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 →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: $3 x+y=8$ $6 x+2 y=16$ Solution: The given equations are $3 x+y=8$...(i) $6 x+2 y=16$...(ii) Putting $x=0$ in equation $(i)$, we get: $\Rightarrow 3 \times 0+y=8$ $\Rightarrow y=8$ $x=0, \quad y=8$ Putting $y=0$ in equations $(i)$ we get: $\Rightarrow 3 x+0=8$ $\Rightarrow x=8 / 3$ $x=8 / 3, \quad y=0$ Use the following table to draw the graph. Draw the graph by plotting the two points $...
Read More →Classify the following polynomials as linear, quadratic, cubic and biquadratic polynomials:
Question: Classify the following polynomials as linear, quadratic, cubic and biquadratic polynomials: 1. $x+x^{2}+4$ 2. $3 x-2$ 3. $2 x+x^{2}$ 4. $3 y$ 5. $t^{2}+1$ f. $7 t^{4}+4 t^{2}+3 t-2$ Solution: Given 1. $x+x^{2}+4$ - it is a quadratic polynomial as its degree is 2 2. $3 x-2$ - it is a linear polynomial as its degree is 1 3. $2 x+x^{2}-$ it is a quadratic polynomial as its degree is 2 4. $3 y$ - it is a linear polynomial as its degree is 1 5. $t^{2}+1$ - it is a quadratic polynomial as it...
Read More →Write the degrees of each of the following polynomials:
Question: Write the degrees of each of the following polynomials: 1. $7 x^{3}+4 x^{2}-3 x+12$ 2. $12-x+2 x^{2}$ 3. $5 y-\sqrt{2}$ 4. $7-7 x^{\circ}$ 5. 0 Solution: Given, to find degrees of the polynomials Degree is highest power in the polynomial 1. $7 x^{3}+4 x^{2}-3 x+12$ the degree is 3 2. $12-x+2 x^{2}$ the degree is 3 3. $5 y-\sqrt{2}$ the degree is 1 4. $7-7 x^{\circ}$ the degree is 0 5. 0 the degree of 0 is not defined...
Read More →Let A = {1, 2, 4, 5} B = {2, 3, 5, 6} C = {4, 5, 6, 7}.
Question: LetA= {1, 2, 4, 5}B= {2, 3, 5, 6}C= {4, 5, 6, 7}. Verify the following identities: (i) $A \cup(B \cap C)=(A \cup B) \cap(A \cup C)$ (ii) $A \cap(B \cup C)=(A \cap B) \cup(A \cap C)$ (iii) $A \cap(B-C)=(A \cap B)-(A \cap C)$ (iv) $A-(B \cup C)=A(A-B) \cap(A-C)$ (v) $A-(B \cap C)=(A-B) \cup(A-C)$ (vi) $A \cap(B \Delta C)=(A \cap B) \Delta(A \cap C)$. Solution: Given: A= {1, 2, 4, 5},B= {2, 3, 5, 6} andC= {4, 5, 6, 7} We have to verify the following identities: (i) $A \cup(B \cap C)=(A \c...
Read More →Write the coefficients of
Question: Write the coefficients of $x^{2}$ in each of the following 1. $17-2 x+7 x^{2}$ 2. $9-12 x+x^{2}$ 3. $\frac{\pi}{6} x^{2}-3 x+4$ 4. $\sqrt{3} x-7$ Solution: Given, to find the coefficients of $x^{2}$ 1. $17-2 x+7 x^{2}$ - the coefficient is 7 2. $9-12 x+x^{2}-$ the coefficient is 0 3. $\frac{\pi}{6} x^{2}-3 x+4$ - the coefficient is $\pi / 6$ 4. $\sqrt{3} x-7$ - the coefficient is 0...
Read More →Let A = {1, 2, 4, 5} B = {2, 3, 5, 6} C = {4, 5, 6, 7}.
Question: LetA= {1, 2, 4, 5}B= {2, 3, 5, 6}C= {4, 5, 6, 7}. Verify the following identities: (i) $A \cup(B \cap C)=(A \cup B) \cap(A \cup C)$ (ii) $A \cap(B \cup C)=(A \cap B) \cup(A \cap C)$ (iii) $A \cap(B-C)=(A \cap B)-(A \cap C)$ (iv) $A-(B \cup C)=A(A-B) \cap(A-C)$ (v) $A-(B \cap C)=(A-B) \cup(A-C)$ (vi) $A \cap(B \Delta C)=(A \cap B) \Delta(A \cap C)$. Solution: Given: A= {1, 2, 4, 5},B= {2, 3, 5, 6} andC= {4, 5, 6, 7} We have to verify the following identities: (i) $A \cup(B \cap C)=(A \c...
Read More →Which of the following expressions are polynomials in one variable and which are not?
Question: Which of the following expressions are polynomials in one variable and which are not? State the reasons for your answers 1. $3 x^{2}-4 x+15$ 2. $y^{2}+2 \sqrt{3}$ 3. $3 \sqrt{x}+\sqrt{2} x$ 4. $x-4 / x$ 5. $x^{12}+y^{2}+t^{50}$ Solution: 1. $3 x^{2}-4 x+15$ it is a polynomial of $x$ 2. $y^{2}+2 \sqrt{3}$ it is a polynomial of $y$ $3 \sqrt{x}+\sqrt{2} x$ it is not a polynomial since the exponent of $3 \sqrt{x}$ is not a positive term 4. $x-4 / x$ - it is not a polynomial since the expon...
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: $x-2 y=5$ $3 x-6 y=15$ Solution: The given equations are $x-2 y=5$...(i) $3 x-6 y=15$...(ii) Putting $x=0$ in equation $(i)$, we get: $\Rightarrow 0-2 y=5$ $\Rightarrow y=-5 / 2$ $x=0, \quad y=-5 / 2$ Putting $y=0$ in equations $(i)$ we get: $\Rightarrow x-2 \times 0=5$ $\Rightarrow x=5$ $x=5, \quad y=0$ Use the following table to draw the graph. Draw the graph by plotting the two points...
Read More →Multiply:
Question: Multiply: (i) $x^{2}+y^{2}+z^{2}-x y+x z+y z$ by $x+y-z$ (ii) $x^{2}+4 y^{2}+z^{2}+2 x y+x z-2 y z b y x-2 y-z$ (iii) $x^{2}+4 y^{2}+2 x y-3 x+6 y+9$ by $(x-2 y+3)$ (iv) $9 x^{2}+25 y^{2}+15 x y+12 x-20 y+16$ by $3 x-5 y+4$ Solution: (i) $x^{2}+y^{2}+z^{2}-x y+x z+y z$ by $x+y-z$ $=\left(x^{2}+y^{2}+z^{2}-x y+x z+y z\right)(x+y-z)$ $=x^{3}+y^{3}+z^{3}-3 x y z$ (ii) $x^{2}+4 y^{2}+z^{2}+2 x y+x z-2 y z b y x-2 y-z$ $x^{2}+(-2 y)^{2}+(-z)^{2}-(-2 y)(-z)-(-z)(x)$ $=x^{3}+(-2 y)^{3}+(-z)^{...
Read More →A window is in the form of rectangle surmounted by a semicircular opening.
Question: A window is in the form of rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening. Solution: Letxandybe the length and breadth of the rectangular window. Radius of the semicircular opening $=\frac{x}{2}$ It is given that the perimeter of the window is 10 m. $\therefore x+2 y+\frac{\pi x}{2}=10$ $\Rightarrow x\left(1+\frac{\pi}{2}\right)+2 y=10$ $\Rightarrow 2 y=10-x\le...
Read More →Find the smallest set A such that
Question: Find the smallest set $A$ such that $A \cup\{1,2\}=\{1,2,3,5,9\}$. Solution: We have to find the smallest set A such that A $[\{1,2\}[\{1,2,3,5,9\}]$The union of the two setsABis the set of all those elements that belong toAor toBor to bothAB.Thus,Amust be {3, 5, 9}....
Read More →Find the smallest set A such that
Question: Find the smallest setAsuch thatA{1,2}={1,2,3,5,9}A1,2=1,2,3,5,9. Solution: We have to find the smallest setAsuch thatA{1,2}={1,2,3,5,9}A1,2=1,2,3,5,9.The union of the two setsABis the set of all those elements that belong toAor toBor to bothAB.Thus,Amust be {3, 5, 9}....
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: $2 x+3 y=6$ $4 x+6 y=12$ Solution: The given equations are $2 x+3 y=6$(i) $4 x+6 y=12$(ii) Putting $x=0$ in equation $(i)$, we get: $\Rightarrow 2 \times 0+3 y=6$ $\Rightarrow y=2$ $x=0, \quad y=2$ Putting $y=0$ in equation $(i,$, we get: $\Rightarrow 2 x+3 x=6$ $\Rightarrow x=3$ $x=3, \quad y=0$ Use the following table to draw the graph. Draw the graph by plotting the two points $A(0,2)...
Read More →Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9},
Question: LetU= {1, 2, 3, 4, 5, 6, 7, 8, 9},A= {2, 4, 6, 8} andB= {2, 3, 5, 7}. Verify that (i) $(A \cup B)^{\prime}=A^{\prime} \cap B$, (ii) $(A \cap B)^{\prime}=A^{\prime} \cup B$ '. Solution: Given: U= {1, 2, 3, 4, 5, 6, 7, 8, 9},A= {2, 4, 6, 8} andB= {2, 3, 5, 7} We have to verify: (i) $(A \cup B)^{\prime}=A^{\prime} \cap B$, LHS $A \cup B=\{2,3,4,5,6,7,8\}$ $(A \cup B)^{\prime}=\{1,9\}$ RHS $A^{\prime}=\{1,3,5,7,9\}$ $B^{\prime}=\{1,4,6,8,9\}$ $A^{\prime} \cap B^{\prime}=\{1,9\}$ LHS = RHS ...
Read More →Find the value of
Question: Find the value of $x^{3}+y^{3}-12 x y+64$ when $x+y=-4$ Solution: $=x^{3}+y^{3}+64-12 x y$ $=x^{3}+y^{3}+4^{3}-3(x)(y)(4)$ $=(x+y+4)\left(x^{2}+y^{2}+4^{2}-x y-y \times 4-4 \times x\right)$ $=(-4+4)\left(x^{2}+y^{2}+16-x y-4 y-4 x\right)$ $[\therefore x+y=-4]=0$ $\therefore x^{3}+y^{3}-12 x y+64=0$...
Read More →Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9},
Question: LetU= {1, 2, 3, 4, 5, 6, 7, 8, 9},A= {2, 4, 6, 8} andB= {2, 3, 5, 7}. Verify that (i) $(A \cup B)^{\prime}=A^{\prime} \cap B$, (ii) $(A \cap B)^{\prime}=A^{\prime} \cup B$ '. Solution: Given: U= {1, 2, 3, 4, 5, 6, 7, 8, 9},A= {2, 4, 6, 8} andB= {2, 3, 5, 7} We have to verify: (i) $(A \cup B)^{\prime}=A^{\prime} \cap B$, LHS $A \cup B=\{2,3,4,5,6,7,8\}$ $(A \cup B)^{\prime}=\{1,9\}$ RHS $A^{\prime}=\{1,3,5,7,9\}$ $B^{\prime}=\{1,4,6,8,9\}$ $A^{\prime} \cap B^{\prime}=\{1,9\}$ LHS = RHS ...
Read More →Find the value
Question: Find the value $2 \sqrt{2} a^{3}+16 \sqrt{2} b^{3}+c^{3}-12 a b c$ Solution: $=(\sqrt{2} a)^{3}+(2 \sqrt{2} b)^{3}+c^{3}-3 \times \sqrt{2} a \times 2 \sqrt{2} b \times c$ $=(\sqrt{2} a+2 \sqrt{2} b+c)\left((\sqrt{2} a)^{2}+(2 \sqrt{2} b)^{2}+c^{2}-(\sqrt{2} a)(2 \sqrt{2} b)-(2 \sqrt{2} b) c-(\sqrt{2} a) c\right)$ $=(\sqrt{2} a+2 \sqrt{2} b+c)\left(2 a^{2}+8 b^{2}+c^{2}-4 a b-2 \sqrt{2} b c-\sqrt{2} a c\right)$ $\therefore 2 \sqrt{2} a^{3}+16 \sqrt{2} b^{3}+c^{3}-12 a b c$ $=(\sqrt{2} a...
Read More →Find the value
Question: Find the value $8 x^{3}-125 y^{3}+216+180 x y$ Solution: $=(2 x)^{3}+(-5 y)^{3}+6^{3}-3 \times(2 x)(-5 y)(6)$ $=(2 x+(-5 y)+6)\left((2 x)^{2}+(-5 y)^{2}+6^{2}-2 x \times(-5 y)-(-5 y)^{6}-6(2 x)\right)$ $=(2 x-5 y+6)\left(4 x^{2}+25 y^{2}+36+10 x y+30 y-12 x\right)$ $\therefore 8 x^{3}-125 y^{3}+216+180 x y$ $=(2 x-5 y+6)\left(4 x^{2}+25 y^{2}+36+10 x y+30 y-12 x\right)$...
Read More →The sum of the perimeter of a circle and square is k, where k is some constant.
Question: The sum of the perimeter of a circle and square isk, wherekis some constant. Prove that the sum of their areas is least when the side of square is double the radius of the circle. Solution: Letrbe the radius of the circle andabe the side of the square. Then, we have: $2 \pi r+4 a=k$ (where $k$ is constant) $\Rightarrow a=\frac{k-2 \pi r}{4}$ The sum of the areas of the circle and the square (A)is given by, $A=\pi r^{2}+a^{2}=\pi r^{2}+\frac{(k-2 \pi r)^{2}}{16}$ $\therefore \frac{d A}{...
Read More →Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9},
Question: Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9},A= {1, 2, 3, 4}, = {2, 4, 6, 8} andC= {3, 4, 5, 6}. Find (i) $A^{\prime}$ (ii) $B^{\prime}$ (iii) $(A \cap C)^{\prime}$ (iv) $(A \cup B)^{\prime}$ (v) $\left(A^{\prime}\right)^{\prime}$ (vi) $(B-C)^{\prime}$ Solution: Given: U = {1, 2, 3, 4, 5, 6, 7, 8, 9},A= {1, 2, 3, 4},B= {2, 4, 6, 8} andC= {3, 4, 5, 6} (i) $A^{\prime}=\{5,6,7,8,9\}$ (ii) $B^{\prime}=\{1,3,5,7,9\}$ (iii) $(A \cap C)^{\prime}=\{1,2,5,6,7,8,9\}$ (iv) $(A \cup B)^{\prime}=\{5,7,9\}$ ...
Read More →Find the value
Question: Find the value $3 \sqrt{3} a^{3}-b^{3}-5 \sqrt{5} c^{3}-3 \sqrt{15} a b c$ Solution: $=(\sqrt{3} a)^{3}+(-b)^{3}-(\sqrt{5} c)^{3}-3(\sqrt{3} a)(-b)(-\sqrt{5} c)$ $=(\sqrt{3} a+(-b)+(-\sqrt{5} c))$ $\left((\sqrt{3} a)^{2}+(-b)^{2}+(-\sqrt{5} c)^{2}-\sqrt{3} a(-b)-(-b)(-\sqrt{5} c)-(-\sqrt{5} c) \sqrt{3} a\right)$ $=(\sqrt{3} a-b-\sqrt{5} c)\left(3 a^{2}+b^{2}+5 c^{2}+\sqrt{3} a b-\sqrt{5} b c+\sqrt{15} a c\right)$ $\therefore 3 \sqrt{3} a^{3}-b^{3}-5 \sqrt{5} c^{3}-3 \sqrt{15} a b c$ $=...
Read More →Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9},
Question: Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9},A= {1, 2, 3, 4}, = {2, 4, 6, 8} andC= {3, 4, 5, 6}. Find (i) $A^{\prime}$ (ii) $B^{\prime}$ (iii) $(A \cap C)^{\prime}$ (iv) $(A \cup B)^{\prime}$ (v) $\left(A^{\prime}\right)^{\prime}$ (vi) $(B-C)^{\prime}$ Solution: Given: U = {1, 2, 3, 4, 5, 6, 7, 8, 9},A= {1, 2, 3, 4},B= {2, 4, 6, 8} andC= {3, 4, 5, 6} (i) $A^{\prime}=\{5,6,7,8,9\}$ (ii) $B^{\prime}=\{1,3,5,7,9\}$ (iii) $(A \cap C)^{\prime}=\{1,2,5,6,7,8,9\}$ (iv) $(A \cup B)^{\prime}=\{5,7,9\}$ ...
Read More →Solve the following systems of equations graphically:
Question: Solve the following systems of equations graphically: $2 x+3 y+5=0$ $3 x-2 y-12=0$ Solution: The given equations are $2 x+3 y+5=0$$.(i)$ $3 x-2 y-12=0$$.(i i)$ Putting $x=0$ in equation $(i)$, we get: $\Rightarrow 2 \times 0+3 y=-5$ $\Rightarrow y=-5 / 3$ $x=0, \quad y=-5 / 3$ Putting $y=0$ in equation $(i)$ we get: $\Rightarrow 2 x+3 \times 0=-5$ $\Rightarrow x=-5 / 2$ $x=-5 / 2, \quad y=0$ Use the following table to draw the graph. Draw the graph by plotting the two points from table...
Read More →Find the value
Question: Find the value $2 \sqrt{2} a^{3}+3 \sqrt{3} b^{3}+c^{3}-3 \sqrt{6 a b c}$ Solution: $=(\sqrt{2} a)^{3}+(\sqrt{3} b)^{3}+c^{3}-3 \times \sqrt{2 a} \times \sqrt{3 b} \times c$ $=(\sqrt{2 a}+\sqrt{3 b}+c)\left((\sqrt{2 a})^{2}+(\sqrt{3 b})^{2}+c^{2}-(\sqrt{2 a})(\sqrt{3 b})-(\sqrt{3 b}) c-(\sqrt{2 a} c)\right)$ $=(\sqrt{2 a}+\sqrt{3 b}+c)\left(2 a^{2}+3 b^{2}+c^{2}-\sqrt{6} a b-\sqrt{3} b c-\sqrt{2} a c\right)$ $\therefore 2 \sqrt{2} a^{3}+3 \sqrt{3} b^{3}+c^{3}-3 \sqrt{6 a b c}$ $=(\sqrt...
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