(cosec θ − cot θ)2 = ?
Question: (cosec cot )2= ? (a) $\frac{1+\cos \theta}{1-\cos \theta}$ (b) $\frac{1-\cos \theta}{1+\cos \theta}$ (c) $\frac{1+\sin \theta}{1-\sin \theta}$ (d) $\frac{1-\sin \theta}{1+\sin \theta}$ Solution: (b) $\frac{1-\cos \theta}{1+\cos \theta}$ $(\operatorname{cosec} \theta-\cot \theta)^{2}$ $=\left(\frac{1}{\sin \theta}-\frac{\cos \theta}{\sin \theta}\right)^{2}$ $=\left(\frac{1-\cos \theta}{\sin \theta}\right)^{2}$ $=\frac{(1-\cos \theta)^{2}}{\sin ^{2} \theta}$ $=\frac{(1-\cos \theta)^{2}}{...
Read More →Rationalise the denominator of the following
Question: Rationalise the denominator of the following (i) $\frac{2}{3 \sqrt{3}}$ (ii) $\frac{\sqrt{40}}{\sqrt{3}}$ (iii) $\frac{3+\sqrt{2}}{4 \sqrt{2}}$ (iv) $\frac{16}{\sqrt{41}-5}$ (v) $\frac{2+\sqrt{3}}{2-\sqrt{3}}$ (vi) $\frac{\sqrt{6}}{\sqrt{2}+\sqrt{3}}$ (vii) $\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}$ (viii) $\frac{3 \sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}$ (ix) $\frac{4 \sqrt{3}+5 \sqrt{2}}{\sqrt{48}+\sqrt{18}}$ Solution: (i) Let $E=\frac{2}{3 \sqrt{3}}$ For rationalising the denominat...
Read More →The volume of a cube is 9261000
Question: The volume of a cube is 9261000 m3. Find the side of the cube. Solution: Volume of a cube is given by: $V=s^{3}$, where $s=$ Side of the cube It is given that the volume of the cube is 9261000 m3; therefore, we have: $s^{3}=9261000$ Let us find the cube root of 9261000 using prime factorisation: $9261000=2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 5 \times 5 \times 5 \times 7 \times 7 \times 7=\{2 \times 2 \times 2\} \times\{3 \times 3 \times 3\} \times\{5 \times 5 \times 5\}...
Read More →Solve this
Question: $\sqrt{\frac{1+\cos A}{1-\cos A}}=?$ (a) cosecA cotA(b) cosecA+ cotA(c) cosecAcotA(d) none of these Solution: $\sqrt{\frac{1+\cos (A)}{1-\cos (A)}}=\sqrt{\frac{1+\cos (A)}{1-\cos (A)} \times \frac{1+\cos (A)}{1+\cos (A)}}=\frac{1+\cos (A)}{\sqrt{1-\cos ^{2}(A)}}$ $=\frac{1+\cos (A)}{\sqrt{\sin ^{2}(A)}}=\frac{1+\cos (A)}{\sin (A)}=\frac{1}{\sin (A)}+\frac{\cos (A)}{\sin (A)}$ $=\operatorname{cosec}(A)+\cot (A)$ Hence, the correct answer is option B....
Read More →Three numbers are in the ratio 1 : 2 : 3. The sum of their cubes is 98784.
Question: Three numbers are in the ratio 1 : 2 : 3. The sum of their cubes is 98784. Find the numbers. Solution: Let the numbers bex, 2xand 3x. Therefore $x^{3}+(2 x)^{3}+(3 x)^{3}=98784$ $\Rightarrow x^{3}+8 x^{3}+27^{3}=98784$ $\Rightarrow 36 x^{3}=98784$ $\Rightarrow x^{3}=\frac{98784}{36}=2744$ $\Rightarrow x^{3}=2744$ $\Rightarrow x=\sqrt[3]{2744}=\sqrt[3]{\{2 \times 2 \times 2\} \times\{7 \times 7 \times 7\}}=2 \times 7=14$ Hence, the numbers are $14,(2 \times 14=28)$ and $(3 \times 14=42)...
Read More →Solve this
Question: $\sqrt{\frac{1-\sin A}{1+\sin A}}=?$ (a) (secA+ tanA)(b) (secA tanA)(c) secAtanA(d) None to these Solution: (b) (secA tanA) $\sqrt{\frac{1-\sin A}{1+\sin A}}$ $=\sqrt{\frac{(1-\sin A)}{(1+\sin A)} \times \frac{(1-\sin A)}{(1-\sin A)}} \quad$ [Multiplying the denominator and numerator by $\left.(1-\sin A)\right]$ $=\frac{(1-\sin A)}{\sqrt{1-\sin ^{2} A}}$ $=\frac{(1+\sin A)}{\sqrt{\cos ^{2} A}}$ $=\frac{(1-\sin A)}{\cos A}$ $=\frac{1}{\cos A}-\frac{\sin A}{\cos A}$ $=\sec A-\tan A$...
Read More →What is the smallest number by which 8192 must be divided so that quotient is a perfect cube?
Question: What is the smallest number by which 8192 must be divided so that quotient is a perfect cube? Also, find the cube root of the quotient so obtained. Solution: On factorising 8192 into prime factors, we get: $8192=2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$ On grouping the factors in triples of equal factors, we get: $8192=\{2 \times 2 \times 2\} \times\{2 \times 2 \times 2\} \times\{2 \times 2 \times 2\} \times\{2 \times...
Read More →Prove that.
Question: Prove that. $\left|\begin{array}{llll}(a+1) (a+2) a+2 1 \\ (a+2) (a+3) a+3 1 \\ (a+3) (a+4) a+4 1\end{array}\right|=-2$ Solution: Let LHS $=\Delta=\mid(a+1)(a+2) \quad a+2 \quad 1$ $(a+2)(a+3) \quad a+3 \quad 1$ $(a+3)(a+4) \quad a+4 \quad 1 \mid$ $\mid(a+1)(a+2)-(a+2)(a+3) \quad(a+2)-(a+3) \quad 0(a+2)(a+3)-(a+3)(a+4) \quad(a+3)-(a+4) \quad 0(a+3)(a+4)$ $(a+4)$ $1 \mid$ [Applying $\mathrm{C}_{1} \rightarrow \mathrm{C}_{1}-\mathrm{C}_{2}$ and $\mathrm{C}_{2} \rightarrow \mathrm{C}_{2}-...
Read More →The value of
Question: The value of $\frac{\tan ^{2} \theta-\sec ^{2} \theta}{\cot ^{2} \theta-\operatorname{cosec}^{2} \theta}$ is (a) 1 (b) $\frac{1}{2}$ (c) $\frac{-1}{2}$ (d) 1 Solution: $\frac{\tan ^{2} \theta-\sec ^{2} \theta}{\cot ^{2} \theta-\operatorname{cosec}^{2} \theta}$ $=\frac{-1}{-1} \quad\left(1+\tan ^{2} \theta=\sec ^{2} \theta\right.$ and $\left.1+\cot ^{2} \theta=\operatorname{cosec}^{2} \theta\right)$ $=1$ Hence, the correct answer is option (d)....
Read More →Multiply 210125 by the smallest number so that the product is a perfect cube.
Question: Multiply 210125 by the smallest number so that the product is a perfect cube. Also, find out the cube root of the product. Solution: On factorising 210125 into prime factors, we get: $210125=5 \times 5 \times 5 \times 41 \times 41$ On grouping the factors in triples of equal factors, we get: $210125=\{5 \times 5 \times 5\} \times 41 \times 41$ It is evident that the prime factors of 210125 cannot be grouped into triples of equal factors such that no factor is left over. Therefore, 2101...
Read More →Find the smallest number which when multiplied with 3600 will make the product a perfect cube.
Question: Find the smallest number which when multiplied with 3600 will make the product a perfect cube. Further, find the cube root of the product. Solution: On factorising 3600 into prime factors, we get: $3600=2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 5 \times 5$ On grouping the factors in triples of equal factors, we get: $3600=\{2 \times 2 \times 2\} \times 2 \times 3 \times 3 \times 5 \times 5$ It is evident that the prime factors of 3600 cannot be grouped into triples of equal...
Read More →Simplify the following
Question: Simplify the following (i) $\sqrt{45}-\sqrt[3]{20}+4 \sqrt{5}$ (ii) $\frac{\sqrt{24}}{8}+\frac{\sqrt{54}}{9}$ (iii) $\sqrt[4]{12} \times \sqrt[7]{6}$ (iv) $4 \sqrt{28}+3 \sqrt{7} \div \sqrt[3]{7}$ (v) $3 \sqrt{3}+2 \sqrt{27}+\frac{7}{\sqrt{3}}$ (vi) $(\sqrt{3}-\sqrt{2})^{2}$ (vii) $\sqrt[4]{81}-8 \sqrt[3]{216}+15 \sqrt[5]{32}+\sqrt{225}$ (viii) $\frac{3}{\sqrt{8}}+\frac{1}{\sqrt{2}}$ (ix) $\frac{2 \sqrt{3}}{3}-\frac{\sqrt{3}}{6}$ Solution: (i) $\sqrt{45}-3 \sqrt{20}+4 \sqrt{5}=\sqrt{3 ...
Read More →Prove that:
Question: Prove that: $\left|\begin{array}{lll}(b+c)^{2} a^{2} b c \\ (c+a)^{2} b^{2} c a \\ (a+b)^{2} c^{2} a b\end{array}\right|=(a-b)(b-c)(c-a)(a+b+c)\left(a^{2}+b^{2}+c^{2}\right)$ Solution: Let LHS $=\Delta=\mid(b+c)^{2} \quad a^{2} \quad b c$ $\begin{array}{lll}(c+a)^{2} b^{2} c a \\ (a+b)^{2} c^{2} a b \mid\end{array}$ $=\mid(b+c)^{2}-(c+a)^{2} \quad a^{2}-b^{2} \quad b c-c a$ $\begin{array}{lll}(\mathrm{c}+\mathrm{a})^{2}-(\mathrm{a}+\mathrm{b})^{2} \mathrm{~b}^{2}-\mathrm{c}^{2} \mathrm...
Read More →Find the cube root of each of the following natural numbers:
Question: Find the cube root of each of the following natural numbers: (i) 343 (ii) 2744 (iii) 4913 (iv) 1728 (v) 35937 (vi) 17576 (vii) 134217728 (viii) 48228544 (ix) 74088000 (x) 157464 (xi) 1157625 (xii) 33698267 Solution: (i)Cube root using units digit: Let us consider 343. The unit digit is 3; therefore, the unit digit in the cube root of 343 is 7. There is no number left after striking out the units, tens and hundreds digits of the given number; therefore, the cube root of 343 is 7. Hence,...
Read More →Show that 0.142857142857...... =1 / 7 .
Question: Show that $0.142857142857 \ldots=1 / 7 .$ Solution: Let x = 0.142857142857 ..(i) On multiplying both sides of Eq. (i) by 1000000, we get 1000000 x = 142857.142857(ii) On subtracting Eq. (i) from Eq. (ii), we get 1000000 x x = (142857.142857) (0.142857..) = 999999 x = 142857 x = 142857/999999 = 1/7 Hence proved....
Read More →If cos 9α = sin α and 9α < 90° then the value of tan 5α is
Question: If cos 9 = sin and 9 90 then the value of tan 5 is (a) $\frac{1}{\sqrt{3}}$ (b) $\sqrt{3}$ (c) 1(d) 0 Solution: $\cos 9 \alpha=\sin \alpha$ $\Rightarrow \cos 9 \alpha=\cos \left(90^{\circ}-\alpha\right)$ $\Rightarrow 9 \alpha=90^{\circ}-\alpha$ $\Rightarrow 10 \alpha=90^{\circ}$ $\Rightarrow 5 \alpha=\frac{90^{\circ}}{2}=45^{\circ}$ $\therefore \tan 5 \alpha=\tan 45^{\circ}=1$ Thus, the value of tan5is 1.Hence, the correct answer is option (c)....
Read More →Express the following in the form
Question: Express the following in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. (i) $0.2$ (ii) $0.888 \ldots$ (iii) $5 . \overline{2}$ (iv) $0 . \overline{001}$ (v) $0.2555 \ldots$ (vi) $0.1 \overline{34}$ (vii) $0.00323232 \ldots \quad$ (viii) $0.404040 \ldots$ Solution: (i) Let $x=0.2=\frac{2}{10}=\frac{1}{5}$ (ii) Let $x=0.888 \ldots \ldots \ldots$ $\ldots$ (i) On multiplying both sides of Eq.(i) by 10 , we get $10 x=8.888$ ... (ii) On subtracting Eq. (i) from Eq. (i...
Read More →If sin θ – cos θ = 0 then the value of (sin4θ + cos4θ) is
Question: If sin cos = 0 then the value of (sin4 + cos4) is (a) $\frac{1}{4}$ (b) $\frac{1}{2}$ (c) $\frac{3}{4}$ (d) 1 Solution: $\sin \theta-\cos \theta=0$ $\Rightarrow \sin \theta=\cos \theta$ $\Rightarrow \frac{\sin \theta}{\cos \theta}=1$ $\Rightarrow \tan \theta=\tan 45^{\circ}$ $\Rightarrow \theta=45^{\circ}$ $\therefore \sin ^{4} \theta+\cos ^{4} \theta$ $=\sin ^{4} 45^{\circ}+\cos ^{4} 45^{\circ}$ $=\left(\frac{1}{\sqrt{2}}\right)^{4}+\left(\frac{1}{\sqrt{2}}\right)^{4}$ $=\frac{1}{4}+\...
Read More →Prove that:
Question: Prove that: $\left|\begin{array}{ccc}z x y \\ z^{2} x^{2} y^{2} \\ z^{4} x^{4} y^{4}\end{array}\right|=\left|\begin{array}{ccc}x y z \\ x^{2} y^{2} z^{2} \\ x^{4} y^{4} z^{4}\end{array}\right|=\left|\begin{array}{ccc}x^{2} y^{2} z^{2} \\ x^{4} y^{4} z^{4} \\ x y z\end{array}\right|=x y z(x-y)(y-z)(z-x)(x+y+z)$ Solution: Let $\Delta_{1}=\mid z \quad x \quad y$ $z^{2} \quad x^{2} \quad y^{2}$ $z^{4} \quad x^{4} \quad y^{4}\left|, \Delta_{2}=\right| x \quad y \quad z$ $x^{2} \quad y^{2} \...
Read More →Find the smallest number that must be subtracted from those of the numbers in question 2 which are not perfect cubes,
Question: Find the smallest number that must be subtracted from those of the numbers in question 2 which are not perfect cubes, to make them perfect cubes. What are the corresponding cube roots? Solution: (i)We have: $\because$ The next number to be subtracted is 91 , which is greater than 5 . $\therefore 130$ is not a perfect cube. However, if we subtract 5 from 130, we will get 0 on performing successive subtraction and the number will become a perfect cube. If we subtract 5 from 130, we get 1...
Read More →(1 + tan θ + sec θ) (1 + cot θ – cosec θ) = ?
Question: (1 + tan + sec ) (1 + cot cosec ) = ?(a) 1(b) 0(c) 1(d) 2 Solution: $(1+\tan \theta+\sec \theta)(1+\cot \theta-\operatorname{cosec} \theta)$ $=1+\cot \theta-\operatorname{cosec} \theta+\tan \theta+\tan \theta \cot \theta-\tan \theta \operatorname{cosec} \theta+\sec \theta+\sec \theta \cot \theta-\sec \theta \operatorname{cosec} \theta$ $=1+\frac{\cos \theta}{\sin \theta}-\frac{1}{\sin \theta}+\frac{\sin \theta}{\cos \theta}+\tan \theta \times \frac{1}{\tan \theta}-\frac{\sin \theta}{\c...
Read More →(sec A + tan A) (1 − sin A) = ?
Question: (secA+ tanA) (1 sinA) = ?(a) sinA(b) cosA(c) secA(d) cosecA Solution: (b) cosA $(\sec A+\tan A)(1-\sin A)$ $=\left(\frac{1}{\cos A}+\frac{\sin A}{\cos A}\right)(1-\sin A)$ $=\left(\frac{1+\sin A}{\cos A}\right)(1-\sin A)$ $=\left(\frac{1-\sin ^{2} A}{\cos A}\right)$ $=\left(\frac{\cos ^{2} A}{\cos A}\right)$ $=\cos A$...
Read More →Prove that:
Question: Prove that: $\left|\begin{array}{ccc}z x y \\ z^{2} x^{2} y^{2} \\ z^{4} x^{4} y^{4}\end{array}\right|=\left|\begin{array}{ccc}x y z \\ x^{2} y^{2} z^{2} \\ x^{4} y^{4} z^{4}\end{array}\right|=\left|\begin{array}{ccc}x^{2} y^{2} z^{2} \\ x^{4} y^{4} z^{4} \\ x y z\end{array}\right|=x y z(x-y)(y-z)(z-x)(x+y+z)$ Solution: Let $\Delta_{1}=\mid z \quad x \quad y$ $z^{2} \quad x^{2} \quad y^{2}$ $z^{4} \quad x^{4} \quad y^{4}\left|, \Delta_{2}=\right| x \quad y \quad z$ $x^{2} \quad y^{2} \...
Read More →Solve this
Question: $\frac{1+\tan ^{2} A}{1+\cot ^{2} A}=?$ (a) 1(b) sec2A(c) tan2A(d) cot2A Solution: $\frac{1+\tan ^{2} A}{1+\cot ^{2} A}$ $=\frac{\sec ^{2} A}{\operatorname{cosec}^{2} A}$ $=\frac{\sin ^{2} A}{\cos ^{2} A} \quad\left(\sec \theta=\frac{1}{\cos \theta}\right.$ and $\left.\operatorname{cosec} \theta=\frac{1}{\sin \theta}\right)$ $=\tan ^{2} A$ Hence, the correct answer is option (c)....
Read More →Using the method of successive subtraction examine whether or not the following numbers are perfect cubes:
Question: Using the method of successive subtraction examine whether or not the following numbers are perfect cubes: (i) 130 (ii) 345 (iii) 792 (iv) 1331 Solution: (i)We have: $\because$ The next number to be subtracted is 91 , which is greater than 5 . $\therefore 130$ is not a perfect cube. (ii)We have: $\because$ The next number to be subtracted is 161 , which is greater than 2 . $\therefore 345$ is not a perfect cube. (iii)We have: $\because$ The next number to be subtracted is 271 , which i...
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