A square board has an area of 144 sq units.
Question: A square board has an area of 144 sq units. How long is each side of the board? (a) 11 units (b) 12 units (c) 13 units (d) 14 units Solution: (b) Given, area of square board $=144 \mathrm{sq}$ units $\therefore$ $(\text { Side })^{2}=144$ $\left[\because\right.$ area of square $\left.=(\text { side })^{2}\right]$ $\Rightarrow$ $(\text { Side })^{2}=(12)^{2}$ $\Rightarrow$ Side $=12$ units Hence, the length of each side of the board is 12 units....
Read More →Find the surface area of a chalk box, whose length,
Question: Find the surface area of a chalk box, whose length, breadth and height are 18 cm, 10 cm and 8 cm respectively. Solution: Length $=18 \mathrm{~cm}$ Breadth $=10 \mathrm{~cm}$ Height $=8 \mathrm{~cm}$ $\therefore$ Total surface area $=2(l b+l h+b h)=2(18 \times 10+18 \times 8+10 \times 8)=2(180+144+80)=808 \mathrm{~cm}^{2}$...
Read More →Find the number of coins, 1.5 cm in diameter and 0.2 cm thick,
Question: Find the number of coins, 1.5 cm in diameter and 0.2 cm thick, to be melted to form a right circular cylinder with a height of 10 cm and a diameter of 4.5 cm. Solution: Volume of the coin $=\pi \mathrm{r}^{2} \mathrm{~h}=\frac{22}{7} \times 0.75 \times 0.75 \times 0.2$ Volume of the cylinder $=\pi \mathrm{r}^{2} \mathrm{~h}=\frac{22}{7} \times 2.25 \times 2.25 \times 10$ No. of coins $=\frac{\text { volume of cylinder }}{\text { volume of coin }}=\frac{2.25 \times 2.25 \times 10}{0.75 ...
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Question: $25 x^{2}-30 x+11=0$ Solution: Given: $25 x^{2}-30 x+11=0$ Solution of a general quadratic equation $a x^{2}+b x+c=0$ is given by: $x=\frac{-b \pm \sqrt{\left(b^{2}-4 a c\right)}}{2 a}$ $\Rightarrow x=\frac{-(-30) \pm \sqrt{(-30)^{2}-(4 \times 25 \times 11)}}{2 \times 25}$ $\Rightarrow x=\frac{30 \pm \sqrt{900-1100}}{50}$ $\Rightarrow x=\frac{30 \pm \sqrt{-200}}{50}$ $\Rightarrow x=\frac{30 \pm 10 \sqrt{2} i}{50}$ $\Rightarrow x=-\frac{30}{50} \pm \frac{10 \sqrt{2}}{50} i$ Ans: $x=-\fr...
Read More →The radius and height of a cylinder are in the ratio 5 : 7
Question: The radius and height of a cylinder are in the ratio 5 : 7 and its volume is 550 cm3. Find its radius and height. Solution: $\frac{\text { Radius }}{\text { height }}=\frac{r}{h}=\frac{5}{7}$ $\Rightarrow r=\frac{5}{7} h$ Now, volume $=\pi \mathrm{r}^{2} \mathrm{~h}=\frac{22}{7} \times \frac{5}{7} \mathrm{~h} \times \frac{5}{7} \mathrm{~h} \times \mathrm{h}=550 \mathrm{~cm}^{3}$ $\therefore h=\sqrt[3]{\frac{550 \times 7 \times 7 \times 7}{22 \times 5 \times 5}}=7 \mathrm{~cm}$ Also, $r...
Read More →The one’s digit of the cube of 23 is
Question: The ones digit of the cube of 23 is (a) 6 (b) 7 (c) 3 (d) 9 Solution: (b) We know that, the cubes of the numbers ending with digits 3 and 7, have 7 and 3 at ones digit, respectively. So, the ones digit of the cube of 23 is 7....
Read More →Which of the following cannot be
Question: Which of the following cannot be a perfect square? (a) 841 (b) 529 (c) 198 (d) All of these Solution: (c) We know that, a number ending with digits 2, 3, 7 or 8 can never be a perfect square. So, 198 cannot be written in the form of a perfect square....
Read More →How many soap cakes each measuring 7 cm × 5 cm × 2.5 cm
Question: How many soap cakes each measuring 7 cm 5 cm 2.5 cm can be placed in a box of size 56 cm 40 cm 25 cm? Solution: Volume of a soap cake $=7 \times 5 \times 2.5=87.5 \mathrm{~cm}^{3}$ Volume of the box $=56 \times 40 \times 25=56000 \mathrm{~cm}^{3}$ No. of soap cakes $=\frac{56000}{87.5}=640$ units $\therefore 640$ cakes of soap can be placed in a box of the given size....
Read More →How many natural numbers
Question: How many natural numbers lie between 52and 62? (a) 9 (b) 10 (c)11 (d) 12 Solution: (b) The natural numbers lying between 52and 62, i.e. between 25 and 36 are 26, 27, 28, 29, 30, 31,32, 33, 34 and 35. Hence, 10 natural numbers lie between 52and 62....
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Question: $\sqrt{5} x^{2}+x+\sqrt{5}=0$ Solution: Given: $\sqrt{5} \mathrm{x}^{2}+\mathrm{x}+\sqrt{5}=0$ Solution of a general quadratic equation $a x^{2}+b x+c=0$ is given by: $x=\frac{-b \pm \sqrt{\left(b^{2}-4 a c\right)}}{2 a}$ $\Rightarrow x=\frac{-1 \pm \sqrt{(1)^{2}-(4 \times \sqrt{5} \times \sqrt{5})}}{2 \times \sqrt{5}}$ $\Rightarrow x=\frac{-1 \pm \sqrt{1-20}}{2 \sqrt{5}}$ $x=\frac{-1 \pm \sqrt{-19}}{2 \sqrt{5}}$ $\Rightarrow \quad x=\frac{-1 \pm \sqrt{19} i}{2 \sqrt{5}}$ $\Rightarrow ...
Read More →Find the volume of a cube whose total surface area is 384 cm
Question: Find the volume of a cube whose total surface area is 384 cm2. Solution: Total surface area $=6 a^{2}$ $\Rightarrow 6 a^{2}=384$ $\Rightarrow a=\sqrt{\frac{384}{6}}=8 \mathrm{~cm}$ $\therefore$ Volume $=a^{3}=512 \mathrm{~cm}^{3}$...
Read More →Which of the following
Question: Which of the following will have 4 at the units place? (a) 142 (b) 622 (c) 272 (d)352 Solution: (b) The unit place of the square of $14=4^{2}=16=6$ The unit place of the square of $62=2^{2}=4$ $\left[\because 2^{2}=4\right]$ The unit place of the square of $27=7^{2}=49=9$ The unit place of the square of $35=5^{2}=25=5$ Clearly, $62^{2}$ has 4 at the unit's place....
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Question: Tick (✓) the correct answer: The ratio of the radii of two cylinders is 2 : 3 and the ratio of their heights is 5 : 3. The ratio of their volumes will be (a) 4 : 9 (b) 9 : 4 (c) 20 : 27 (d) 27 : 20 Solution: (c) 20:27 We have the following: $\frac{r_{1}}{r_{2}}=\frac{2}{3}$ $\frac{h_{1}}{h_{2}}=\frac{5}{3}$ $\therefore \frac{V_{1}}{V_{2}}=\frac{\pi \mathrm{r}_{1}{ }^{2} \mathrm{~h}_{1}}{\pi \mathrm{r}_{2}{ }^{2} \mathrm{~h}_{2}}=\frac{20}{27}$...
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Question: Tick (✓) the correct answer: The diameter of a cylinder is 14 cm and its curved surface area is 220 cm2. The volume of the cylinder is (a) 770 cm3 (b) 1000 cm3 (c) 1540 cm3 (d) 6622 cm3 Solution: (a) $770 \mathrm{~cm}^{3}$ Diameter $=14 \mathrm{~cm}$ Radius $=7 \mathrm{~cm}$ Now, curved surface area $=2 \pi \mathrm{rh}=220 \mathrm{~cm}^{2}$ $\Rightarrow h=\frac{220 \times 7}{2 \times 22 \times 7}=5 \mathrm{~cm}$ $\therefore$ Volume $=\pi \mathrm{r}^{2} \mathrm{~h}=\frac{22}{7} \times 7...
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Question: $x^{2}+3 x+5=0$ Solution: Given: $x^{2}+3 x+5=0$ Solution of a general quadratic equation $a x^{2}+b x+c=0$ is given by: $x=\frac{-b \pm \sqrt{\left(b^{2}-4 a c\right)}}{2 a}$ $\Rightarrow x=\frac{-3 \pm \sqrt{(3)^{2}-(4 \times 1 \times 5)}}{2 \times 1}$ $\Rightarrow x=\frac{-3 \pm \sqrt{9-20}}{2}$ $\Rightarrow \quad x=\frac{-3 \pm \sqrt{-11}}{2}$ $\Rightarrow x=\frac{-3 \pm \sqrt{11 i}}{2}$ $x=-\frac{3}{2} \pm \frac{\sqrt{11}}{2} i$ Ans: $x=-\frac{3}{2}+\frac{\sqrt{11}}{2} i$ and $x=-...
Read More →A number ending in 9 will
Question: A number ending in 9 will have the units place of its square as (a) 3 (b) 9 (c) 1 (d) 6 Solution: (c) We know that, if a number is ending in 1 or 9 in the unit's place, then its square ends in 1 . The number ending in 9 , will have the unit's place of its square as 1 . $[\because 9 \times 9=81]$...
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Question: Tick (✓) the correct answer: The height of a cylinder is 14 cm and its curved surface area is 264 cm2. The volume of the cylinder is (a) 308 cm3 (b) 396 cm3 (c) 1232 cm3 (d) 1848 cm3 Solution: (b) $396 \mathrm{~cm}^{3}$ Here, curved surface area $=2 \pi \mathrm{rh}=264 \mathrm{~cm}^{3}$ $\Rightarrow r=\frac{264 \times 7}{2 \times 22 \times 14}=3 \mathrm{~cm}$ $\therefore$ Volume $=\pi r^{2} \mathrm{~h}=\frac{22}{7} \times 3 \times 3 \times 14=396 \mathrm{~cm}^{3}$...
Read More →Which of the following is a square
Question: Which of the following is a square of an even number? (a) 144 (b) 169 (c) 441 (d) 625 Solution: (a) Here, $144=(12)^{2}$ Similarly, $\quad 169=(13)^{2}$ $441=(21)^{2}$ $625=(25)^{2}$ Thus, 144 is a square of an even number. Alternate Method We know that, square of an even number is always an even number. Hence, 169, 441 and 625 are not even numbers. So, only 144 is an even number, which is the square of 12....
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Question: $2 x^{2}-4 x+3=0$ Solution: Given: $2 x^{2}-4 x+3=0$ Solution of a general quadratic equation $a x^{2}+b x+c=0$ is given by: $x=\frac{-b \pm \sqrt{\left(b^{2}-4 a c\right)}}{2 a}$ $\Rightarrow x=\frac{-(-4) \pm \sqrt{(-4)^{2}-(4 \times 2 \times 3)}}{2 \times 2}$ $\Rightarrow x=\frac{4 \pm \sqrt{16-24}}{4}$ $\Rightarrow x=\frac{4 \pm \sqrt{-8}}{4}$ $\Rightarrow x=\frac{4 \pm 2 \sqrt{2} i}{4}$ $\Rightarrow x=\frac{4}{4} \pm \frac{2 \sqrt{2}}{4} i$ $\Rightarrow \quad x=1 \pm \frac{i}{\sqr...
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Question: Tick (✓) the correct answer: The height of a cylinder is 80 cm and the diameter of its base is 7 cm. The whole surface area of the cylinder is (a) 1837 cm2 (b) 1760 cm2 (c) 1942 cm2 (d) 3080 cm2 Solution: (a) $1837 \mathrm{~cm}^{2}$ Diameter $=7 \mathrm{~cm}$ Radius $=3.5 \mathrm{~cm}$ Height $=80 \mathrm{~cm}$ $\therefore$ Total surface area $=2 \pi \mathrm{r}(\mathrm{r}+\mathrm{h})=2 \times \frac{22}{7} \times 3.5(3.5+80)=22(83.5)=1837 \mathrm{~cm}^{2}$...
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Question: $2 x^{2}-4 x+3=0$ Solution: Given: $2 x^{2}-4 x+3=0$ Solution of a general quadratic equation $a x^{2}+b x+c=0$ is given by: $x=\frac{-b \pm \sqrt{\left(b^{2}-4 a c\right)}}{2 a}$ $\Rightarrow x=\frac{-(-4) \pm \sqrt{(-4)^{2}-(4 \times 2 \times 3)}}{2 \times 2}$ $\Rightarrow x=\frac{4 \pm \sqrt{16-24}}{4}$ $\Rightarrow x=\frac{4 \pm \sqrt{-8}}{4}$ $\Rightarrow x=\frac{4 \pm 2 \sqrt{2} i}{4}$ $\Rightarrow x=\frac{4}{4} \pm \frac{2 \sqrt{2}}{4} i$ $\Rightarrow \quad x=1 \pm \frac{i}{\sqr...
Read More →196 is the square of
Question: 196 is the square of (a) 11 (b) 12 (c) 14 (d) 16 Solution: (c) Square of 11 = 11 x 11 = 121 Square of 12 = 12 x 12 = 144 Square of 14 = 14 x 14 = 196 Clearly, 196 is the square of 14...
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Question: Tick (✓) the correct answer: The height of a cylinder is 14 cm and its diameter is 10 cm. The volume of the cylinder is (a) 1100 cm3 (b) 3300 cm3 (c) 3500 cm3 (d) 7700 cm3 Solution: (a) $1100 \mathrm{~cm}^{3}$ Volume $=\pi r^{2} \mathrm{~h}=\frac{22}{7} \times 5 \times 5 \times 14=1100 \mathrm{~cm}^{3}$...
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Question: Tick (✓) the correct answer: 66 cm3of silver is drawn into a wire 1 mm in diameter. The length of the wire will be (a) 78 m (b) 84 m (c) 96 m (d) 108 m Solution: (b) 84 m Length $=\frac{\text { volume }}{\pi r^{2} 2}=\frac{66 \times 7}{22 \times 0.05 \times 0.05}=8400 \mathrm{~cm}=84 \mathrm{~m}$...
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Question: $x^{2}+2 x+2=0$ Solution: Given: $x^{2}+2 x+2=0$ Solution of a general quadratic equation $a x^{2}+b x+c=0$ is given by: $x=\frac{-b \pm \sqrt{\left(b^{2}-4 a c\right)}}{2 a}$ $\Rightarrow x=\frac{-2 \pm \sqrt{(2)^{2}-(4 \times 1 \times 2)}}{2 \times 1}$ $\Rightarrow x=\frac{-2 \pm \sqrt{4-8}}{2}$ $\Rightarrow \quad x=\frac{-2 \pm \sqrt{-4}}{2}$ $\Rightarrow x=\frac{-2 \pm 2 i}{2}$ $\Rightarrow x=-\frac{2}{2} \pm \frac{2}{2} i$ $\Rightarrow^{x}=-1 \pm i$ Ans: $x=-1+i$ and $x=-1-i$...
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