If sin A + sin2 A = 1, then (cos2 A + cos4 A) = ?
Question: If sinA+ sin2A= 1, then (cos2A+ cos4A) = ? (a) $\frac{1}{2}$ (b) 1(c) 2(d) 3 Solution: (b) 1 $\sin A+\sin ^{2} A=1$ $=\sin A=1-\sin ^{2} A$ $=\sin A=\cos ^{2} A \quad\left(\because 1-\sin ^{2} A\right)$ $=\sin ^{2} A=\cos ^{4} A \quad$ (Squaring both sides) $=1-\cos ^{2} A=\cos ^{4} A$ $=\cos ^{4} A+\cos ^{2} A=1$...
Read More →Find the cube roots of the following numbers by successive subtraction of numbers:
Question: Find the cube roots of the following numbers by successive subtraction of numbers: 1, 7, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397, ... (i) 64 (ii) 512 (iii) 1728 Solution: (i)We have: $\because$ Subtraction is performed 4 times. $\therefore \sqrt[3]{64}=4$ (ii)We have: $\because$ Subtraction is performed 8 times. $\therefore \sqrt[3]{512}=8$ (iii)We have: $\because$ Subtraction is performed 12 times. $\therefore \sqrt[3]{1728}=12$...
Read More →If (sec A – tan A) = x then prove that
Question: If $(\sec A-\tan A)=x$ then prove that $\frac{1+x^{2}}{1-x^{2}}=\operatorname{cosec} A$. Solution: Given: $\sec A-\tan A=x \quad \ldots \ldots$ (1) We know $\sec ^{2} A-\tan ^{2} A=1$ $\Rightarrow(\sec A+\tan A)(\sec A-\tan A)=1 \quad\left[a^{2}-b^{2}=(a-b)(a+b)\right]$ $\Rightarrow(\sec A+\tan A) x=1 \quad[$ From (1) $]$ $\Rightarrow \sec A+\tan A=\frac{1}{x} \quad \ldots(2)$ Adding (1) and (2), we get $\sec A-\tan A+\sec A+\tan A=x+\frac{1}{x}$ $\Rightarrow 2 \sec A=\frac{x^{2}+1}{x}...
Read More →Represent geometrically the following numbers on the number line
Question: Represent geometrically the following numbers on the number line (i) $\sqrt{4} .5$ (ii) $\sqrt{5} .6$ (iii) $\sqrt{8.1}$ (iv) $\sqrt{2} .3$ Thinking Process (i) Firstly, we draw a line segment $A B$ of length equal to the number inside the root and extend it to $C$ such that $B C=1$ (ii) Draw a semi-circle with centre $O$ ( $O$ is the mid-point of $A C$ ) and radius $O A$ (iii) Now, draw a perpendicular line from $B$ to cut the semi-circle atD. (iv) Further, draw an arc with centre $B$...
Read More →Find which of the following numbers are cubes of rational numbers:
Question: Find which of the following numbers are cubes of rational numbers: (i) $\frac{27}{64}$ (ii) $\frac{125}{128}$ (iii) 0.001331 (iv) 0.04 Solution: (i)We have: $\frac{27}{64}=\frac{3 \times 3 \times 3}{8 \times 8 \times 8}=\frac{3^{3}}{8^{3}}=\left(\frac{3}{8}\right)^{3}$ Therefore, $\frac{27}{64}$ is a cube of $\frac{3}{8}$. (ii)We have: $\frac{125}{128}=\frac{5 \times 5 \times 5}{2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2}=\frac{5^{3}}{2^{3} \times 2^{3} \times 2}$ It is ev...
Read More →Prove that:
Question: Prove that: $\left|\begin{array}{lll}1 a b c \\ 1 b c a \\ 1 c a b\end{array}\right|=\left|\begin{array}{lll}1 a a^{2} \\ 1 b b^{2} \\ 1 c c^{2}\end{array}\right|$ Solution: Let $\quad$ LHS $=\Delta=\mid \begin{array}{lll}1 a b c\end{array}$ $\begin{array}{lll}1 b c a \\ 1 c a b\end{array}$ $=\frac{1}{\mathrm{abc}} \mid \begin{array}{lll}a a^{2} a b c\end{array}$ $b \quad b^{2} \quad b c a$ $c \quad c^{2} \quad a b c \mid \quad$ [Applying $\mathrm{R}_{1} \rightarrow \mathrm{a} \mathrm{...
Read More →Prove the following
Question: Locate $\sqrt{5}, \sqrt{10}$ and $\sqrt{1} 7$ on the number line. Solution: (i) Here, $5=2^{2}+1^{2}$ So, draw a right angled $\triangle O A B$, in which $O A=2$ units and $A B=1$ unit and $\angle O A B=90^{\circ}$ By using Pythagoras theorem, we get $O B=\sqrt{O A^{2}+A B^{2}}$ $=\sqrt{2^{2}+1^{2}}=\sqrt{4+1}=\sqrt{5}$ Taking $O B=\sqrt{5}$ as radius and point $O$ as centre, draw an arc which meets the number line at point $P$ on the positive side of it. Hence, it is clear that point ...
Read More →If (cosec A + cot A) = m then prove that
Question: If $(\operatorname{cosec} A+\cot A)=m$ then prove that $\frac{m^{2}-1}{m^{2}+1}=\cos \theta$. Solution: Given: $\operatorname{cosec} A+\cot A=m$ ....(1) We know $\operatorname{cosec}^{2} A-\cot ^{2} A=1$ $\Rightarrow(\operatorname{cosec} A-\cot A)(\operatorname{cosec} A+\cot A)=1 \quad\left[a^{2}-b^{2}=(a-b)(a+b)\right]$ $\Rightarrow(\operatorname{cosec} A-\cot A) m=1 \quad[$ From $(1)]$ $\Rightarrow \operatorname{cosec} A-\cot A=\frac{1}{m} \quad \ldots \ldots(2)$ Adding (1) and (2), ...
Read More →Find the cube of:
Question: Find the cube of: (i) $\frac{7}{9}$ (ii) $-\frac{8}{11}$ (iii) $\frac{12}{7}$ (iv) $-\frac{13}{8}$ (v) $2 \frac{2}{5}$ (vi) $3 \frac{1}{4}$ (vii) $0.3$ (viii) $1.5$ (ix) $0.08$ (x) $2.1$ Solution: (i) $\because\left(\frac{m}{n}\right)^{3}=\frac{m^{3}}{n^{3}}$ $\therefore\left(\frac{7}{9}\right)^{3}=\frac{7^{3}}{9^{3}}=\frac{7 \times 7 \times 7}{9 \times 9 \times 9}=\frac{343}{729}$ (ii) $\because\left(-\frac{m}{n}\right)^{3}=-\frac{m^{3}}{n^{3}}$ $\therefore\left(-\frac{8}{11}\right)^{...
Read More →Prove that:
Question: Prove that: $\left|\begin{array}{ccc}a a+b a+2 b \\ a+2 b a a+b \\ a+b a+2 b a\end{array}\right|=9(a+b) b^{2}$ Solution: Let LHS $=\Delta=\mid \begin{array}{lll}a a+b a+2 b\end{array}$ $\begin{array}{lll}a+2 b a a+b \\ a+b a+2 b a \mid\end{array}$ $\Delta=\mid \begin{array}{lll}3 a+3 b 3 a+3 b 3 a+3 b\end{array}$ $\begin{array}{ccr}a+2 b a a+b \\ a+b a+2 b a \mid\end{array}$ $\left[\right.$ Applying $\left.\mathrm{R}_{1} \rightarrow \mathrm{R}_{1}+\mathrm{R}_{2}+\mathrm{R}_{3}\right]$ ...
Read More →Represent the following numbers
Question: Represent the following numbers on the number line $7,7.2,-3 / 2$ and $-12 / 5$. Solution: Firstly, we draw a number line whose mid-point is 0 . Marks a positive numbers on right hand side of 0 and negative numbers on left hand side of 0 . (i)Number 7 is a positive number. So we mark a number 7 on the right hand side of 0, which is a 7 units distance from zero. (ii)Number 7.2 is a positive number. So, we mark a number 7.2 on the right hand side of 0, which is a 7.2 units distance from ...
Read More →If (sec θ + tan θ ) = p then show that (sec θ – tan θ)
Question: If $(\sec \theta+\tan \theta)=p$ then show that $(\sec \theta-\tan \theta)=\frac{1}{p}$. Hence, show that $\cos \theta=\frac{2 p}{\left(p^{2}+1\right)}$ and $\sin \theta=\frac{p^{2}-1}{p^{2}+1}$. Solution: Given: $\sec \theta+\tan \theta=p \quad \ldots$ (1) We know $\sec ^{2} \theta-\tan ^{2} \theta=1$ $\Rightarrow(\sec \theta-\tan \theta)(\sec \theta+\tan \theta)=1$ $\Rightarrow(\sec \theta-\tan \theta) p=1 \quad[$ From $(1)]$ $\Rightarrow \sec \theta-\tan \theta=\frac{1}{p} \quad \ld...
Read More →Insert a rational number and an irrational number between the following
Question: Insert a rational number and an irrational number between the following (i) 2 and 3 (ii) 0 and $0.1$ (iii) $1 / 3$ and $1 / 2$ (iv) $-2 / 5$ and $-1 / 2$ (v) $0.15$ and $0.16$ (vi) $\sqrt{2}$ and $\sqrt{3}$ (vii) $2.357$ and $3.121$ (viii) 0001 and 001 (ix) $3.623623$ and $0.484848$ (x) $3.375289$ and $6.375738$ Solution: We know that, there are infinitely many rational and irrational values between any two numbers. (i)A rational number between 2 and 3 is 2.1.To find an irrational numb...
Read More →Show that the following integers are cubes of negative integers.
Question: Show that the following integers are cubes of negative integers. Also, find the integer whose cube is the given integer. (i) 5832 (ii) 2744000 Solution: In order to check if a negative number is a perfect cube, first check if the corresponding positive integer is a perfect cube. Also, for any positive integer $m,-m^{3}$ is the cube of $-m$. (i)On factorising 5832 into prime factors, we get: $5832=2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3$ On grouping the...
Read More →Find three rational numbers between
Question: Find three rational numbers between (i) $-1$ and $-2$ (ii) $0.1$ and $0.11$ (iii) $5 / 7$ and $6 / 7$ (iv) $1 / 4$ and $1 / 5$ Thinking Process Use the concept that three rational numbers between $x$ and $y$ are $x+d, x+2 d$ and $x+3 d$, where $d=(y-x) /(n+1)$ , $xy$ and $n=3$. Solution: (i) Let $y=-1$ and $x=-2$ Here $x$ y and we have to find three rational numbers, so $n=3$. $\because \quad d=\frac{y-x}{n+1}=\frac{-1+2}{3+1}=\frac{1}{4}$ Since, the three rational numbers between $x$ ...
Read More →Prove that:
Question: Prove that: $\left|\begin{array}{ccc}1 b+c b^{2}+c^{2} \\ 1 c+a c^{2}+a^{2} \\ 1 a+b a^{2}+b^{2}\end{array}\right|=(a-b)(b-c)(c-a)$ Solution: Let LHS $=\Delta=\mid \begin{array}{lll}1 b+c b^{2}+c^{2}\end{array}$ $\begin{array}{lcc}1 c+a c^{2}+a^{2}\end{array}$ $\begin{array}{lll}1 a+b a^{2}+b^{2}\end{array}$ $\left[\right.$ Applying $\mathrm{R}_{1} \rightarrow \mathrm{R}_{1}-\mathrm{R}_{2}$ and $\left.\mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-\mathrm{R}_{3}\right]$ $=\mid \begin{array}...
Read More →If (sec θ – tan θ)
Question: If $(\sec \theta-\tan \theta)=\sqrt{2} \tan \theta$ then prove that $(\sec \theta+\tan \theta)=\sqrt{2} \sec \theta$. Solution: $\sec \theta-\tan \theta=\sqrt{2} \tan \theta$ Squaring on both sides, we get $(\sec \theta-\tan \theta)^{2}=(\sqrt{2} \tan \theta)^{2}$ $\Rightarrow \sec ^{2} \theta+\tan ^{2} \theta-2 \sec \theta \tan \theta=2 \tan ^{2} \theta$ $\Rightarrow \sec ^{2} \theta-\tan ^{2} \theta=2 \sec \theta \tan \theta$ $\Rightarrow(\sec \theta-\tan \theta)(\sec \theta+\tan \th...
Read More →Which of the following numbers are cubes of negative integers
Question: Which of the following numbers are cubes of negative integers (i) 64 (ii) 1056 (iii) 2197 (iv) 2744 (v) 42875 Solution: In order to check if a negative number is a perfect cube, first check if the corresponding positive integer is a perfect cube. Also, for any positive integer $m,-m^{3}$ is the cube of $-m$. (i) On factorising 64 into prime factors, we get: $64=2 \times 2 \times 2 \times 2 \times 2 \times 2$ On grouping the factors in triples of equal factors, we get: $64=\{2 \times 2 ...
Read More →Prove that:
Question: Prove that: $\left|\begin{array}{ccc}1 b+c b^{2}+c^{2} \\ 1 c+a c^{2}+a^{2} \\ 1 a+b a^{2}+b^{2}\end{array}\right|=(a-b)(b-c)(c-a)$ Solution: Let LHS $=\Delta=\mid \begin{array}{lll}1 b+c b^{2}+c^{2}\end{array}$ $\begin{array}{lcc}1 c+a c^{2}+a^{2}\end{array}$ $\begin{array}{lll}1 a+b a^{2}+b^{2}\end{array}$ $\left[\right.$ Applying $\mathrm{R}_{1} \rightarrow \mathrm{R}_{1}-\mathrm{R}_{2}$ and $\left.\mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-\mathrm{R}_{3}\right]$ $=\mid \begin{array}...
Read More →Solve this
Question: If $(\cos \theta-\sin \theta)=\sqrt{2} \sin \theta$ then prove that $(\cos \theta+\sin \theta)=\sqrt{2} \cos \theta$. Solution: $\cos \theta-\sin \theta=\sqrt{2} \sin \theta$ Squaring on both sides, we get $(\cos \theta-\sin \theta)^{2}=(\sqrt{2} \sin \theta)^{2}$ $\Rightarrow \cos ^{2} \theta+\sin ^{2} \theta-2 \sin \theta \cos \theta=2 \sin ^{2} \theta$ $\Rightarrow \cos ^{2} \theta-\sin ^{2} \theta=2 \sin \theta \cos \theta$ $\Rightarrow(\cos \theta-\sin \theta)(\cos \theta+\sin \th...
Read More →Solve this
Question: If $m=(\cos \theta-\sin \theta)$ and $n=(\cos \theta+\sin \theta)$, then show that $\sqrt{\frac{m}{n}}+\sqrt{\frac{n}{m}}=\frac{2}{\sqrt{1-\tan ^{2} \theta}}$. Solution: $\mathrm{LHS}=\sqrt{\frac{m}{n}}+\sqrt{\frac{n}{m}}$ $=\frac{\sqrt{m}}{\sqrt{n}}+\frac{\sqrt{n}}{\sqrt{m}}$ $=\frac{m+n}{\sqrt{m n}}$ $=\frac{(\cos \theta-\sin \theta)+(\cos \theta+\sin \theta)}{\sqrt{(\cos \theta-\sin \theta)(\cos \theta+\sin \theta)}}$ $=\frac{2 \cos \theta}{\sqrt{\cos ^{2} \theta-\sin ^{2} \theta}}$...
Read More →Prove that:
Question: Prove that: $\left|\begin{array}{ccc}a-b-c 2 a 2 a \\ 2 b b-c-a 2 b \\ 2 c 2 c c-a-b\end{array}\right|=(a+b+c)^{3}$ Solution: Let LHS $=\Delta=\mid a-b-c \quad 2 a \quad 2 a$ $\begin{array}{lll}2 b b-c-a 2 b\end{array}$ $\begin{array}{lll}2 c 2 c c-a-b \mid\end{array}$ $\Rightarrow \Delta=\mid a+b+c \quad a+b+c \quad a+b+c$ $\begin{array}{ccc}2 b b-c-a 2 b \\ 2 c 2 c c-a-b \mid\end{array}$ $\left[\right.$ Applying $\left.\mathrm{R}_{1} \rightarrow \mathrm{R}_{1}+\mathrm{R}_{2}+\mathrm{...
Read More →Which of the following numbers are cubes of negative integers
Question: Which of the following numbers are cubes of negative integers (i) 64 (ii) 1056 (iii) 2197 (iv) 2744 (v) 42875 Solution: In order to check if a negative number is a perfect cube, first check if the corresponding positive integer is a perfect cube. Also, for any positive integerm,m3is the cube ofm. (i) On factorising 64 into prime factors, we get: $64=2 \times 2 \times 2 \times 2 \times 2 \times 2$ On grouping the factors in triples of equal factors, we get: $64=\{2 \times 2 \times 2\} \...
Read More →If x = (sec A + sin A) and y = (sec A – sin A), prove that
Question: If $x=(\sec A+\sin A)$ and $y=(\sec A-\sin A)$, prove that $\left(\frac{2}{x+y}\right)^{2}+\left(\frac{x-y}{2}\right)^{2}=1$ Solution: Given:x= secA+ sinA.....(1)y= secA sinA.....(2)Adding (1) and (2), we get $x+y=\sec A+\sin A+\sec A-\sin A$ $\Rightarrow 2 \sec A=x+y$ $\Rightarrow \sec A=\frac{x+y}{2}$ $\Rightarrow \frac{1}{\sec A}=\frac{2}{x+y}$ $\Rightarrow \cos A=\frac{2}{x+y} \quad \ldots \ldots(3)$ Subtracting (2) from (1), we get $x-y=\sec A+\sin A-\sec A+\sin A$ $\Rightarrow 2 ...
Read More →Prove that:
Question: Prove that: $\left|\begin{array}{ccc}a+b+2 c a b \\ c b+c+2 a b \\ c a c+a+2 b\end{array}\right|=2(a+b+c)^{3}$ Solution: Let LHS $=\Delta=\mid a+b+2 c \quad a \quad b$ $\begin{array}{ccc}c b+c+2 a b \\ c a c+a+2 b\end{array}$ $b \quad 2 a+2 b+2 c \quad b+c+2 a \quad b 2 a+2 b+2 c \quad a \quad c+a+2 b \mid$ $\left[\right.$ Applying $\left.\mathrm{C}_{1} \rightarrow \mathrm{C}_{1}+\mathrm{C}_{2}+\mathrm{C}_{3}\right]$ $=2(\mathrm{a}+\mathrm{b}+\mathrm{c}) \mid 1 \quad a \quad b$ $1 \qua...
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