Here is a table which gives the information
Question: Here is a table which gives the information about the total rainfall for several months compared to the average monthly rains of a town. Write each decimal in the form of rational number p/q. Solution: (i) May 2.6924 cm = 26924/10000 [∵by the decimal removing method] By dividing both numerator and denominator by 4 we get, = 6731/2500 cm (ii) June 0.6096 cm = 0.6096/10000 [∵by the decimal removing method] By dividing both numerator and denominator by 16 we get, = 381/625 cm (iii) July -...
Read More →Solve this
Question: If $\left(\frac{z-1}{z+1}\right)$ is purely an imaginary number and $z \neq-1$ then find the value of |z|. Solution: Given: $\frac{z-1}{z+1}$ is purely imaginary number Let $z=x+i y$ So, $\frac{z-1}{z+1}=\frac{x+i y-1}{x+i y+1}$ $=\frac{(x-1)+i y}{(x+1)+i y}$ Now, rationalizing the above by multiply and divide by the conjugate of [(x + 1) + iy] $=\frac{(x-1)+i y}{(x+1)+i y} \times \frac{(x+1)-i y}{(x+1)-i y}$ $=\frac{[(x-1)+i y][(x+1)-i y]}{[(x+1)+i y][(x+1)-i y]}$ Using $(a-b)(a+b)=\l...
Read More →Roller Coaster at an amusement park is 2/3m high.
Question: Roller Coaster at an amusement park is 2/3m high. If a new roller coaster is built that is 3/5 times the height of the existing coaster, what will be the height of the new roller coaster? Solution: From the question it is given that, Height of the roller coaster at an amusement park = 2/3 m Height of the new roller coaster is about to build = 3/5 times the height of the existing Coaster = (2/3) (3/5) = (2/1) (1/5) = (2/5) m...
Read More →In the given figure, ABCD is a trapezium in which AD||BC,
Question: In the given figure,ABCDis a trapezium in whichAD||BC,ABC= 90,AD= 16 cm,AC= 41 cm andBC= 40 cm. Find the area of the trapezium. Solution: $\angle \mathrm{ABC}=90^{\circ}$ From the right $\Delta \mathrm{ABC}$, we have: $\mathrm{AB}^{2}=\left(\mathrm{AC}^{2}-\mathrm{BC}^{2}\right)$ $\Rightarrow \mathrm{AB}^{2}=\left\{\left(41^{2}\right)-\left(40^{2}\right)\right\}$ $\Rightarrow \mathrm{AB}^{2}=(1681-1600)$ $\Rightarrow \mathrm{AB}^{2}=81$ $\Rightarrow \mathrm{AB}=\sqrt{81}$ $\Rightarrow ...
Read More →The overall width in cm of several
Question: The overall width in cm of several wide-screen televisions are 97.28 cm,cm,cm and 97.94 cm. Express these numbers as rational numbers in the form p/q and arrange the widths in ascending order. Solution: From the question, The overall width in cm of several wide screen television are, 97.28 cm = 9728/100 [∵by the decimal removing method] By dividing both numerator and denominator by 4 we get, = 2432/25cm cm = by converting mixed fraction into improper fraction we get, = 886/9 cm cm = by...
Read More →The table shows the portion of some
Question: The table shows the portion of some common materials that are recycled. (a) Is the rational number expressing the amount of paper recycled more than or less than ? (b) Which items have a recycled amount less than ? (c) Is the quantity of aluminium cans recycled more (or less) than half of the quantity of aluminium cans? (d) Arrange the rate of recycling the materials from the greatest to the smallest. Solution: (a) Is the rational number expressing the amount of paper recycled more tha...
Read More →The length of the fence of a trapezium-shaped field ABCD is 130 m
Question: The length of the fence of a trapezium-shaped fieldABCDis 130 m and sideABis perpendicular to each of the parallel sidesADandBC. IfBC= 54 m,CD= 19 m andAD= 42 m, find the area of the field. Solution: Length of the side $\mathrm{AB}=(130-(54+19+42)) \mathrm{m}$ $=15 \mathrm{~m}$ Area of the trapezium $-$ shaped field $=\left\{\frac{1}{2} \times(\mathrm{AD}+\mathrm{BC}) \times \mathrm{AB}\right\}$ $=\left\{\frac{1}{2} \times(42+54) \times 15\right\} \mathrm{m}^{2}$ $=\left(\frac{1}{2} \t...
Read More →Prove that
Question: If $(1+i) z=(1-i)^{\bar{z}}$ then prove that $z=-i \bar{z}$. Solution: Let z = x + iy Then $\bar{z}=x-i y$ Now, Given: $(1+i) z=(1-i) \bar{z}$ Therefore, $(1+i)(x+i y)=(1-i)(x-i y)$ $x+i y+x i+i^{2} y=x-i y-x i+i^{2} y$ We know that $i^{2}=-1$, therefore, $x+i y+i x-y=x-i y-i x-y$ $2 x i+2 y i=0$ $x=-y$ Now, as $x=-y$ $z=-\bar{Z}$ Hence, Proved....
Read More →Four friends had a competition to see
Question: Four friends had a competition to see how far could they hop on one foot. The table given shows the distance covered by each. (a) How farther did Soni hop than Nancy? (b) What is the total distance covered by Seema and Megha? (c) Who walked farther, Nancy or Megha? Solution: The LCM of the denominators 25, 32, 40 and 20 is 800 1/25 = [(132)/ (2532)] = (32/800) (1/32) = [(125)/ (3225)] = (25/800) (1/40) = [(120)/ (4020)] = (20/800) (1/20) = [(140)/ (2040)] = (40/800) Then, (a) Soni hop ...
Read More →In a trapezium-shaped field, one of the parallel sides is twice the other.
Question: In a trapezium-shaped field, one of the parallel sides is twice the other. If the area of the field is 9450 m2and the perpendicular distance between the two parallel sides is 84 m, find the length of the longer of the parallel sides. Solution: Let the lengths of the parallel sides be $x \mathrm{~cm}$ and $2 x \mathrm{~cm}$. Area of trapezium $=\left\{\frac{1}{2} \times(x+2 x) \times 84\right\} \mathrm{m}^{2}$ $=\left(\frac{1}{2} \times 3 x \times 84\right) \mathrm{m}^{2}$ $=(42 \times ...
Read More →Prove that
Question: If $z=(2-3 i)$, prove that $z^{2}-4 z+13=0$ and hence deduce that $4 z^{3}-3 z^{2}+$169=0$ Solution: Given: $z=2-3 i$ To Prove: $z^{2}-4 z+13=0$ Taking LHS, $z^{2}-4 z+13$ Putting the value of $z=2-3 i$, we get $(2-3 i)^{2}-4(2-3 i)+13$ $=4+9 i^{2}-12 i-8+12 i+13$ $=9(-1)+9$ $=-9+9$ $=0$ $=\mathrm{RHS}$ Hence, $z^{2}-4 z+13=0$ Now, we have to deduce $4 z^{3}-3 z^{2}+169$ Now, we will expand $4 z^{3}-3 z^{2}+169$ in this way so that we can use the above equation i.e. $z^{2}-4 z+13$ $=4 ...
Read More →One fruit salad recipe requires ½ cup of sugar.
Question: One fruit salad recipe requires cup of sugar. Another recipe for the same fruit salad requires 2 tablespoons of sugar. If 1 tablespoon is equivalent to 1/16 cup, how much more sugar does the first recipe require? Solution: From the question it is given that, One fruit salad recipe requires = cup of sugar Sugar required for another salad = 2 (1/16) = 2/16 cup Hence, the required sugar = (2/16) = (8 2)/16 = 6/16 = 3/8 cup of sugar....
Read More →Find the product of additive inverse
Question: Find the product of additive inverse and multiplicative inverse of 1/3. Solution: Additive inverse of -1/3 = 1/3 Multiplicative inverse of -1/3 = -3/1 Then, The product of additive inverse and multiplicative inverse of 1/3 = 1/3 (-3) = -1...
Read More →Find the sum of additive inverse
Question: Find the sum of additive inverse and multiplicative inverse of 7. Solution: Additive inverse of 7 = 7 Multiplicative inverse of 7 = 1/7 Then, Sum of additive inverse and multiplicative inverse of 7 = -7 + (1/7) = (-49 + 1)/7 = 48/7...
Read More →The area of a trapezium is 180 cm
Question: The area of a trapezium is 180 cm2and its height is 9 cm. If one of the parallel sides is longer than the other by 6 cm, find the two parallel sides. Solution: Let the lengths of the parallel sides be $x \mathrm{~cm}$ and $(x+6) \mathrm{cm}$. Now, Area of trapezium $=\left\{\frac{1}{2} \times(x+x+6) \times 9\right\} \mathrm{cm}^{2}$ $=\left(\frac{1}{2} \times(2 x+6) \times 9\right) \mathrm{cm}^{2}$ $=4.5(2 x+6) \mathrm{cm}^{2}$ $=(9 x+27) \mathrm{cm}^{2}$ Area of trapezium $=180 \mathr...
Read More →On a winter day the temperature
Question: On a winter day the temperature at a place in Himachal Pradesh was 16C. Convert it in degree Fahrenheit (oF) by using the formula. (C/5) = (F 32)/9 Solution: Given, a winter day the temperature at a place in Himachal Pradesh was 16C. Formula, (C/5) = (F 32)/9 Where, C = -16o Then, (-16o/5) = (F 32)/9 (-16o/5) 9 = F -32 (-144/5) = F 32 F = 32 (144/5) F = (160 144)/5 F = 16/5 F = 3.2oF...
Read More →The area of a trapezium is 405 cm
Question: The area of a trapezium is 405 cm2. Its parallel sides are in the ratio 4 : 5 and the distance between them is 18 cm. Find the length of each of the parallel sides. Solution: Let the lengths of the parallel sides of the trapezium be $4 x \mathrm{~cm}$ and $5 x \mathrm{~cm}$, respectively. Now, Area of trapezium $=\left\{\frac{1}{2} \times(4 x+5 x) \times 18\right\} \mathrm{cm}^{2}$ $=\left(\frac{1}{2} \times 9 x \times 18\right) \mathrm{cm}^{2}$ $=81 x \mathrm{~cm}^{2}$ Area of trapezi...
Read More →Find the real values of x and y for which the complex number
Question: Find the real values of $x$ and $y$ for which the complex number $\left(-3+i y x^{2}\right)$ and $\left(x^{2}+y+4 i\right)$ are conjugates of each other. Solution: Let $z_{1}=-3+i y x^{2}$ So, the conjugate of $Z_{1}$ is $\overline{z_{1}}=-3-i y x^{2}$ And $z 2=x^{2}+y+4 i$ So, the conjugate of z2 is $\bar{z}_{2}=x^{2}+y-4 i$ Given that: $\overline{z_{1}}=z_{2} \ z_{1}=\bar{z}_{2}$ Firstly, consider $\overline{z_{1}}=z_{2}$ $-3-i y x^{2}=x^{2}+y+4 i$ $\Rightarrow x^{2}+y+4 i+i y x^{2}=...
Read More →Write the following rational numbers in the descending order.
Question: Write the following rational numbers in the descending order. (8/7), (-9/8), (-3/2), 0, (2/5) Solution: The LCM of the denominators 7, 8, 2 and 5 is 280 8/7 = [(840)/ (740)] = (320/280) (-9/8) = [(-935)/ (835)] = (-315/280) (-3/2) = [(-3140)/ (2140)] = (-420/280) (2/5) = [(256)/ (5656)] = (112/280) Now, 320 112 0 -315 -420 Hence, ⇒ 8/7 2/5 0 -9/8 -3/2 Descending order 8/7, 2/5, 0, -9/8, -3/2...
Read More →A field is in the form of a trapezium.
Question: A field is in the form of a trapezium. Its area is 1586 m2and the distance between its parallel sides is 26 m. If one of the parallel sides is 84 m, find the other. Solution: Let the length of the required side be $x \mathrm{~cm}$. Now, Area of trapezium $=\left\{\frac{1}{2} \times(84+x) \times 26\right\} \mathrm{m}^{2}$ $=(1092+13 x) \mathrm{m}^{2}$ Area of trapezium $=1586 \mathrm{~m}^{2}$ (Given) $\therefore 1092+13 x=1586$ $\Rightarrow 13 x=(1586-1092)$ $\Rightarrow 13 x=494$ $\Rig...
Read More →5½ metres long rope is cut into 12 equal pieces.
Question: 5 metres long rope is cut into 12 equal pieces. What is the length of each piece? Solution: From the question it is given that, The length of rope = 5 m = 11/2 m The total number of pieces = 12 Let us assume the length of one piece of rope be y. So, 12y = 11/2 m y = (11/2) (1/12) y = 11/24 The length of one piece of rope 11/24....
Read More →From a rope 40 metres long, pieces of equal size are cut.
Question: From a rope 40 metres long, pieces of equal size are cut. If the length of one piece is 10/3 metre, find the number of such pieces. Solution: From the question it is given that, The length of rope = 40 m The length of one piece of rope = 10/3 Let us assume the total number of pieces be y. So, (10/3) y = 40 y = (40 3)/10 y = 120/10 y = 12 pieces The number of pieces cut from the rope are 12....
Read More →The area of a trapezium is 1080 cm
Question: The area of a trapezium is 1080 cm2. If the lengths of its parallel sides be 55 cm and 35 cm, find the distance between them. Solution: Let the distance between the parallel sides be $x$. Now, Area of trapezium $=\left\{\frac{1}{2} \times(55+35) \times x\right\} \mathrm{cm}^{2}$ $=\left(\frac{1}{2} \times 90 \times x\right) \mathrm{cm}^{2}$ $=45 x \mathrm{~cm}^{2}$ Area of the trapezium $=1080 \mathrm{~cm}^{2}$ (Given) $\therefore 45 x=1080$ $\Rightarrow x=\frac{1080}{45}$ $\Rightarrow...
Read More →Find two rational numbers
Question: Find two rational numbers whose absolute value is 1/5. Solution: 1/5 and -1/5 are the rational number whose absolute value is 1/5....
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