Let A = {–1, 2, 3} and
Question: Let A = {1, 2, 3} and B = {1, 3}. Determine (i) A B (ii) B A (iii) B B (iv) A A Solution: According to the question, A = {1, 2, 3} and B = {1, 3} (i) A B {1, 2, 3} {1, 3} So, A B = {(1, 1), (1, 3), (2, 1), (2, 3), (3, 1), (3, 3)} Hence, the Cartesian product = {(1, 1), (1, 3), (2, 1), (2, 3), (3, 1), (3, 3)} (ii) B A. {1, 3} {1, 2, 3} So, B A = {(1, 1), (1, 2), (1, 3), (3, 1), (3, 2), (3, 3)} Hence, the Cartesian product = {(1, 1), (1, 2), (1, 3), (3, 1), (3, 2), (3, 3)} (iii) B B {1, ...
Read More →Find the length of the perpendicular from the origin to each of the following lines :
Question: Find the length of the perpendicular from the origin to each of the following lines : (i) $7 x+24 y=50$ (ii) $4 x+3 y=9$ (iii) $x=4$ Solution: Given: Point (0,0) and line 7x + 24y = 50 To find: The length of the perpendicular from the origin to the line $7 x+24 y=50$ Formula used: We know that the length of the perpendicular from P (m,n) to the line ax + by + c = 0 is given by, $D=\frac{|a m+b n+c|}{\sqrt{a^{2}+b^{2}}}$ The given equation of the line is $7 x+24 y-50=0$ Here $m=0$ and $...
Read More →Suppose A1, A2, …, A30 are thirty sets each
Question: Suppose A1, A2, , A30are thirty sets each having 5 elements and B1, B2, , Bn are n sets each with 3 elements, let $\bigcup_{i=1}^{30} A_{i}=\bigcup_{j=1}^{n} B_{j}=S$ and each element of S belongs to exactly 10 of the Ais and exactly 9 of the B,S. then n is equal toA. 15B. 3C. 45D. 35 Solution: According to the question, $U_{i=1}^{30} A_{i}=U_{j=1}^{n} B_{j}=S$ Since elements are not repeating, number of elements in A1A2A3A30= 30 5 Now, since each element is used 10 times We get, 10 S ...
Read More →Show that
Question: Show that $f(x)=\log _{a} x, 0a1$ is a decreasing function for all $x0$. Solution: Given:- Function $f(x)=\log _{a} x, 0a1$ 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 $f^{\prime}(x)0$ for all $x \in(a, b)$, then $f(x)$ is decreasing on $(a, b)$ Algorithm:- (i) Obtain the function and put it equal to $f(x)$ (ii) Find $f^{\prime}(x)$ (iii) Put $f^{\p...
Read More →In a town of 10,000 families it was found
Question: In a town of 10,000 families it was found that 40% families buy newspaper A, 20% families buy newspaper B, 10% families buy newspaper C, 5% families buy A and B, 3% buy B and C and 4% buy A and C. If 2% families buy all the three newspapers. Find (a) The number of families which buy newspaper A only. (b) The number of families which buy none of A, B and C Solution: According to the question, Total number of families = 10,000 Number of families buying newspaper A = n(A) = 40% Number of ...
Read More →In a survey of 200 students of a school,
Question: In a survey of 200 students of a school, it was found that 120 study Mathematics, 90 study Physics and 70 study Chemistry, 40 study Mathematics and Physics, 30 study Physics and Chemistry, 50 study Chemistry and Mathematics and 20 none of these subjects. Find the number of students who study all the three subjects. Solution: According to the question, Total number of students = n(U) = 200 Number of students who study Mathematics = n(M) = 120 Number of students who study Physics = n(P) ...
Read More →Show that
Question: Show that $f(x)=e^{\frac{1}{x}}, x \neq 0$ is a decreasing function for all $x \neq 0$. Solution: Given:- Function $\mathrm{f}(\mathrm{x})=\mathrm{e}^{\frac{1}{x}}$ 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 $f^{\prime}(x)0$ for all $x \in(a, b)$, then $f(x)$ is decreasing on $(a, b)$ Algorithm:- (i) Obtain the function and put it equal to $f(x)$ (...
Read More →In a class of 60 students,
Question: In a class of 60 students, 25 students play cricket and 20 students play tennis, and 10 students play both the games. Find the number of students who play neither? Solution: According to the question, Total number of students = 60 Students who play cricket = 25 Students who play tennis = 20 Students who play both the games = 10 To find:number of students who play neither Let the total number of students = S Let the number of students who play cricket = C Let the number of students who ...
Read More →Find the distance of the point
Question: Find the distance of the point (4, 2) from the line joining the points (4, 1) and (2, 3) Solution: Given: Point (4,2) and the line joining the points (4, 1) and (2, 3) To find: The distance of the point $(4,2)$ from the line joining the points $(4,1)$ and $(2,3)$ Formula used: The equation of the line joining the points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$ is given by $\frac{y-y_{1}}{y_{2}-y_{1}}=\frac{x-x_{1}}{x_{2}-x_{1}}$ Here $x_{1}=4 y_{1}=1$ and $x_{2}=2 y_...
Read More →Let A, B and C be sets.
Question: Let A, B and C be sets. Then show that A (B C) = (A B) (A C) Solution: According to the question, A, B and C are three given sets To prove:A(BC) = (AB)(AC) Let xA(BC) ⇒xA and x(BC) ⇒xA and (xB or xC) ⇒(xA and xB) or (xA and xC) ⇒xABor xAC ⇒x(AB)(AC) ⇒A(BC)(AB)(AC) (i) Let y(AB)(AC) ⇒yABor xAC ⇒(yA and yB) or (yA and yC) ⇒yA and (yB or yC) ⇒yA and y(BC) ⇒yA(BC) ⇒(AB)(AC)A(BC) (ii) We know that: PQ and QP⇒P = Q From equations (i) and (ii), we have, A(BC) = (AB)(AC) Hence Proved...
Read More →Prove the following
Question: Let, $T=\left\{x \mid \frac{x+5}{x-7}-5=\frac{4 x-40}{13-x}\right\}$ Is T an empty set? Justify your answer. Solution: According to the question, $T=\left\{x \mid \frac{x+5}{x-7}-5=\frac{4 x-40}{13-x}\right\}$ To check whetherT is an empty set or not, We solve, $\frac{x+5}{x-7}-5=\frac{4 x-40}{13-x}$ $\Rightarrow \frac{x+5-5(x-7)}{x-7}=\frac{4 x-40}{13-x}$ $\Rightarrow \frac{x+5-5 x+35}{x-7}=\frac{4 x-40}{13-x}$ $\Rightarrow \frac{-4 x+40}{x-7}=\frac{4 x-40}{13-x}$ $\Rightarrow \frac{-...
Read More →Find the distance of the point
Question: Find the distance of the point $(2,3)$ from the line $y=4$. Solution: Given: Point (2,3) and line y = 4 To find: The distance of the point (2, 3) from the line y = 4 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 equation of the line is $y-4=0$ Here $m=2$ and $n=3, a=0, b=1, c=-4$ $D=\frac{|1(3)-4|}{\sqrt{0^{2}+1^{2}}}$ $D=\frac{|3-4|}{\sqrt{0+1}}=\frac{|-1|}{\sqrt{1}}=1$ $D=1$ Th...
Read More →For all sets A and B,
Question: For all sets A and B, (A B) B = A B Solution: According to the question, There are two sets A and B To prove:(AB) B = A B L.H.S = (AB) B Since,A B = AB, we get, = (AB)B Since,Distributive property of set: (AB)(AC) = A(BC), we get, = (AB)(BB) Since,AA = Φ, we get, = (AB)Φ =AB Since,A B = AB, we get, = A B = R.H.S Hence Proved...
Read More →For all sets A and B, A – (A ∩ B) = A – B
Question: For all sets A and B, A (A B) = A B Solution: According to the question, There are two sets A and B To prove:A (AB) = A B L.H.S = A (AB) Since, A B = AB, we get, = A(AB) = A(AB) Since, (AB) = AB, we get, = A(AB) Since, Distributive property of set ⇒ (AB)(AC) = A(BC), we get, = (AA)(AB) Since, AA = Φ, we get, = Φ(AB) =AB Since, A B = AB, we get, = A B = R.H.S Hence Proved...
Read More →For all sets A and B, A – (A – B) = A ∩ B
Question: For all sets A and B, A (A B) = A B Solution: According to the question, There are two sets A and B To prove:A (A B) = AB L.H.S = A (A B) Since, A B = AB, we get, = A (AB) = A(AB) Since, (AB) = AB, we get, = A[A(B)] Since, (B) = B, we get, = A(AB) Since, distributive property of set ⇒ (AB)(AC) = A(BC), we get, = (AA)(AB) Since, AA = Φ, we get, = Φ(AB) =AB = R.H.S Hence Proved...
Read More →Find the distance of the point
Question: Find the distance of the point $(-4,3)$ from the line $4(x+5)=3(y-6)$ Solution: Given: Point $(-4,3)$ and line $4(x+5)=3(y-6)$ To find: The distance of the point $(-4,3)$ from the line $4(x+5)=3(y-6)$ Formula used: We know that the distance between a point $P(m, n)$ and a line ax $+b y+$ $c=0$ is given by, $D=\frac{|\mathrm{am}+\mathrm{bn}+\mathrm{c}|}{\sqrt{\mathrm{a}^{2}+\mathrm{b}^{2}}}$ The equation of the line is $4 x+20=3 y-18$ $4 x-3 y+38=0$ Here $m=-4$ and $n=3, a=4, b=-3, c=38...
Read More →Show that
Question: Show that $f(x)=e^{2 x}$ is increasing on $R$. Solution: Given:- Function $f(x)=e^{2 x}$ 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 $f^{\prime}(x)0$ for all $x \in(a, b)$, then $f(x)$ is decreasing on $(a, b)$ Algorithm:- (i) Obtain the function and put it equal to $f(x)$ (ii) Find $f^{\prime}(x)$ (iii) Put $f^{\prime}(x)0$ and solve this inequatio...
Read More →For all sets A and B,
Question: For all sets A and B, A (B A) = A B Solution: According to the question, There are two sets A and B To prove:A(B A) = AB L.H.S = A(B A) Since,A B = AB, we get, = A(BA) Since, distributive property of set ⇒ (AB)(AC) = A(BC), we get, = (AB)(AA) Since, AA = U, we get, = (AB)U =AB = R.H.S Hence Proved...
Read More →For all sets A, B and C,
Question: For all sets A, B and C, if A C and B C, then A B C Solution: True According to the question, There are three sets A, B and C To check:if AC and BC, then ABC is true or false Let xAB ⇒xA or xC ⇒xC or xC {∵AC and BC} ⇒xC ⇒ABC Hence, the given statement for all sets A, B and C, if A C and B C, then A B C is true...
Read More →For all sets A, B and C, if A ⊂ B,
Question: For all sets A, B and C, if A B, then A C B C Solution: True According to the question, There are three sets A, B and C To check:if AB, then ACBC is true or false Let xAC ⇒xA or xC ⇒xB or xC {∵AB} ⇒xBC ⇒ACBC Hence, the given statement for all sets A, B and C, if A B, then A C B C is true...
Read More →For all sets A, B and C,
Question: For all sets A, B and C, if A B, then A C B C Solution: True According to the question, There are three sets A, B and C To check:if AB, then ACBC is true or false Let xAC ⇒xA and xC ⇒xB and xC {∵AB} ⇒xBC ⇒ACBC Hence, the given statement for all sets A, B and C, if A B, then A C B C is true....
Read More →For all sets A and B,
Question: For all sets A and B, (A B) (A B) = A Solution: True According to the question, There are two sets A and B To check:(A B)(AB) = A is true or false L.H.S = (A B)(AB) Since, A B = AB, We get, = (AB)(AB) Using distributive property of set: We get, (AB)(AC) = A(BC) = A(BB) = AU = A = R.H.S Hence, the given statement for all sets A and B, (A B) (A B) = A is true...
Read More →For all sets A, B and C,
Question: For all sets A, B and C, show that (A B) (A C) = A (B C) Solution: According to the question, There are three sets A, B and C To show: (A B)(A C) = A (BC) Let x(A B)(A C) ⇒x(A B) and x(A C) ⇒(xA and xB) and (xA and xC) ⇒xA and (xB and xC) ⇒xA and x(BC) ⇒xA (BC) ⇒(A B)(A C)A (BC) (i) Let yA (BC) ⇒yA and y(BC) ⇒yA and (yB and yC) ⇒(yA and yB) and (yA and yC) ⇒y(A B) and y(A C) ⇒y(A B)(A C) ⇒A (BC)(A B)(A C) (ii) We know that, If PQ and QP Then,P = Q Therefore, from equations (i) and (ii)...
Read More →If Y = {1, 2, 3,…, 10}, and a represents any element of Y,
Question: If Y = {1, 2, 3,, 10}, and a represents any element of Y, write the following sets, containing all the elements satisfying the given conditions. (i) aY but a2Y (ii) a + 1 = 6, aY (iii) a is less than 6 and a Y Solution: (i)According to the question, Y = {1, 2, 3,, 10} where a represents any element of Y Y = {1, 2, 3,, 10} 12= 1, 22= 4, 32= 9 1, 4, 9Y⇒1, 2, 3 do not satisfy given condition Hence, {a: aY and a2Y} = {4, 5, 6, 7, 8, 9, 10} (ii)According to the question, Y = {1, 2, 3,, 10} ...
Read More →Find the intervals in which
Question: Find the intervals in which $f(x)=\sin x-\cos x$, where $0x2 \pi$ is increasing or decreasing. Solution: Given:- Function $f(x)=\sin x-\cos x, 0x2 \pi$ 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 $f^{\prime}(x)0$ for all $x \in(a, b)$, then $f(x)$ is decreasing on $(a, b)$ Algorithm:- (i) Obtain the function and put it equal to $f(x)$ (ii) Find $f^{...
Read More →