After rationalising the denominator
Question: After rationalising the denominator of $\frac{7}{3 \sqrt{3}-2 \sqrt{2}}$, we get the denominator as (a) 13 (b) 19 (c) 5 (d) 35 Solution: $\frac{7}{3 \sqrt{3}-2 \sqrt{2}}=\frac{7}{3 \sqrt{3}-2 \sqrt{2}} \cdot \frac{3 \sqrt{3}+2 \sqrt{2}}{3 \sqrt{3}+2 \sqrt{2}}$ [multiplying numerator and denominator by $3 \sqrt{3}+2 \sqrt{2}$ ] $=\frac{7(3 \sqrt{3}+2 \sqrt{2})}{(3 \sqrt{3})^{2}-(2 \sqrt{2})^{2}}$ [using identity $(a-b)(a+b)=a^{2}-b^{2}$ ] $=\frac{7(3 \sqrt{3}+2 \sqrt{2})}{27-8}$ $\left[...
Read More →Solve this
Question: $(1+\tan \theta+\cot \theta)(\sin \theta-\cos \theta)=\left(\frac{\sec \theta}{\operatorname{cosec}^{2} \theta}-\frac{\operatorname{cosec} \theta}{\sec ^{2} \theta}\right)$ Solution: $\mathrm{LHS}=(1+\tan \theta+\cot \theta)(\sin \theta-\cos \theta)$ $=\sin \theta+\tan \theta \sin \theta+\cot \theta \sin \theta-\cos \theta-\tan \theta \cos \theta-\cot \theta \cos \theta$ $=\sin \theta+\tan \theta \sin \theta+\frac{\cos \theta}{\sin \theta} \times \sin \theta-\cos \theta-\frac{\sin \the...
Read More →Prove the following
Question: $\frac{1}{\sqrt{9}-\sqrt{8}}$ is equal to (a) $\frac{1}{2}(3-2 \sqrt{2})$ (b) $\frac{1}{3+2 \sqrt{2}}$ (c) $3-2 \sqrt{2}$ (d) $3+2 \sqrt{2}$ Solution: $\frac{1}{\sqrt{9}-\sqrt{8}}=\frac{1}{3-2 \sqrt{2}}=\frac{1}{3-2 \sqrt{2}} \cdot \frac{3+2 \sqrt{2}}{3+2 \sqrt{2}}$ $[\because \sqrt{8}=\sqrt{2 \times 2 \times 2}=2 \sqrt{2}]$ [multiplying numerator and denominator by $3+2 \sqrt{2}]$ $=\frac{3+2 \sqrt{2}}{9-(2 \sqrt{2})^{2}}$ [using identity $(a-b)(a+b)=a^{2}-b^{2}$ ] $=\frac{3+2 \sqrt{2...
Read More →The number obtained on rationalising
Question: The number obtained on rationalising the denominator of $\frac{1}{\sqrt{7}-2}$ is (a) $\frac{\sqrt{7}+2}{3}$ (b) $\frac{\sqrt{7}-2}{3}$ (c) $\frac{\sqrt{7}+2}{5}$ (d) $\frac{\sqrt{7}+2}{45}$ Solution: (a) $\frac{1}{\sqrt{7}-2}=\frac{1}{\sqrt{7}-2} \cdot \frac{\sqrt{7}+2}{\sqrt{7}+2} \quad$ [multiplying numerator and denominator by $\left.\sqrt{7}+2\right]$ $=\frac{\sqrt{7}+2}{(\sqrt{7})^{2}-(2)^{2}}=\frac{\sqrt{7}+2}{7-4}=\frac{\sqrt{7}+2}{3} \quad$ [using identity $(a-b)(a+b)=a^{2}-b^...
Read More →Prove that:
Question: Prove that: $\left|\begin{array}{ccc}a b c \\ a-b b-c c-a \\ b+c c+a a+b\end{array}\right|=a^{3}+b^{3}+c^{3}-3 a b c$ Solution: $\Delta=\left|\begin{array}{ccc}a b c \\ a-b b-c c-a \\ b+c c+a a+b\end{array}\right|$ $=\left|\begin{array}{ccc}a b c \\ a-b b-c c-a \\ a+b+c c+a+b a+b+c\end{array}\right| \quad$ [Applying $R_{3} \rightarrow R_{3}+R_{2}$ ] $=\left(\begin{array}{c}a+b+c \\ \end{array}\right)\left|\begin{array}{ccc}a b c \\ a-b b-c c-a \\ 1 1 1\end{array}\right| \quad$ [Taking ...
Read More →Show that
Question: If $\Delta=\left|\begin{array}{lll}1 x x^{2} \\ 1 y y^{2} \\ 1 z z^{2}\end{array}\right|, \Delta_{1}=\left|\begin{array}{ccc}1 1 1 \\ y z z x x y \\ x y z\end{array}\right|$, then prove that $\Delta+\Delta_{1}=0$. Solution: $\Delta+\Delta_{1}=\left|\begin{array}{lll}1 x x^{2} \\ 1 y y^{2} \\ 1 z z^{2}\end{array}\right|+\left|\begin{array}{ccc}1 1 1 \\ y z z x x y \\ x y z\end{array}\right|$ $=\left|\begin{array}{lll}1 x x^{2} \\ 1 y y^{2} \\ 1 z z^{2}\end{array}\right|+\left|\begin{arr...
Read More →Evaluate the following:
Question: Evaluate the following: $\left|\begin{array}{ccc}a+x y z \\ x a+y z \\ x y a+z\end{array}\right|$ Solution: Let $\Delta=\left|\begin{array}{ccc}a+x y z \\ x a+y z \\ x y a+z\end{array}\right|$ $\Delta=\left|\begin{array}{ccc}a+x y z \\ x a+y z \\ x y a+z\end{array}\right|$ $=\left|\begin{array}{ccc}a+x+y+z y z \\ a+x+y+z a+y z \\ a+x+y+z y a+z\end{array}\right|$ [Applying $C_{1} \rightarrow C_{1}+C_{2}+C_{3}$ ] $=(a+x+y+z)\left|\begin{array}{ccc}1 y z \\ 1 a+y z \\ 1 y a+z\end{array}\r...
Read More →Prove the following
Question: $\sqrt{10}, \sqrt{15}$ is equal to (a) $6 \sqrt{5}$ (b) $5 \sqrt{6}$ (c) $7 \sqrt{5}$ (d) $10 \sqrt{5}$ Solution: (b) $\sqrt{10}, \sqrt{15}=\sqrt{2} .5 \sqrt{3} .5=\sqrt{2} \sqrt{5} \sqrt{3} \sqrt{5}==5 \sqrt{6}$...
Read More →The value of
Question: The value of $2 \sqrt{3}+\sqrt{3}$ is (a) $2 \sqrt{6}$ (b) 6 (c) $3 \sqrt{3}$ (d) $4 \sqrt{6}$ Solution: $2 \sqrt{3}+\sqrt{3}=\sqrt{3}(2+1)=3 \sqrt{3}$...
Read More →The value of 1.999... in the form of
Question: The value of $1.999$... in the form of $\mathrm{p} / \mathrm{q}$, where $p$ and $q$ are integers and (a) $\frac{19}{10}$ (b) $\frac{1999}{1000}$ (c) 2 (d) $\frac{1}{9}$ Solution: (c) Let $x=1.999 . . .$ Now, $\quad 10 x=19.999 \ldots$ On subtracting Eq. (i) from Eq. (ii), we get $10 x-x=(19.999 \ldots)-(1.9999 \ldots)$ $\Rightarrow \quad 9 x=18$ $\therefore$ $x=\frac{18}{9}=2$...
Read More →A rational number between
Question: A rational number between $\sqrt{2}$ and $\sqrt{3}$ is (a) $\frac{\sqrt{2}+\sqrt{3}}{2}$ (b) $\frac{\sqrt{2} \cdot \sqrt{3}}{2}$ (c) $1.5$ (d) $1.8$ Solution: (c) A rational number between $(\sqrt{2}$ and $\sqrt{3})$ i.e., $1.414$ and $1.732$. (a) $\frac{\sqrt{2}+\sqrt{3}}{2}$, which is an irrational number, so it is not a solution. (b) $\frac{\sqrt{2} \cdot \sqrt{3}}{2}=\frac{\sqrt{6}}{2}$, which is an irrational number, so it is not a solution. Now, $1.5$ and $1.8$ both are the ratio...
Read More →Which of the following is irrational?
Question: Which of the following is irrational? (a) $0.14$ (b) $0.14 \overline{16}$ (c) $0 . \overline{1416}$ (d) $0.4014001400014$ Solution: (d) An irrational number is non-terminating non-recurring which is $0.4014001400014 \ldots .$ Here, $0.14$ is terminating and $0.14 \overline{6}, . \overline{1416}$ are non-terminating recurring....
Read More →Which of the following is irrational?
Question: Which of the following is irrational? (a) $\sqrt{\frac{4}{9}}$ (b) $\frac{\sqrt{12}}{\sqrt{3}}$ (c) $\sqrt{7}$ (d) $\sqrt{81}$ Solution: (c) $\sqrt{\frac{4}{9}}=\frac{2}{3}$ (rational) $\frac{\sqrt{12}}{\sqrt{3}}=\frac{2 \sqrt{3}}{\sqrt{3}}=2$ (rational) $\sqrt{81}=9$ (rational) but $\sqrt{7}$ is an irrational number. Hence, $\sqrt{7}$ is an irrational number....
Read More →The decimal expansion of the number
Question: The decimal expansion of the number $\sqrt{2}$ is (a) a finite decimal (b) $1.41421$ (c) non-terminating recurring (d) non-terminating non-recurring Solution: The decimal expansion of the number $\sqrt{2}$ is non-terminating non-recurring. Because $\sqrt{2}$ is an irrational number. Also, we know that an irrational number is non-terminating non-recurring....
Read More →The product of any two irrational numbers is
Question: The product of any two irrational numbers is (a) always an irrational number (b) always a rational number (c) always an integer (d) sometimes rational, sometimes irrational Solution: (d) We know that, the product of any two irrational numbers is sometimes rational and sometimes irrational. e.g., $\sqrt{2} \times \sqrt{2}=2$ (rational) and $\sqrt{2} \times \sqrt{3}=\sqrt{6}$ (irrational)...
Read More →Decimal representation of a rational number cannot be
Question: Decimal representation of a rational number cannot be (a) terminating (b) non-terminating (c) non-terminating repeating (d) non-terminating non-repeating Solution: (d) Decimal representation of a rational number cannot be non-terminating non-repeating because the decimal expansion of rational number is either terminating or non-terminating recurring (repeating)....
Read More →Between two rational numbers
Question: Between two rational numbers (a) there is no rational number (b) there is exactly one rational number (c) there are infinitely many rational numbers (d) there are only rational numbers and no irrational numbers Solution: (c) Between two rational numbers, there are infinitely many rational numbers. e.g., $\frac{3}{5}$ and $\frac{4}{5}$ are two rational numbers, then $\frac{31}{50}, \frac{32}{50}, \frac{33}{50}, \frac{34}{50}, \frac{35}{50}, \ldots$ are infinite rational numbers between ...
Read More →Every rational number is
Question: Every rational number is (a) a natural number (b) an integer (c) a real number (d) a whole number Solution: (c) Since, real numbers are the combination of rational and irrational numbers. Hence, every rational number is a real number....
Read More →Find six rational numbers between 3 and 4 .
Question: Find six rational numbers between 3 and 4 . Solution: There can be infinitely many rationals between 3 and 4 , one way is to take them $3=\frac{21}{7}$ and $4=\frac{28}{7} .$ $(\because 6+1=7)$ First rational number between 3 and 4 $q_{1}=\left(\right.$ rational number between $\frac{21}{7}$ and $\left.\frac{28}{7}\right)=\frac{\frac{21}{7}+\frac{28}{7}}{2}=\frac{\frac{49}{7}}{2}=\frac{7}{2}$ $\therefore \quad \frac{21}{7}\frac{7}{2}\frac{28}{7}$ Second rational number between 3 and 4 ...
Read More →Is zero a rational number?
Question: Is zero a rational number? Can you write it in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$ ? Solution: Yes, write $\frac{0}{1}$ (where 0 and 1 are integers and $q=1$ which is not equal to zero)....
Read More →Evaluate the following:
Question: Evaluate the following: $\left|\begin{array}{ccc}0 x y^{2} x z^{2} \\ x^{2} y 0 y z^{2} \\ x^{2} z z y^{2} 0\end{array}\right|$ Solution: Let $\Delta=\left|\begin{array}{ccc}0 x y^{2} x z^{2} \\ x^{2} y 0 y z^{2} \\ x^{2} z z y^{2} 0\end{array}\right|$ $\Delta=\left|\begin{array}{ccc}0 x y^{2} x z^{2} \\ x^{2} y 0 y z^{2} \\ x^{2} z z y^{2} 0\end{array}\right|$ $=x^{2} y^{2} z^{2}\left|\begin{array}{ccc}0 x x \\ y 0 y \\ z z 0\end{array}\right| \quad\left[\right.$ Taking $x^{2}$ common...
Read More →Evaluate the following:
Question: Evaluate the following: $\left|\begin{array}{lll}x 1 1 \\ 1 x 1 \\ 1 1 x\end{array}\right|$ Solution: Let $\Delta=\left|\begin{array}{lll}x 1 1 \\ 1 x 1 \\ 1 1 x\end{array}\right|$ $\Delta=\left|\begin{array}{lll}x 1 1 \\ 1 x 1 \\ 1 1 x\end{array}\right|$ $=\left|\begin{array}{ccc}x-1 1-x 0 \\ 1 x 1 \\ 0 1-x x-1\end{array}\right| \quad\left[\right.$ Applying $R_{1} \rightarrow R_{1}-R_{2}$ and $\left.R_{3} \rightarrow R_{3}-R_{2}\right]$ $=(x-1)^{2}\left|\begin{array}{ccc}1 -1 0 \\ 1 x...
Read More →Evaluate:
Question: Evaluate: $\left|\begin{array}{lll}a b c \\ c a b \\ b c a\end{array}\right|$ Solution: $\Delta=\left|\begin{array}{lll}a b c \\ c a b \\ b c a\end{array}\right|$ $=a\left(a^{2}-b c\right)-b\left(c a-b^{2}\right)+c\left(c^{2}-b a\right)$ $=a^{3}-a b c-b c a+b^{3}+c^{3}-a b c$ $=a^{3}+b^{3}+c^{3}-3 a b c=(a+b+c)\left(a^{2}+b^{2}+c^{2}-a b-b c-c a\right)$...
Read More →Evaluate :
Question: Evaluate : $\left|\begin{array}{ccc}x+\lambda x x \\ x x+\lambda x \\ x x x+\lambda\end{array}\right|$ Solution: $\Delta=\left|\begin{array}{ccc}x+\lambda x x \\ x x+\lambda x \\ x x x+\lambda\end{array}\right|$ $=\left|\begin{array}{rrr}\lambda 0 x \\ -\lambda \lambda x \\ 0 -\lambda x+\lambda\end{array}\right| \quad\left[\right.$ Applying $\left.C_{1} \rightarrow C_{1}-C_{2}, C_{2} \rightarrow C_{2}-C_{3}\right]$ $=\left|\begin{array}{ccc}\lambda 0 x \\ -\lambda 0 2 x+\lambda \\ 0 -\...
Read More →(a) A cube of side 5cm is immersed in
Question: (a) A cube of side $5 \mathrm{~cm}$ is immersed in water and then in saturated salt solution. In which case will it experience a greater buoyant force. If each side of the cube is reduced to $4 \mathrm{~cm}$ and then immersed in water, what will be the effect on the buoyant force experienced by the cubeas compared to the first case for water. Give reason for each case. (CBSE 2012) (b) A ball weighing $4 \mathrm{~kg}$ of density $4000 \mathrm{~kg} \mathrm{~m}^{-3}$ is completely immerse...
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