For any two sets A and B, prove that:
Question: For any two sets $A$ and $B$, prove that: $A \cap B=\phi \Rightarrow A \subseteq B$. Solution: Let $a \in A \Rightarrow a \notin B \quad(\because A \cap B=\phi)$. $\Rightarrow a \in B$ Thus, $a \in A$ and $a \in B^{\prime} \Rightarrow A \subseteq B$ '....
Read More →Show that the altitude of the right circular cone of maximum volume
Question: Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius $r$ is $\frac{4 r}{3}$. Solution: A sphere of fixed radius (r)is given. LetRandhbe the radius and the height of the cone respectively. The volume (V)of the cone is given by, $V=\frac{1}{3} \pi R^{2} h$ Now, from the right triangle BCD, we have: $\mathrm{BC}=\sqrt{r^{2}-R^{2}}$ $\therefore h=r+\sqrt{r^{2}-R^{2}}$ $\begin{aligned} \therefore V =\frac{1}{3} \pi R^{2}\left(r+\sqr...
Read More →Find sets A, B and C such that
Question: Find sets $A, B$ and $C$ such that $A \cap B, A \cap C$ and $B \cap C$ are non-empty sets and $A \cap B \cap C=\phi$. Solution: Let us consider the following sets,A = {5, 6, 10 } B = {6,8,9} C = {9,10,11} Clearly, $A \cap B=\{6\}$ $B \cap C=\{9\}, A \cap C=\{10\}$ and $\mathrm{A} \cap \mathrm{B} \cap \mathrm{C}=\phi$ It means that, $A \cap B, B \cap C$ and $A \cap C$ are non empty sets and $\mathrm{A} \cap \mathrm{B} \cap \mathrm{C}=\phi$...
Read More →For any two sets, prove that:
Question: For any two sets, prove that: (i) $A \cup(A \cap B)=A$ (ii) $A \cap(A \cup B)=A$ Solution: (i) $\mathrm{LHS}=A \cup(A \cap B)$ $\Rightarrow \mathrm{LHS}=(A \cup A) \cap(A \cup B)$ $\Rightarrow \mathrm{LHS}=A \cap(A \cup B) \quad(\because A \subset A \cup B)$ $\Rightarrow \mathrm{LHS}=A=\mathrm{RHS}$ (ii) $\mathrm{LHS}=A \cap(A \cup B)$ $\Rightarrow \mathrm{LHS}=(A \cap A) \cup(A \cap B)$ $\Rightarrow \mathrm{LHS}=A \cup(A \cap B)$ $\Rightarrow \mathrm{LHS}=A=\mathrm{RHS}$...
Read More →For any two sets, prove that:
Question: For any two sets, prove that: (i) $A \cup(A \cap B)=A$ (ii) $A \cap(A \cup B)=A$ Solution: (i) $\mathrm{LHS}=A \cup(A \cap B)$ $\Rightarrow \mathrm{LHS}=(A \cup A) \cap(A \cup B)$ $\Rightarrow \mathrm{LHS}=A \cap(A \cup B) \quad(\because A \subset A \cup B)$ $\Rightarrow \mathrm{LHS}=A=\mathrm{RHS}$ (ii) $\mathrm{LHS}=A \cap(A \cup B)$ $\Rightarrow \mathrm{LHS}=(A \cap A) \cup(A \cap B)$ $\Rightarrow \mathrm{LHS}=A \cup(A \cap B)$ $\Rightarrow \mathrm{LHS}=A=\mathrm{RHS}$...
Read More →Find the absolute maximum and minimum values of the function f given by
Question: Find the absolute maximum and minimum values of the functionfgiven by $f(x)=\cos ^{2} x+\sin x, x \in[0, \pi]$ Solution: $f(x)=\cos ^{2} x+\sin x$ $f^{\prime}(x)=2 \cos x(-\sin x)+\cos x$ $=-2 \sin x \cos x+\cos x$ Now, $f^{\prime}(x)=0$ $\Rightarrow 2 \sin x \cos x=\cos x \Rightarrow \cos x(2 \sin x-1)=0$ $\Rightarrow \sin x=\frac{1}{2}$ or $\cos x=0$ $\Rightarrow x=\frac{\pi}{6}$, or $\frac{\pi}{2}$ as $x \in[0, \pi]$ Now, evaluating the value of $f$ at critical points $x=\frac{\pi}{...
Read More →Show graphically that each one of the following systems of equations is in-consistent (i.e. has no solution) :
Question: Show graphically that each one of the following systems of equations is in-consistent (i.e. has no solution) : $x-2 y=6$ $3 x-6 y=0$ Solution: The given equations are $x-2 y=6$$.(i)$ $3 x-6 y=0$..(ii) Putting $x=0$ in equation $(i)$, we get: $\Rightarrow 0-2 y=6$ $\Rightarrow y=-3$ $\Rightarrow x=0, \quad y=-3$ Putting $y=0$ in equation $(i)$ we get: $\Rightarrow x-2 \times 0=6$ $\Rightarrow x=6$ $\Rightarrow x=6, \quad y=0$ Use the following table to draw the graph. The graph of $(i)$...
Read More →For three sets A, B and C, show that
Question: For three setsA,BandC, show that (i) $A \cap B=A \cap C$ need not imply $B=C$. (ii) $A \subset B \Rightarrow C-B \subset C-A$ Solution: (i) Let A = {2, 4, 5, 6}, B = {6, 7, 8, 9} and C = {6, 10, 11, 12,13} So, $A \cap B=\{6\}$ and $A \cap C=\{6\}$ Hence, $A \cap B=A \cap C$ but $B \neq C$ (ii) Let $z \in C-B \quad \ldots(1)$ $\Rightarrow z \in C$ and $z \notin B$ $\Rightarrow z \in C$ and $z \notin A \quad[\because A \subset B]$ $\Rightarrow z \in C-A$ ...(2) From $(1)$ and $(2)$, we g...
Read More →Find the points at which the function
Question: Find the points at which the function $f$ given by $f(x)=(x-2)^{4}(x+1)^{3}$ has (i) local maxima (ii) local minima (ii) point of inflexion Solution: The given function is $f(x)=(x-2)^{4}(x+1)^{3}$. $\begin{aligned} \therefore f^{\prime}(x) =4(x-2)^{3}(x+1)^{3}+3(x+1)^{2}(x-2)^{4} \\ =(x-2)^{3}(x+1)^{2}[4(x+1)+3(x-2)] \\ =(x-2)^{3}(x+1)^{2}(7 x-2) \end{aligned}$ Now, $f^{\prime}(x)=0 \Rightarrow x=-1$ and $x=\frac{2}{7}$ or $x=2$ Now, for values of $x$ close to $\frac{2}{7}$ and to the...
Read More →If f(x) = 2x3 − 13x2
Question: If $f(x)=2 x^{3}-13 x^{2}+17 x+12$, Find 1. $f(2)$ 2. $f(-3)$ 3. $f(0)$ Solution: The given polynomial is $f(x)=2 x^{3}-13 x^{2}+17 x+12$ 1. $f(2)$ we need to substitute the ' 2 ' in $f(x)$ $f(2)=2(2)^{3}-13(2)^{2}+17(2)+12$ $=(2 * 8)-(13 * 4)+(17 * 2)+12$ $=16-52+34+12$ $=10$ therefore $f(2)=10$ 2. $f(-3)$ we need to substitute the '(-3)' in f(x) $f(-3)=2(-3)^{3}-13(-3)^{2}+17(-3)+12$ $=\left(2^{*}-27\right)-\left(13^{*} 9\right)-\left(17^{*} 3\right)+12$ $=-54-117-51+12$ = -210 there...
Read More →For any two sets A and B, show that the following statements are equivalent:
Question: For any two setsAandB, show that the following statements are equivalent: (i) $A \subset B$ (ii) $A-B=\phi$ (iii) $A \cup B=B$ (iv) $A \cap B=A$. Solution: We have that the following statements are equivalent: (i) $A \subset B$ (ii) $A-B=\phi$ (iii) $A \cup B=B$ (iv) $A \cap B=A$. Proof: Let $A \subset B$ Let $x$ be an arbitary element of $(A-B)$. NOW, $x \in(A-B)$ $\Rightarrow x \in A \ x \notin B \quad$ (Which is contradictory) Also, $\because A \subset B$ $\Rightarrow A-B \subseteq ...
Read More →For any two sets A and B, show that the following statements are equivalent:
Question: For any two setsAandB, show that the following statements are equivalent: (i) $A \subset B$ (ii) $A-B=\phi$ (iii) $A \cup B=B$ (iv) $A \cap B=A$. Solution: We have that the following statements are equivalent: (i) $A \subset B$ (ii) $A-B=\phi$ (iii) $A \cup B=B$ (iv) $A \cap B=A$. Proof: Let $A \subset B$ Let $x$ be an arbitary element of $(A-B)$. NOW, $x \in(A-B)$ $\Rightarrow x \in A \ x \notin B \quad$ (Which is contradictory) Also, $\because A \subset B$ $\Rightarrow A-B \subseteq ...
Read More →Show graphically that each one of the following systems of equations is in-consistent (i.e. has no solution) :
Question: Show graphically that each one of the following systems of equations is in-consistent (i.e. has no solution) : $3 x-5 y=20$ $6 x-10 y=-40$ Solution: The given equations are $3 x-5 y=20$$(i)$ $6 x-10 y=-4$$\ldots \ldots .(i i)$ Putting $x=0$ in equation $(i)$, we get: $\Rightarrow 3 \times 0-5 y=20$ $\Rightarrow y=-4$ $x=0, \quad y=-4$ Putting $y=0$ in equation $(i)$ we get $\Rightarrow 3 x-5 \times 0=20$ $\Rightarrow x=20 / 3$ $x=20 / 3, \quad y=0$ Use the following table to draw the g...
Read More →A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle.
Question: A point on the hypotenuse of a triangle is at distanceaandbfrom the sides of the triangle. Show that the minimum length of the hypotenuse is $\left(a^{\frac{2}{3}}+b^{\frac{2}{3}}\right)^{\frac{3}{2}}$ Solution: LetΔABC be right-angled at B. Let AB =xand BC =y. Let P be apoint on the hypotenuse of the triangle such that P is at a distance ofaandbfrom the sides AB and BC respectively. LetC =. We have, $\mathrm{AC}=\sqrt{x^{2}+y^{2}}$ Now, PC =bcosec And, AP =asec AC = AP + PC $\Rightarr...
Read More →Give one example each of a binomial of degree 25, and of a monomial of degree 100
Question: Give one example each of a binomial of degree 25, and of a monomial of degree 100 Solution: Given, to write the examples for binomial and monomial with the given degrees Example of a binomial with degree $25-7 x^{35}-5$ Example of a monomial with degree $100-2 t^{100}$...
Read More →Identify constant, linear, quadratic abd cubic polynomial from the following polynomials :
Question: Identify constant, linear, quadratic abd cubic polynomial from the following polynomials : 1. $f(x)=0$ 2. $g(x)=2 x^{3}-7 x+4$ 3. $h(x)=-3 x+1 / 2$ 4. $p(x)=2 x^{2}-x+4$ 5. $\mathrm{q}(\mathrm{x})=4 \mathrm{x}+3$ 6. $r(x)=3 x^{3}+4 x^{2}+5 x-7$ Solution: Given, 1. $f(x)=0$ as 0 is constant, it is a constant variable 2. $g(x)=2 x^{3}-7 x+4$ since the degree is 3 , it is a cubic polynomial 3. $h(x)=-3 x+1 / 2$ since the degree is 1 , it is a linear polynomial 4. $p(x)=2 x^{2}-x+4$ since ...
Read More →Show graphically that each one of the following systems of equations has infinitely many solutions:
Question: Show graphically that each one of the following systems of equations has infinitely many solutions: $x-2 y+11=0$ $3 x-6 y+33=0$ Solution: The given equations are $x-2 y+11=0$$\ldots \ldots \ldots(i)$ $3 x-6 y+33=0$(ii) Putting $x=0$ in equation $(i)$, we get: $\Rightarrow 0-2 y=-11$ $\Rightarrow y=11 / 2$ $x=0, \quad y=11 / 2$ Putting $y=0$ in equation $(i)$ we get: $\Rightarrow x-2 \times=-11$ $\Rightarrow x=-11$ $x=-11, \quad y=0$ Use the following table to draw the graph. Draw the g...
Read More →Identify the polynomials in the following:
Question: Identify the polynomials in the following: 1. $f(x)=4 x^{3}-x^{2}-3 x+7$ 2. b. $g(x)=2 x^{3}-3 x^{2}+\sqrt{x}-1$ 3. $p(x)=\frac{2}{3} x^{2}+\frac{7}{4} x+9$ 4. $q(x)=2 x^{2}-3 x+4 / x+2$ 5. $h(x)=x^{4}-x^{3 / 2}+x-1$ 6. $f(x)=2+3 x+4 x$ Solution: Given 1. $f(x)=4 x^{3}-x^{2}-3 x+7$ it is a polynomial 2. b. $g(x)=2 x^{3}-3 x^{2}+\sqrt{x}-1$ it is not a polynomial since the exponent of $\sqrt{x}$ is a negative integer 3. $p(x)=\frac{2}{3} x^{2}+\frac{7}{4} x+9$ it is a polynomial as it h...
Read More →Show graphically that each one of the following systems of equations has infinitely many solutions:
Question: Show graphically that each one of the following systems of equations has infinitely many solutions: $x-2 y+11=0$ $3 x-6 y+33=0$ Solution: The given equations are $x-2 y+11=0$$\ldots \ldots \ldots(i)$ $3 x-6 y+33=0$(ii) Putting $x=0$ in equation $(i)$, we get: $\Rightarrow 0-2 y=-11$ $\Rightarrow y=11 / 2$ $x=0, \quad y=11 / 2$ Putting $y=0$ in equation $(i)$ we get: $\Rightarrow x-2 \times=-11$ $\Rightarrow x=-11$ $x=-11, \quad y=0$ Use the following table to draw the graph. Draw the g...
Read More →For any two sets A and B, prove that
Question: For any two setsAandB, prove that (i) $B \subset A \cup B$ (ii) $A \cap B \subset A$ (iii) $A \subset B \Rightarrow A \cap B=A$ Solution: (i) For allxBxAorxBxA B (Definition of union of sets)BA B(ii) For allxABxAandx B (Definition of intersection of sets)xAABA(iii)LetAB. We need to proveAB=A.For allxAxAandx B (AB)xA BAA BAlso,ABAThus,AA B andABAAB=A[Proved in (ii)]AB⇒AB=A...
Read More →For any two sets A and B, prove that
Question: For any two setsAandB, prove that (i) $B \subset A \cup B$ (ii) $A \cap B \subset A$ (iii) $A \subset B \Rightarrow A \cap B=A$ Solution: (i) For allxBxAorxBxA B (Definition of union of sets)BA B(ii) For allxABxAandx B (Definition of intersection of sets)xAABA(iii)LetAB. We need to proveAB=A.For allxAxAandx B (AB)xA BAA BAlso,ABAThus,AA B andABAAB=A[Proved in (ii)]AB⇒AB=A...
Read More →If U = {2, 3, 5, 7, 9} is the universal set
Question: IfU= {2, 3, 5, 7, 9} is the universal set andA= {3, 7},B= {2, 5, 7, 9}, then prove that: (i) $(A \cup B)^{\prime}=A^{\prime} \cap B^{\prime}$ (ii) $(A \cap B)^{\prime}=A^{\prime} B^{\prime}$. Solution: Given: U= {2, 3, 5, 7, 9} A= {3, 7} B= {2, 5, 7, 9}To prove : (i) $(A \cup B)^{\prime}=A^{\prime} \cap B^{\prime}$ (ii) $(A \cap B)^{\prime}=A^{\prime} \cup B$, Proof :(i) LHS: $(A \cup B)=\{2,3,5,7,9\}$ $(A \cup B)^{\prime}=\phi$ RHS $A^{\prime}=\{2,5,9\}$ $B^{\prime}=\{3\}$ $A^{\prime}...
Read More →If U = {2, 3, 5, 7, 9} is the universal set
Question: IfU= {2, 3, 5, 7, 9} is the universal set andA= {3, 7},B= {2, 5, 7, 9}, then prove that: (i) $(A \cup B)^{\prime}=A^{\prime} \cap B^{\prime}$ (ii) $(A \cap B)^{\prime}=A^{\prime} B^{\prime}$. Solution: Given: U= {2, 3, 5, 7, 9} A= {3, 7} B= {2, 5, 7, 9}To prove : (i) $(A \cup B)^{\prime}=A^{\prime} \cap B^{\prime}$ (ii) $(A \cap B)^{\prime}=A^{\prime} \cup B$, Proof :(i) LHS: $(A \cup B)=\{2,3,5,7,9\}$ $(A \cup B)^{\prime}=\phi$ RHS $A^{\prime}=\{2,5,9\}$ $B^{\prime}=\{3\}$ $A^{\prime}...
Read More →Classify the following polynomials as polynomials in one variables, two - variables etc:
Question: Classify the following polynomials as polynomials in one variables, two - variables etc: 1. $x^{2}-x y+7 y^{2}$ 2. $x^{2}-2 t x+7 t^{2}-x+t$ 3. $t^{3}-3 t^{2}+4 t-5$ 4. $x y+y z+z x$ Solution: Given 1. $x^{2}-x y+7 y^{2}$ it is a polynomial in two variables $x$ and $y$ 2. $x^{2}-2 t x+7 t^{2}-x+t$ it is a polynomial in two variables $x$ and $t$ 3. $t^{3}-3 t^{2}+4 t-5$ it is a polynomial in one variable $t$ 4. $x y+y z+z x$ it is a polynomial in 3 variables in $x, y$ and $z$...
Read More →If U = {2, 3, 5, 7, 9} is the universal set
Question: IfU= {2, 3, 5, 7, 9} is the universal set andA= {3, 7},B= {2, 5, 7, 9}, then prove that: (i) $(A \cup B)^{\prime}=A^{\prime} \cap B^{\prime}$ (ii) $(A \cap B)^{\prime}=A^{\prime} B^{\prime}$. Solution: Given: U= {2, 3, 5, 7, 9} A= {3, 7} B= {2, 5, 7, 9}To prove : (i) $(A \cup B)^{\prime}=A^{\prime} \cap B^{\prime}$ (ii) $(A \cap B)^{\prime}=A^{\prime} \cup B$, Proof :(i) LHS: $(A \cup B)=\{2,3,5,7,9\}$ $(A \cup B)^{\prime}=\phi$ RHS $A^{\prime}=\{2,5,9\}$ $B^{\prime}=\{3\}$ $A^{\prime}...
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