If m and M respectively denote the minimum and maximum values
Question: If $m$ and $M$ respectively denote the minimum and maximum values of $f(x)=(x+1)^{2}+3$ in the interval $[-3,1]$, then the ordered pair $(m, M)=$_____________ Solution: The given function is $f(x)=(x+1)^{2}+3, x \in[-3,1]$. $f(x)=(x+1)^{2}+3$ Differentiating both sides with respect tox, we get $f^{\prime}(x)=2(x+1)$ For maxima or minima, $f^{\prime}(x)=0$ $f^{\prime}(x)=0$ $\Rightarrow 2(x+1)=0$ $\Rightarrow x+1=0$ $\Rightarrow x=-1$ Now, $f^{\prime \prime}(x)=20$ So,x=1 is the point o...
Read More →Match the following and choose
Question: Match the following and choose the correct option: Options a. A-v, B-iv, C-ii, D-i, E-iii b. A-iv, B-iii, C-v, D-ii, E-i c. A-iv, B-iii, C-v, D-i, E-ii d. A-iv, B-iii, C-ii, D-v, E-i Solution: Option (a)A-v, B-iv, C-ii, D-i, E-iii is the answer....
Read More →Which of the following is a defining
Question: Which of the following is a defining characteristic of living organisms? a. Growth b. Ability to make sound c. Reproduction d. Response to external stimuli Solution: Option (d)Response to external stimuliis the answer...
Read More →The number that exceeds its square by
Question: The number that exceeds its square by the greatest amount is _______________. Solution: Let the number bex. The square of the number is $x^{2}$. Let $f(x)=x-x^{2}$. Now, we need to find the value of $x$ for which $f(x)$ is maximum. $f(x)=x-x^{2}$ Differentiating both sides with respect tox, we get $f^{\prime}(x)=1-2 x$ For maxima or minima, $f^{\prime}(x)=0$ $\Rightarrow 1-2 x=0$ $\Rightarrow x=\frac{1}{2}$ Now, $f^{\prime \prime}(x)=-20$ So, $x=\frac{1}{2}$ is the point of local maxim...
Read More →All living organisms are linked
Question: All living organisms are linked to one another because of a. They have the common genetic material of the same type b. They share common genetic material but to varying degrees c. All have common cellular organization d. All of the above Solution: Option (b)They share common genetic material but to varying degreesis the answer...
Read More →The number that exceeds its square by
Question: The number that exceeds its square by the greatest amount is _______________. Solution: Let the number bex. The square of the number is $x^{2}$. Let $f(x)=x-x^{2}$. Now, we need to find the value of $x$ for which $f(x)$ is maximum. $f(x)=x-x^{2}$ Differentiating both sides with respect tox, we get $f^{\prime}(x)=1-2 x$ For maxima or minima, $f^{\prime}(x)=0$ $\Rightarrow 1-2 x=0$ $\Rightarrow x=\frac{1}{2}$ Now, $f^{\prime \prime}(x)=-20$ So, $x=\frac{1}{2}$ is the point of local maxim...
Read More →A taxonomic key is one of the taxonomic tools
Question: A taxonomic key is one of the taxonomic tools in the identification and classification of plants and animals. It is used in the preparation of a. Monographs b. Flora c. Both a b d. None of these Solution: Option (c)Both a b is the answer....
Read More →Botanical gardens and zoological parks have
Question: Botanical gardens and zoological parks have a. Collection of endemic living species only b. Collection of exotic living species only c. Collection of endemic and exotic living species d. Collection of only local plants and animals Solution: Option (c)Collection of endemic and exotic living species is the answer....
Read More →Evaluate the following limits:
Question: Evaluate the following limits: $\lim _{x \rightarrow 0} \frac{1-\cos m x}{1-\cos n x}$ Solution: To Find: Limits NOTE: First Check the form of imit. Used this method if the limit is satisfying any one from 7 indeterminate form. In this Case, indeterminate Form is $\frac{0}{0}$ Formula used: $\lim _{x \rightarrow 0} \frac{1-\cos x}{x^{2}}=\frac{1}{2}$ Divide numerator and denominator by $m^{2}$ and $n^{2}$, we have So, by using the above formula, we have $\lim _{x \rightarrow 0} \frac{1...
Read More →The taxonomic unit ‘Phylum’ in the classification
Question: The taxonomic unit Phylum in the classification of animals is equivalent to which hierarchical level in classification of plants a. Class c. Orderc. Division d. Family Solution: Option (c)Orderc. Division is the answer....
Read More →Genus represents
Question: Genus represents a. An individual plant or animal b. A collection of plants or animals c. A group of closely related species of plants or animals d. None of these Solution: Option (c)A group of closely related species of plants or animals is the answer....
Read More →Evaluate the following limits:
Question: Evaluate the following limits: $\lim _{x \rightarrow 0} \frac{(1-\cos 4 x)}{(1-\cos 6 x)}$ Solution: To Find: Limits NOTE: First Check the form of imit. Used this method if the limit is satisfying any one from 7 indeterminate form. In this Case, indeterminate Form is $\frac{0}{0}$ Formula used: $\lim _{x \rightarrow 0} \frac{1-\cos x}{x^{2}}=\frac{1}{2}$ Divide numerator and denominator by $x^{2}$, we have So, by using the above formula, we have $\lim _{x \rightarrow 0} \frac{1-\cos 4 ...
Read More →The term ‘systematics’ refers to:
Question: The term systematics refers to: a. Identification and study of organ systems of plants and animals b. Identification and preservation of plants and animals c. Diversity of kinds of organisms and their relationship d. Study of habitats of organisms and their classificationSolution: Solution: Option (c)Diversity of kinds of organisms and their relationship is the answer....
Read More →Evaluate the following limits:
Question: Evaluate the following limits: $\lim _{x \rightarrow 0} \frac{1-\cos 2 x}{3 \tan ^{2} x}$ Solution: To Find: Limits NOTE: First Check the form of imit. Used this method if the limit is satisfying any one from 7 indeterminate form. In this Case, indeterminate Form is $\frac{0}{0}$ Formula used: $\lim _{x \rightarrow 0} \frac{1-\cos x}{x^{2}}=\frac{1}{2}$ and $\lim _{x \rightarrow 0} \frac{\tan x}{x}=1$ Divide numerator and denominator by $x^{2}$, we have So, by using the above formula, ...
Read More →If x and y are two real numbers
Question: Ifxandyare two real numbers such thatx 0 andxy= 1. The the minimum value ofx + yis ________________. Solution: It is given that,xandyare two real numbers such thatx 0 andxy= 1. Let $S=x+y$ Now, $x y=1 \Rightarrow y=\frac{1}{x}$ $\therefore S=x+y=x+\frac{1}{x}$ Differentiating both sides with respect tox, we get $\frac{d S}{d x}=1-\frac{1}{x^{2}}$ For maxima or minima, $\frac{d S}{d x}=0$ $\Rightarrow 1-\frac{1}{x^{2}}=0$ $\Rightarrow x^{2}=1$ $\Rightarrow x=1 \quad(x0)$ Now, $\frac{d^{...
Read More →Which of the following ‘suffixes’ used for
Question: Which of the following suffixes used for units of classification in plants indicates a taxonomic category of family. a. Ales b. Onae c. Aceae d. Ae Solution: Option (c)Aceaeis the answer....
Read More →If x and y are two real numbers
Question: Ifxandyare two real numbers such thatx 0 andxy= 1. The the minimum value ofx + yis ________________. Solution: It is given that,xandyare two real numbers such thatx 0 andxy= 1. Let $S=x+y$ Now, $x y=1 \Rightarrow y=\frac{1}{x}$ $\therefore S=x+y=x+\frac{1}{x}$ Differentiating both sides with respect tox, we get $\frac{d S}{d x}=1-\frac{1}{x^{2}}$ For maxima or minima, $\frac{d S}{d x}=0$ $\Rightarrow 1-\frac{1}{x^{2}}=0$ $\Rightarrow x^{2}=1$ $\Rightarrow x=1 \quad(x0)$ Now, $\frac{d^{...
Read More →Evaluate the following limits:
Question: Evaluate the following limits: $\lim _{x \rightarrow 0} \frac{1-\cos x}{\sin ^{2} 2 x}$ Solution: To Find: Limits NOTE: First Check the form of imit. Used this method if the limit is satisfying any one from 7 indeterminate form. In this Case, indeterminate Form is $\frac{0}{0}$ Formula used: $\lim _{x \rightarrow 0} \frac{1-\cos x}{x^{2}}=\frac{1}{2}$ and $\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$ Divide numerator and denominator by $x^{2}$, we have So, by using the above formula, we...
Read More →As we go from species to kingdom in a taxonomic
Question: As we go from species to kingdom in a taxonomic hierarchy, the number of common characteristics a. Will decrease b. Will increase c. Remain the same d. May increase or decrease Solution: Option (a) Will decreaseis the answer....
Read More →Evaluate the following limits:
Question: Evaluate the following limits: $\lim _{x \rightarrow 0} \frac{1-\cos 3 x}{x^{2}}$ Solution: To Find: Limits NOTE: First Check the form of imit. Used this method if the limit is satisfying any one from 7 indeterminate form. In this Case, indeterminate Form is $\frac{0}{0}$ Formula used: $\lim _{x \rightarrow 0} \frac{1-\cos x}{x^{2}}=\frac{1}{2}$ So, by using the above formula, we have $\lim _{x \rightarrow 0} \frac{1-\cos 3 x}{x^{2}}=\lim _{x \rightarrow 0} \frac{9[1-\cos 3 x]}{(3 x)^{...
Read More →Evaluate the following limits:
Question: Evaluate the following limits: $\lim _{x \rightarrow 0} \frac{1-\cos x}{\sin ^{2} x}$ Solution: To Find: Limits NOTE: First Check the form of imit. Used this method if the limit is satisfying any one from 7 indeterminate form. In this Case, indeterminate Form is $\frac{0}{0}$ [NOTE: $1-\cos x=2 \sin ^{2}(x / 2)$ ] Formula used: $\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$ So, by using the above formula, we have $\lim _{x \rightarrow 0} \frac{1-\cos x}{\sin ^{2} x}=\lim _{x \rightarrow ...
Read More →If the sum of two non-zero numbers is 4 ,
Question: If the sum of two non-zero numbers is 4 , then the minimum value of the sum of their reciprocals is_______________ Solution: Let the two numbers be $x$ and $4-x(x \neq 0,4)$. SupposeSbe the sum of their reciprocals. $\therefore S=\frac{1}{x}+\frac{1}{4-x}, x \neq 0,4$ Differentiating both sides with respect tox, we get $\frac{d S}{d x}=-\frac{1}{x^{2}}-\frac{1}{(4-x)^{2}} \times(-1)$ $\Rightarrow \frac{d S}{d x}=-\frac{1}{x^{2}}+\frac{1}{(4-x)^{2}}$ For maxima or minima, $\frac{d S}{d ...
Read More →If the sum of two non-zero numbers is 4 ,
Question: If the sum of two non-zero numbers is 4 , then the minimum value of the sum of their reciprocals is_______________ Solution: Let the two numbers be $x$ and $4-x(x \neq 0,4)$. SupposeSbe the sum of their reciprocals. $\therefore S=\frac{1}{x}+\frac{1}{4-x}, x \neq 0,4$ Differentiating both sides with respect tox, we get $\frac{d S}{d x}=-\frac{1}{x^{2}}-\frac{1}{(4-x)^{2}} \times(-1)$ $\Rightarrow \frac{d S}{d x}=-\frac{1}{x^{2}}+\frac{1}{(4-x)^{2}}$ For maxima or minima, $\frac{d S}{d ...
Read More →Evaluate the following limits:
Question: Evaluate the following limits: $\lim _{x \rightarrow 0} \frac{\tan (x / 2)}{3 x}$ Solution: To Find: Limits NOTE: First Check the form of imit. Used this method if the limit is satisfying any one from 7 indeterminate form. In this Case, indeterminate Form is $\frac{0}{0}$ Formula used: $\lim _{x \rightarrow 0} \frac{\operatorname{tanx}}{x}=1$ So, by using the above formula, we have $\lim _{x \rightarrow 0} \frac{\tan (x / 2)}{3 x}=\lim _{x \rightarrow 0} \frac{\tan (x / 2)}{6(x / 2)}=\...
Read More →Evaluate the following limits:
Question: Evaluate the following limits: $\lim _{x \rightarrow 0} \frac{\sin (x / 4)}{x}$ Solution: To Find: Limits NOTE: First Check the form of imit. Used this method if the limit is satisfying any one from 7 indeterminate form. In this Case, indeterminate Form is $\frac{0}{0}$ Formula used: $\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$ So, by using the above formula, we have $\lim _{x \rightarrow 0} \frac{\sin (x / 4)}{x}=\lim _{x \rightarrow 0} \frac{\sin (x / 4)}{4(x / 4)}=\frac{1}{4}$ There...
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