Find the cubes of:
Question: Find the cubes of: (i) 11 (ii) 12 (iii) 21 Solution: (i) Cube of $-11$ is given as: $(-11)^{3}=-11 \times-11 \times-11=-1331$ Thus, the cube of 11 is $(-1331)$. (ii) Cube of $-12$ is given as: $(-12)^{3}=-12 \times-12 \times-12=-1728$ Thus, the cube of $-12$ is $(-1728)$. (iii) Cube of $-21$ is aiven as: $(-21)^{3}=-21 \times-21 \times-21=-9261$ Thus, the cube of $-21$ is $(-9261)$....
Read More →Find which of the variables x, y, z
Question: Find which of the variables $x, y, z$ and $u$ represent rational numbers and which irrational numbers. (i) $x^{2}=5$ (ii) $y^{2}=9$ (iii) $z^{2}=0.04$ (iv) $u^{2}=17 / 4$ Solution: (i) Given, $x^{2}=5$ On taking square root both sides we get $x=\pm \sqrt{5}$ [irrational number] (ii) Given, $y^{2}=9$ On taking square root both sides, we get $y=\pm \sqrt{9}=\pm 3$ [rational number] (iii) Given, $z^{2}=0.04$ On taking square root both sides, we get $z=\sqrt{0.04}=\sqrt{\frac{4}{100}}=\fra...
Read More →Find the length of a side of a sqiare,
Question: Find the length of a side of a sqiare, whose area is equal to the area of a rectangle with sides 240 m and 70 m. Solution: The area of the rectangle $=240 \mathrm{~m} \times 70 \mathrm{~m}=16800 \mathrm{~m}^{2}$ Given that the area of the square is equal to the area of the rectangle. Hence, the area of the square will also be 16800 m2. The length of one side of a square is the square root of its area. $\therefore \sqrt{16800}=\sqrt{2 \times 2 \times 2 \times 2 \times 2 \times 3 \times ...
Read More →Find the length of a side of a sqiare,
Question: Find the length of a side of a sqiare, whose area is equal to the area of a rectangle with sides 240 m and 70 m. Solution: The area of the rectangle $=240 \mathrm{~m} \times 70 \mathrm{~m}=16800 \mathrm{~m}^{2}$ Given that the area of the square is equal to the area of the rectangle. Hence, the area of the square will also be 16800 m2. The length of one side of a square is the square root of its area. $\therefore \sqrt{16800}=\sqrt{2 \times 2 \times 2 \times 2 \times 2 \times 3 \times ...
Read More →Classify the following numbers as rational or irrational with justification
Question: Classify the following numbers as rational or irrational with justification (i) $\sqrt{196}$ (ii) $3 \sqrt{18}$ (iii) $\sqrt{\frac{9}{27}}$ (iv) $\frac{\sqrt{28}}{\sqrt{343}}$ (v) $-\sqrt{0.4}$ (vi) $\frac{\sqrt{12}}{\sqrt{75}}$ (vii) $0.5918$ (viii) $(1+\sqrt{5})-(4+\sqrt{5})$ (ix) $10.124124 \ldots$ (x) $1.010010001 \ldots$ Thinking Process To classify, use the definition a rational number is in the form of $\mathrm{p} / \mathrm{q}$, where $p$ and $q$ are integers and $q \neq 0$ and ...
Read More →if (cosec θ − sin θ) = a3 and (sec θ − cos θ) = b3,
Question: If $(\operatorname{cosec} \theta-\sin \theta)=a^{3}$ and $(\sec \theta-\cos \theta)=b^{3}$, prove that $a^{2} b^{2}\left(a^{2}+b^{2}\right)=1$. Solution: We have $(\operatorname{cosec} \theta-\sin \theta)=a^{3}$ $=a^{3}=\left(\frac{1}{\sin \theta}-\sin \theta\right)$ $=a^{3}=\frac{\left(1-\sin ^{2} \theta\right)}{\sin \theta}=\frac{\cos ^{2} \theta}{\sin \theta}$ $\therefore a=\frac{\cos ^{\frac{2}{3}} \theta}{\sin ^{\frac{1}{3}} \theta}$ Again, $(\sec \theta-\cos \theta)=b^{3}$ $=b^{3...
Read More →The area of a square field is 325 m
Question: The area of a square field is 325 m2. Find the approximate length of one side of the field. Solution: The length of one side of the square field will be the square root of 325. $\therefore \sqrt{325}=\sqrt{5 \times 5 \times 13}$ $=5 \times \sqrt{13}$ $=5 \times 3.605$ $=18.030$ Hence, the length of one side of the field is 18.030 m....
Read More →Using square root table, find the square root
Question: Using square root table, find the square root11.11 Solution: We have: $\sqrt{11}=3.317$ and $\sqrt{12}=3.464$ Their difference is 0.1474. Thus, for the difference of 1 (12-11), the difference in the values of the square roots is 0.1474. For the difference of 0.11, the difference in the values of the square roots is: $0.11 \times 0.1474=0.0162$ $\therefore \sqrt{11.11}=3.3166+0.0162=3.328 \approx 3.333$...
Read More →Using square root table, find the square root
Question: Using square root table, find the square root1110 Solution: $\sqrt{1110}=\sqrt{2} \times \sqrt{3} \times \sqrt{5} \times \sqrt{37}$ $=1.414 \times 1.732 \times 2.236 \times 6.083 \quad$ (Using the table to find all the square roots) $=33.312$...
Read More →Using square root table, find the square root
Question: Using square root table, find the square root110 Solution: $\sqrt{110}=\sqrt{2} \times \sqrt{5} \times \sqrt{11}$ $=1.414 \times 2.236 \times 3.317 \quad$ (Using the square root table to find all the square roots) $=10.488$...
Read More →State whether the following statements are true or false?
Question: State whether the following statements are true or false? Justify your answer. (i) $\frac{\sqrt{2}}{3}$ is a rational number. (ii) There are infinitely many integers between any two integers. (iii) Number of rational numbers between 15 and 18 is finite. (iv) There are numbers which cannot be written in the form $\frac{p}{q}$, $q \neq 0, p$ and $q$ both are integers. (v) The Square of an irrational number is always rational. (vi) $\frac{\sqrt{12}}{\sqrt{3}}$ is not a rational number as ...
Read More →Using square root table, find the square root
Question: Using square root table, find the square root Solution: We have to find $\sqrt{21.97}$. From the square root table, we have: $\sqrt{21}=\sqrt{3} \times \sqrt{7}=4.583$ and $\sqrt{22}=\sqrt{2} \times \sqrt{11}=4.690$ Their difference is 0.107. Thus, for the difference of 1 (2221), the difference in the values of the square roots is 0.107. For the difference of 0.97, the difference in the values of their square roots is: $0.107 \times 0.97=0.104$ $\therefore \sqrt{21.97}=4.583+0.104 \app...
Read More →If (cot θ + tan θ) = m and (sec θ − cos θ) = n,
Question: If $(\cot \theta+\tan \theta)=m$ and $(\sec \theta-\cos \theta)=n$, prove that $\left(m^{2} n\right)^{2 / 3}-\left(m n^{2}\right)^{2 / 3}=1$ Solution: We have $(\cot \theta+\tan \theta)=m$ and $(\sec \theta-\cos \theta)=n$ Now, $m^{2} n=\left[(\cot \theta+\tan \theta)^{2}(\sec \theta-\cos \theta)\right]$ $=\left[\left(\frac{1}{\tan \theta}+\tan \theta\right)^{2}\left(\frac{1}{\cos \theta}-\cos \theta\right)\right]$ $=\frac{\left(1+\tan ^{2} \theta\right)^{2}}{\tan ^{2} \theta} \times \...
Read More →Using square root table, find the square root
Question: Using square root table, find the square root13.21 Solution: From the square root table, we have: $\sqrt{13}=3.606$ and $\sqrt{14}=\sqrt{2} \times \sqrt{7}=3.742$ Their difference is 0.136. Thus, for the difference of 1 (1413), the difference in the values of the square roots is 0.136. For the difference of 0.21, the difference in the values of their square roots is: $0.136 \times 0.21=0.02856$ $\therefore \sqrt{13.21}=3.606+0.02856 \approx 3.635$...
Read More →Let x be rational and y be irrational. I
Question: Let $x$ be rational and $y$ be irrational. Is $x y$ necessarily irrational? Justify your answer by an example. Solution: No, $(x y)$ is necessarily an irrational only when $x \neq 0$. Let $x$ be a non-zero rational and $y$ be an irrational. Then, we have to show that $x y$ be an irrational. If possible, let $x y$ be a rational number. Since, quotient of two non-zero rational number is a rational number. So, $(x y / x)$ is a rational number $=y$ is a rational number. But, this contradic...
Read More →Using square root table, find the square root
Question: Using square root table, find the square root $\frac{101}{169}$ Solution: $\sqrt{\frac{101}{169}}=\frac{\sqrt{101}}{\sqrt{169}}$ The square root of 101 is not listed in the table. This is because the table lists the square roots of all the numbers below 100. Hence, we have to manipulate the number such that we get the square root of a number less than 100. This can be done in the following manner: $\sqrt{101}=\sqrt{1.01 \times 100}=\sqrt{1.01} \times 10$ Now, we have to find the square...
Read More →Let x and y be rational and irrational numbers,
Question: Let $x$ and $y$ be rational and irrational numbers, respectively. Is $x+y$ necessarily an irrational number? Give an example in support of your answer. Solution: Yes, $(x+y)$ is necessarily an irrational number. e.g., Let $\quad x=2, y=\sqrt{3}$ Then, $x+y=2+\sqrt{3}$ If possible, let $x+y=2+\sqrt{3}$ be a rational number. Consider $a=2+\sqrt{3}$ On squaring both sides, we get $a^{2}=(2+\sqrt{3})^{2}$ [using identity $(a+b)^{2}=a^{2}+b^{2}+2 a b$ ] $\Rightarrow$ $a^{2}=2^{2}+(\sqrt{3})...
Read More →Prove that:
Question: Prove that: $\left|\begin{array}{lll}a+b b+c c+a \\ b+c c+a a+b \\ c+a a+b b+c\end{array}\right|=2\left|\begin{array}{lll}a b c \\ b c a \\ c a b\end{array}\right|$ Solution: Let $\Delta=\mid \begin{array}{lll}a+b b+c c+a\end{array}$ $b+c \quad c+a \quad a+b$ $c+a \quad a+b \quad b+c \mid$ Using the property of determinants that if each element of a row or column is expressed as the sum of two or more quantities, the determinant is expressed as the sum of two or more determinants, we g...
Read More →Using square root table, find the square root
Question: Using square root table, find the square root $\frac{57}{169}$ Solution: $\sqrt{\frac{57}{169}}=\frac{\sqrt{3} \times \sqrt{19}}{\sqrt{169}}$ $\frac{1.732 \times 4.3589}{13} \quad$ (using the square root table to find $\sqrt{3}$ and $\sqrt{19}$ ) 0.581...
Read More →Using square root table, find the square root
Question: Using square root table, find the square root $\frac{99}{144}$ Solution: $\sqrt{\frac{99}{144}}=\frac{\sqrt{3 \times 3 \times 11}}{\sqrt{144}}$ $=\frac{3 \sqrt{11}}{12}$ $=\frac{3 \times 3.3166}{12} \quad$ (using the square root table to find $\sqrt{11}$ ) =0.829...
Read More →If (tan θ + sin θ) = m and (tan θ − sin θ) = n,
Question: If (tan + sin ) =mand (tan sin ) =n, prove that (m2n2)2= 16mn. Solution: We have $(\tan \theta+\sin \theta)=m$ and $(\tan \theta-\sin \theta)=n$ Now, LHS $=\left(m^{2}-n^{2}\right)^{2}$ $=\left[(\tan \theta+\sin \theta)^{2}-(\tan \theta-\sin \theta)^{2}\right]^{2}$ $=\left[\left(\tan ^{2} \theta+\sin ^{2} \theta+2 \tan \theta \sin \theta\right)-\left(\tan ^{2} \theta+\sin ^{2} \theta-2 \tan \theta \sin \theta\right)\right]^{2}$ $=\left[\left(\tan ^{2} \theta+\sin ^{2} \theta+2 \tan \th...
Read More →Which of the following is equal to
Question: Which of the following is equal to $X$ ? (a) $x^{\frac{12}{7}}-x^{\frac{5}{7}}$ (b) $\sqrt[12]{\left(x^{4}\right)^{\frac{1}{3}}}$ (c) $\left(\sqrt{x^{3}}\right)^{\frac{2}{3}}$ (d) $x^{\frac{12}{7}} \times x^{\frac{7}{12}}$ Solution: (c) (a) $x^{\frac{12}{7}}-x^{\frac{5}{7}}=x^{\frac{5}{7}+1}-x^{\frac{5}{7}}$ $=x^{\frac{5}{7}} \cdot x-x^{\frac{5}{7}}$ $\left[\because a^{m+n}=a^{m} a^{n}\right]$ (b) $\sqrt[12]{\left(x^{4}\right)^{\frac{1}{3}}}=\left(\left(x^{4}\right)^{\frac{1}{3}}\right...
Read More →Using square root table, find the square root
Question: Using square root table, find the square root4955 Solution: On prime factorisation: 4955 is equal to $5 \times 991$, which means that $\sqrt{4955}=\sqrt{5} \times \sqrt{11}$. The square root of 991 is not listed in the table; it lists the square roots of all the numbers below 100. Hence, we have to manipulate the number such that we get the square root of a number less than 100. This can be done in the following manner: $\sqrt{4955}=\sqrt{49.55 \times 100}=\sqrt{49.55} \times 10$ Now, ...
Read More →If x = b sec3θ and y = a tan3θ, prove that
Question: If $x=b \sec ^{3} \theta$ and $y=a \tan ^{3} \theta$, prove that $\left(\frac{x}{b}\right)^{2 / 3}-\left(\frac{y}{a}\right)^{2 / 3}=1$ Solution: $x=b \sec ^{3} \theta$ $\Rightarrow \sec ^{3} \theta=\frac{x}{b}$ $\Rightarrow \sec \theta=\left(\frac{x}{b}\right)^{\frac{1}{3}} \quad \ldots \ldots(1)$ Also, $y=a \tan ^{3} \theta$ $\Rightarrow \tan ^{3} \theta=\frac{y}{a}$ $\Rightarrow \tan \theta=\left(\frac{y}{a}\right)^{\frac{1}{3}} \quad \ldots \ldots(2)$ We know $\sec ^{2} \theta-\tan ...
Read More →Using square root table, find the square root
Question: Using square root table, find the square root4192 Solution: $\sqrt{4192}=\sqrt{2 \times 2 \times 2 \times 2 \times 2 \times 131}$ $=2 \times 2 \sqrt{2} \times \sqrt{131}$ The square root of 131 is not listed in the table. Hence, we have to apply long division to find it.Substituting the values: $=2 \times 2 \times 11.4455 \quad$ (using the table to find $\sqrt{2}$ ) = 64.75...
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