If sin θ + cos θ = 2–√ cos (90°−θ), find cot θ.

Question: If $\sin \theta+\cos \theta=\sqrt{2} \cos \left(90^{\circ}-\theta\right)$, find $\cot \theta$. Solution: Given: $\sin \theta+\cos \theta=\sqrt{2} \cos \left(90^{\circ}-\theta\right)$ We have to find the value of $\cot \theta$. Now, $\sin \theta+\cos \theta=\sqrt{2} \cos \left(90^{\circ}-\theta\right)$ $\Rightarrow \sin \theta+\cos \theta=\sqrt{2} \sin \theta$ (since, $\cos \left(90^{\circ}-\theta\right)=\sin \theta$ ) $\Rightarrow \cos \theta=(\sqrt{2}-1) \sin \theta$ $\Rightarrow \fra...

The product of two irrational number is

Question: The product of two irrational number is(a) always irrational(b) always rational(c) always an integer(d) sometimes rational and sometimes irrational Solution: (d) sometimes rational and sometimes irrational For example: $\sqrt{2}$ is an irrational number, when it is multiplied with itself it results into 2 , which is a rational number. $\sqrt{2}$ when multiplied with $\sqrt{3}$, which is also an irrational number, results into $\sqrt{6}$, which is an irrational number....

An experiment succeeds twice as often as it fails.

Question: An experiment succeeds twice as often as it fails. Find the probability that in the next six trials, there will be at least 4 successes. Solution: The probability of success is twice the probability of failure. Let the probability of failure bex. Probability of success = 2x $x+2 x=1$ $\Rightarrow 3 x=1$ $\Rightarrow x=\frac{1}{3}$ $\therefore 2 x=\frac{2}{3}$ Let $p=\frac{1}{3}$ and $q=\frac{2}{3}$ Let X be the random variable that represents the number of successes in six trials. By b...

Which of the following numbers is irrational?

Question: Which of the following numbers is irrational? (a) $\sqrt{\frac{4}{9}}$ (b) $\frac{\sqrt{1250}}{\sqrt{8}}$ (c) $\sqrt{8}$ (d) $\frac{\sqrt{24}}{\sqrt{6}}$ Solution: Since, $\sqrt{\frac{4}{9}}=\frac{2}{3}$, which is a rational number, $\frac{\sqrt{1250}}{\sqrt{8}}=\sqrt{\frac{1250}{8}}=\sqrt{\frac{625}{4}}=\frac{25}{2}$, which is a rational number, $\sqrt{8}=2 \sqrt{2}$, which is an irrational number, and $\frac{\sqrt{24}}{\sqrt{6}}=\sqrt{\frac{24}{6}}=\sqrt{4}=2$, which is a rational nu...

Question: $\frac{1}{1.4}+\frac{1}{4.7}+\frac{1}{7.10}+\ldots+\frac{1}{(3 n-2)(3 n+1)}=\frac{n}{3 n+1}$ Solution: LetP(n) be the given statement. Now, $P(n)=\frac{1}{1.4}+\frac{1}{4.7}+\frac{1}{7.10}+\ldots+\frac{1}{(3 n-2)(3 n+1)}=\frac{n}{3 n+1}$ Step 1; $P(1)=\frac{1}{1.4}=\frac{1}{4}=\frac{1}{3 \times 1+1}$ Hence, $P(1)$ is true. Step 2: Let $P(m)$ be true. i.e., $\frac{1}{1.4}+\frac{1}{4.7}+\ldots+\frac{1}{(3 m-2)(3 m+1)}=\frac{m}{3 m+1}$ Now, $P(m)=\frac{1}{1.4}+\frac{1}{4.7}+\ldots+\frac{1...

If a leap year is selected at random, what is the chance that it will contain 53 Tuesdays?

Question: If a leap year is selected at random, what is the chance that it will contain 53 Tuesdays? Solution: In a leap year, there are 366 days i.e., 52 weeks and 2 days. In 52 weeks, there are 52 Tuesdays. Therefore, the probability that the leap year will contain 53 Tuesdays is equal to the probability that the remaining 2 days will be Tuesdays. The remaining 2 days can be Monday and Tuesday Tuesday and Wednesday Wednesday and Thursday Thursday and Friday Friday and Saturday Saturday and Sun...

How many digits are there in the repeating block of digits in the decimal expansion of

Question: How many digits are there in the repeating block of digits in the decimal expansion of $\frac{17}{7} ?$ (a) 16 (b) 6 (c) 26 (d) 7 Solution: $\because \frac{17}{7}=2 . \overline{428571}$ So, there are 6 digits in the repeating block of digits in the decimal expansion of $\frac{17}{7}$. Hence, the correct option is (b)....

A die is thrown again and again until three sixes are obtained. Find the probability of obtaining the third six in the sixth throw of the die.

Question: A die is thrown again and again until three sixes are obtained. Find the probability of obtaining the third six in the sixth throw of the die. Solution: The probability of getting a six in a throw of die is $\frac{1}{6}$ and not getting a six is $\frac{5}{6}$. Let $p=\frac{1}{6}$ and $q=\frac{5}{6}$ The probability that the 2 sixes come in the first five throws of the die is ${ }^{5} C_{2}\left(\frac{1}{6}\right)^{2}\left(\frac{5}{6}\right)^{3}=\frac{10 \times(5)^{3}}{(6)^{5}}$ $\there...

If cosec θ=1312, find the value of 2 sin θ−3 cos θ4 sin θ−9 cos θ.

Question: If $\operatorname{cosec} \theta=\frac{13}{12}$, find the value of $\frac{2 \sin \theta-3 \cos \theta}{4 \sin \theta-9 \cos \theta}$. Solution: Given: $\operatorname{cosec} \theta=\frac{13}{12}$ We have to find the value of the expression $\frac{2 \sin \theta-3 \cos \theta}{4 \sin \theta-9 \cos \theta}$. Now, $\operatorname{cosec} \theta=\frac{13}{12}$ $\Rightarrow \sin \theta=\frac{1}{\operatorname{cosec} \theta}=\frac{1}{\frac{13}{12}}=\frac{12}{13}$ $\cos \theta=\sqrt{1-\sin ^{2} \th...

Which of the following is an irrational number?

Question: Which of the following is an irrational number? (a) $\sqrt{23}$ (b) $\sqrt{225}$ (c) $0.3799$ (d) $7 . \overline{478}$ Solution: Since, $\sqrt{225}=15$, which is an integer, 0.3799 is a number with terminating decimal expansion, and 7. $\overline{478}$ is a number with non-terminating recurring decimal expansion Also, 23 is a prime number. So, $\sqrt{23}$ is an irrational number. $\quad[\because \sqrt{n}$ is always an irrational number, if $n$ is a prime number. $]$ Hence, the correct ...

Question: $\frac{1}{2.5}+\frac{1}{5.8}+\frac{1}{8.11}+\ldots+\frac{1}{(3 n-1)(3 n+2)}=\frac{n}{6 n+4}$ Solution: LetP(n) be the given statement. Now, $P(n)=\frac{1}{2.5}+\frac{1}{5.8}+\frac{1}{8.11}+\ldots+\frac{1}{(3 n-1)(3 n+2)}=\frac{n}{6 n+4}$ Step 1: $P(1)=\frac{1}{2.5}=\frac{1}{10}=\frac{1}{6+4}$ Hence, $P(1)$ is true. Step 2; Let $P(m)$ be true. Then, $\frac{1}{2.5}+\frac{1}{5.8}+\frac{1}{8.11}+\ldots+\frac{1}{(3 m-1)(3 m+2)}=\frac{m}{6 m+4}$ To prove : $P(m+1)$ is true. i. e., $\frac{1}{...

If 3–√ tan θ=3 sin θ, find the value of sin2θ − cos2θ.

Question: If $\sqrt{3} \tan \theta=3 \sin \theta$, find the value of $\sin ^{2} \theta-\cos ^{2} \theta$. Solution: Given: $\sqrt{3} \tan \theta=3 \sin \theta$ We have to find the value of $\sin ^{2} \theta-\cos ^{2} \theta$. $\sqrt{3} \tan \theta=3 \sin \theta$ $\Rightarrow \sqrt{3} \frac{\sin \theta}{\cos \theta}=3 \sin \theta$ $\Rightarrow \cos \theta=\frac{\sqrt{3}}{3}=\frac{1}{\sqrt{3}}$ Therefore, $\sin ^{2} \theta-\cos ^{2} \theta=1-\cos ^{2} \theta-\cos ^{2} \theta \quad\left(\right.$ si...

Decimal expansion of

Question: Decimal expansion of $\sqrt{2}$ is (a) a finite decimal(b) 1.4121(c) non-terminating recurring(d) non-terminating, non-recurring Solution: (c) a non-terminating and non-repeating decimal Because $\sqrt{2}$ is an irrational number, its decimal expansion is non-terminating and non-repeating....

1 + 3 + 5 + ... + (2n − 1) =

Question: $1+3+5+\ldots+(2 n-1)=n^{2}$ i.e., the sum of first $n$ odd natural numbers is $n^{2}$. Solution: LetP(n) be the given statement. Now, $P(n)=1+3+5+\ldots+(2 n-1)=n^{2}$ Step 1: $P(1)=1=1^{2}$ Hence, $P(1)$ is true. Step 2; Let $P(m)$ be true. Then: $1+3+5+\ldots+(2 m-1)=m^{2}$ To prove : $P(m+1)$ is true. i. e., $1+3+5+\ldots+\{2(m+1)-1\}=(m+1)^{2}$ $\Rightarrow 1+3+5+\ldots+(2 m+1)=(m+1)^{2}$ Now, we have; $1+3+5+\ldots+(2 m-1)=m^{2}$ $\Rightarrow 1+3+\ldots+(2 m-1)+(2 m+1)=m^{2}+2 m+...

Question: $\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\ldots+\frac{1}{n(n+1)}=\frac{n}{n+1}$ Solution: LetP(n) be the given statement. Now, $P(n)=\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\ldots+\frac{1}{n(n+1)}=\frac{n}{n+1}$ Step 1: $P(1)=\frac{1}{1.2}=\frac{1}{2}=\frac{1}{1+1}$ Hence, $P(1)$ is true. Step 2: Let $P(m)$ be true. Then, $\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\ldots+\frac{1}{m(m+1)}=\frac{m}{m+1}$ We shall now prove that $P(m+1)$ is true. i. e., $\frac{1}{1.2}+\frac{1}{2.3}+\f...

1 + 3 + 3

Question: $1+3+3^{2}+\ldots+3^{n-1}=\frac{3^{n}-1}{2}$ Solution: LetP(n) be the given statement. Now, $P(n)=1+3+3^{2}+\ldots+3^{n-1}=\frac{3^{n}-1}{2}$ Step 1: $P(1)=1=\frac{3^{1}-1}{2}=\frac{2}{2}=1$ Hence, $P(1)$ is true. Step 2: Let $P(m)$ is true. then, $1+3+3^{2}+\ldots+3^{m-1}=\frac{3^{m}-1}{2}$ We shall prove that $P(m+1)$ is true. That is, $1+3+3^{2}+\ldots+3^{m}=\frac{3^{m+1}-1}{2}$ Now, we have: $1+3+3^{2}+\ldots+3^{m-1}=\frac{3^{m}-1}{2}$ $\Rightarrow 1+3+3^{2}+\ldots+3^{m-1}+3^{m}=\f...

Solve this

Question: is(a) a rational number(b) an integer(c) an irrational number(d) a whole number Solution: Since, $\pi$ has a non-terminating non-recurring decimal expansion. So, $\pi$ is an irrational number. Hence, the correct option is (c)....

Choose the rational number which does not lie between

Question: Choose the rational number which does not lie between $-\frac{2}{3}$ and $-\frac{1}{5}$. (a) $-\frac{3}{10}$ (b) $\frac{3}{10}$ (c) $-\frac{1}{4}$ (d) $-\frac{7}{20}$ Solution: We have, $-\frac{2}{3}=-\frac{2 \times 20}{3 \times 20}=-\frac{40}{60}$ and $-\frac{1}{5}=-\frac{1 \times 12}{5 \times 12}=-\frac{12}{60}$ And $,-\frac{3}{10}=-\frac{3 \times 6}{10 \times 6}=-\frac{18}{60}, \frac{3}{10}=\frac{3 \times 6}{10 \times 6}=\frac{18}{60},-\frac{1}{4}=-\frac{1 \times 15}{4 \times 15}=-\...

If 3cosθ = 1, find the value of 6 sin2 θ+tan2 θ4 cos θ

Question: If $3 \cos \theta=1$, find the value of $\frac{6 \sin ^{2} \theta+\tan ^{2} \theta}{4 \cos \theta}$ Solution: Given: $3 \cos \theta=1$ We have to find the value of the expression $\frac{6 \sin ^{2} \theta+\tan ^{2} \theta}{4 \cos \theta}$ We have, $3 \cos \theta=1$ $\Rightarrow \cos \theta=\frac{1}{3}$ $\sin \theta=\sqrt{1-\cos ^{2} \theta}=\sqrt{1-\left(\frac{1}{3}\right)^{2}}=\frac{\sqrt{8}}{3}$ $\tan \theta=\frac{\sin \theta}{\cos \theta}=\frac{\frac{\sqrt{8}}{3}}{\frac{1}{3}}=\sqrt...

12 + 22 + 32 + ... + n

Question: $1^{2}+2^{2}+3^{2}+\ldots+\mathrm{n}^{2}=\frac{n(n+1)(2 n+1)}{6}$ Solution: LetP(n) be the given statement. Now, $P(n)=1^{2}+2^{2}+3^{2}+\ldots+n^{2}=\frac{n(n+1)(2 n+1)}{6}$ Step 1 : $P(1)=1^{2}=\frac{1(1+1)(2+1)}{6}=\frac{6}{6}=1$ Hence, $P(1)$ is true. Step 2 : Let $P(m)$ be true. Then, $1^{2}+2^{2}+\ldots+m^{2}=\frac{m(m+1)(2 m+1)}{6}$ We shall now prove that $P(m+1)$ is true. i. e., $1^{2}+2^{2}+3^{2}+\ldots+(m+1)^{2}=\frac{(m+1)(m+2)(2 m+3)}{6}$ Now, $P(m)=1^{2}+2^{2}+3^{2}+\ldot...

In a hurdle race, a player has to cross 10 hurdles.

Question: In a hurdle race, a player has to cross 10 hurdles. The probability that he will clear each hurdle is $\frac{5}{6}$. What is the probability that he will knock down fewer than 2 hurdles? Solution: Letpandqrespectively be the probabilities that the player will clear and knock down the hurdle. $\therefore p=\frac{5}{6}$ $\Rightarrow q=1-p=1-\frac{5}{6}=\frac{1}{6}$ Let X be the random variable that represents the number of times the player will knock down the hurdle. Therefore, by binomi...

A rational number equivalent to

Question: A rational number equivalent to $\frac{7}{19}$ is (a) $\frac{17}{119}$ (b) $\frac{14}{57}$ (c) $\frac{21}{38}$ (d) $\frac{21}{57}$ Solution: Since, $\frac{7}{19}=\frac{7 \times 3}{19 \times 3}=\frac{21}{57}$ Hence, the correct option is (d)....

If cot θ=3–√, find the value of cosec2 θ+cot2 θcosec2 θ−sec2 θ.

Question: If $\cot \theta=\sqrt{3}$, find the value of $\frac{\cos e c^{2} \theta+\cot ^{2} \theta}{\operatorname{cosec}^{2} \theta-\sec ^{2} \theta}$. Solution: Given: $\cot \theta=\sqrt{3}$ We have to find the value of the expression $\frac{\operatorname{cosec}^{2} \theta+\cot ^{2} \theta}{\operatorname{cosec}^{2} \theta-\sec ^{2} \theta}$. We know that, $\cot \theta=\sqrt{3} \Rightarrow \cot ^{2} \theta=3$ $\operatorname{cosec}^{2} \theta=1+\cot ^{2} \theta=1+(\sqrt{3})^{2}=4$ $\sec ^{2} \the...

1 + 2 + 3 + ... + n =

Question: $1+2+3+\ldots+n=\frac{n(n+1)}{2}$ i.e. the sum of the first $n$ natural numbers is $\frac{n(n+1)}{2}$ Solution: Let P(n) be the given statement. Now, $\mathrm{P}(n)=1+2+3+\ldots+n=\frac{n(n+1)}{2}$ Step1: $P(1)=1=\frac{1(1+1)}{2}=1$ Hence, $P(1)$ is true. Step 2: Let $P(m)$ be true. Then, $1+2+3+\ldots+m=\frac{m(m+1)}{2}$ We shall now prove that $P(m+1)$ is true. i. e., $1+2+\ldots+(m+1)=\frac{(m+1)(m+2)}{2}$ Now, $1+2+\ldots+m=\frac{m(m+1)}{2}$ $\Rightarrow 1+2+\ldots m+m+1=\frac{m(m+...

If cosec A=2–√, find the value of 2 sin2 A+3 cot2 A4(tan2 A−cos2 A)

Question: If $\operatorname{cosec} A=\sqrt{2}$, find the value of $\frac{2 \sin ^{2} A+3 \cot ^{2} A}{4\left(\tan ^{2} A-\cos ^{2} A\right)}$. Solution: Given: $\operatorname{cosec} A=\sqrt{2}$ We have to find the value of the expression $\frac{2 \sin ^{2} A+3 \cot ^{2} A}{4\left(\tan ^{2} A-\cos ^{2} A\right)}$ We know that, $\operatorname{cosec} A=\sqrt{2}$ $\Rightarrow \sin A=\frac{1}{\operatorname{cosec} A}=\frac{1}{\sqrt{2}}$ $\cos A=\sqrt{1-\sin ^{2} A}=\sqrt{1-\left(\frac{1}{\sqrt{2}}\rig...