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...
Read More →From what you have learnt, can you tell whether enzymes are bigger or DNA is bigger in molecular size?
Question: From what you have learnt, can you tell whether enzymes are bigger or DNA is bigger in molecular size? How did you know? Solution: Enzymes are smaller in size than DNA molecules. This is because DNA contains genetic information for the development and functioning of all living organisms. It contains instructions for the synthesis of proteins and DNA molecules. On the other hand, enzymes are proteins which are synthesised from a small stretch of DNA known as genes, which are involved in...
Read More →Make a chart (with diagrammatic representation) showing a restriction enzyme,
Question: Make a chart (with diagrammatic representation) showing a restriction enzyme, the substrate DNA on which it acts, the site at which it cuts DNA and the product it produces. Solution: The name of the restriction enzyme is Bam H 1....
Read More →Can you list 10 recombinant proteins which are used in medical practice?
Question: Can you list 10 recombinant proteins which are used in medical practice? Find out where they are used as therapeutics (use the internet). Solution: Recombinant proteins are obtained from the recombinant DNA technology. This technology involves the transfer of specific genes from an organism into another organism using vectors and restriction enzymes as molecular tools. Ten recombinant proteins used in medical practice are...
Read More →Prove that
Question: Prove that $\sqrt{3}+\sqrt{4}$ is an irrational number. Solution: Let us assume that $\sqrt{3}+\sqrt{4}$ is rational .Then, there exist positive co primes $a$ and $b$ such that $\sqrt{3}+\sqrt{4}=\frac{a}{b}$ $\sqrt{4}=\frac{a}{b}-\sqrt{3}$ $(\sqrt{4})^{2}=\left(\frac{a}{b}-\sqrt{3}\right)^{2}$ $4-3=\left(\frac{a}{b}\right)^{2}-\frac{2 a \sqrt{3}}{b}$ $4-3=\left(\frac{a}{b}\right)^{2}-\frac{2 a \sqrt{3}}{b}$ $1=\left(\frac{a}{b}\right)^{2}-\frac{2 a \sqrt{3}}{b}$ $\left(\frac{a}{b}\rig...
Read More →Find the equation of the circle with centre (1, 1) and radius
Question: Find the equation of the circle with centre $(1,1)$ and radius $\sqrt{2}$ 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 $(h, k)=(1,1)$ and radius $(r)=\sqrt{2}$. Therefore, the equation of the circle is $(x-1)^{2}+(y-1)^{2}=(\sqrt{2})^{2}$ $x^{2}-2 x+1+v^{2}-2 y+1=2$ $x^{2}+y^{2}-2 x-2 y=0$...
Read More →Differentiate the function with respect to x.
Question: Differentiate the function with respect tox. $(x+3)^{2} \cdot(x+4)^{3} \cdot(x+5)^{4}$ Solution: Let $y=(x+3)^{2} \cdot(x+4)^{3} \cdot(x+5)^{4}$ Taking logarithm on both the sides, we obtain $\log y=\log (x+3)^{2}+\log (x+4)^{3}+\log (x+5)^{4}$ $\Rightarrow \log y=2 \log (x+3)+3 \log (x+4)+4 \log (x+5)$ Differentiating both sides with respect tox, we obtain $\frac{1}{y} \cdot \frac{d y}{d x}=2 \cdot \frac{1}{x+3} \cdot \frac{d}{d x}(x+3)+3 \cdot \frac{1}{x+4} \cdot \frac{d}{d x}(x+4)+4...
Read More →Find the equation of the circle with centreand radius
Question: Find the equation of the circle with centre $\left(\frac{1}{2}, \frac{1}{4}\right)$ and radius $\frac{1}{12}$ 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 $(h, k)=\left(\frac{1}{2}, \frac{1}{4}\right)$ and radius $(r)=\frac{1}{12}$. Therefore, the equation of the circle is $\left(x-\frac{1}{2}\right)^{2}+\left(y-\frac{1}{4}\right)^{2}=\left(\frac{1}{12}\right)^{2}$ $x^{2}-x+\frac{1}{4}+y^{2}-\frac{y}{2}+...
Read More →If p, q are prime positive integers,
Question: If $p, q$ are prime positive integers, prove that $\sqrt{p}+\sqrt{q}$ is an irrational number. Solution: Let us assume that $\sqrt{p}+\sqrt{q}$ is rational. Then, there exist positive co primes $a$ and $b$ such that $\sqrt{p}+\sqrt{q}=\frac{a}{b}$ $\sqrt{p}=\frac{a}{b}-\sqrt{q}$ $(\sqrt{p})^{2}=\left(\frac{a}{b}-\sqrt{q}\right)^{2}$ $p=\left(\frac{a}{b}\right)^{2}-\frac{2 a \sqrt{q}}{b}+q$ $p-q=\left(\frac{a}{b}\right)^{2}-\frac{2 a \sqrt{q}}{b}$ $p-q=\left(\frac{a}{b}\right)^{2}-\frac...
Read More →Differentiate the function with respect to x.
Question: Differentiate the function with respect tox. $(x+3)^{2} \cdot(x+4)^{3} \cdot(x+5)^{4}$ Solution: Let $y=(x+3)^{2} \cdot(x+4)^{3} \cdot(x+5)^{4}$ Taking logarithm on both the sides, we obtain $\log y=\log (x+3)^{2}+\log (x+4)^{3}+\log (x+5)^{4}$ $\Rightarrow \log y=2 \log (x+3)+3 \log (x+4)+4 \log (x+5)$ Differentiating both sides with respect tox, we obtain $\frac{1}{y} \cdot \frac{d y}{d x}=2 \cdot \frac{1}{x+3} \cdot \frac{d}{d x}(x+3)+3 \cdot \frac{1}{x+4} \cdot \frac{d}{d x}(x+4)+4...
Read More →How do biofertilisers enrich the fertility of the soil?
Question: How do biofertilisers enrich the fertility of the soil? Solution: Bio-fertilizers are living organisms which help in increasing the fertility of soil. It involves the selection of beneficial micro-organisms that help in improving plant growth through the supply of plant nutrients. These are introduced to seeds, roots, or soil to mobilize the availability of nutrients by their biological activity. Thus, they are extremely beneficial in enriching the soil with organic nutrients. Many spe...
Read More →Arrange the following in the decreasing order (most important first) of their importance, for the welfare of human society.
Question: Arrange the following in the decreasing order (most important first) of their importance, for the welfare of human society. Give reasons for your answer. Biogas, Citric acid, Penicillin and Curd Solution: The order of arrangement of products according to their decreasing importance is: Penicillin- Biogas Citric acid Curd Penicillin is the most important product for the welfare of human society. It is an antibiotic, which is used for controlling various bacterial diseases. The second mo...
Read More →Prove that
Question: Prove that $\sqrt{5}+\sqrt{3}$ is irrational. Solution: Let us assume that $\sqrt{5}+\sqrt{3}$ is rational .Then, there exist positive co primes $a$ and $b$ such that $\sqrt{5}+\sqrt{3}=\frac{a}{b}$ $\sqrt{5}=\frac{a}{b}-\sqrt{3}$ $(\sqrt{5})^{2}=\left(\frac{a}{b}-\sqrt{3}\right)^{2}$ $5=\left(\frac{\mathrm{a}}{\mathrm{b}}\right)^{2}-\frac{2 a \sqrt{3}}{b}+3$ $\Rightarrow \quad 5-3=\left(\frac{\mathrm{a}}{\mathrm{b}}\right)^{2}-\frac{2 a \sqrt{3}}{b}$ $\Rightarrow \quad 2=\left(\frac{\...
Read More →Find out the role of microbes in the following and discuss it with your teacher.
Question: Find out the role of microbes in the following and discuss it with your teacher. (a)Single cell protein (SCP) (b)Soil Solution: (a)Single cell protein (SCP) A single cell protein is a protein obtained from certain microbes, which forms an alternate source of proteins in animal feeds. The microbes involved in the preparation of single cell proteins are algae, yeast, or bacteria. These microbes are grown on an industrial scale to obtain the desired protein. For example,Spirulinacan be gr...
Read More →Find the equation of the circle with centre (–2, 3) and radius 4
Question: Find the equation of the circle with centre (2, 3) and radius 4 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) = (2, 3) and radius (r) = 4. Therefore, the equation of the circle is $(x+2)^{2}+(y-3)^{2}=(4)^{2}$ $x^{2}+4 x+4+y^{2}-6 y+9=16$ $x^{2}+y^{2}+4 x-6 y-3=0$...
Read More →Differentiate the function with respect to x.
Question: Differentiate the function with respect tox. $x^{x}-2^{\sin x}$ Solution: Let $y=x^{x}-2^{\sin x}$ Also, let $x^{x}=u$ and $2^{\sin x}=v$ $\therefore y=u-v$ $\Rightarrow \frac{d y}{d x}=\frac{d u}{d x}-\frac{d v}{d x}$ $u=x^{X}$ Taking logarithm on both the sides, we obtain $\log u=x \log x$ Differentiating both sides with respect tox, we obtain $\frac{1}{u} \frac{d u}{d x}=\left[\frac{d}{d x}(x) \times \log x+x \times \frac{d}{d x}(\log x)\right]$ $\Rightarrow \frac{d u}{d x}=u\left[1...
Read More →Find out the name of the microbes from
Question: Find out the name of the microbes from which Cyclosporin A (an immunosuppressive drug) and Statins (blood cholesterol lowering agents) are obtained. Solution:...
Read More →Find the equation of the circle with centre (0, 2) and radius 2
Question: Find the equation of the circle with centre (0, 2) and radius 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) = (0, 2) and radius (r) = 2. Therefore, the equation of the circle is $(x-0)^{2}+(y-2)^{2}=2^{2}$ $x^{2}+y^{2}+4-4 y=4$ $x^{2}+y^{2}-4 y=0$...
Read More →Three water samples namely river water,
Question: Three water samples namely river water, untreated sewage water and secondary effluent discharged from a sewage treatment plant were subjected to BOD test. The samples were labelled A, B and C; but the laboratory attendant did not note which was which. The BOD values of the three samples A, B and C were recorded as 20mg/L, 8mg/L and 400mg/L, respectively. Which sample of the water is most polluted? Can you assign the correct label to each assuming the river water is relatively clean? So...
Read More →Show that the following numbers are irrational.
Question: Show that the following numbers are irrational. (i) $\frac{1}{\sqrt{2}}$ (ii) $7 \sqrt{5}$ (iii) $6+\sqrt{2}$ (iv) $3-\sqrt{5}$ Solution: (i) Let us assume that $\frac{1}{\sqrt{2}}$ is rational .Then, there exist positive co primes $a$ and $b$ such that $\frac{1}{\sqrt{2}}=\frac{\mathrm{a}}{\mathrm{b}}$ $\frac{1}{\sqrt{2}}=\left(\frac{a}{b}\right)^{2}$ $\Rightarrow \frac{1}{2}=\frac{a^{2}}{b^{2}}$ $\Rightarrow b^{2}=2 a^{2}$ $\Rightarrow 2 \mid b^{2}\left(\because 2 \mid 2 a^{2}\right)...
Read More →Show that the following numbers are irrational.
Question: Show that the following numbers are irrational. (i) $\frac{1}{\sqrt{2}}$ (ii) $7 \sqrt{5}$ (iii) $6+\sqrt{2}$ (iv) $3-\sqrt{5}$ Solution: (i) Let us assume that $\frac{1}{\sqrt{2}}$ is rational .Then, there exist positive co primes $a$ and $b$ such that $\frac{1}{\sqrt{2}}=\frac{\mathrm{a}}{\mathrm{b}}$ $\frac{1}{\sqrt{2}}=\left(\frac{a}{b}\right)^{2}$ $\Rightarrow \frac{1}{2}=\frac{a^{2}}{b^{2}}$ $\Rightarrow b^{2}=2 a^{2}$ $\Rightarrow 2 \mid b^{2}\left(\because 2 \mid 2 a^{2}\right)...
Read More →Microbes can be used to decrease the use of chemical fertilisers and pesticides. Explain how this can be accomplished.
Question: Microbes can be used to decrease the use of chemical fertilisers and pesticides. Explain how this can be accomplished. Solution: Microbes play an important role in organic farming, which is done without the use of chemical fertilizers and pesticides. Bio-fertilizers are living organisms which help increase the fertility of soil. It involves the selection of beneficial micro-organisms that help in improving plant growth through the supply of plant nutrients. Bio-fertilizers are introduc...
Read More →The hypotenuse of a right angled triangle has its ends at the points (1, 3) and (−4, 1).
Question: The hypotenuse of a right angled triangle has its ends at the points (1, 3) and (4, 1).Find the equation of the legs (perpendicular sides) of the triangle. Solution: Let A(1,3) and B(4,1) be the coordinates of the end points of the hypotenuse.Now, plotting the line segment joining the points A(1,3) and B(4,1) on the coordinate plane, we will get two right triangles with AB as the hypotenuse. Now from the diagram, it is clear that the point of intersection of the other two legs of the r...
Read More →Differentiate the function with respect to x.
Question: Differentiate the function with respect tox. $(\log x)^{\cos x}$ Solution: Let $y=(\log x)^{\cos x}$ Taking logarithm on both the sides, we obtain $\log y=\cos x \cdot \log (\log x)$ Differentiating both sides with respect tox, we obtain $\frac{1}{y} \cdot \frac{d y}{d x}=\frac{d}{d x}(\cos x) \times \log (\log x)+\cos x \times \frac{d}{d x}[\log (\log x)]$ $\Rightarrow \frac{1}{y} \cdot \frac{d y}{d x}=-\sin x \log (\log x)+\cos x \times \frac{1}{\log x} \cdot \frac{d}{d x}(\log x)$ $...
Read More →Do you think microbes can also be used as source of energy? If yes, how?
Question: Do you think microbes can also be used as source of energy? If yes, how? Solution: Yes, microbes can be used as a source of energy. Bacteria such asMethane bacteriumis used for the generation ofgobargas or biogas. The generation of biogas is an anaerobic process in a biogas plant, which consists of a concrete tank (1015 feet deep) with sufficient outlets and inlets. The dung is mixed with water to form the slurry and thrown into the tank. The digester of the tank is filled with numerou...
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