Differentiate the functions with respect to x.
Question: Differentiate the functions with respect tox. $\frac{\sin (a x+b)}{\cos (c x+d)}$ Solution: The given function is $f(x)=\frac{\sin (a x+b)}{\cos (c x+d)}=\frac{g(x)}{h(x)}$, where $g(x)=\sin (a x+b)$ and $h(x)=\cos (c x+d)$ $\therefore f^{\prime}=\frac{g^{\prime} h-g h^{\prime}}{h^{2}}$ Consider $g(x)=\sin (a x+b)$ Let $u(x)=a x+b, v(t)=\sin t$ Then, $(v o u)(x)=v(u(x))=v(a x+b)=\sin (a x+b)=g(x)$ gis a composite function of two functions,uandv. Put $t=u(x)=a x+b$ $\frac{d v}{d t}=\fra...
Read More →Depending upon the chemical nature of the template
Question: Depending upon the chemical nature of the template (DNA or RNA) and the nature of nucleic acids synthesised from it (DNA or RNA), list the types of nucleic acid polymerases. Solution: There are two different types of nucleic acid polymerases. (1)DNA-dependent DNA polymerases (2)DNA-dependent RNA polymerases The DNA-dependent DNA polymerases use a DNA template for synthesizing a new strand of DNA, whereas DNA-dependent RNA polymerases use a DNA template strand for synthesizing RNA....
Read More →For any positive integer n,
Question: For any positive integern, prove thatn3ndivisible by 6. Solution: To Prove: For any positive integern,n3nis divisible by 6. Proof: Letnbe any positive integer. $\Rightarrow n^{3}-n=(n-1)(n)(n+1)$ Since any positive integer is of the form 6qor 6q+ 1 or 6q+ 2 or 6q+ 3 or 6q+ 4, 6q+ 5 Ifn= 6q Then, $(n-1) n(n+1)=(6 q-1) 6 q(6 q+1)$ which is divisble by 6 Ifn= 6q+ 1 Then, $(n-1) n(n+1)=(6 q)(6 q+1)(6 q+2)$ $\Rightarrow$ which is divisble by 6 Ifn= 6q+ 2 Then, $(n-1) n(n+1)=(6 q+1)(6 q+2)(6...
Read More →Prove that the line through the point (x1, y1)
Question: Prove that the line through the point $\left(x_{1}, y_{1}\right)$ and parallel to the line $A x+B y+C=0$ is $A\left(x-x_{1}\right)+B\left(y-y_{1}\right)=0$. Solution: The slope of line $\mathrm{Ax}+\mathrm{B} y+\mathrm{C}=0$ or $y=\left(\frac{-\mathrm{A}}{\mathrm{B}}\right) x+\left(\frac{-\mathrm{C}}{\mathrm{B}}\right)$ is $m=-\frac{\mathrm{A}}{\mathrm{B}}$ It is known that parallel lines have the same slope. $\therefore$ Slope of the other line $=m=-\frac{\mathrm{A}}{\mathrm{B}}$ The ...
Read More →Which property of DNA double helix led Watson and Crick to hypothesise semi-conservative mode of DNA replication? Explain.
Question: Which property of DNA double helix led Watson and Crick to hypothesise semi-conservative mode of DNA replication? Explain. Solution: Watson and Crick observed that the two strands of DNA are anti-parallel and complementary to each other with respect to their base sequences. This type of arrangement in DNA molecule led to the hypothesis that DNA replication is semi-conservative. It means that the double stranded DNA molecule separates and then, each of the separated strand acts as a tem...
Read More →Prove that the product of three consecutive positive integer is divisible by 6.
Question: Prove that the product of three consecutive positive integer is divisible by 6. Solution: To Prove: the product of three consecutive positive integers is divisible by 6. Proof: Letnbe any positive integer. Since any positive integer is of the form 6qor 6q+ 1 or 6q+ 2 or 6q+ 3 or 6q+ 4, 6q+ 5 Ifn= 6q $\Rightarrow n(n+1)(n+2)=6 q(6 q+1)(6 q+2)$, which is divisible by 6 Ifn= 6q+ 1 $\Rightarrow n(n+1)(n+2)=(6 q+1)(6 q+2)(6 q+3)$ $\Rightarrow n(n+1)(n+2)=6(6 q+1)(3 q+1)(2 q+1)$ Which is div...
Read More →The line through the points (h, 3) and (4, 1) intersects the line
Question: The line through the points $(h, 3)$ and $(4,1)$ intersects the line $7 x-9 y-19=0$. at right angle. Find the value of $h$. Solution: The slope of the line passing through points $(h, 3)$ and $(4,1)$ is $m_{1}=\frac{1-3}{4-h}=\frac{-2}{4-h}$ The slope of line $7 x-9 y-19=0$ or $y=\frac{7}{9} x-\frac{19}{9}$ is $m_{2}=\frac{7}{9}$ It is given that the two lines are perpendicular. $\therefore m_{1} \times m_{2}=-1$ $\Rightarrow\left(\frac{-2}{4-h}\right) \times\left(\frac{7}{9}\right)=-1...
Read More →Differentiate the functions with respect to x.
Question: Differentiate the functions with respect tox. $\sec (\tan (\sqrt{x}))$ Solution: Let $f(x)=\sec (\tan \sqrt{x}), u(x)=\sqrt{x}, v(t)=\tan t$, and $w(s)=\sec s$ Then, $($ wovou $)(x)=w[v(u(x))]=w[v(\sqrt{x})]=w(\tan \sqrt{x})=\sec (\tan \sqrt{x})=f(x)$ Thus,fis a composite function of three functions,u, v, andw. Put $s=v(t)=\tan t$ and $t=u(x)=\sqrt{x}$ Then, $\frac{d w}{d s}=\frac{d}{d s}(\sec s)=\sec s \tan s=\sec (\tan t) \cdot \tan (\tan t) \quad[s=\tan t]$ $=\sec (\tan \sqrt{x}) \c...
Read More →If the sequence of the coding strand in a transcription unit is written as follows:
Question: If the sequence of the coding strand in a transcription unit is written as follows: 5'-ATGCATGCATGCATGCATGCATGCATGC-3' Write down the sequence of mRNA. Solution: If the coding strand in a transcription unit is 5 ATGCATGCATGCATGCATGCATGCATGC-3 Then, the template strand in 3 to 5 direction would be 3 TACGTACGTACGTACGTACGTACGTACG-5 It is known that the sequence of mRNA is same as the coding strand of DNA. However, in RNA, thymine is replaced by uracil. Hence, the sequence of mRNA will be ...
Read More →Prove that the square of any positive integer of the form 5q + 1 is of the same form.
Question: Prove that the square of any positive integer of the form 5q+ 1 is of the same form. Solution: To Prove: that the square of a positive integer of the form 5q+ 1 is of the same form Proof: Since positive integernis of the form 5q+ 1 Ifn= 5q+ 1 Then $n^{2}=(5 q+1)^{2}$ $\Rightarrow n^{2}=(5 q)^{2}+(1)^{2}+2(5 q)(1)$ $\Rightarrow n^{2}=25 q^{2}+1+10 q$ $\Rightarrow n^{2}=25 q^{2}+10 q+1$ $\Rightarrow n^{2}=5\left(5 q^{2}+2 q\right)+1$ $\Rightarrow n^{2}=5 m+1$ (where $m=\left(5 q^{2}+2 q\...
Read More →If the sequence of one strand of DNA is written as follows:
Question: If the sequence of one strand of DNA is written as follows: 5'-ATGCATGCATGCATGCATGCATGCATGC-3' Write down the sequence of complementary strand in 5'3'direction Solution: The DNA strands are complementary to each other with respect to base sequence. Hence, if the sequence of one strand of DNA is 5'- ATGCATGCATGCATGCATGCATGCATGC 3 Then, the sequence of complementary strand indirection will be 3'- TACGTACGTACGTACGTACGTACGTACG 5 Therefore, the sequence of nucleotides on DNA polypeptide ind...
Read More →Find angles between the lines
Question: Find angles between the lines$\sqrt{3} x+y=1$ and $x+\sqrt{3} y=1$ Solution: The given lines are $\sqrt{3} x+y=1$ and $x+\sqrt{3} y=1$. $y=-\sqrt{3} x+1$ $\ldots(1)$ and $y=-\frac{1}{\sqrt{3}} x+\frac{1}{\sqrt{3}}$$\ldots(2)$ The slope of line (1) is $m_{1}=-\sqrt{3}$, while the slope of line (2) is $m_{2}=-\frac{1}{\sqrt{3}}$. The acute angle i.e., $\theta$ between the two lines is given by $\tan \theta=\left|\frac{m_{1}-m_{2}}{1+m_{1} m_{2}}\right|$ $\tan \theta=\left|\frac{-\sqrt{3}...
Read More →Prove that if a positive integer is of the form 6q + 5,
Question: Prove that if a positive integer is of the form 6q+ 5, then it is of the form 3q+ 2 for some integerq, but not conversely. Solution: To Prove: that if a positive integer is of the form 6q+ 5 then it is of the form 3q+ 2 for some integerq, but not conversely. Proof: Letn= 6q+ 5 Since any positive integer n is of the form of 3kor 3k+ 1, 3k+ 2 Ifq= 3k Then, $n=6 q+5$ $\Rightarrow \quad n=18 k+5(q=3 k)$ $\Rightarrow \quad n=3(6 k+1)+2$ $\Rightarrow \quad n=3 m+2($ where $m=(6 k+1))$ Ifq= 3...
Read More →If a double stranded DNA has 20 per cent of cytosine,
Question: If a double stranded DNA has 20 per cent of cytosine, calculate the per cent of adenine in the DNA. Solution: According to Chargaffs rule, the DNA molecule should have an equal ratio of pyrimidine (cytosine and thymine) and purine (adenine and guanine). It means that the number of adenine molecules is equal to thymine molecules and the number of guanine molecules is equal to cytosine molecules. % A = % T and % G = % C If dsDNA has 20% of cytosine, then according to the law, it would ha...
Read More →Group the following as nitrogenous bases and nucleosides:
Question: Group the following as nitrogenous bases and nucleosides: Adenine, Cytidine, Thymine, Guanosine, Uracil and Cytosine. Solution: Nitrogenous bases present in the list are adenine, thymine, uracil, and cytosine. Nucleosides present in the list are cytidine and guanosine....
Read More →Differentiate the functions with respect to x.
Question: Differentiate the functions with respect tox. $\sin (a x+b)$ Solution: Let $f(x)=\sin (a x+b), u(x)=a x+b$, and $v(t)=\sin t$ Then, $($ vou $)(x)=v(u(x))=v(a x+b)=\sin (a x+b)=f(x)$ Thus,fis a composite function of two functions,uandv. Putt=u(x) =ax+b Therefore, $\frac{d v}{d t}=\frac{d}{d t}(\sin t)=\cos t=\cos (a x+b)$ $\frac{d t}{d x}=\frac{d}{d x}(a x+b)=\frac{d}{d x}(a x)+\frac{d}{d x}(b)=a+0=a$ Hence, by chain rule, we obtain $\frac{d f}{d x}=\frac{d v}{d t} \cdot \frac{d t}{d x}...
Read More →Find equation of the line perpendicular to the line x – 7y + 5 = 0 and having x intercept 3.
Question: Find equation of the line perpendicular to the line $x-7 y+5=0$ and having $x$ intercept 3 . Solution: The given equation of line is $x-7 y+5=0$. Or, $y=\frac{1}{7} x+\frac{5}{7}$, which is of the form $y=m x+c$ $\therefore$ Slope of the given line $=\frac{1}{7}$ The slope of the line perpendicular to the line having a slope of $\frac{1}{7}$ is $m=-\frac{1}{\left(\frac{1}{7}\right)}=-7$ The equation of the line with slope $-7$ and $x$-intercept 3 is given by $y=m(x-d)$ $\Rightarrow y=-...
Read More →Prove that the square of any positive integer is of the form 5q, 5q + 1, 5q + 4 for some integer q.
Question: Prove that the square of any positive integer is of the form 5q, 5q+ 1, 5q+ 4 for some integerq. Solution: To Prove: that the square of any positive integer is of the form 5qor 5q+ 1, 5q+ 4 for some integerq. Proof: Since positive integernis of the form of 5qor 5q+ 1, 5q+ 4 Ifn= 5q Then, $n^{2}=(5 q)^{2}$ $\Rightarrow \quad n^{2}=25 q^{2}$ $\Rightarrow \quad n^{2}=5(5 q)$ $\Rightarrow \quad n^{2}=5 m($ where $m=5 q)$ Ifn= 5q+ 1 Then, $n^{2}=(5 q+1)^{2}$b $\Rightarrow \quad n^{2}=(5 q)^...
Read More →Mention any two autosomal genetic disorders with their symptoms.
Question: Mention any two autosomal genetic disorders with their symptoms. Solution: Two autosomal genetic disorders are as follows. 1. Sickle cell Anaemia It is an autosomal linked recessive disorder, which is caused by point mutation in the beta-globin chain of haemoglobin pigment of the blood. The disease is characterized by sickle shaped red blood cells, which are formed due to the mutant haemoglobin molecule. The disease is controlled by HbAand HbSallele. The homozygous individuals with gen...
Read More →Differentiate the functions with respect to x.
Question: Differentiate the functions with respect tox. $\cos (\sin x)$ Solution: Let $f(x)=\cos (\sin x), u(x)=\sin x$, and $v(t)=\cos t$ Then, $(v o u)(x)=v(u(x))=v(\sin x)=\cos (\sin x)=f(x)$ Thus,fis a composite function of two functions. Putt=u(x) = sinx $\therefore \frac{d v}{d t}=\frac{d}{d t}[\cos t]=-\sin t=-\sin (\sin x)$ $\frac{d t}{d x}=\frac{d}{d x}(\sin x)=\cos x$ By chain rule, $\frac{d f}{d x}=\frac{d v}{d t} \cdot \frac{d t}{d x}=-\sin (\sin x) \cdot \cos x=-\cos x \sin (\sin x)...
Read More →Find equation of the line parallel to the line 3x – 4y + 2 = 0 and passing through the point (–2, 3).
Question: Find equation of the line parallel to the line $3 x-4 y+2=0$ and passing through the point $(-2,3)$. Solution: The equation of the given line is $3 x-4 y+2=0$ or $y=\frac{3 x}{4}+\frac{2}{4}$ or $y=\frac{3}{4} x+\frac{1}{2}$, which is of the form $y=m x+c$ $\therefore$ Slope of the given line $=\frac{3}{4}$ It is known that parallel lines have the same slope. $\therefore$ Slope of the other line $=m=\frac{3}{4}$ Now, the equation of the line that has a slope of $\frac{3}{4}$ and passes...
Read More →Who had proposed the chromosomal theory of inheritance?
Question: Who had proposed the chromosomal theory of inheritance? Solution: Sutton and Boveri proposed the chromosomal theory of inheritance in 1902. They linked the inheritance of traits to the chromosomes....
Read More →What is point mutation? Give one example.
Question: What is point mutation? Give one example. Solution: Point mutation is a change in a single base pair of DNA by substitution, deletion, or insertion of a single nitrogenous base. An example of point mutation is sickle cell anaemia. It involves mutation in a single base pair in the beta-globin chain of haemoglobin pigment of the blood. Glutamic acid in short arm of chromosome II gets replaced with valine at the sixth position....
Read More →Prove that the square of any positive integer is of the form 4q or 4q + 1 for some integer q.
Question: Prove that the square of any positive integer is of the form 4qor 4q+1 for some integerq. Solution: To Prove: that the square of any positive integer is of the form 4qor 4q+ 1 for some integerq. Proof: Since positive integernis of the form of 2qor 2q+ 1 Ifn= 2q Then, $n^{2}=(2 q)^{2}$ $\Rightarrow \quad n^{2}=4 q^{2}$ $\Rightarrow \quad n^{2}=4 m\left(\right.$ where $\left.m=q^{2}\right)$' Ifn= 2q+ 1 Then, $n^{2}=(2 q+1)^{2}$ $\Rightarrow n^{2}=(2 q)^{2}+4 q+1$ $\Rightarrow n^{2}=4 q^{...
Read More →Explain the following terms with example
Question: Explain the following terms with example (a)Co-dominance (b)Incomplete dominance Solution: (a)Co-dominance Co-dominance is the phenomenon in which both the alleles of a contrasting character are expressed in heterozygous condition. Both the alleles of a gene are equally dominant. ABO blood group in human beings is an example of co-dominance. The blood group character is controlled by three sets of alleles, namely, IA, IB, and i. The alleles, IAand IB, are equally dominant and are said ...
Read More →