If X = {1, 2, 3}, if n represents any member of X,
Question: If X = {1, 2, 3}, if n represents any member of X, write the following sets containing all numbers represented by: (i) 4n (ii) n + 6 (iii)n/2 (iv) n 1 Solution: (i)According to the question, X = {1, 2, 3} where n represents any member of X X = {1, 2, 3} {4n | nx} = {41, 42, 43} ={4, 8, 12} (ii)According to the question, X = {1, 2, 3} where n represents any member of X X = {1, 2, 3} {n + 6 | nx} = {1 + 6, 2 + 6, 3 + 6} ={7, 8, 9} (iii)According to the question, X = {1, 2, 3} where n rep...
Read More →Given that N = {1, 2, 3,…, 100}.
Question: Given that N = {1, 2, 3,, 100}. Then write (i) the subset of N whose elements are even numbers.(ii) the subset of N whose element are perfect square numbers. Solution: We know that, AsetA is asubset of a setB, if A is contained inside B. Hence, all elements of A are also elements of B. (i)According to the question, N = {1, 2, 3 ,, 100} Hence, subset of N whose elements are even numbers = {2, 4, 6, 8,,100} (ii)According to the question, N = {1, 2, 3 ,, 100} Hence, subset of N whose elem...
Read More →If A and B are subsets of the universal set U,
Question: If A and B are subsets of the universal set U, then show that (i) AAB (ii) AB⇔AB = B (iii) (A B) A Solution: (i)According to the question, A and B are two subsets To prove:AAB Proof: Let xA ⇒xA or xB ⇒xAB ⇒AAB Hence Proved (ii)According to the question, A and B are two subsets To prove:AB⇔AB = B Proof: Let xAB ⇒xA or xB Since,AB, we get, ⇒xB ⇒ABB (i) We know that, BAB (ii) From equations (i) and (ii), We get, AB = B Now, Let yA ⇒yAB Since,AB = B, we get, ⇒yB } ⇒AB So, AB⇔AB = B Hence P...
Read More →Given L = {1, 2, 3, 4}, M = {3, 4, 5, 6}
Question: Given L = {1, 2, 3, 4}, M = {3, 4, 5, 6} and N = {1, 3, 5}. Verify that L(MN) = (LM)(LN). Solution: According to the question, L = {1, 2, 3, 4}, M = {3, 4, 5, 6} and N = {1, 3, 5} To verify: L (MN) = (L M)(L N) M = {3, 4, 5, 6}, N = {1, 3, 5}⇒MN = {1, 3, 4, 5, 6} L = {1, 2, 3, 4} and MN = {1, 3, 4, 5, 6} ⇒L (MN) = {2}(i) L = {1, 2, 3, 4} and M = {3, 4, 5, 6}⇒L M = {1, 2} L = {1, 2, 3, 4} and N = {1, 3, 5}⇒L N = {2, 4} L M = {1, 2} and L N = {2, 4} ⇒(L M)(L N) = {2}(ii) From equations (...
Read More →State which of the following statements are
Question: State which of the following statements are true and which are false. Justify your answer. (i) 35{x | x has exactly four positive factors}. (ii) 128{y | the sum of all the positive factors of y is 2y} (iii) 3{x | x4 5x3+ 2x2 112x + 6 = 0} (iv) 496 {y | the sum of all the positive factors of y is 2y}. Solution: (i)True According to the question, 35{x | x has exactly four positive factors} The possible positive factors of 35 = 1, 5, 7, 35 35 belongs to given set Since, 35 has exactly fou...
Read More →If Y = {x | x is a positive factor of the number 2p – 1 (2p – 1),
Question: If Y = {x | x is a positive factor of the number 2p 1(2p 1), where 2p 1 is a prime number}. Write Y in the roaster form. Solution: According to the question, Y = {x | x is a positive factor of the number 2p 1(2p 1), where 2p 1 is a prime number}. Roster form of given set, Only possible positive factors of a prime number p are 1 and p itself. Possible factors of 2p 1(2p 1) are all possible factors of 2p 1and 2p 1 individually. Possible factors of 2p 1are 20, 21 2p 1and that of 2p 1 are ...
Read More →Write the following sets in the roaster form:
Question: Write the following sets in the roaster form: (i) $D=\left\{t \mid t^{3}=t, t \in R\right\}$ (ii) $E=\left\{w \mid \frac{w-2}{w+3}=3, w \in \mathbf{R}\right\}$ (iii) $\mathrm{F}=\left\{\mathrm{x} \mid \mathrm{x}^{4}-5 \mathrm{x}^{2}+6=0, \mathrm{x} \in \mathbf{R}\right\}$ Solution: (i)According to the question, D = {t | t3= t, tR} Roster form, t3= t ⇒t3 t = 0 ⇒t(t2 1) = 0 ⇒t(t 1)(t + 1) = 0 ⇒t = 0, -1 or 1 Hence,D = {-1, 0, 1} (ii) According to the question, $E=\left\{w \mid \frac{w-2}...
Read More →Determine the values of x for which the function
Question: Determine the values of $x$ for which the function $f(x)=x^{2}-6 x+9$ is increasing or decreasing. Also, find the coordinates of the point on the curve $y=x^{2}-6 x+9$ where the normal is parallel to the liney $=x+5$. Solution: Given:- Function $f(x)=x^{2}-6 x+9$ and a line parallel to $y=x+5$ Theorem:- Let $f$ be a differentiable real function defined on an open interval $(a, b)$. (i) If $f^{\prime}(x)0$ for all $x \in(a, b)$, then $f(x)$ is increasing on $(a, b)$ (ii) If $f^{\prime}(...
Read More →Write the following sets in the roaster form:
Question: Write the following sets in the roaster form: (i) A = {x : xR, 2x + 11 = 15}(ii) B = {x | x2= x, xR}(iii) C = {x | x is a positive factor of a prime number p} Solution: (i)According to the question, A = {x : xR, 2x + 11 = 15} Roster form, 2x + 11 = 15 ⇒2x = 15 11 ⇒2x = 4 ⇒x = 2 Hence,A = {2} (ii)According to the question, B = {x | x2= x, xR} Roster form, x2= x ⇒x2 x = 0 ⇒x(x 1) = 0 ⇒x = 0 or 1 Hence,B = {0, 1} (iii)According to the question, C = {x | x is a positive factor of a prime n...
Read More →Find the distance of the point
Question: Find the distance of the point $(-2,3)$ from the line $12 x=5 y+13$ Solution: Given: Point $(-2,3)$ and line $12 x-5 y=13$ To find: The distance of the point $(-2,3)$ from the line $12 x-5 y=13$ Formula used: We know that the distance between a point $P(m, n)$ and a line $a x+b y+$ $c=0$ is given by, $D=\frac{|a m+b n+c|}{\sqrt{a^{2}+b^{2}}}$ The given equation of the line is $12 x-5 y-13=0$ Here $m=-2$ and $n=3, a=12, b=-5, c=-13$ $D=\frac{|12(-2)-5(3)-13|}{\sqrt{12^{2}+5^{2}}}$ $D=\f...
Read More →Find the intervals in which the following functions are increasing or decreasing.
Question: Find the intervals in which the following functions are increasing or decreasing. $f(x)=\log (2+x)-\frac{2 x}{2+x}$ Solution: Given:- Function $f(x)=\log (2+x)-\frac{2 x}{2+x}$ Theorem:- Let $f$ be a differentiable real function defined on an open interval $(a, b)$. (i) If $f^{\prime}(x)0$ for $a l l x \in(a, b)$, then $f(x)$ is increasing on $(a, b)$ (ii) If $f^{\prime}(x)0$ for all $x \in(a, b)$, then $f(x)$ is decreasing on $(a, b)$ Algorithm:- (i) Obtain the function and put it equ...
Read More →Find the distance of the point (3, -5) from the line
Question: Find the distance of the point $(3,-5)$ from the line $3 x-4 y=27$ Solution: Given: Point (3,-5) and line 3x 4y = 27 To find: The distance of the point $(3,-5)$ from the line $3 x-4 y=27$ Formula used: We know that the distance between a point P(m,n) and a line ax + by + c = 0 is given by, $D=\frac{|a m+b n+c|}{\sqrt{a^{2}+b^{2}}}$ The equation of the line is $3 x-4 y-27=0$ Here $m=3$ and $n=-5, a=3, b=-4, c=-27$ $D=\frac{|3(3)-4(-5)-27|}{\sqrt{3^{2}+4^{2}}}$ $D=\frac{|9+20-27|}{\sqrt{...
Read More →Assertion (A): If BOD level of water in a reservoir is less
Question: Assertion (A): If BOD level of water in a reservoir is less than 5 ppm it is highly polluted. Reason (R): High biological oxygen demand means a low activity of bacteria in water. (i) Both A and R are correct and R is the correct explanation of A. (ii) Both A and R are correct but R is not the correct explanation of A. (iii) Both A and R are not correct. (iv) A is not correct but R is correct. Solution: Option (iii) is the answer....
Read More →Reduce each of the following equations to normal form :
Question: Reduce each of the following equations to normal form : (i) $x+y-2=0$ (ii) $\mathrm{x}+\mathrm{y}+\sqrt{2}=0$ (iii) $x+5=0$ (iv) $2 y-3=0$ (v) $4 x+3 y-9=0$ Solution: $\Rightarrow x+y=2$ If the equation is in the form of ax + by = c, to get into the normal form we should divide it by $\sqrt{a^{2}+b^{2}}$ so now Divide by $\sqrt{1^{2}+1^{2}}=\sqrt{2}$ $\Rightarrow \frac{x}{\sqrt{2}}+\frac{y}{\sqrt{2}}=\frac{2}{\sqrt{2}}$ $\Rightarrow \frac{x}{\sqrt{2}}+\frac{y}{\sqrt{2}}=\sqrt{2}$ This ...
Read More →Assertion (A): Excessive use of chlorinated synthetic
Question: Assertion (A): Excessive use of chlorinated synthetic pesticides causes soil and water pollution. Reason (R): Such pesticides are non-biodegradable. (i) Both A and R are correct and R is the correct explanation of A. (ii) Both A and R are correct but R is not the correct explanation of A. (iii) Both A and R are not correct. (iv) A is not correct but R is correct Solution: Option (i) is the answer....
Read More →Assertion (A): Ozone is destroyed by solar radiation
Question: Assertion (A): Ozone is destroyed by solar radiation in the upper stratosphere. Reason (R): Thinning of the ozone layer allows excessive UV radiations to reach the surface of the earth. (i) Both A and R are correct and R is the correct explanation of A. (ii) Both A and R are correct but R is not the correct explanation of A. (iii) Both A and R are not correct. (iv) A is not correct but R is correct. Solution: Option (iv) A is not correct but R is correct is the answer....
Read More →Assertion (A): Carbon dioxide is one of the important greenhouse gases
Question: Assertion (A): Carbon dioxide is one of the important greenhouse gases. Reason (R): It is largely produced by respiratory function of animals and plants. (i) Both A and R are correct and R is the correct explanation of A. (ii) Both A and R are correct but R is not the correct explanation of A. (iii) Both A and R are not correct. (iv) A is not correct but R is correct. Solution: Option (ii)Both A and R are correct but R is not the correct explanation of A is the answer....
Read More →Assertion (A): Photochemical smog is oxidising in nature.
Question: Assertion (A): Photochemical smog is oxidising in nature. Reason (R): Photochemical smog contains NO2 and O3, which are formed during the sequence of reactions. (i) Both A and R are correct and R is the correct explanation of A. (ii) Both A and R are correct but R is not the correct explanation of A. (iii) Both A and R are not correct. (iv) A is not correct but R is correct. Solution: Option (i) Both A and R are correct and R is the correct explanation of Ais correct....
Read More →Assertion (A): The pH of acid rain is less than 5.6.
Question: Assertion (A): The pH of acid rain is less than 5.6. Reason (R): Carbon dioxide present in the atmosphere dissolves in rain water and forms carbonic acid. (i) Both A and R are correct and R is the correct explanation of A. (ii) Both A and R are correct but R is not the correct explanation of A. (iii) Both A and R are not correct. (iv) A is not correct but R is correct. Solution: Option (ii)Both A and R are correct but R is not the correct explanation of A is correct....
Read More →Assertion (A): Greenhouse effect was observed in houses used to grow
Question: Assertion (A): Greenhouse effect was observed in houses used to grow plants and these are made of green glass. Reason (R): Greenhouse name has been given because glasshouses are made of green glass. (i) Both A and R are correct and R is the correct explanation of A. (ii) Both A and R are correct but R is not the correct explanation of A. (iii) Both A and R are not correct. (iv) A is not correct but R is correct. Solution: Option (iii)Both A and R are not correct is correct....
Read More →Match the pollutants given in Column
Question: Match the pollutants given in Column I with their effects given in Column II. Solution: (i) are a and d (ii) is c (iii) is a (iv) is b...
Read More →Match the activity given in Column I
Question: Match the activity given in Column I with the type of pollution created by it given in Column II. Solution: (i) is e (ii) is d (iii) is a (iv) is b (v) is c...
Read More →Find the intervals in which the following functions are increasing or decreasing.
Question: Find the intervals in which the following functions are increasing or decreasing. $f(x)=\frac{3}{2} x^{4}-4 x^{3}-45 x^{2}+51$ Solution: Given:- Function $f(x)=\frac{3}{2} x^{4}-4 x^{3}-45 x^{2}+51$ Theorem:- Let $\mathrm{f}$ be a differentiable real function defined on an open interval $(\mathrm{a}, \mathrm{b})$. (i) If $f^{\prime}(x)0$ for all $x \in(a, b)$, then $f(x)$ is increasing on $(a, b)$ (ii) If $f^{\prime}(x)0$ for all $x \in(a, b)$, then $f(x)$ is decreasing on $(a, b)$ Alg...
Read More →Match the pollutant(s)
Question: Match the pollutant(s) in Column I with the effect(s) in Column II. Solution: (i) is d (ii) is e (iii) is a (iv) is c (v) is b...
Read More →Match the terms given in Column
Question: Match the terms given in Column I with the compounds given in Column II. Solution: (i) are c and d (ii) are e and d (iii) is b (iv) is a...
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