Find the real values of x and y for which:
Question: Find the real values of x and y for which: $x+4 y i=i x+y+3$ Solution: Given: x + 4yi = ix + y + 3 or $x+4 y i=i x+(y+3)$ Comparing the real parts, we get $x=y+3$ Or $x-y=3 \ldots$ (i) Comparing the imaginary parts, we get $4 y=x \ldots$ (ii) Putting the value of $x=4 y$ in eq. (i), we get $4 y-y=3$ $\Rightarrow 3 y=3$ $\Rightarrow y=1$ Putting the value of y = 1 in eq. (ii), we get x = 4(1) = 4 Hence, the value of x = 4 and y = 1...
Read More →Mark (✓) against the correct answer:
Question: Mark (✓) against the correct answer: How many diagonals are there in a hexagon? (a) 6 (b) 8 (c) 9 (d) 10 Solution: (c) 9 Hexagon has six sides. Number of diagonals $=\frac{n(n-3)}{2} \quad$ (where $n$ is the number of side $s$ ) $=\frac{6(6-3)}{2}$ $=9$...
Read More →Find the real values of x and y for which:
Question: Find the real values of x and y for which: $(x+i y)(3-2 i)=(12+5 i)$ Solution: $x(3-2 i)+i y(3-2 i)=12+5 i$ $\Rightarrow 3 x-2 i x+3 i y-2 i^{2} y=12+5 i$ $\Rightarrow 3 x+i(-2 x+3 y)-2(-1) y=12+5 i\left[\because i^{2}=-1\right]$ $\Rightarrow 3 x+i(-2 x+3 y)+2 y=12+5 i$ $\Rightarrow(3 x+2 y)+i(-2 x+3 y)=12+5 i$ Comparing the real parts, we get $3 x+2 y=12 \ldots(i)$ Comparing the imaginary parts, we get $-2 x+3 y=5 \ldots$ (ii) Solving eq. (i) and (ii) to find the value of $x$ and $y$ ...
Read More →2/5 of total number of students of a school come
Question: 2/5 of total number of students of a school come by car while of students come by bus to school. All the other students walk to school of which 1/3 walk on their own and the rest are escorted by their parents. If 224 students come to school walking on their own, how many students study in that school? Solution: Let us assume total number of students in the school be x. From the question it is given that, The number of students come by car = (2/5) x The number of students come by bus = ...
Read More →Mark (✓) against the correct answer:
Question: Mark (✓) against the correct answer: The bisectors of two adjacent angles of a parallelogram intersect at (a) 30 (b) 45 (c) 60 (d) 90 Solution: (d) 90 We know that the opposite sides and the angles in a parallelogram are equal. Also, its adjacent sides are supplementary, i.e. sum of the sides is equal to 180. Now, the bisectors of these angles form a triangle, whose two angles are: $\frac{A}{2}$ and $\frac{B}{2}$ or $\frac{A}{2}=\left(90-\frac{A}{2}\right)$ $\frac{\angle A}{2}+90-\frac...
Read More →1/6 of the class students are above average,
Question: 1/6 of the class students are above average, are average and rest are below average. If there are 48 students in all, how many students are below average in the class? Solution: From the question it is given that, Number of students in the class are above average = 1/6 Number of students in the class are average = Number of students in the class are below average = 1 ((1/6) + ()) = 1 ((2 + 3)/12) = 1 (5/12) = (12 5)/12 = 7/12 students. So, the number of students in the class = 48 Then,...
Read More →Mark (✓) against the correct answer:
Question: Mark (✓) against the correct answer: In a squarePQRS, ifPQ= (2x+ 3) cm andQR= (3x 5) cm then (a)x= 4 (b)x= 5 (c)x= 6 (d)x= 8 Solution: (d)x= 8All sides of a square are equal. $P Q=Q R$ $(2 x+3)=(3 x-5)$ $=2 x-3 x=-5-3$ $=x=8 \mathrm{~cm}$...
Read More →Find the real values of x and y for which:
Question: Find the real values of x and y for which: $(1-i) x+(1+i) y=1-3 i$ Solution: $(1-i) x+(1+i) y=1-3 i$ $x-i x+y+i y=1-3 i$ $\Rightarrow(x+y)-i(x-y)=1-3 i$ Comparing the real parts, we get $x+y=1 \ldots$ (i) Comparing the imaginary parts, we get $x-y=-3 \ldots$ (ii) Solving eq. (i) and (ii) to find the value of $x$ and $y$ Adding eq. (i) and (ii), we get $x+y+x-y=1+(-3)$ $\Rightarrow 2 x=1-3$ $\Rightarrow 2 x=-2$ $\Rightarrow x=-1$ Putting the value of x = -1 in eq. (i), we get $(-1)+y=1$...
Read More →Prove the following
Question: $117 \frac{1}{3}$m long rope is cut into equal pieces measuring m each. How many such small pieces are these? Solution: From the question it is given that, The length of the rope =m = (117 3 + 1)/3 = 352/3m Then length of each piece measures =m = 22/3 m So, the number of pieces of the rope = total length of the rope/ length of each piece = (352/3)/ (22/3) = (352/3) (3/22) = (16/1) (1/1) = 16 Hence, number of small pieces cut from them long rope is 16...
Read More →Mark (✓) against the correct answer:
Question: Mark (✓) against the correct answer: The length of a rectangle is 8 cm and each of its diagonals measures 10 cm. The breadth of the rectangle is (a) 5 cm (b) 6 cm (c) 7 cm (d) 9 cm Solution: (b) $6 \mathrm{~cm}$ Let the breadth of the rectangle be $x \mathrm{~cm}$. Diagonal $=10 \mathrm{~cm}$ Length $=8 \mathrm{~cm}$ The rectangle is divided into two right triangles. Diagonal $^{2}=$ Length $^{2}+$ Breadth $^{2}$ $10^{2}=8^{2}+x^{2}$ $100-64=x^{2}$ $x^{2}=36$ $x=6 \mathrm{~cm}$ Breadth...
Read More →Mark (✓) against the correct answer:
Question: Mark (✓) against the correct answer: The angles of quadrilateral are in the ratio 1 : 3 : 7 : 9. The measure of the largest angle is (a) 63 (b) 72 (c) 81 (d) none of these Solution: (d) none of the these Let the angles be $(x)^{\circ},(3 x)^{\circ},(7 x)^{\circ}$ and $(9 x)^{\circ}$. Sum of the angles of the quadrilateral is $360^{\circ}$. $x+3 x+7 x+9 x=360$ $20 x=360$ $x=18$ Angles : $(3 x)^{\circ}=(3 \times 18)=54^{\circ}$ $(7 x)^{\circ}=(7 \times 18)^{\circ}=126^{\circ}$ $(9 x)^{\c...
Read More →Find the real values of x and y for which:
Question: Find the real values of x and y for which: $(x+i y)(3-2 i)=(12+5 i)$ Solution: $x(3-2 i)+i y(3-2 i)=12+5 i$ $\Rightarrow 3 x-2 i x+3 i y-2 i^{2} y=12+5 i$ $\Rightarrow 3 x+i(-2 x+3 y)-2(-1) y=12+5 i\left[\because i^{2}=-1\right]$ $\Rightarrow 3 x+i(-2 x+3 y)+2 y=12+5 i$ $\Rightarrow(3 x+2 y)+i(-2 x+3 y)=12+5 i$ Comparing the real parts, we get $3 x+2 y=12 \ldots(i)$ Comparing the imaginary parts, we get $-2 x+3 y=5 \ldots$ (ii) Solving eq. (i) and (ii) to find the value of x and y Mult...
Read More →7/11 of all the money in Hamid’s bank account is ₹ 77,000.
Question: 7/11 of all the money in Hamids bank account is ₹ 77,000. How much money does Hamid have in his bank account? Solution: From the question, it is given that 7/11 of all the money in Hamids bank account = ₹ 77,000 Now, let us assume money in Hamids bank account be ₹ x. Then, (7/11) (x) = 77,000 x = 77,000/ (7/11) x = 77000 (11/7) x = 11000 (11/1) x = 121000 The total money in Hamids bank account is ₹ 121000....
Read More →If 16 shirts of equal size can be made out of 24m
Question: If 16 shirts of equal size can be made out of 24m of cloth, how much cloth is needed for making one shirt? Solution: From the question it is given that, The 16 shirts are made out of= 24m of cloth. Cloth needed for making one shirt = 24/16 m of cloth = 3/2m of cloth i.e. 1.5m So, Cloth is needed for making one shirt is 1.5m....
Read More →Mark (✓) against the correct answer:
Question: Mark (✓) against the correct answer: Two opposite angles of a parallelogram are (3x 2) and (50 x). The measures of all its angles are (a) 97, 83, 97, 83 (b) 37, 143, 37, 143 (c) 76, 104, 76, 104 (d) none of these Solution: (b) 37o, 143o, 37o143o Opposite angles of a parallelogram are equal. $\therefore 3 x-2=50-x$ $\Rightarrow 3 x+x=50+2$ $\Rightarrow 4 x=52$ $\Rightarrow x=13$ Therefore, the first and the second angles are: $(3 x-2)^{\circ}=(2 \times 13-2)^{\circ}=37^{\circ}$ $(50-x)^...
Read More →A train travels 1445/2 km in 17/2 hours.
Question: A train travels 1445/2 km in 17/2 hours. Find the speed of the train in km/h. Solution: From the question it is given that, Distance travelled by train = 1445/2 km Time taken by the train to cover distance 1445/2 = 17/2 hours The speed of the train = (1445/2) (17/2) = (1445/2) (2/17) = (85/1) (1/1) = 85 km/h The speed of the train is 85 km/h....
Read More →Find the real values of x and y for which:
Question: Find the real values of x and y for which: $(1-i) x+(1+i) y=1-3 i$ Solution: $(1-i) x+(1+i) y=1-3 i$ $\Rightarrow x-i x+y+i y=1-3 i$ $\Rightarrow(x+y)-i(x-y)=1-3 i$ Comparing the real parts, we get $x+y=1 \ldots(i)$ Comparing the imaginary parts, we get $x-y=-3 \ldots$ (ii) Solving eq. (i) and (ii) to find the value of x and y Adding eq. (i) and (ii), we get $x+y+x-y=1+(-3)$ $\Rightarrow 2 x=1-3$ $\Rightarrow 2 x=-2$ $\Rightarrow x=-1$ Putting the value of x = -1 in eq. (i), we get $(-...
Read More →The cost of 19/4 metres of wire is ₹ 171/2.
Question: The cost of 19/4 metres of wire is ₹171/2. Find the cost of one metre of the wire. Solution: From the question it is given that, The cost of 19/4 meters of wire is = ₹ 171/2 Then, cost of one meter of wire = (171/2) (19/4) = (171/2) (4/19) = (9/1) (2/1) = 18/1 = ₹ 18 The cost of one meter of wire is ₹ 18....
Read More →The diagonals of a rhombus are 16 cm and 12 cm.
Question: The diagonals of a rhombus are 16 cm and 12 cm. Find the length of each side of the rhombus. Solution: All the sides of a rhombus are equal in length. The diagonals of a rhombus intersect at90∘90∘. The diagonal and the side of a rhombus form right triangles. $\ln \triangle A O B:$ $A B^{2}=A O^{2}+O B^{2}$ $=8^{2}+6^{2}$ $=64+36$ $=100$ $A B=10 \mathrm{~cm}$ Therefore, the length of each side of the rhombus is $10 \mathrm{~cm}$....
Read More →The diagonals of a rhombus are 16 cm and 12 cm.
Question: The diagonals of a rhombus are 16 cm and 12 cm. Find the length of each side of the rhombus. Solution: All the sides of a rhombus are equal in length. The diagonals of a rhombus intersect at90∘90∘. The diagonal and the side of a rhombus form right triangles. $\ln \triangle A O B:$ $A B^{2}=A O^{2}+O B^{2}$ $=8^{2}+6^{2}$ $=64+36$ $=100$ $A B=10 \mathrm{~cm}$ Therefore, the length of each side of the rhombus is $10 \mathrm{~cm}$....
Read More →Identify the rational number that does not belong
Question: Identify the rational number that does not belong with the other three. Explain your reasoning (-5/11), (-1/2), (-4/9), (-7/3) Solution: The rational number that does not belong with the other three is -7/3 as it is smaller than 1 whereas rest of the numbers are greater than 1....
Read More →Solve this
Question: If $z_{1}=(2-i)$ and $z_{2}=(1+i)$, find $\left|\frac{z_{1}+z_{2}+1}{z_{1}-z_{2}+i}\right|$. Solution: Given: $z_{1}=(2-i)$ and $z_{2}=(1+i)$ To find: $\left|\frac{z_{1}+z_{2}+1}{z_{1}-z_{2}+i}\right|$ Consider, $\left|\frac{z_{1}+z_{2}+1}{z_{1}-z_{2}+i}\right|$ Putting the value of $z_{1}$ and $z_{2}$, we get $=\left|\frac{2-i+1+i+1}{2-i-(1+i)+i}\right|$ $=\left|\frac{4}{2-i-1-i+i}\right|$ $=\left|\frac{4}{1-i}\right|$ Now, rationalizing by multiply and divide by the conjugate of $1-\...
Read More →Prove the following
Question: simplify (a) (32/5) + (23/11) (22/15) (b) (3/7) (28/15) (14/5) (c) (3/7) + (-2/21) (-5/6) (d) (7/8) + (1/6) (1/12) Solution: (a) (32/5) + (23/11) (22/15) = (32/5) + (23/1) (2/15) = (32/5) + (46/15) = (96 + 46)/15 = 142/15 (b) (3/7) (28/15) (14/5) = (3/7) (28/15) (14/5) = (1/1) (4/5) (14/5) = (4/5) (14/5) = (4/5) (5/14) = (2/1) (1/7) = 2/7 (c) (3/7) + (-2/21) (-5/6) = (3/7) (2/21) (-5/6) = (3/7) (1/21) (-5/3) = (3/7) (-5/63) = (3/7) + (5/63) = (27 + 5)/63 = 32/63 (d) (7/8) + (1/6) (1/12...
Read More →Prove that the diagonals of a rhombus bisect each other at right angles.
Question: Prove that the diagonals of a rhombus bisect each other at right angles. Solution: Rhombus is a parallelogram. Consider: $\Delta A O B$ and $\Delta C O D$ $\angle O A B=\angle O C D \quad$ (alternate angle) $\angle O D C=\angle O B A \quad$ (alternate angle) $\angle D O C=\angle A O B \quad$ (vertically opposite angles) $\Delta A O B \cong C O B$ $\therefore A O=C O$ $O B=O D$ Therefore, the diagonals bisects at O. Now, let us prove that the diagonals intersect each other at right angl...
Read More →The sides of a rectangle are in the ratio 4 : 5 and its perimeter is 180 cm.
Question: The sides of a rectangle are in the ratio 4 : 5 and its perimeter is 180 cm. Find its sides. Solution: Let the length be $4 x \mathrm{~cm}$ and the breadth be $5 x \mathrm{~cm}$. Perimeter of the rectangle $=180 \mathrm{~cm}$ Perimeter of the rectangle $=2(l+b)$ $2(l+b)=180$ $\Rightarrow 2(4 x+5 x)=180$ $\Rightarrow 2(9 x)=180$ $\Rightarrow 18 x=180$ $\Rightarrow x=10$ $\therefore$ Length $=4 x \mathrm{~cm}=4 \times 10=40 \mathrm{~cm}$ Breadth $=5 x \mathrm{~cm}=5 \times 10=50 \mathrm{...
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