Which of the following is an irrational number?
Question: Which of the following is an irrational number?(a) 3.14(b) 3.141414...(c) 3.14444...(d) 3.141141114... Solution: (d) 3.141141114...Because 3.141141114... is neither a repeating decimal nor a terminating decimal, it is an irrational number....
Read More →If cot θ=13√, find the value of 1−cos2 θ2−sin2 θ.
Question: If $\cot \theta=\frac{1}{\sqrt{3}}$, find the value of $\frac{1-\cos ^{2} \theta}{2-\sin ^{2} \theta}$. Solution: Given: $\cot \theta=\frac{1}{\sqrt{3}}$ We have to find the value of the expression $\frac{1-\cos ^{2} \theta}{2-\sin ^{2} \theta}$ We know that, $1+\cot ^{2} \theta=\operatorname{cosec}^{2} \theta$ $\Rightarrow \operatorname{cosec}^{2} \theta=1+\left(\frac{1}{\sqrt{3}}\right)^{2}$ $\Rightarrow \operatorname{cosec}^{2} \theta=\frac{4}{3}$ Using the identity $\sin ^{2} \thet...
Read More →An urn contains 25 balls of which 10 balls bear a mark ‘X’ and the remaining 15 bear a mark ‘Y’.
Question: An urn contains 25 balls of which 10 balls bear a mark X and the remaining 15 bear a mark Y. A ball is drawn at random from the urn, its mark is noted down and it is replaced. If 6 balls are drawn in this way, find the probability that (i) all will bear X mark. (ii) not more than 2 will bear Y mark. (iii) at least one ball will bear Y mark (iv) the number of balls with X mark and Y mark will be equal. Solution: Total number of balls in the urn = 25 Balls bearing mark X = 10 Balls beari...
Read More →The decimal expansion that a rational number cannot have is
Question: The decimal expansion that a rational number cannot have is (a) $0.25$ (b) $0.25 \overline{28}$ (c) $0 . \overline{2528}$ (d) $0.5030030003$ Solution: As, any number which have a terminating or non-terminating recurring decimal expansion is a rational number.So, 0.5030030003... which is non-termintaing non-recurring decimal expansion is not a rational number.Hence, the correct option is (d)....
Read More →The decimal representation of an irrational number is
Question: The decimal representation of an irrational number is(a) always terminating(b) either terminating or repeating(c) either terminating or non-repeating(d) neither terminating nor repeating Solution: (d) neither terminating nor repeatingAs per the definition of irrational numbers, these are neither terminating nor repeating decimals....
Read More →If tan θ=125, find the value of 1+sin θ1−sin θ.
Question: If $\tan \theta=\frac{12}{5}$, find the value of $\frac{1+\sin \theta}{1-\sin \theta}$. Solution: Given: $\tan \theta=\frac{12}{5}$ We have to find the value of the expression $\frac{1+\sin \theta}{1-\sin \theta}$. $A C=\sqrt{A B^{2}+B C^{2}}$ $=\sqrt{12^{2}+5^{2}}$ $=13$ $\Rightarrow \sin \theta=\frac{12}{13}$ Therefore, $\frac{1+\sin \theta}{1-\sin \theta}=\frac{1+\frac{12}{13}}{1-\frac{12}{13}}$ $=25$ Hence, the value of the given expression is 25....
Read More →The decimal representation of a rational number is
Question: The decimal representation of a rational number is(a) always terminating(b) either terminating or repeating(c) either terminating or non-repeating(d) neither terminating nor repeating Solution: (b) either terminating or repeatingAs per the definition of rational numbers, they are either repeating or terminating decimals....
Read More →If tan θ=34, find the value of 1−cos θ1+cos θ.
Question: If $\tan \theta=\frac{3}{4}$, find the value of $\frac{1-\cos \theta}{1+\cos \theta}$. Solution: Given: $\tan \theta=\frac{3}{4}$ We have to find the value of the expression $\frac{1-\cos \theta}{1+\cos \theta}$. From the above figure, we have $A C=\sqrt{A B^{2}+B C^{2}}$ $=\sqrt{3^{2}+4^{2}}$ $=5$ $\Rightarrow \cos \theta=\frac{4}{5}$ Therefore, $\frac{1-\cos \theta}{1+\cos \theta}=\frac{1-\frac{4}{5}}{1+\frac{4}{5}}$ $=\frac{1}{9}$ Hence, the value of the given expression is $\frac{1...
Read More →Between any two rational numbers there
Question: Between any two rational numbers there(a) is no rational number(b) is exactly one rational numbers(c) are infinitely many rational numbers(d) is no irrational number Solution: (c) are infinitely many rational numbersBecause the range between any two rational numbers can be easily divided into any number of divisions, there can be an infinite number of rational numbers between any two rational numbers....
Read More →Every rational number is
Question: Every rational number is(a) a natural number(b) a whole number(c) an integer(d) a real number Solution: (d) a real numberEvery rational number is a real number, as every rational number can be easily expressed on the real number line...
Read More →If tan θ=12√, find the value of cosec2 θ−sec2 θcosec2 θ+cot2 θ.
Question: If $\tan \theta=\frac{1}{\sqrt{2}}$, find the value of $\frac{\operatorname{cosec}^{2} \theta-\sec ^{2} \theta}{\operatorname{cosec}^{2} \theta+\cot ^{2} \theta}$. Solution: Given: $\tan \theta=\frac{1}{\sqrt{2}}$ We have to find the value of the expression $\frac{\operatorname{cosec}^{2} \theta-\sec ^{2} \theta}{\operatorname{cosec}^{2} \theta+\cot ^{2} \theta}$ We know that, $1+\cot ^{2} \theta=\operatorname{cosec}^{2} \theta$ $\Rightarrow \operatorname{cosec}^{2} \theta-\cot ^{2} \t...
Read More →Which of the following is a rational number?
Question: Which of the following is a rational number? (a) $\sqrt{2}$ (b) $\sqrt{23}$ (c) $\sqrt{225}$ (d) $0.1010010001$ Solution: (c) $\sqrt{225}$ Because 225 is a square of 15 , i.e., $\sqrt{225}=15$, and it can be expressed in the $\frac{p}{q}$ form, it is a rational number....
Read More →Suppose that 90% of people are right-handed.
Question: Suppose that 90% of people are right-handed. What is the probability that at most 6 of a random sample of 10 people are right-handed? Solution: A person can be either right-handed or left-handed. It is given that 90% of the people are right-handed. $\therefore p=\mathrm{P}($ right-handed $)=\frac{9}{10}$ $q=\mathrm{P}($ left-handed $)=1-\frac{9}{10}=\frac{1}{10}$ Using binomial distribution, the probability that more than 6 people are right-handed is given by, $\sum_{r=7}^{10}{ }^{10} ...
Read More →Every point on a number line represents
Question: Every point on a number line represents(a) a rational number(b) a natural number(c) an irrational number(d) a unique number Solution: As, all rational numbers, all natural numbers and all irrational numbers can be represented on a nuumber line in an unique way.So, every point on a number line represents a unique number.Hence, the correct option is (d)....
Read More →If sin θ=12√, find all other trigonometric ratios of angle θ.
Question: If $\sin \theta=\frac{1}{\sqrt{2}}$, find all other trigonometric ratios of angle $\theta$ Solution: Given: $\sin \theta=\frac{1}{\sqrt{2}}$ We have to find all the trigonometric ratios. We have the following right angle triangle. Base $=\sqrt{\text { Hypotenuse }^{2}-\text { Perpendicular }^{2}}$ $\Rightarrow B C=\sqrt{A C^{2}-A B^{2}}$ $\Rightarrow B C=\sqrt{(\sqrt{2})^{2}-1^{2}}$ $\Rightarrow B C=1$ $\cos \theta=\frac{B C}{A C}=\frac{1}{\sqrt{2}}$ $\operatorname{cosec} \theta=\frac{...
Read More →If secx cos5x + 1 = 0,
Question: If $\sec x \cos 5 x+1=0$, where $0x \leq \frac{\pi}{2}$, find the value of $x$. Solution: The given equation is secxcos5x+ 1 = 0. Now, $\sec x \cos 5 x+1=0$ $\Rightarrow \frac{\cos 5 x}{\cos x}+1=0$ $\Rightarrow \cos 5 x+\cos x=0$ $\Rightarrow 2 \cos 3 x \cos 2 x=0$ $\Rightarrow \cos 3 x=0$ or $\cos 2 x=0$ $\Rightarrow 3 x=(2 n+1) \frac{\pi}{2}, n \in \mathbf{Z}$ or $2 x=(2 m+1) \frac{\pi}{2}, m \in \mathbf{Z}$ $\Rightarrow x=(2 n+1) \frac{\pi}{6}$ or $x=(2 m+1) \frac{\pi}{4}$ Puttingn...
Read More →Suppose that 5% of men and 0.25% of women have grey hair.
Question: Suppose that 5% of men and 0.25% of women have grey hair. A haired person is selected at random. What is the probability of this person being male? Assume that there are equal number of males and females. Solution: It is given that 5% of men and 0.25% of women have grey hair. Therefore, percentage of people with grey hair = (5 + 0.25) % = 5.25% $\therefore$ Probability that the selected haired person is a male $=\frac{5}{5.25}=\frac{20}{21}$...
Read More →Two rational numbers between
Question: Two rational numbers between $\frac{2}{3}$ and $\frac{5}{3}$ are (a) $\frac{1}{6}$ and $\frac{2}{6}$ (b) $\frac{1}{2}$ and $\frac{2}{1}$ (c) $\frac{5}{6}$ and $\frac{7}{6}$ (d) $\frac{2}{3}$ and $\frac{4}{3}$ Solution: We have, $\frac{2}{3}=\frac{2 \times 2}{3 \times 2}=\frac{4}{6}$ and $\frac{5}{3}=\frac{5 \times 2}{3 \times 2}=\frac{10}{6}$ And,$\frac{1}{2}=\frac{1 \times 3}{2 \times 3}=\frac{3}{6}$ and $\frac{2}{1}=\frac{2 \times 6}{1 \times 6}=\frac{12}{6}$ Also, $\frac{2}{3}=\frac...
Read More →If secx cos5x + 1 = 0,
Question: If $\sec x \cos 5 x+1=0$, where $0x \leq \frac{\pi}{2}$, find the value of $x$. Solution: The given equation is secxcos5x+ 1 = 0. Now, $\sec x \cos 5 x+1=0$ $\Rightarrow \frac{\cos 5 x}{\cos x}+1=0$ $\Rightarrow \cos 5 x+\cos x=0$ $\Rightarrow 2 \cos 3 x \cos 2 x=0$ $\Rightarrow \cos 3 x=0$ or $\cos 2 x=0$ $\Rightarrow 3 x=(2 n+1) \frac{\pi}{2}, n \in \mathbf{Z}$ or $2 x=(2 m+1) \frac{\pi}{2}, m \in \mathbf{Z}$ $\Rightarrow x=(2 n+1) \frac{\pi}{6}$ or $x=(2 m+1) \frac{\pi}{4}$ Puttingn...
Read More →A couple has two children,
Question: A couple has two children, (i) Find the probability that both children are males, if it is known that at least one of the children is male. (ii) Find the probability that both children are females, if it is known that the elder child is a female. Solution: If a couple has two children, then the sample space is S = {(b,b), (b,g), (g,b), (g,g)} (i) Let E and F respectively denote the events that both children are males and at least one of the children is a male. $\therefore \mathrm{E} \c...
Read More →If 2sin
Question: If $2 \sin ^{2} x=3 \cos x$, where $0 \leq x \leq 2 \pi$, then find the value of $x$. Solution: The given equation is $2 \sin ^{2} x=3 \cos x$. Now, $2 \sin ^{2} x=3 \cos x$ $\Rightarrow 2\left(1-\cos ^{2} x\right)=3 \cos x$ $\Rightarrow 2 \cos ^{2} x+3 \cos x-2=0$ $\Rightarrow(2 \cos x-1)(\cos x+2)=0$ $\Rightarrow \cos x=\frac{1}{2}$ or $\cos x=-2$ But, $\cos x=-2$ is not possible. $\quad(-1 \leq \cos x \leq 1)$ $\therefore \cos x=\frac{1}{2}=\cos \frac{\pi}{3}$ $\Rightarrow x=2 n \pi...
Read More →A rational number between –3 and 3 is
Question: A rational number between 3 and 3 is(a) 0(b) 4.3(c) 3.4(d) 1.101100110001... Solution: Since,4.3 3.4 3 0 1.101100110001... 3But1.101100110001... is an irrational numberSo, the rational number between3 and 3 is 0.Hence, the correct option is (a)....
Read More →Which of the following is a rational number?
Question: Which of the following is a rational number? (a) $1+\sqrt{3}$ (b) $\pi$ (c) $2 \sqrt{3}$ (d) 0 Solution: Since, the sum and product of a rational and an irrational is always irrational. So, $1+\sqrt{3}$ and $2 \sqrt{3}$ are irrational numbers. Also, is an irrational number.And, 0 is an integer.So, 0 is a rational number.Hence, the correct option is (d)....
Read More →If 3tan
Question: If $3 \tan \left(x-15^{\circ}\right)=\tan \left(x+15^{\circ}\right), 0x90^{\circ}$, find $\theta$. Solution: Given: $3 \tan \left(x-15^{\circ}\right)=\tan \left(x+15^{\circ}\right)$ $\Rightarrow \frac{\tan \left(x+15^{\circ}\right)}{\tan \left(x-15^{\circ}\right)}=3$ Applying componendo and dividendo, we have $\frac{\tan \left(x+15^{\circ}\right)+\tan \left(x-15^{\circ}\right)}{\tan \left(x+15^{\circ}\right)-\tan \left(x-15^{\circ}\right)}=\frac{3+1}{3-1}$ $\Rightarrow \frac{\frac{\sin...
Read More →If 3tan
Question: If $3 \tan \left(x-15^{\circ}\right)=\tan \left(x+15^{\circ}\right), 0x90^{\circ}$, find $\theta$. Solution: Given: $3 \tan \left(x-15^{\circ}\right)=\tan \left(x+15^{\circ}\right)$ $\Rightarrow \frac{\tan \left(x+15^{\circ}\right)}{\tan \left(x-15^{\circ}\right)}=3$ Applying componendo and dividendo, we have $\frac{\tan \left(x+15^{\circ}\right)+\tan \left(x-15^{\circ}\right)}{\tan \left(x+15^{\circ}\right)-\tan \left(x-15^{\circ}\right)}=\frac{3+1}{3-1}$ $\Rightarrow \frac{\frac{\sin...
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