a human cell? Consult your teacher. Solution: The molar concentration of human DNA in a human diploid cell is as follows: $\Rightarrow$ Total number of chromosomes $\times 6.023 \times 10^{23}$ $\Rightarrow 46 \times 6.023 \times 10^{23}$ $\Rightarrow 2.77 \times 10^{18}$ Moles Hence, the molar concentration of DNA in each diploid cell in humans is $2.77 \times 10^{23}$ moles....
Read More →Differentiate the function with respect to x.
Question: Differentiate the function with respect tox. $\left(x+\frac{1}{x}\right)^{x}+x^{\left(1+\frac{1}{x}\right)}$ Solution: Let $y=\left(x+\frac{1}{x}\right)^{x}+x^{\left(1+\frac{1}{x}\right)}$ Also, let $u=\left(x+\frac{1}{x}\right)^{x}$ and $v=x^{\left(1+\frac{1}{x}\right)}$ $\therefore y=u+v$ $\Rightarrow \frac{d y}{d x}=\frac{d u}{d x}+\frac{d v}{d x}$ ...(1) Then, $u=\left(x+\frac{1}{x}\right)^{x}$ $\Rightarrow \log u=\log \left(x+\frac{1}{x}\right)^{x}$ $\Rightarrow \log u=x \log \lef...
Read More →Find the equation of the circle with centre (–a, –b) and radius
Question: Find the equation of the circle with centre (a, b) and radius$\sqrt{a^{2}-b^{2}}$ Solution: The equation of a circle with centre (h,k) and radiusris given as $(x-h)^{2}+(y-k)^{2}=r^{2}$ It is given that centre $(h, k)=(-a,-b)$ and radius $(r)=\sqrt{a^{2}-b^{2}}$. Therefore, the equation of the circle is $(x+a)^{2}+(y+b)^{2}=\left(\sqrt{a^{2}-b^{2}}\right)^{2}$ $x^{2}+2 a x+a^{2}+y^{2}+2 b y+b^{2}=a^{2}-b^{2}$ $x^{2}+y^{2}+2 a x+2 b y+2 b^{2}=0$...
Read More →Prove that following numbers are irrationals:
Question: Prove that following numbers are irrationals: (i) $\frac{2}{\sqrt{7}}$ (ii) $\frac{3}{2 \sqrt{5}}$ (iii) $4+\sqrt{2}$ (iv) $5 \sqrt{2}$ Solution: (i) Let us assume that $\frac{2}{\sqrt{7}}$ is rational . Then, there exist positive co primes a and b such that $\frac{2}{\sqrt{7}}=\frac{a}{b}$ $\sqrt{7}=\frac{2 b}{a}$ $\sqrt{7}$ is rational number which is a contradication. Hence $\frac{2}{\sqrt{7}}$ is irrational (ii) Let us assume that $\frac{3}{2 \sqrt{5}}$ is rational .Then, there exi...
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