The following data gives the number of children in 40 families:
Question: The following data gives the number of children in 40 families: 1, 2, 6, 5, 1, 5, 1, 3, 2, 6, 2, 3, 4, 2, 0, 4, 4, 3, 2, 2, 0, 0, 1, 2, 2, 4, 3, 2, 1, 0, 5, 1, 2, 4, 3, 4, 1, 6, 2, 2. Represent it in the form of a frequency distribution. Solution: Arranging the dates in ascending order, we get:0, 0 ,0 , 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,2,2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6There are:4 families with no children7 families with 1 child12 families with 2...
Read More →The next two numbers in the number pattern
Question: The next two numbers in the number pattern 1, 4, 9,16, 25, are (a) 35, 48 (b) 36, 49 (c) 36, 48 (d) 35, 49 Solution: (b) We have, 1,4, 9,16, 25, . The number pattern can be written as(1)2, (2)2, (3)2, (4)2, (5)2 Hence, the next two numbers are (6)2and (7)2, i.e. 36 and 49....
Read More →A dice was thrown 30 times and the following outcomes were noted:
Question: A dice was thrown 30 times and the following outcomes were noted: 2, 1, 2, 4, 6, 1, 2, 3, 6, 5, 4, 4, 3, 1, 1, 3, 1, 1, 5, 6, 6, 2, 2, 3, 4, 2, 5, 5, 6, 4. Prepare a frequency table. Solution: Arranging the outcomes in increasing order, we get:1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6Now, we have:1 was thrown 6 times2 was thrown 6 times3 was thrown 4 times4 was thrown 5 times5 was thrown 4 times6 was thrown 5 timesFrequency distribution ta...
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Question: $3 x^{2}-4 x+\frac{20}{3}=0$ Solution: Given: $3 x^{2}-4 x+\frac{20}{3}=0$ Multiplying both the sides by 3 we get, $9 x^{2}-12 x+20=0$ Solution of a general quadratic equation $a x^{2}+b x+c=0$ is given by: $x=\frac{-b \pm \sqrt{\left(b^{2}-4 a c\right)}}{2 a}$ $\Rightarrow x=\frac{-(-12) \pm \sqrt{(-12)^{2}-(4 \times 9 \times 20)}}{2 \times 9}$ $\Rightarrow x=\frac{12 \pm \sqrt{144-720}}{18}$ $\Rightarrow x=\frac{12 \pm \sqrt{-576}}{18}$ $\Rightarrow x=\frac{12 \pm 24 i}{18}$ $x=\frac...
Read More →A dice was thrown 30 times and the following outcomes were noted:
Question: A dice was thrown 30 times and the following outcomes were noted: 2, 1, 2, 4, 6, 1, 2, 3, 6, 5, 4, 4, 3, 1, 1, 3, 1, 1, 5, 6, 6, 2, 2, 3, 4, 2, 5, 5, 6, 4. Prepare a frequency table. Solution: Arranging the outcomes in increasing order, we get:1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6Now, we have:1 was thrown 6 times2 was thrown 6 times3 was thrown 4 times4 was thrown 5 times5 was thrown 4 times6 was thrown 5 timesFrequency distribution ta...
Read More →The number of members in 20 families are given below:
Question: The number of members in 20 families are given below: 4, 6, 5, 5, 4, 6, 3, 3, 5, 5, 3, 5, 4, 4, 6, 7, 3, 5, 5, 7. Prepare a frequency distribution of the data. Solution: First, we will arrange the data in increasing order. 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 7, 7 Frequency table of the above data:/spanbr data-mce-bogus="1"/ppimg src="https://www.esaral.com/qdb/uploads/2022/02/07/image32754.png" alt=""...
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Question: $3 x^{2}+5=7 x$ Solution: Given: $3 x^{2}+5=7 x$ $\Rightarrow 3 x^{2}-7 x+5=0$ Solution of a general quadratic equation $a x^{2}+b x+c=0$ is given by: $x=\frac{-b \pm \sqrt{\left(b^{2}-4 a c\right)}}{2 a}$ $\Rightarrow x=\frac{-(-7) \pm \sqrt{(-7)^{2}-(4 \times 3 \times 5)}}{2 \times 3}$ $\Rightarrow x=\frac{7 \pm \sqrt{49-60}}{6}$ $\Rightarrow \quad x=\frac{7 \pm \sqrt{-11}}{6}$ $\Rightarrow x=\frac{7 \pm \sqrt{11} i}{6}$ $\Rightarrow x=\frac{7}{6} \pm \frac{\sqrt{11}}{6} i$ Ans: $x=\...
Read More →Fill in the blanks.
Question: Fill in the blanks. (i) Ifl,b,hbe the length, breadth and height of a cuboid, then its whole surface area = (.......) sq units. (ii) Ifl,b,hbe the length, breadth and height of a cuboid, then its lateral surface area = (.......) sq units. (iii) If each side of a cube isa, then its lateral surface area is ....... sq units. (iv) Ifris the radius of the base andhbe the height of a cylinder, then its volume is (.......) cubic units. (v) Ifris the radius of the base andhbe the height of a c...
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Question: $17 x^{2}-8 x+1=0$ Solution: Given: $17 x^{2}-8 x+1=0$ Solution of a general quadratic equation $a x^{2}+b x+c=0$ is given by: $x=\frac{-b \pm \sqrt{\left(b^{2}-4 a c\right)}}{2 a}$ $\Rightarrow x=\frac{-(-8) \pm \sqrt{(-8)^{2}-(4 \times 17 \times 1)}}{2 \times 17}$ $x=\frac{8 \pm \sqrt{64-68}}{34}$ $\Rightarrow x=\frac{8 \pm \sqrt{-4}}{34}$ $\Rightarrow x=\frac{8 \pm 2 i}{34}$ $\Rightarrow x=\frac{8}{34} \pm \frac{2}{34} i$ Ans: $x=\frac{4}{17}+\frac{1}{17} i$ and $x=\frac{4}{17}-\fra...
Read More →Mark (✓) against the correct answer:
Question: Mark (✓) against the correct answer: The surface area of a cube is 384 cm2. Its volume is (a) 512 cm3 (b) 256 cm3 (c) 384 cm3 (d) 320 cm3 Solution: (a) $512 \mathrm{~cm}^{3}$ Surface area $=6 a^{2}$ $\Rightarrow 6 a^{2}=384$ $\Rightarrow a=\sqrt{\frac{384}{6}}=\sqrt{64}=8 \mathrm{~cm}$ $\therefore$ Volume $=a^{3}=8^{3}=512 \mathrm{~cm}^{3}$...
Read More →Mark (✓) against the correct answer:
Question: Mark (✓) against the correct answer: The length, breadth and height of a cuboid are in the ratio 3 : 4 : 6 and its volume is 576 cm3. The whole surface area of the cuboid is (a) 216 cm2 (b) 324 cm2 (c) 432 cm2 (d) 460 cm2 Solution: (c) $432 \mathrm{sq} \mathrm{cm}$ Volume $=l b h=3 x \times 4 x \times 6 x=72 x^{3}=576 \mathrm{~cm}^{3}$ $\Rightarrow x=\sqrt[3]{\frac{576}{72}}=2$ $\therefore$ Total surface area $=2(l b+b h+l h)=2(3 x 4 x+4 x 6 x+3 x 6 x)=2(48+96+72)=432 \mathrm{~cm}^{2}$...
Read More →The hypotenuse of a right angled triangle
Question: The hypotenuse of a right angled triangle with its legs of lengths 3x x 4x is (a) 5X (b )7x (c) 16x (d) 25x Solution: (a) Given, lengths of the legs of right angled triangle are $3 x$ and $4 x$. Now, hypotenuse $=\sqrt{(3 x)^{2}+(4 x)^{2}}$ [using Pythagoras theorem] $=\sqrt{9 x^{2}+16 x^{2}}$ $=\sqrt{25 x^{2}}=5 x$...
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Question: Mark (✓) against the correct answer: The dimensions of a cuboid are 8 m 6 m 4 m. Its lateral surface area is (a) 210 m2 (b) 105 m2 (c) 160 m2 (d) 240 m2 Solution: Lateral surface area $=2((l+b) \times h)=2((8+6) \times 4)=2(56)=112 \mathrm{~m}^{2}$...
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Question: Mark (✓) against the correct answer: A cuboid having dimensions 16 m 11 m 8 m is melted to form a cylinder of radius 4 m. What is the height of the cylinder? (a) 28 m (b) 14 m (c) 21 m (d) 32 m Solution: (a) $28 \mathrm{~m}$ Volume of the cuboid $=16 \times 11 \times 8=1408 \mathrm{~m}^{3}$ Volume of the cylinder $=\pi \mathrm{r}^{2} \mathrm{~h}=1408 \mathrm{~m}^{3}$ $\therefore h=\frac{1408 \times 7}{22 \times 4 \times 4}=28 \mathrm{~m}$...
Read More →Which of the following numbers is a perfect cube?
Question: Which of the following numbers is a perfect cube? (a) 243 (b) 216 (c) 392 (d) 8640 Solution: (b) For option (a) We have, 243 Resolving 243 into prime factors, we have 243= 3 x 3 x 3 x 3 x 3 Grouping the factors in triplets of equal factors, we get 243 = (3 x 3 x 3) x 3 x 3 Clearly, in grouping, the factors in triplets of equal factors, we are left with two factors 3 x 3. Therefore, 243 is not a perfect cube. For option (b) We have, 216 Resolving 216 into prime factqrs, we have 216 = 2 ...
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Question: Mark (✓) against the correct answer: the area of the base of a circular cylinder is 35 cm2and its height is 8 cm. The volume of the cylinder is (a) 140 cm3 (b) 280 cm3 (c) 420 cm3 (d) 210 cm3 Solution: (b) $280 \mathrm{~cm}^{3}$ Area $=35 \mathrm{~cm}^{2}$ Height $=8 \mathrm{~cm}$ $\therefore$ Volume $=$ base area $\times$ height $=35 \times 8=280 \mathrm{~cm}^{3}$...
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Question: $2 x^{2}-\sqrt{3} x+1=0$ Solution: Given: $2 x^{2}-\sqrt{3} x+1=0$ Solution of a general quadratic equation $a x^{2}+b x+c=0$ is given by: $x=\frac{-b \pm \sqrt{\left(b^{2}-4 a c\right)}}{2 a}$ $\Rightarrow x=\frac{-(-\sqrt{3}) \pm \sqrt{(-\sqrt{3})^{2}-(4 \times 2 \times 1)}}{2 \times 2}$ $\Rightarrow x=\frac{\sqrt{3} \pm \sqrt{3-8}}{4}$ $\Rightarrow x=\frac{\sqrt{3} \pm \sqrt{-5}}{4}$ $\Rightarrow \quad x=\frac{\sqrt{3} \pm \sqrt{5} i}{4}$ $\Rightarrow \quad x=\frac{\sqrt{3}}{4} \pm ...
Read More →The sum of first n odd natural numbers is
Question: The sum of first n odd natural numbers is (a) 2n+1 (b) n2 (c) n21 (d) n2+1 Solution: (b) Sum of first $n$ odd natural numbers $=\Sigma(2 n-1)=2 \Sigma n-n$ $=\frac{2 \times n(n+1)}{2}-n=n(n+1)-n=n^{2}+n-n=n^{2}$...
Read More →The sum of successive odd numbers
Question: The sum of successive odd numbers 1, 3, 5, 7, 9, 11, 13 and 15 is (a) 61 (b) 64 (c) 49 (d) 36 Solution: (b) We know that, the sum of first n odd natural numbers is n2. Given odd numbers are 1,3, 5, 7, 9,11,13 and 15. So, number of odd numbers, n = 8 The sum of given odd numbers =n2= (8)2= 64...
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Question: $27 x^{2}+10 x+1=0$ Solution: Given: $27 x^{2}+10 x+1=0$ Solution of a general quadratic equation $a x^{2}+b x+c=0$ is given by: $x=\frac{-b \pm \sqrt{\left(b^{2}-4 a c\right)}}{2 a}$ $\Rightarrow x=\frac{-10 \pm \sqrt{(10)^{2}-(4 \times 27 \times 1)}}{2 \times 27}$ $\Rightarrow x=\frac{-10 \pm \sqrt{100-108}}{54}$ $\Rightarrow x=\frac{-10 \pm \sqrt{-8}}{54}$ $\Rightarrow \quad x=\frac{-10 \pm 2 \sqrt{2} i}{54}$ $\Rightarrow \quad x=-\frac{10}{54} \pm \frac{2 \sqrt{2}}{54} i$ Ans: $x=-...
Read More →If one member of a Pythagorean
Question: If one member of a Pythagorean triplet is $2 \mathrm{~m}$, then the other two members are (a) $m, m^{2}+1$ (b) $m^{2}+1, m^{2}-1$ (c) $m^{2}, m^{2}-1$ (a) $m, m^{2}+1$ (b) $m^{2}+1, m^{2}-1$ (c) $m^{2}, m^{2}-1$ (d) $m^{2}, m+1$ Solution: $2 m=4$ $\Rightarrow \quad m=2$ $m^{2}+1=2^{2}+1=4+1=5$ and $\quad m^{2}-1=2^{2}-1=4-1=3$ Now. $\quad 3^{2}+4^{2}=5^{2}$ $\Rightarrow \quad 9+16=25 .$ $\Rightarrow \quad 25=25$ So, 3,4 and 5 are pythagorean triplets....
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Question: $8 x^{2}+2 x+1=0$ Solution: Given: $8 x^{2}+2 x+1=0$ Solution of a general quadratic equation $a x^{2}+b x+c=0$ is given by: $x=\frac{-b \pm \sqrt{\left(b^{2}-4 a c\right)}}{2 a}$ $\Rightarrow x=\frac{-2 \pm \sqrt{(2)^{2}-(4 \times 8 \times 1)}}{2 \times 8}$ $\Rightarrow x=\frac{-2 \pm \sqrt{4-32}}{16}$ $\Rightarrow x=\frac{-2 \pm \sqrt{-28}}{16}$ $\Rightarrow x=\frac{-2 \pm 2 \sqrt{7} i}{16}$ $\Rightarrow x=-\frac{2}{16} \pm \frac{2 \sqrt{7}}{16} i$ Ans: $x=-\frac{1}{8}+\frac{\sqrt{7}...
Read More →Which letter best represents the location of
Question: Which letter best represents the location of $\sqrt{25}$ on a number line? (a) $A$ (b) $B$ (c) $C$ (d) $D$ Solution: (c) We have, $\sqrt{25}=5$ Therefore, 5 at letter $C$ represents the best location of $\sqrt{25}$ on number line....
Read More →Mark (✓) against the correct answer:
Question: Mark (✓) against the correct answer: The circumference of the circular base of a cylinder is 44 cm and its height is 15 cm. The volume of the cylinder is (a) 1155 cm3 (b) 2310 cm3 (c) 770 cm3 (d) 1540 cm3 Solution: (b) $2310 \mathrm{~cm}^{3}$ Height $=15 \mathrm{~cm}$ Circumference $=2 \pi \mathrm{r}=44 \mathrm{~cm}$ $\therefore r=\frac{44 \times 7}{2 \times 22}=7 \mathrm{~cm}$ $\therefore$ Volume $=\pi r^{2} \mathrm{~h}=\frac{22}{7} \times 7 \times 7 \times 15=2310 \mathrm{~cm}^{3}$...
Read More →The curved surface area of a cylindrical pillar is 264 m
Question: The curved surface area of a cylindrical pillar is 264 m2and its volume is 924 m3. Find the diameter and height of the pillar. Solution: Curved surface area $=2 \pi \mathrm{rh}=264 \mathrm{~m}^{2}$ $\therefore r=\frac{264}{2 \pi \mathrm{h}}=\frac{132}{\pi \mathrm{h}} m$ Volume $=\pi \mathrm{r}^{2} \mathrm{~h}=\pi \times \frac{132}{\pi \mathrm{h}} \times \frac{132}{\pi \mathrm{h}} \times \mathrm{h}=924 \mathrm{~m}^{3}$ $\therefore h=\frac{132 \times 132 \times 7}{22 \times 924}=6 \mathr...
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