Find the distance between parallel lines
Question: Find the distance between parallel lines (i) $15 x+8 y-34=0$ and $15 x+8 y+31=0$ (ii) $I(x+y)+p=0$ and $I(x+y)-r=0$ Solution: It is known that the distance $(d)$ between parallel lines $A x+B y+C_{1}=0$ and $A x+B y+C_{2}=0$ is given by $d=\frac{\left|C_{1}-C_{2}\right|}{\sqrt{A^{2}+B^{2}}}$. (i) The given parallel lines are $15 x+8 y-34=0$ and $15 x+8 y+31=0$. Here, $A=15, B=8, C_{1}=-34$, and $C_{2}=31$. Therefore, the distance between the parallel lines is $d=\frac{\left|C_{1}-C_{2}...
Read More →Find the points on the x-axis,
Question: Find the points on thex-axis,whose distances from the line $\frac{x}{3}+\frac{y}{4}=1$ are 4 units. Solution: The given equation of line is $\frac{x}{3}+\frac{y}{4}=1$ or, $4 x+3 y-12=0$ $\ldots(1)$ On comparing equation (1) with general equation of lineAx+By+C= 0, we obtainA= 4,B= 3, andC= 12. Let (a, 0) be the point on thex-axis whose distance from the given line is 4 units. It is known that the perpendicular distance $(d)$ of a line $A x+B y+C=0$ from a point $\left(x_{1}, y_{1}\rig...
Read More →Differentiate the functions with respect to x.
Question: Differentiate the functions with respect tox. $\sin \left(x^{2}+5\right)$ Solution: Let $f(x)=\sin \left(x^{2}+5\right), u(x)=x^{2}+5$, and $v(t)=\sin t$ Then, $($ vou $)=v(u(x))=v\left(x^{2}+5\right)=\tan \left(x^{2}+5\right)=f(x)$ Thus,fis a composite of two functions. Put $t=u(x)=x^{2}+5$ Then, we obtain $\frac{d v}{d t}=\frac{d}{d t}(\sin t)=\cos t=\cos \left(x^{2}+5\right)$ $\frac{d t}{d x}=\frac{d}{d x}\left(x^{2}+5\right)=\frac{d}{d x}\left(x^{2}\right)+\frac{d}{d x}(5)=2 x+0=2 ...
Read More →Prove that the square of any positive integer is of the form 3m or,
Question: Prove that the square of any positive integer is of the form 3mor, 3m+ 1 but not of the form 3m+ 2. Solution: To Prove: that the square of an positive integer is of the form 3mor 3m+ 1 but not of the form 3m+ 2. Proof: Since positive integernis of the form of 3q, 3q+ 1 and 3q+ 2 Ifn= 3q $\Rightarrow n^{2}=(3 q)^{2}$ $\Rightarrow n^{2}=9 q^{2}$ $\Rightarrow n^{2}=3\left(3 q^{2}\right)$ $\Rightarrow n^{2}=3 m\left(m=3 q^{2}\right)$ Ifn= 3q+ 1 Then, $n^{2}=(3 q+1)^{2}$ $\Rightarrow n^{2}=...
Read More →A child has blood group O. If the father has blood group A and mother
Question: A child has blood group O. If the father has blood group A and mother blood group B, work out the genotypes of the parents and the possible genotypes of the other offsprings. Solution: The blood group characteristic in humans is controlled by three set of alleles, namely, IA, IB, and i. The alleles, IAand IB, are equally dominant whereas allele, i, is recessive to the other alleles. The individuals with genotype, IAIAand IAi, have blood group A whereas the individuals with genotype, IB...
Read More →How is sex determined in human beings?
Question: How is sex determined in human beings? Solution: Human beings exhibit male heterogamy. In humans, males (XY) produce two different types of gametes, X and Y. The human female (XX) produces only one type of gametes containing X chromosomes. The sex of the baby is determined by the type of male gamete that fuses with the female gamete. If the fertilizing sperm contains X chromosome, then the baby produced will be a girl and if the fertilizing sperm contains Y chromosome, then the baby pr...
Read More →Show that any positive odd integer is of the form
Question: Show that any positive odd integer is of the form 6q+ 1 or, 6q+ 3 or, 6q+ 5, whereqis some integer. Solution: To Show: That any positive odd integer is of the form 6q+ 1 or 6q+ 3 or 6q+ 5 whereqis any some integer. Proof: Letabe any odd positive integer andb= 6. Then, there exists integersqandrsuch that a= 6q+r, 0r 6 (by division algorithm) ⇒a= 6qor 6q+ 1 or 6q+ 2 or 6q+ 3 or 6q+ 4 But 6qor 6q+ 2 or 6q+ 4 are even positive integers So, $a=6 q+1$ or $6 q+3$ or $6 q+5$ Hence it is proved...
Read More →Find the distance of the point (–1, 1) from the line 12(x + 6)
Question: Find the distance of the point $(-1,1)$ from the line $12(x+6)=5(y-2)$. Solution: The given equation of the line is 12(x+ 6) = 5(y 2). ⇒ 12x+ 72 = 5y 10 ⇒12x 5y+ 82 = 0 (1) On comparing equation (1) with general equation of lineAx+By+C= 0, we obtainA= 12,B= 5, andC= 82. It is known that 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 $d=\frac{\left|A x_{1}+B y_{1}+C\right|}{\sqrt{A^{2}+B^{2}}}$. The given point is $\left(x_{...
Read More →What is pedigree analysis?
Question: What is pedigree analysis? Suggest how such an analysis, can be useful. Solution: Pedigree analysis is a record of occurrence of a trait in several generations of a family. It is based on the fact that certain characteristic features are heritable in a family, for example, eye colour, skin colour, hair form and colour, and other facial characteristics. Along with these features, there are other genetic disorders such as Mendelian disorders that are inherited in a family, generation aft...
Read More →Show that any positive odd integer is of the form
Question: Show that any positive odd integer is of the form 6q+ 1 or, 6q+ 3 or, 6q+ 5, whereqis some integer. Solution: To Show: That any positive odd integer is of the form 6q+ 1 or 6q+ 3 or 6q+ 5 whereqis any some integer. Proof: Letabe any odd positive integer andb= 6. Then, there exists integersqandrsuch that a= 6q+r, 0r 6 (by division algorithm) ⇒a= 6qor 6q+ 1 or 6q+ 2 or 6q+ 3 or 6q+ 4 But 6qor 6q+ 2 or 6q+ 4 are even positive integers...
Read More →Briefly mention the contribution of T.H. Morgan in genetics.
Question: Briefly mention the contribution of T.H. Morgan in genetics. Solution: Morgans work is based on fruit flies (Drosophila melanogaster). He formulated the chromosomal theory of linkage. He defined linkage as the co-existence of two or more genes in the same chromosome and performed dihybrid crosses inDrosophilato show that linked genes are inherited together and are located on X-chromosome. His experiments have also proved that tightly linked genes show very low recombination while loose...
Read More →Show that the square of an odd positive integer is of the form 8q + 1, for some integer q.
Question: Show that the square of an odd positive integer is of the form 8q+ 1, for some integerq. Solution: To Prove: that the square of an odd positive integer is of the form 8q+ 1, for some integerq. Proof: Since any positive integernis of the form 4m+ 1 and 4m+ 3 Ifn= 4m+ 1 $\Rightarrow n^{2}=(4 m+1)^{2}$ $\Rightarrow n^{2}=(4 m)^{2}+8 m+1$ $\Rightarrow n^{2}=16 m^{2}+8 m+1$ $\Rightarrow n^{2}=8 m(2 m+1)+1$ $\Rightarrow n^{2}+8 q+1(q=m(2 m+1))$ Ifn= 4m+ 3 $\Rightarrow n^{2}=(4 m+3)^{2}$ $\Ri...
Read More →Find all the points of discontinuity of f defined by
Question: Find all the points of discontinuity of $f$ defined by $f(x)=|x|-|x+1|$. Solution: The given function is $f(x)=|x|-|x+1|$ Thetwo functions,gandh, are defined as $g(x)=|x|$ and $h(x)=|x+1|$ Then,f=gh Thecontinuity ofgandhis examined first. $g(x)=|x|$ can be written as $g(x)= \begin{cases}-x, \text { if } x0 \\ x, \text { if } x \geq 0\end{cases}$ Clearly,gis defined for all real numbers. Letcbe a real number. Case I: If $c0$, then $g(c)=-c$ and $\lim _{x \rightarrow} g(x)=\lim _{x \righ...
Read More →Two heterozygous parents are crossed.
Question: Two heterozygous parents are crossed. If the two loci are linked what would be the distribution of phenotypic features in F1generation for a dihybrid cross? Solution: Linkage is defined as the coexistence of two or more genes in the same chromosome. If the genes are situated on the same chromosome and lie close to each other, then they are inherited together and are said to be linked genes. For example, a cross between yellow body and white eyes and wild type parent in aDrosophilawill ...
Read More →Reduce the following equations into normal form.
Question: Reduce the following equations into normal form. Find their perpendicular distances from the origin and angle between perpendicular and the positivex-axis. (i) $x-\sqrt{3} y+8=0$ (ii) $y-2=0$ (iii) $x-y=4$ Solution: (i) The given equation is $x-\sqrt{3} y+8=0$. Itcan be reduced as: $x-\sqrt{3} y=-8$ $\Rightarrow-x+\sqrt{3} y=8$ On dividing both sides by $\sqrt{(-1)^{2}+(\sqrt{3})^{2}}=\sqrt{4}=2$, we obtain $-\frac{x}{2}+\frac{\sqrt{3}}{2} y=\frac{8}{2}$ $\Rightarrow\left(-\frac{1}{2}\...
Read More →When a cross in made between tall plants with yellow seeds (TtYy) and tall plant with green seed (TtYy),
Question: When a cross in made between tall plants with yellow seeds (TtYy) and tall plant with green seed (TtYy), what proportions of phenotype in the offspring could be expected to be (a)Tall and green. (b)Dwarf and green. Solution: A cross between tall plant with yellow seeds and tall plant with green seeds will produce (a)three tall and green plants (b)one dwarf and green plant...
Read More →If a and b are two odd positive integers such that a > b,
Question: If $a$ and $b$ are two odd positive integers such that $ab$, then prove that one of the two numbers $\frac{a+b}{2}$ and $\frac{a-b}{2}$ is odd and the other is even. Solution: Given: Ifaandbare two odd positive integers such thatab. To Prove: That one of the two numbers $\frac{a+b}{2}$ and $\frac{a-b}{2}$ is odd and the other is even. To Prove: That one of the two numbersandis odd and the other is even. Proof: Letaandbbe any odd odd positive integer such thatab. Since any positive inte...
Read More →Using a Punnett square, work out the distribution of phenotypic features in the first filial generation after
Question: Using a Punnett square, work out the distribution of phenotypic features in the first filial generation after a cross between a homozygous female and a heterozygous male for a single locus. Solution: In guinea pigs, heterozygous male with black coat colour (Bb) is crossed with the female having white coat colour (bb). The male will produce two types of gametes, B and b, while the female will produce only one kind of gamete, r. The genotypic and phenotypic ratio in the progenies of F1ge...
Read More →Define and design a test − cross?
Question: Define and design a test cross? Solution: Test cross is a cross between an organism with unknown genotype and a recessive parent. It is used to determine whether the individual is homozygous or heterozygous for a trait. If the progenies produced by a test cross show 50% dominant trait and 50% recessive trait, then the unknown individual is heterozygous for a trait. On the other hand, if the progeny produced shows dominant trait, then the unknown individual is homozygous for a trait....
Read More →Prove that the product of two consecutive positive integers is divisible by 2.
Question: Prove that the product of two consecutive positive integers is divisible by 2. Solution: To Prove: that the product of two consecutive integers is divisible by 2. Proof: Letn 1 andnbe two consecutive positive integers. Then their product isn(n 1) =n2n We know that every positive integer is of the form 2qor 2q+ 1 for some integerq. So letn= 2q So,n2n= (2q)2 (2q) $\Rightarrow n^{2}-n=(2 q)^{2}-2 q$ $\Rightarrow n^{2}-n=4 q^{2}-2 q$ $\Rightarrow n^{2}-n=2 q(2 q-1)$ $\Rightarrow n^{2}-n=2 ...
Read More →Explain the Law of Dominance using a monohybrid cross.
Question: Explain the Law of Dominance using a monohybrid cross. Solution: Mendels law of dominance states that a dominant allele expresses itself in a monohybrid cross and suppresses the expression of recessive allele. However, this recessive allele for a character is not lost and remains hidden or masked in the progenies of F1generation and reappears in the next generation. For example, when pea plants with round seeds (RR) are crossed with plants with wrinkled seeds (rr), all seeds in F1gener...
Read More →A diploid organism is heterozygous for 4 loci, how many types of gametes can be produced?
Question: A diploid organism is heterozygous for 4 loci, how many types of gametes can be produced? Solution: Locus is a fixed position on a chromosome, which is occupied by a single or more genes. Heterozygous organisms contain different alleles for an allelic pair. Hence, a diploid organism, which is heterozygous at four loci, will have four different contrasting characters at four different loci. For example, if an organism is heterozygous at four loci with four characters, say Aa, Bb, Cc, Dd...
Read More →Examine that
Question: Examine that $\sin |x|$ is a continuous function. Solution: Let $f(x)=\sin |x|$ This functionfis defined for every real number andfcan be written as the composition of two functions as, $f=g \circ h$, where $g(x)=|x|$ and $h(x)=\sin x$ $[\because(g o h)(x)=g(h(x))=g(\sin x)=|\sin x|=f(x)]$ It has to be proved first that $g(x)=|x|$ and $h(x)=\sin x$ are continuous functions. $g(x)=|x|$ can be written as $g(x)= \begin{cases}-x, \text { if } x0 \\ x, \text { if } x \geq 0\end{cases}$ Clea...
Read More →Differentiate between the following −
Question: Differentiate between the following (a)Dominance and Recessive (b)Homozygous and Heterozygous (c)Monohybrid and Dihybrid. Solution: (a)Dominance and Recessive (b) Homozygous and Heterozygous (c) Monohybrid and Dihybrid...
Read More →Mention the advantages of selecting pea plant for experiment by Mendel.
Question: Mention the advantages of selecting pea plant for experiment by Mendel. Solution: Mendel selected pea plants to carry out his study on the inheritance of characters from parents to offspring. He selected a pea plant because of the following features. (a)Peas have many visible contrasting characters such as tall/dwarf plants, round/wrinkled seeds, green/yellow pod, purple/white flowers, etc. (b)Peas have bisexual flowers and therefore undergo self pollination easily. Thus, pea plants pr...
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