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Question: Find $\frac{d y}{d x}$ : $2 x+3 y=\sin y$ Solution: The given relationship is Differentiating this relationship with respect tox, we obtain $\frac{d}{d x}(2 x)+\frac{d}{d x}(3 y)=\frac{d}{d x}(\sin y)$ $\Rightarrow 2+3 \frac{d y}{d x}=\cos y \frac{d y}{d x} \quad$ [By using chain rule] $\Rightarrow 2=(\cos y-3) \frac{d y}{d x}$ $\therefore \frac{d y}{d x}=\frac{2}{\cos y-3}$...
Read More →In the triangle ABC with vertices A (2, 3), B (4, –1) and C (1, 2),
Question: In the triangle ABC with vertices A (2, 3), B (4,1) and C (1, 2), find the equation and length of altitude from the vertex A. Solution: Let AD be the altitude of triangle ABC from vertex A. Accordingly, $A D \perp B C$ The equation of the line passing through point (2, 3) and having a slope of 1 is (y 3) = 1(x 2) ⇒xy+ 1 = 0 ⇒yx= 1 Therefore, equation of the altitude from vertex A =yx= 1. Length of AD = Length of the perpendicular from A (2, 3) to BC The equation of BC is $(y+1)=\frac{2...
Read More →Briefly describe the following:
Question: Briefly describe the following: (a) Transcription (b) Polymorphism (c) Translation (d) Bioinformatics Solution: (a)Transcription Transcription is the process of synthesis of RNA from DNA template. A segment of DNA gets copied into mRNA during the process. The process of transcription starts at the promoter region of the template DNA and terminates at the terminator region. The segment of DNA between these two regions is known as transcription unit. The transcription requires RNA polyme...
Read More →Find :
Question: Find $\frac{d y}{d x}$ : $2 x+3 y=\sin x$ Solution: The given relationship is Differentiating this relationship with respect tox, we obtain $\frac{d}{d x}(2 x+3 y)=\frac{d}{d x}(\sin x)$ $\Rightarrow \frac{d}{d x}(2 x)+\frac{d}{d x}(3 y)=\cos x$ $\Rightarrow 2+3 \frac{d y}{d x}=\cos x$ $\Rightarrow 3 \frac{d y}{d x}=\cos x-2$ $\therefore \frac{d y}{d x}=\frac{\cos x-2}{3}$...
Read More →Find the HCF of the following pairs of integers and express it as a linear combination of them.
Question: Find the HCF of the following pairs of integers and express it as a linear combination of them. (i) 963 and 657 (ii) 592 and 252 (iii) 506 and 1155 (iv) 1288 and 575 Solution: (i) We need to find the H.C.F. of 963 and 657 and express it as a linear combination of 963 and 657. By applying Euclid's division lemma $963=657 \times 1+306$ Since remainder $\neq 0$, apply division lemma on divisor 657 and remainder 306 $657=306 \times 2+45$ Since remainder $\neq 0$, apply division lemma on di...
Read More →Find the HCF of the following pairs of integers and express it as a linear combination of them.
Question: Find the HCF of the following pairs of integers and express it as a linear combination of them. (i) 963 and 657 (ii) 592 and 252 (iii) 506 and 1155 (iv) 1288 and 575 Solution: (i) We need to find the H.C.F. of 963 and 657 and express it as a linear combination of 963 and 657. By applying Euclid's division lemma $963=657 \times 1+306$ Since remainder $\neq 0$, apply division lemma on divisor 657 and remainder 306 $657=306 \times 2+45$ Since remainder $\neq 0$, apply division lemma on di...
Read More →If p and q are the lengths of perpendiculars from the origin to the lines
Question: If $p$ and $q$ are the lengths of perpendiculars from the origin to the lines $x \cos \theta-y \sin \theta=k \cos 2 \theta$ and $x \sec \theta+y \operatorname{cosec} \theta=k$, respectively, prove that $p^{2}+4 q^{2}=k^{2}$ Solution: The equations of given lines are $x \cos \theta-y \sin \theta=k \cos 2 \theta \ldots$ (1) $x \sec \theta+y \operatorname{cosec} \theta=k$ (2) The perpendicular distance $(d)$ of a line $A x+B y+C=0$ from a point $\left(x_{1}, y_{1}\right)$ is given by $d=\...
Read More →Prove that the greatest integer function defined by
Question: Prove that the greatest integer function defined by $f(x)=[x], 0x3$ is not differentiable atx= 1 andx= 2. Solution: The given function $f$ is $f(x)=[x], 0x3$ It is known that a functionfis differentiable at a pointx=cin its domain if both$\lim _{h \rightarrow 0} \frac{f(c+h)-f(c)}{h}$ and $\lim _{h \rightarrow 0^{\circ}} \frac{f(c+h)-f(c)}{h}$ are finite and equal. To check the differentiability of the given function atx= 1, consider the left hand limit offatx= 1 $\lim _{h \rightarrow ...
Read More →What is DNA fingerprinting? Mention its application.
Question: What is DNA fingerprinting? Mention its application. Solution: DNA fingerprinting is a technique used to identify and analyze the variations in various individuals at the level of DNA. It is based on variability and polymorphism in DNA sequences. Application (1)It is used in forensic science to identify potential crime suspects. (2)It is used to establish paternity and family relationships. (3)It is used to identify and protect the commercial varieties of crops and livestock. (4)It is ...
Read More →Why is the Human Genome project called a mega project?
Question: Why is the Human Genome project called a mega project? Solution: Human genome project was considered to be a mega project because it had a specific goal to sequence every base pair present in the human genome. It took around 13 years for its completion and got accomplished in year 2006. It was a large scale project, which aimed at developing new technology and generating new information in the field of genomic studies. As a result of it, several new areas and avenues have opened up in ...
Read More →Explain (in one or two lines) the function of the followings:
Question: Explain (in one or two lines) the function of the followings: (a)Promoter (b)tRNA (c)Exons Solution: (a)Promoter Promoter is a region of DNA that helps in initiating the process of transcription. It serves as the binding site for RNA polymerase. (b)tRNA tRNA or transfer RNA is a small RNA that reads the genetic code present on mRNA. It carries specific amino acid to mRNA on ribosome during translation of proteins. (c)Exons Exons are coding sequences of DNA in eukaryotes that transcribe...
Read More →In the medium where E. coli was growing, lactose was added, which induced the lac operon.
Question: In the medium whereE. coliwas growing, lactose was added, which induced thelacoperon. Then, why does lac operon shut down some time after addition of lactose in the medium? Solution: Lacoperon is a segment of DNA that is made up of three adjacent structural genes, namely, an operator gene, a promoter gene, and a regulator gene. It works in a coordinated manner to metabolize lactose into glucose and galactose. Inlacoperon, lactose acts as an inducer. It binds to the repressor and inacti...
Read More →Prove that the function f given by
Question: Prove that the functionfgiven by $f(x)=|x-1|, x \in \mathbf{R}$ is notdifferentiable at $x=1$. Solution: The given function is $f(x)=|x-1|, x \in \mathbf{R}$ It is known that a functionfis differentiable at a pointx=cin its domain if both $\lim _{h \rightarrow 0^{-}} \frac{f(c+h)-f(c)}{h}$ and $\lim _{h \rightarrow 0^{+}} \frac{f(c+h)-f(c)}{h}$ are finite and equal. To check the differentiability of the given function atx= 1, consider the left hand limit offatx= 1 $\lim _{h \rightarrow...
Read More →The perpendicular from the origin to the line y = mx + c meets it at the point
Question: The perpendicular from the origin to the liney = mx + cmeets it at the point (1, 2). Find the values ofmandc. Solution: The given equation of line isy = mx + c. It is given that the perpendicular from the origin meets the given line at (1, 2). Therefore, the line joining the points (0, 0) and (1, 2) is perpendicular to the given line. $\therefore$ Slope of the line joining $(0,0)$ and $(-1,2)=\frac{2}{-1}=-2$ The slope of the given line ism. $\therefore m \times-2=-1 \quad$ [The two li...
Read More →Differentiate the functions with respect to x.
Question: Differentiate the functions with respect tox. $\cos (\sqrt{x})$ Solution: Let $f(x)=\cos (\sqrt{x})$ Also, let $u(x)=\sqrt{x}$ And, $v(t)=\cos t$ Then, $(v o u)(x)=v(u(x))$ $=v(\sqrt{x})$ $=\cos \sqrt{x}$ $=f(x)$ Clearly,fis a composite function of two functions,uandv, such that Then, $\frac{d t}{d x}=\frac{d}{d x}(\sqrt{x})=\frac{d}{d x}\left(x^{\frac{1}{2}}\right)=\frac{1}{2} x^{-\frac{1}{2}}$ $=\frac{1}{2 \sqrt{x}}$ And, $\frac{d v}{d t}=\frac{d}{d t}(\cos t)=-\sin t$ $=-\sin (\sqrt...
Read More →Define HCF of two positive integers and find the HCF of the following pairs of numbers:
Question: Define HCF of two positive integers and find the HCF of the following pairs of numbers: (i) 32 and 54 (ii) 18 and 24 (iii) 70 and 30 (iv) 56 and 88 (v) 475 and 495 (vi) 75 and 243. (vii) 240 and 6552 (viii) 155 and 1385 (ix) 100 and 190 (x) 105 and 120 Solution: (i) We need to find H.C.F. of 32 and 54. By applying division lemma $54=32 \times 1+22$ Since remainder $\neq 0$, apply division lemma on 32 and remainder 22 $32=22 \times 1+10$ Since remainder $\neq 0$, apply division lemma on...
Read More →Define HCF of two positive integers and find the HCF of the following pairs of numbers:
Question: Define HCF of two positive integers and find the HCF of the following pairs of numbers: (i) 32 and 54 (ii) 18 and 24 (iii) 70 and 30 (iv) 56 and 88 (v) 475 and 495 (vi) 75 and 243. (vii) 240 and 6552 (viii) 155 and 1385 (ix) 100 and 190 (x) 105 and 120 Solution: (i) We need to find H.C.F. of 32 and 54. By applying division lemma $54=32 \times 1+22$ Since remainder $\neq 0$, apply division lemma on 32 and remainder 22 $32=22 \times 1+10$ Since remainder $\neq 0$, apply division lemma on...
Read More →Differentiate the functions with respect to x.
Question: Differentiate the functions with respect tox. $2 \sqrt{\cot \left(x^{2}\right)}$ Solution: $\frac{d}{d x}\left[2 \sqrt{\cot \left(x^{2}\right)}\right]$ $=2 \cdot \frac{1}{2 \sqrt{\cot \left(x^{2}\right)}} \times \frac{d}{d x}\left[\cot \left(x^{2}\right)\right]$ $=\sqrt{\frac{\sin \left(x^{2}\right)}{\cos \left(x^{2}\right)}} \times-\operatorname{cosec}^{2}\left(x^{2}\right) \times \frac{d}{d x}\left(x^{2}\right)$ $=-\sqrt{\frac{\sin \left(x^{2}\right)}{\cos \left(x^{2}\right)}} \times...
Read More →Find the coordinates of the foot of perpendicular from the point (–1, 3)
Question: Find the coordinates of the foot of perpendicular from the point (1, 3)to the line $3 x-4 y-16=0$. Solution: Let (a,b) be the coordinates of the foot of the perpendicular from the point (1, 3) to the line 3x 4y 16 = 0. Slope of the line joining $(-1,3)$ and $(a, b), m_{1}=\frac{b-3}{a+1}$ Slope of the line $3 x-4 y-16=0$ or $y=\frac{3}{4} x-4, m_{2}=\frac{3}{4}$ Since these two lines are perpendicular, $m_{1} m_{2}=-1$ $\therefore\left(\frac{b-3}{a+1}\right) \times\left(\frac{3}{4}\rig...
Read More →List two essential roles of ribosome during translation.
Question: List two essential roles of ribosome during translation. Solution: The important functions of ribosome during translation are as follows. (a)Ribosome acts as the site where protein synthesis takes place from individual amino acids. It is made up of two subunits. The smaller subunit comes in contact with mRNA and forms a protein synthesizing complex whereas the larger subunit acts as an amino acid binding site. (b)Ribosome acts as a catalyst for forming peptide bond. For example, 23sr-R...
Read More →Differentiate between the followings:
Question: Differentiate between the followings: (a)Repetitive DNA and Satellite DNA (b)mRNA and tRNA (c)Template strand and Coding strand Solution: (a)Repetitive DNA and satellite DNA (b) mRNA and tRNA (c) Template strand and coding strand...
Read More →Find the equation of the right bisector of the line segment joining the points
Question: Find the equation of the right bisector of the line segment joining the points (3, 4) and (1, 2). Solution: The right bisector of a line segment bisects the line segment at $90^{\circ}$. The end-points of the line segment are given as $A(3,4)$ and $B(-1,2)$. Accordingly, mid-point of $A B=\left(\frac{3-1}{2}, \frac{4+2}{2}\right)=(1,3)$ Slope of $A B=\frac{2-4}{-1-3}=\frac{-2}{-4}=\frac{1}{2}$ $\therefore$ Slope of the line perpendicular to $\mathrm{AB}=\frac{1}{\left(\frac{1}{2}\right...
Read More →Differentiate the functions with respect to x.
Question: Differentiate the functions with respect tox. $\cos x^{3} \cdot \sin ^{2}\left(x^{5}\right)$ Solution: The given function is $\cos x^{3} \cdot \sin ^{2}\left(x^{5}\right)$. $\frac{d}{d x}\left[\cos x^{3} \cdot \sin ^{2}\left(x^{5}\right)\right]=\sin ^{2}\left(x^{5}\right) \times \frac{d}{d x}\left(\cos x^{3}\right)+\cos x^{3} \times \frac{d}{d x}\left[\sin ^{2}\left(x^{5}\right)\right]$\ $=\sin ^{2}\left(x^{5}\right) \times\left(-\sin x^{3}\right) \times \frac{d}{d x}\left(x^{3}\right)...
Read More →Two lines passing through the point (2, 3) intersects each other at an angle of 60°.
Question: Two lines passing through the point $(2,3)$ intersects each other at an angle of $60^{\circ}$. If slope of one line is 2 , find equation of the other line. Solution: It is given that the slope of the first line, $m_{1}=2$. Let the slope of the other line be $m_{2}$. The angle between the two lines is $60^{\circ}$. $\therefore \tan 60^{\circ}=\left|\frac{m_{1}-m_{2}}{1+m_{1} m_{2}}\right|$ $\Rightarrow \sqrt{3}=\left|\frac{2-m_{2}}{1+2 m_{2}}\right|$ $\Rightarrow \sqrt{3}=\pm\left(\frac...
Read More →How did Hershey and Chase differentiate between DNA and protein in their experiment while proving that DNA is the genetic material?
Question: How did Hershey and Chase differentiate between DNA and protein in their experiment while proving that DNA is the genetic material? Solution: Hershey and Chase worked with bacteriophage andE.colito prove that DNA is the genetic material. They used different radioactive isotopes to label DNA and protein coat of the bacteriophage. They grew some bacteriophages on a medium containing radioactive phosphorus (32P) to identify DNA and some on a medium containing radioactive sulphur (35S) to ...
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