Prove the following trigonometric identities.

Question: Prove the following trigonometric identities. $\frac{1+\tan ^{2} \theta}{1+\cot ^{2} \theta}=\left(\frac{1-\tan \theta}{1-\cot \theta}\right)^{2}=\tan ^{2} \theta$ Solution: We have to prove $\frac{1+\tan ^{2} \theta}{1+\cot ^{2} \theta}=\left(\frac{1-\tan \theta}{1-\cot \theta}\right)^{2}=\tan ^{2} \theta$ Consider the expression $\frac{1+\tan ^{2} \theta}{1+\cot ^{2} \theta}=\frac{1+\tan ^{2} \theta}{1+\frac{1}{\tan ^{2} \theta}}$ $=\frac{1+\tan ^{2} \theta}{\frac{\tan ^{2} \theta+1}...

A bag contains 4 red and 4 black balls, another bag contains 2 red and 6 black balls.

Question: A bag contains 4 red and 4 black balls, another bag contains 2 red and 6 black balls. One of the two bags is selected at random and a ball is drawn from the bag which is found to be red. Find the probability that the ball is drawn from the first bag. Solution: Let E1and E2be the events of selecting first bag and second bag respectively $\therefore \mathrm{P}\left(\mathrm{E}_{1}\right)=\mathrm{P}\left(\mathrm{E}_{2}\right)=\frac{1}{2}$ Let A be the event of getting a red ball. $\Rightar...

In a ∆ABC, if a = 8, b = 9 and 3 cos C = 2,

Question: In a ∆ABC, ifa= 8,b= 9 and 3 cosC= 2, thenC= ____________. Solution: a= 8 ,b= 9 3 cosC= 2 $\Rightarrow \cos C=\frac{2}{3}$ also using, cosine formula $\cos C=\frac{a^{2}+b^{2}-c^{2}}{2 a b}$ $\Rightarrow \frac{2}{3}=\frac{8^{2}+9^{2}-c^{2}}{2 \times 8 \times 9}$ $\Rightarrow 4 \times 8 \times 3=8^{2}+9^{2}-c^{2}$ $\Rightarrow C^{2}=8^{2}+9^{2}-12 \times 8$ = 49 i. e $C=7$...

It being given that

Question: It being given that $\sqrt{3}=1.732, \sqrt{5}=2.236, \sqrt{6}=2.449$ and $\sqrt{10}=3.162$, find to three places of decimal, the value of each of the following. (i) $\frac{1}{\sqrt{6}+\sqrt{5}}$ (ii) $\frac{6}{\sqrt{5}+\sqrt{3}}$ (iii) $\frac{1}{4 \sqrt{3}-3 \sqrt{5}}$ (iv) $\frac{3+\sqrt{5}}{3-\sqrt{5}}$ (v) $\frac{1+2 \sqrt{3}}{2-\sqrt{3}}$ (vi) $\frac{\sqrt{5}+\sqrt{2}}{\sqrt{5}-\sqrt{2}}$ Solution: (i) $\frac{1}{\sqrt{6}+\sqrt{5}}$ $=\frac{1}{\sqrt{6}+\sqrt{5}} \times \frac{\sqrt{6...

In a ∆ABC, if a = 8, b = 9 and 3 cos C = 2,

Question: In a ∆ABC, ifa= 8,b= 9 and 3 cosC= 2, thenC= ____________. Solution: a= 8 ,b= 9 3 cosC= 2 $\Rightarrow \cos C=\frac{2}{3}$ also using, cosine formula $\cos C=\frac{a^{2}+b^{2}-c^{2}}{2 a b}$ $\Rightarrow \frac{2}{3}=\frac{8^{2}+9^{2}-c^{2}}{2 \times 8 \times 9}$ $\Rightarrow 4 \times 8 \times 3=8^{2}+9^{2}-c^{2}$ $\Rightarrow C^{2}=8^{2}+9^{2}-12 \times 8$ = 49 i. e $C=7$...

In a ∆ABC, if

Question: In a ∆ABC, ifc2sinAsinB=ab, thenA+B= ______________. Solution: Since $c^{2} \sin A \sin B=a b$ $\Rightarrow c^{2}=\frac{a b}{\sin A \sin B}$ $=\frac{a}{\sin A} \times \frac{b}{\sin B}$ $c^{2}=\left(\frac{a}{\sin A}\right)^{2} \quad\left(\right.$ since by sine rule $\left.\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\right)$ $\Rightarrow c=\frac{a}{\sin A}=\frac{b}{\sin B}$ $\Rightarrow a=c \sin A$ and $b=c \sin B$ i. e $\sin A=\frac{a}{c}$ and $\sin B=\frac{b}{c}$ $\Rightarrow \De...

An urn contains 5 red and 5 black balls.

Question: An urn contains 5 red and 5 black balls. A ball is drawn at random, its colour is noted and is returned to the urn. Moreover, 2 additional balls of the colour drawn are put in the urn and then a ball is drawn at random. What is the probability that the second ball is red? Solution: The urn contains 5 red and 5 black balls. Let a red ball be drawn in the first attempt. $\therefore P($ drawing a red ball $)=\frac{5}{10}=\frac{1}{2}$ If two red balls are added to the urn, then the urn con...

In a ∆ABC, if

Question: In a $\Delta A B C$, if $\frac{\cos A}{a}+\frac{\cos B}{b}+\frac{\cos C}{c}=k\left(a^{2}+b^{2}+c^{2}\right)$, then $k=$ Solution: Using cosine formula $\cos A=\frac{b^{2}+c^{2}-a^{2}}{2 b c}, \cos B=\frac{a^{2}+c^{2}-b^{2}}{2 a c}$ and $\cos C=\frac{a^{2}+b^{2}-c^{2}}{2 a b}$ L. H. S. $=\frac{\cos A}{a}+\frac{\cos B}{b}+\frac{\cos C}{c}$ $=\frac{b^{2}+c^{2}-a^{2}}{2 b c} \times \frac{1}{a}+\frac{a^{2}+c^{2}-b^{2}}{2 a c} \times \frac{1}{b}+\frac{a^{2}+b^{2}-c^{2}}{2 a b} \times \frac{1...

Two events A and B will be independent, if

Question: Two events A and B will be independent, if (A) $A$ and $B$ are mutually exclusive (B) $\mathrm{P}\left(\mathrm{A}^{\prime} \mathrm{B}^{\prime}\right)=[1-\mathrm{P}(\mathrm{A})][1-\mathrm{P}(\mathrm{B})]$ (C) $P(A)=P(B)$ (D) $P(A)+P(B)=1$ Solution: Two events A and B are said to be independent, if P(AB) = P(A) P(B) Consider the result given in alternativeB. $P\left(A^{\prime} B^{\prime}\right)=[1-P(A)][1-P(B)]$ $\Rightarrow P\left(A^{\prime} \cap B^{\prime}\right)=1-P(A)-P(B)+P(A) \cdot...

Prove the following trigonometric identities.

Question: Prove the following trigonometric identities. $\frac{(1+\sin \theta)^{2}+(1-\sin \theta)^{2}}{2 \cos ^{2} \theta}=\frac{1+\sin ^{2} \theta}{1-\sin ^{2} \theta}$ Solution: We have to prove that $\frac{(1+\sin \theta)^{2}+(1-\sin \theta)^{2}}{2 \cos ^{2} \theta}=\frac{1+\sin ^{2} \theta}{1-\sin ^{2} \theta}$. We know that, $\sin ^{2} \theta+\cos ^{2} \theta=1$ So, $\frac{(1+\sin \theta)^{2}+(1-\sin \theta)^{2}}{2 \cos ^{2} \theta}=\frac{\left(1+2 \sin \theta+\sin ^{2} \theta\right)+\left...

In a ∆ABC, if ∠C

Question: In a $\triangle A B C$, if $\angle C=\frac{\pi}{2}, \angle A=\frac{\pi}{6}, c=20$, then $a=$_____________________ Solution: In ∆ABC $\angle C=\frac{\pi}{2}, \angle A=\frac{\pi}{6}$ $\Rightarrow \angle B=\frac{\pi}{3}$ (By angle sum property $A+B+C=\pi$ ) using sina formula, $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$ $\Rightarrow \frac{a}{\sin \frac{\pi}{6}}=\frac{20}{s \text { in } \frac{\pi}{2}}(\because c=20$ given $)$ $\Rightarrow a=\sin \frac{\pi}{6} \times 20(\because \s...

In a ∆ABC, if ∠C

Question: In a $\triangle A B C$, if $\angle C=\frac{\pi}{2}, \angle A=\frac{\pi}{6}, c=20$, then $a=$_____________________ Solution: In ∆ABC $\angle C=\frac{\pi}{2}, \angle A=\frac{\pi}{6}$ $\Rightarrow \angle B=\frac{\pi}{3}$ (By angle sum property $A+B+C=\pi$ ) using sina formula, $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$ $\Rightarrow \frac{a}{\sin \frac{\pi}{6}}=\frac{20}{s \text { in } \frac{\pi}{2}}(\because c=20$ given $)$ $\Rightarrow a=\sin \frac{\pi}{6} \times 20(\because \s...

In a ∆ABC, if ∠C = 60°, a = 47 cm and b = 94 cm,

Question: In a ∆ABC, if C = 60,a= 47 cm andb= 94 cm, thenc2= ____________. Solution: Given C= 60∘,a= 47 ,b= 94In ∆ABC Using $\cos C=\frac{a^{2}+b^{2}-c^{2}}{2 a b}$ i.e $\cos 60^{\circ}=\frac{(47)^{2}+(94)^{2}-C^{2}}{2 \times 47 \times 94}$ i.e $2 \times \frac{1}{2} \times 47 \times 94=(47)^{2}+(94)^{2}-C^{2}$ i.e $C^{2}=(47)^{2}+(94)^{2}-47 \times 47 \times 2$ $=-(47)^{2}+(94)^{2}$ $\therefore C^{2}=8836-22049=6627$...

Find rational numbers a and b such that

Question: Find rational numbersaandbsuch that (i) $\frac{\sqrt{2}-1}{\sqrt{2}+1}=a+b \sqrt{2}$ (ii) $\frac{2-\sqrt{5}}{2+\sqrt{5}}=a \sqrt{5}+b$ (iii) $\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}=a+b \sqrt{6}$ (iv) $\frac{5+2 \sqrt{3}}{7+4 \sqrt{3}}=a+b \sqrt{3}$ Solution: (i) $\frac{\sqrt{2}-1}{\sqrt{2}+1}$ $=\frac{\sqrt{2}-1}{\sqrt{2}+1} \times \frac{\sqrt{2}-1}{\sqrt{2}-1}$ $=\frac{(\sqrt{2}-1)^{2}}{(\sqrt{2})^{2}-1^{2}}$ $=\frac{2+1-2 \sqrt{2}}{2-1}$ $=3-2 \sqrt{2}$ $\therefore \frac{\sqrt{2...

The probability of obtaining an even prime number on each die, when a pair of dice is rolled is

Question: The probability of obtaining an even prime number on each die, when a pair of dice is rolled is (A) 0 (B) $\frac{1}{3}$ (C) $\frac{1}{12}$ (D) $\frac{1}{36}$ Solution: When two dice are rolled, the number of outcomes is 36. The only even prime number is 2. Let E be the event of getting an even prime number on each die. $\therefore E=\{(2,2)\}$ $\Rightarrow P(E)=\frac{1}{36}$ Therefore, the correct answer is D....

Prove the following trigonometric identities.

Question: Prove the following trigonometric identities. $\frac{1+\sin \theta}{\cos \theta}+\frac{\cos \theta}{1+\sin \theta}=2 \sec \theta$ Solution: We have to prove $\frac{1+\sin \theta}{\cos \theta}+\frac{\cos \theta}{1+\sin \theta}-2 \sec \theta$ We know that, $\sin ^{2} \theta+\cos ^{2} \theta=1$ Multiplying the denominator and numerator of the second term by $(1-\sin \theta)$, we have $\frac{1+\sin \theta}{\cos \theta}+\frac{\cos \theta}{1+\sin \theta}=\frac{1+\sin \theta}{\cos \theta}+\fr...

The angles A, B, C of a ∆ABC are in AP and the sides a, b,

Question: The anglesA,B,Cof a ∆ABCare in AP and the sidesa,b,care in G.P. Ifa2+c2= b2, then = ____________. Solution: SinceA,B,CareA.P $\Rightarrow 2 B=A+C$ Since $A+B+C=\pi$ (By angle sum property) $\Rightarrow 3 B=\pi$ i. e $B=\frac{\pi}{3}$ ....(1) Also, Sincea,b,care ing.p $\Rightarrow b^{2}=a c$ ...(2) Using $\cos B=\frac{a^{2}+c^{2}-b^{2}}{2 a c}$ i.e $\cos \frac{\pi}{3}=\frac{a^{2}+c^{2}-a c}{2 a c} \quad$ from (1) and (2). $\Rightarrow \frac{1}{2}=\frac{a^{2}+c^{2}-a c}{2 a c}$ $\Rightar...

The angles A, B, C of a ∆ABC are in AP and the sides a, b,

Question: The anglesA,B,Cof a ∆ABCare in AP and the sidesa,b,care in G.P. Ifa2+c2= b2, then = ____________. Solution: SinceA,B,CareA.P $\Rightarrow 2 B=A+C$ Since $A+B+C=\pi$ (By angle sum property) $\Rightarrow 3 B=\pi$ i. e $B=\frac{\pi}{3}$ ....(1) Also, Sincea,b,care ing.p $\Rightarrow b^{2}=a c$ ...(2) Using $\cos B=\frac{a^{2}+c^{2}-b^{2}}{2 a c}$ i.e $\cos \frac{\pi}{3}=\frac{a^{2}+c^{2}-a c}{2 a c} \quad$ from (1) and (2). $\Rightarrow \frac{1}{2}=\frac{a^{2}+c^{2}-a c}{2 a c}$ $\Rightar...

Prove the following trigonometric identities.

Question: Prove the following trigonometric identities. $\frac{1}{1+\sin A}+\frac{1}{1-\sin A}=2 \sec ^{2} A$ Solution: We have to prove $\frac{1}{1+\sin A}+\frac{1}{1-\sin A}=2 \sec ^{2} A$ We know that, $\sin ^{2} A+\cos ^{2} A=1$ So, $\frac{1}{1+\sin A}+\frac{1}{1-\sin A}=\frac{(1-\sin A)+(1+\sin A)}{(1+\sin A)(1-\sin A)}$ $=\frac{1-\sin A+1+\sin A}{1-\sin ^{2} A}$ $=\frac{2}{\cos ^{2} A}$ $=2 \sec ^{2} A$...

In a hostel, 60% of the students read Hindi newspaper,

Question: In a hostel, 60% of the students read Hindi newspaper, 40% read English newspaper and 20% read both Hindi and English news papers. A student is selected at random. (a) Find the probability that she reads neither Hindi nor English news papers. (b) If she reads Hindi news paper, find the probability that she reads English news paper. (c) If she reads English news paper, find the probability that she reads Hindi news paper. Solution: Let H denote the students who read Hindi newspaper and ...

Prove the following trigonometric identities.

Question: Prove the following trigonometric identities. $\frac{\cos ^{2} \theta}{\sin \theta}-\operatorname{cosec} \theta+\sin \theta=0$ Solution: We have to prove $\frac{\cos ^{2} \theta}{\sin \theta}-\operatorname{cosec} \theta+\sin \theta=0$ We know that, $\sin ^{2} \theta+\cos ^{2} \theta=1$ So, $\frac{\cos ^{2} \theta}{\sin \theta}-\operatorname{cosec} \theta+\sin \theta=\left(\frac{\cos ^{2} \theta}{\sin \theta}-\operatorname{cosec} \theta\right)+\sin \theta$ $=\left(\frac{\cos ^{2} \theta...

In a triangle ABC, if a = 2,

Question: In a triangle $A B C$, if $a=2, b=4$ and $A+B=\frac{2 \pi}{3}$, then area of $\triangle A B C$ is ______________________ Solution: Since $A+B=\frac{2 \pi}{3}$ given and $A+B+C=\pi \quad$ (Angle sum property) $\Rightarrow C=\pi-\frac{2 \pi}{3}$ $C=\frac{\pi}{3}$ Since area of triangle $\mathrm{ABC}=\frac{1}{2} a b \sin C$ $=\frac{1}{2} \times 2 \times 4 \times \sin \frac{\pi}{3} \quad(\therefore a=2$ and $b=4$ given $)$ $=4 \times \frac{\sqrt{3}}{2}$ $=3 \sqrt{3}$ sq. units...

Prove the following trigonometric identities.

Question: Prove the following trigonometric identities. (i) $\cot \theta-\tan \theta=\frac{2 \cos ^{2} \theta-1}{\sin \theta \cos \theta}$ (ii) $\tan \theta-\cot \theta=\frac{2 \sin ^{2} \theta-1}{\sin \theta \cos \theta}$ Solution: (i) We have to prove $\cot \theta-\tan \theta=\frac{2 \cos ^{2} \theta-1}{\sin \theta \cos \theta}$ We know that, $\sin ^{2} \theta+\cos ^{2} \theta=1$ So, $\cot \theta-\tan \theta=\frac{\cos \theta}{\sin \theta}-\frac{\sin \theta}{\cos \theta}$ $=\frac{\cos ^{2} \th...

It being given that

Question: It being given that $\sqrt{2}=1.414, \sqrt{3}=1.732, \sqrt{5}=2.236$ and $\sqrt{10}=3.162$, find the value of three places of decimals, of each of the following. (i) $\frac{2}{\sqrt{5}}$ (ii) $\frac{2-\sqrt{3}}{\sqrt{3}}$ (iii) $\frac{\sqrt{10}-\sqrt{5}}{\sqrt{2}}$ Solution: (i) $\frac{2}{\sqrt{5}}$ $=\frac{2}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}}$ $=\frac{2 \sqrt{5}}{5}$ $=\frac{2 \times 2.236}{5}$ $=0.894$ (ii) $\frac{2-\sqrt{3}}{\sqrt{3}}$ $=\frac{2-\sqrt{3}}{\sqrt{3}} \times \...

Question: In a hostel, 60% of the students read Hindi newspaper, 40% read English newspaper and 20% read both Hindi and English news papers. A student is selected at random. (a) Find the probability that she reads neither Hindi nor English news papers. (b) If she reads Hindi news paper, find the probability that she reads English news paper. (c) If she reads English news paper, find the probability that she reads Hindi news paper. Solution: Let H denote the students who read Hindi newspaper and ...