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Question: If $\left(\frac{1-i}{1+i}\right)^{100}=(a+i b)$, find the values of $a$ and $b$. Solution: Given: $a+i b=\left(\frac{1-i}{1+i}\right)^{100}$ Consider the given equation, $a+i b=\left(\frac{1-i}{1+i}\right)^{100}$ Now, we rationalize $=\left(\frac{1-i}{1+i} \times \frac{1-i}{1-i}\right)^{100}$ [Here, we multiply and divide by the conjugate of 1 + i] $=\left(\frac{(1-i)^{2}}{(1+i)(1-i)}\right)^{100}$ $=\left(\frac{1+i^{2}-2 i}{(1+i)(1-i)}\right)^{100}$ Using $(a+b)(a-b)=\left(a^{2}-b^{2}...
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Question: Tick (✓) the correct answer: The length and breadth of a rectangle are in the ratio 4 : 3. If the diagonal measures 25 cm then the perimeter of the rectangle is (a) 56 cm (b) 60 cm (c) 70 cm (d) 80 cm Solution: (c) 70 cm Let $A B C D$ be a rectangle and let the diagonal $A C$ be $25 \mathrm{~cm}$, length $A B$ be $4 x \mathrm{~cm}$ and breadth $B C$ be $3 x$ cm. Each angle of a rectangle is a right angle. $\therefore \angle A B C=90^{\circ}$ From the right $\Delta A B C:$ $A C^{2}=A B^...
Read More →Verify the property x + y = y + x of rational numbers by taking
Question: Verify the property x + y = y + x of rational numbers by taking (a) x = , y = (b) x = -2/3, y = -5/6 (c) x = -3/7, y = 20/21 (d) x = -2/5, y = 9/10 Solution: (a) x = , y = In the question is given to verify the property = x + y = y + x Where, x = , y = Then, + = + LHS = + = (1 + 1)/2 = 2/2 = 1 RHS = + = (1 + 1)/2 = 2/2 = 1 By comparing LHS and RHS LHS = RHS 1 = 1 Hence x + y = y + x (b) x = -2/3, y = -5/6 Solution:- In the question is given to verify the property = x + y = y + x Where,...
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Question: Tick (✓) the correct answer: The diagonals do not necessarily intersect at right angles in a (a) parallelogram (b) rectangle (c) rhombus (d) kite Solution: (a) parallelogram In a parallelogram, the diagonals do not necessarily intersect at right angles....
Read More →Tick (✓) the correct answer
Question: Tick (✓) the correct answer Two adjacent angles of a parallelogram are (2x+ 25) and (3x 5). The value ofxis (a) 28 (b) 32 (c) 36 (d) 42 Solution: (b) 32 We know that the sum of adjacent angles of a parallelogram is $180^{\circ}$. $\Rightarrow 2 x+25+3 x-5=180$ $\Rightarrow 5 x+20=180$ $\Rightarrow 5 x=180-20$ $\Rightarrow 5 x=160$ $\Rightarrow x=\frac{160}{5}$ Therefore, the value of $x$ is 32 ....
Read More →Give one example each to show that the rational numbers
Question: Give one example each to show that the rational numbers are closed under addition, subtraction and multiplication. Are rational numbers closed under division? Give two examples in support of your answer. Solution: Rational numbers are closed under addition:- Example:- 5/4 + 1/2 The LCM of the denominators 4 and 2 is 4 (5/4) = [(51)/ (41)] = (5/4) and (1/2) = [(12)/ (42)] = (1/4) Then, = 5/4 + = (5 + 1)/ 4 = 6/4 = 3/2 is a rational number Rational numbers are closed under subtraction:- ...
Read More →Find the multiplicative inverse of each of the following:
Question: Find the multiplicative inverse of each of the following: $\frac{(1+i)(1+2 i)}{(1+3 i)}$ Solution: Given: $\frac{(1+i)(1+2 i)}{(1+3 i)}$ To find: Multiplicative inverse We know that Multiplicative Inverse of $z=z^{-1}$ $=\frac{1}{z}$ Putting $z=\frac{(1+i)(1+2 i)}{(1+3 i)}$ So, Multiplicative inverse of $\frac{(1+i)(1+2 i)}{(1+3 i)}=\frac{1}{\frac{(1+i)(1+2 i)}{(1+3 i)}}$ $=\frac{(1+3 i)}{(1+i)(1+2 i)}$ We solve the above equation $=\frac{1+3 i}{1(1)+1(2 i)+i(1)+i(2 i)}$ $=\frac{1+3 i}...
Read More →Tick (✓) the correct answer
Question: Tick (✓) the correct answer The lengths of the diagonals of a rhombus are 16 cm and 12 cm. The length of each side of the rhombus is (a) 8 cm (b) 9 cm (c) 10 cm (d) 12 cm Solution: (c) $10 \mathrm{~cm}$ Let $A B C D$ be a rhombus. $L$ et $A C$ and $B D$ be the diagonals of the rhombus intersecting at a point $O$. $A C=16 \mathrm{~cm}$ $B D=12 \mathrm{~cm}$ We know that the diagonals of a rhombus bisect each other at right angles. $\therefore A O=\frac{1}{2} A C$ $=\left(\frac{1}{2} \ti...
Read More →Verify – (-x) = x for
Question: Verify (-x) = x for (i) x = 3/5 (ii) x = -7/9 (iii) x = 13/-15 Solution: (i) x = 3/5 x = -3/5 (-x) = (-3/5) X = 3/5 (ii) x = -7/9 x = (-7/9) -x = 7/9 (-x) = (7/9) X = 7/9 (iii) x = 13/-15 x = (-13/15) -x = 13/15 (-x) = (13/15) X = -13/15...
Read More →Tick (✓) the correct answer
Question: Tick (✓) the correct answer The two diagonals are not necessarily equal in a (a) rectangle (b) square (c) rhombus (d) isosceles trapezium Solution: (c) rhombus In a rhombus, the two diagonals are not necessarily equal....
Read More →Which of the following statements are true and which are false?
Question: Which of the following statements are true and which are false? (i) The diagonals of a parallelogram are equal. (ii) The diagonals of a rectangle are perpendicular to each other. (iii) The diagonals of a rhombus are equal. (iv) Every rhombus is a kite. (v) Every rectangle is a square. (vi) Every square is a a parallelogram. (vii) Every square is a rhombus. (viii) Every rectangle is a parallelogram. (ix) Every parallelogram is a rectangle. (x) Every rhombus is a parallelogram. Solution:...
Read More →Using suitable rearrangement
Question: Using suitable rearrangement and find the sum: (a) (4/7) + (-4/9) + (3/7) + (-13/9) (b) -5 + (7/10) + (3/7) + (-3) + (5/45) + (-4/5) Solution: First rearrange the rational numbers and add the numbers with same denominator. = (4/7) + (3/7) (4/9) (13/9) = ((4 + 3)/7) ((4 + 13)/9) = (7/7) (17/9) = 1 (17/9) = (9 17)/9 = -8/9 (b) -5 + (7/10) + (3/7) + (-3) + (5/45) + (-4/5) = -5 + (-3) + (7/10) + (-4/5) + (3/7) + (5/14) = 8 + [(7-8)/10] + [(6 + 5)/14] = 8 (1/10) + (11/14) LCM of 1, 10 and 1...
Read More →Find the multiplicative inverse of each of the following:
Question: Find the multiplicative inverse of each of the following: $\frac{(2+3 i)}{(1+i)}$ Solution: Given: $\frac{2+3 i}{1+i}$ To find: Multiplicative inverse We know that Multiplicative Inverse of $z=z^{-1}$ $=\frac{1}{z}$ Putting $\mathrm{z}=\frac{2+3 i}{1+i}$ So, Multiplicative inverse of $\frac{2+3 i}{1+i}=\frac{1}{\frac{2+3 i}{1+i}}=\frac{1+i}{2+3 i}$ Now, rationalizing by multiply and divide by the conjugate of (2+3i) $=\frac{1+i}{2+3 i} \times \frac{2-3 i}{2-3 i}$ $=\frac{(1+i)(2-3 i)}{...
Read More →Select those which can be written as a rational number
Question: Select those which can be written as a rational number with denominator 4 in their lowest form: (7/8), (64/16), (36/-12), (-16/17), (5/-4), (140/28) Solution: Rational number with denominator 4 in their lowest form are, 64/16 = 16/4, 36/-12 = 12/-4, 5/-4, 140/28 =20/4...
Read More →Name each of the following parallelograms.
Question: Name each of the following parallelograms. (i) The diagonals are equal and the adjacent sides are unequal. (ii) The diagonals are equal and the adjacent sides are equal. (iii) The diagonals are unequal and the adjacent sides are equal. (iv) All the sides are equal and one angle is 60. (v) All the sides are equal and one angle is 90. (vi) All the angles are equal and the adjacent sides are unequal. Solution: (i) The diagonals are equal and the adjacent sides are unequal. Hence, the give...
Read More →Solve the following: Select the rational numbers
Question: Solve the following: Select the rational numbers from the list which are also the integers. 9/4, 8/4, 7/4, 6/4, 9/3, 8/3, 7/3, 6/3, 5/2, 4/2, 3/1, 3/2, 1/1, 0/1, -1/1, -2/1, -3/2, -4/2, -5/2, -6/2 Solution: The rational number from the given list which also the integers are, 8/4 = 2, 9/3 = 3, 6/3 = 2, 4/2 = 2, 3/1 = 3, 1/1 = 1, 0/1 = 0, -1/1 = -1, -2/1 = -2, -4/2 = -2, -6/2 = -3...
Read More →The sides of a rectangle are in the ratio 5 : 4 and its perimeter is 90 cm.
Question: The sides of a rectangle are in the ratio 5 : 4 and its perimeter is 90 cm. Find its length and breadth. Solution: Let the length of two sides of the rectangle be $5 x \mathrm{~cm}$ and $4 x \mathrm{~cm}$, respectively. Then, its perimeter $=2(5 x+4 x) \mathrm{cm}$ $=18 x \mathrm{~cm}$ $\therefore 18 x=90$ $\Rightarrow x=\frac{90}{18}$ $\Rightarrow x=5$ Length of one side $\Rightarrow(5 \times 5) \mathrm{cm}=25 \mathrm{~cm}$ Length of the other side $\Rightarrow(4 \times 5) \mathrm{cm}...
Read More →Rational numbers can be added
Question: Rational numbers can be added (or multiplied) in any order (-4/5) (-6/5) = (-6/5) (-4/5) Solution: True. The arrangements of given rational number is as per the commutative law under multiplication. i.e. a b = b c...
Read More →Find the multiplicative inverse of each of the following:
Question: Find the multiplicative inverse of each of the following: (2 + 5i) Solution: Given: 2 + 5i To find: Multiplicative inverse We know that, Multiplicative Inverse of $z=z^{-1}$ $=\frac{1}{Z}$ Putting z = 2 + 5i So, Multiplicative inverse of $2+5 \mathrm{i}=\frac{1}{2+5 \mathrm{i}}$ Now, rationalizing by multiply and divide by the conjugate of (2+5i) $=\frac{1}{2+5 i} \times \frac{2-5 i}{2-5 i}$ $=\frac{2-5 i}{(2+5 i)(2-5 i)}$ Using $(a-b)(a+b)=\left(a^{2}-b^{2}\right)$ $=\frac{2-5 i}{(2)^...
Read More →The rational numbers
Question: The rational numbers $-\frac{1}{2}$ and $-\frac{-}{5} 2$ are on the opposite sides of zero on the number line. Solution: True Positive rational number and negative rational number remain on opposite sides of zero on the number line....
Read More →In the given figure ABCD is a square.
Question: In the given figureABCDis a square. Find the measure ofCAD. Solution: Refer to the figure given in the book. In $\Delta A D C:$ $D A=D C$ (all sides of a square are equal) $\Rightarrow \angle A C D=\angle C A D$ Let $\angle A C D=\angle C A D=x^{\circ} \quad$ [Angle opposite to the equal sides are equal] $x+x+90=180 \quad$ [since the sum of the angles of a triangle is $180^{\circ}$ ] $\Rightarrow 2 x+90=180$ $\Rightarrow 2 x=90$ $\Rightarrow x=\frac{90}{2}$ $\Rightarrow x=45$ $\therefo...
Read More →The lengths of the diagonals of a rhombus are 16 cm and 12 cm respectively.
Question: The lengths of the diagonals of a rhombus are 16 cm and 12 cm respectively. Find the length of each of its sides. Solution: Let $A B C D$ be a rhombus. $L$ et $A C$ and $B D$ be the diagonals of the rhombus intersecting at a point $O$. Let $A C=16 \mathrm{~cm}$ $\mathrm{BD}=12 \mathrm{~cm}$ We know that the diagonals of a rhombus bisect each other at right angles. $\therefore A O=\frac{1}{2} A C$ $\quad=\left(\frac{1}{2} \times 16\right) \mathrm{cm}$ $\quad=8 \mathrm{~cm}$ $B O=\frac{1...
Read More →Find the multiplicative inverse of each of the following:
Question: Find the multiplicative inverse of each of the following: $(1-\sqrt{3} i)$ Solution: Given: $(1-i \sqrt{3})$ To find: Multiplicative inverse We know that, Multiplicative Inverse of $z=z^{-1}$ $=\frac{1}{Z}$ Putting $z=1-i \sqrt{3}$ So, Multiplicative inverse of $1-\mathrm{i} \sqrt{3}=\frac{1}{1-\mathrm{i} \sqrt{3}}$ Now, rationalizing by multiply and divide by the conjugate of $(1-\mathrm{i} \sqrt{3})$ $=\frac{1}{1-i \sqrt{3}} \times \frac{1+i \sqrt{3}}{1+i \sqrt{3}}$ $=\frac{1+i \sqrt...
Read More →0 is whole number but
Question: 0 is whole number but it is not a rational number. Solution: False0 is a whole number and also a rational number....
Read More →In the adjacent figure, ABCD is a parallelogram and line segments AE and CF bisect the angles A and C respectively.
Question: In the adjacent figure,ABCDis a parallelogram and line segmentsAEandCFbisect the anglesAandCrespectively. Show thatAE||CF. Solution: Refer to the figure of the book. $\angle \mathrm{A}=\angle \mathrm{C}$ (opposite angles of $a$ parallelogram are equal) $\Rightarrow \frac{1}{2} \angle \mathrm{A}=\frac{1}{2} \angle \mathrm{C}$ $=\angle \mathrm{EAD}=\angle \mathrm{FCB}$ ( $A E$ and $C F$ bisect the angles $A$ and $C$, respectively) In $\Delta A D E$ and $\Delta C B F$ : $\angle B=\angle D...
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