Two groups are competing for the position on the board of directors of a corporation.
Question: Two groups are competing for the position on the board of directors of a corporation. The probabilities that the first and the second groups will win are 0.6 and 0.4 respectively. Further, if the first group wins, the probability of introducing a new product is 0.7 and the corresponding probability is 0.3 if the second group wins. Find the probability that the new product introduced was by the second group. Solution: Let E1and E2be the respective events that the first group and the sec...
Read More →solve this
Question: If $x=3+2 \sqrt{2}$, check whether $x+\frac{1}{x}$ is rational or irrational. Solution: $x=3+2 \sqrt{2} \quad \ldots .(1)$ $\Rightarrow \frac{1}{x}=\frac{1}{3+2 \sqrt{2}}$ $\Rightarrow \frac{1}{x}=\frac{1}{3+2 \sqrt{2}} \times \frac{3-2 \sqrt{2}}{3-2 \sqrt{2}}$ $\Rightarrow \frac{1}{x}=\frac{3-2 \sqrt{2}}{3^{2}-(2 \sqrt{2})^{2}}$ $\Rightarrow \frac{1}{x}=\frac{3-2 \sqrt{2}}{9-8}$ $\Rightarrow \frac{1}{a}=3-2 \sqrt{2}$......(2) Adding (1) and (2), we get $x+\frac{1}{x}=3+2 \sqrt{2}+3-2 ...
Read More →Answer each of the following questions in one word or one sentence or as per exact requirement of the question.
Question: Answer each of the following questions in one word or one sentence or as per exact requirement of the question. In any $\triangle \mathrm{ABC}$, find the value of $\sum a(\sin B-\sin C)$. Solution: Using sine rule, we have $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=k$ $\Rightarrow a=k \sin A, b=k \sin B, c=k \sin C$ $\therefore \sum a(\sin B-\sin C)$ $=\sum k \sin A(\sin B-\sin C)$ $=k \sum \sin A(\sin B-\sin C)$ $=k[\sin A(\sin B-\sin C)+\sin B(\sin C-\sin A)+\sin C(\sin A-\s...
Read More →Answer each of the following questions in one word or one sentence or as per exact requirement of the question.
Question: Answer each of the following questions in one word or one sentence or as per exact requirement of the question. In any $\triangle \mathrm{ABC}$, find the value of $\sum a(\sin B-\sin C)$. Solution: Using sine rule, we have $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=k$ $\Rightarrow a=k \sin A, b=k \sin B, c=k \sin C$ $\therefore \sum a(\sin B-\sin C)$ $=\sum k \sin A(\sin B-\sin C)$ $=k \sum \sin A(\sin B-\sin C)$ $=k[\sin A(\sin B-\sin C)+\sin B(\sin C-\sin A)+\sin C(\sin A-\s...
Read More →Simplify
Question: Simplify $\frac{\sqrt{13}-\sqrt{11}}{\sqrt{13}+\sqrt{11}}+\frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}-\sqrt{11}}$. Solution: $\frac{\sqrt{13}-\sqrt{11}}{\sqrt{13}+\sqrt{11}}+\frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}-\sqrt{11}}$ $=\frac{\sqrt{13}-\sqrt{11}}{\sqrt{13}+\sqrt{11}} \times \frac{\sqrt{13}-\sqrt{11}}{\sqrt{13}-\sqrt{11}}+\frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}-\sqrt{11}} \times \frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}+\sqrt{11}}$ $=\frac{(\sqrt{13}-\sqrt{11})^{2}}{(\sqrt{13})^{2}-(\sqrt{11})^{2...
Read More →Answer each of the following questions in one word or one sentence or as per exact requirement of the question.
Question: Answer each of the following questions in one word or one sentence or as per exact requirement of the question. In any triangle ABC, find the value of $a \sin (B-C)+b \sin (C-A)+c \sin (A-B)$ Solution: Using sine rule, we have $a \sin (B-C)+b \sin (C-A)+c \sin (A-B)$ $=k \sin A \sin (B-C)+k \sin B \sin (C-A)+k \sin C \sin (A-B)$ $=k \sin [\pi-(B+C)] \sin (B-C)+k \sin [\pi-(C+A)] \sin (C-A)+k \sin [\pi-(A+B)] \sin (A-B)$ $=k[\sin (B+C) \sin (B-C)+\sin (C+A) \sin (C-A)+\sin (A+B) \sin (A...
Read More →Answer each of the following questions in one word or one sentence or as per exact requirement of the question.
Question: Answer each of the following questions in one word or one sentence or as per exact requirement of the question. In any triangle ABC, find the value of $a \sin (B-C)+b \sin (C-A)+c \sin (A-B)$ Solution: Using sine rule, we have $a \sin (B-C)+b \sin (C-A)+c \sin (A-B)$ $=k \sin A \sin (B-C)+k \sin B \sin (C-A)+k \sin C \sin (A-B)$ $=k \sin [\pi-(B+C)] \sin (B-C)+k \sin [\pi-(C+A)] \sin (C-A)+k \sin [\pi-(A+B)] \sin (A-B)$ $=k[\sin (B+C) \sin (B-C)+\sin (C+A) \sin (C-A)+\sin (A+B) \sin (A...
Read More →Answer each of the following questions in one word or one sentence or as per exact requirement of the question.
Question: Answer each of the following questions in one word or one sentence or as per exact requirement of the question. If in a $\Delta \mathrm{ABC}, \frac{\cos A}{a}=\frac{\cos B}{b}=\frac{\cos C}{c}$, then find the measures of angles $A, B, C$. Solution: In ∆ABC, $\frac{\cos A}{a}=\frac{\cos B}{b}=\frac{\cos C}{c}$ $\Rightarrow \frac{\cos A}{k \sin A}=\frac{\cos B}{k \sin B}=\frac{\cos C}{k \sin C} \quad$ (Using sine rule) $\Rightarrow \cot A=\cot B=\cot C$ $\Rightarrow A=B=C$ $\Rightarrow \...
Read More →Answer each of the following questions in one word or one sentence or as per exact requirement of the question.
Question: Answer each of the following questions in one word or one sentence or as per exact requirement of the question. If in a $\Delta \mathrm{ABC}, \frac{\cos A}{a}=\frac{\cos B}{b}=\frac{\cos C}{c}$, then find the measures of angles $A, B, C$. Solution: In ∆ABC, $\frac{\cos A}{a}=\frac{\cos B}{b}=\frac{\cos C}{c}$ $\Rightarrow \frac{\cos A}{k \sin A}=\frac{\cos B}{k \sin B}=\frac{\cos C}{k \sin C} \quad$ (Using sine rule) $\Rightarrow \cot A=\cot B=\cot C$ $\Rightarrow A=B=C$ $\Rightarrow \...
Read More →Answer each of the following questions in one word or one sentence or as per exact requirement of the question.
Question: Answer each of the following questions in one word or one sentence or as per exact requirement of the question. If the sides of a triangle are proportional to $2, \sqrt{6}$ and $\sqrt{3}-1$, find the measure of its greatest angle. Solution: Let $\triangle \mathrm{ABC}$ be the triangle such that $a=2, b=\sqrt{6}$ and $c=\sqrt{3}-1$. Clearly, $bac$. Then, $\angle B$ is the greatest angle of $\triangle A B C$. (Greatest side has greatest angle opposite to it) Using cosine formula, we have...
Read More →Answer each of the following questions in one word or one sentence or as per exact requirement of the question.
Question: Answer each of the following questions in one word or one sentence or as per exact requirement of the question. If the sides of a triangle are proportional to $2, \sqrt{6}$ and $\sqrt{3}-1$, find the measure of its greatest angle. Solution: Let $\triangle \mathrm{ABC}$ be the triangle such that $a=2, b=\sqrt{6}$ and $c=\sqrt{3}-1$. Clearly, $bac$. Then, $\angle B$ is the greatest angle of $\triangle A B C$. (Greatest side has greatest angle opposite to it) Using cosine formula, we have...
Read More →Find the values of a and b if
Question: Find the values of a and b if $\frac{7+3 \sqrt{5}}{3+\sqrt{5}}-\frac{7-3 \sqrt{5}}{3-\sqrt{5}}=a+b \sqrt{5}$ Solution: $\frac{7+3 \sqrt{5}}{3+\sqrt{5}}-\frac{7-3 \sqrt{5}}{3-\sqrt{5}}$ $=\frac{7+3 \sqrt{5}}{3+\sqrt{5}} \times \frac{3-\sqrt{5}}{3-\sqrt{5}}-\frac{7-3 \sqrt{5}}{3-\sqrt{5}} \times \frac{3+\sqrt{5}}{3+\sqrt{5}}$ $=\frac{7(3-\sqrt{5})+3 \sqrt{5}(3-\sqrt{5})}{3^{2}-(\sqrt{5})^{2}}-\frac{7(3+\sqrt{5})-3 \sqrt{5}(3+\sqrt{5})}{3^{2}-(\sqrt{5})^{2}}$ $=\frac{21-7 \sqrt{5}+9 \sqrt...
Read More →Answer each of the following questions in one word or one sentence or as per exact requirement of the question.
Question: Answer each of the following questions in one word or one sentence or as per exact requirement of the question. In ∆ABC, ifa= 8,b= 10,c= 12 andC=A, find the value of. Solution: Using cosine rule, we have $\cos A=\frac{b^{2}+c^{2}-a^{2}}{2 b c}$ $\Rightarrow \cos A=\frac{10^{2}+12^{2}-8^{2}}{2 \times 10 \times 12}$ $\Rightarrow \cos A=\frac{100+144-64}{240}$ $\Rightarrow \cos A=\frac{180}{240}=\frac{3}{4} \quad \ldots .(1)$ Now, using sine rule, we have $\frac{a}{\sin A}=\frac{c}{\sin C...
Read More →Answer each of the following questions in one word or one sentence or as per exact requirement of the question.
Question: Answer each of the following questions in one word or one sentence or as per exact requirement of the question. In ∆ABC, ifa= 8,b= 10,c= 12 andC=A, find the value of. Solution: Using cosine rule, we have $\cos A=\frac{b^{2}+c^{2}-a^{2}}{2 b c}$ $\Rightarrow \cos A=\frac{10^{2}+12^{2}-8^{2}}{2 \times 10 \times 12}$ $\Rightarrow \cos A=\frac{100+144-64}{240}$ $\Rightarrow \cos A=\frac{180}{240}=\frac{3}{4} \quad \ldots .(1)$ Now, using sine rule, we have $\frac{a}{\sin A}=\frac{c}{\sin C...
Read More →Answer each of the following questions in one word or one sentence or as per exact requirement of the question.
Question: Answer each of the following questions in one word or one sentence or as per exact requirement of the question. In ∆ABC, ifa= 8,b= 10,c= 12 andC=A, find the value of. Solution: Using cosine rule, we have $\cos A=\frac{b^{2}+c^{2}-a^{2}}{2 b c}$ $\Rightarrow \cos A=\frac{10^{2}+12^{2}-8^{2}}{2 \times 10 \times 12}$ $\Rightarrow \cos A=\frac{100+144-64}{240}$ $\Rightarrow \cos A=\frac{180}{240}=\frac{3}{4} \quad \ldots .(1)$ Now, using sine rule, we have $\frac{a}{\sin A}=\frac{c}{\sin C...
Read More →A factory has two machines A and B.
Question: A factory has two machines A and B. Past record shows that machine A produced 60% of the items of output and machine B produced 40% of the items. Further, 2% of the items produced by machine A and 1% produced by machine B were defective. All the items are put into one stockpile and then one item is chosen at random from this and is found to be defective. What is the probability that was produced by machine B? Solution: Let E1and E2be the respective events of items produced by machines ...
Read More →Answer each of the following questions in one word or one sentence or as per exact requirement of the question.
Question: Answer each of the following questions in one word or one sentence or as per exact requirement of the question. In a $\triangle \mathrm{ABC}$, if $\sin A$ and $\sin B$ are the roots of the equation $c^{2} x^{2}-c(a+b) x+a b=0$, then find $\angle C .$ Solution: It is given that $\sin A$ and $\sin B$ are the roots of the equation $c^{2} x^{2}-c(a+b) x+a b=0$. $\therefore \sin A+\sin B=-\frac{-c(a+b)}{c^{2}} \quad$ (Sum of roots $=-\frac{b}{a}$ ) $\Rightarrow \sin A+\sin B=\frac{a+b}{c}$ ...
Read More →Answer each of the following questions in one word or one sentence or as per exact requirement of the question.
Question: Answer each of the following questions in one word or one sentence or as per exact requirement of the question. In a $\triangle \mathrm{ABC}$, if $\sin A$ and $\sin B$ are the roots of the equation $c^{2} x^{2}-c(a+b) x+a b=0$, then find $\angle C .$ Solution: It is given that $\sin A$ and $\sin B$ are the roots of the equation $c^{2} x^{2}-c(a+b) x+a b=0$. $\therefore \sin A+\sin B=-\frac{-c(a+b)}{c^{2}} \quad$ (Sum of roots $=-\frac{b}{a}$ ) $\Rightarrow \sin A+\sin B=\frac{a+b}{c}$ ...
Read More →Prove that
Question: Prove that (i) $\frac{1}{3+\sqrt{7}}+\frac{1}{\sqrt{7}+\sqrt{5}}+\frac{1}{\sqrt{5}+\sqrt{3}}+\frac{1}{\sqrt{3}+1}=1$ (ii) $\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\frac{1}{\sqrt{4}+\sqrt{5}}+\frac{1}{\sqrt{5}+\sqrt{6}}+\frac{1}{\sqrt{6}+\sqrt{7}}+\frac{1}{\sqrt{7}+\sqrt{8}}+\frac{1}{\sqrt{8}+\sqrt{9}}=2$ Solution: (i) $\frac{1}{3+\sqrt{7}}+\frac{1}{\sqrt{7}+\sqrt{5}}+\frac{1}{\sqrt{5}+\sqrt{3}}+\frac{1}{\sqrt{3}+1}$ $=\frac{1}{3+\sqrt{7}} \times \fr...
Read More →An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers.
Question: An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probability of accidents are 0.01, 0.03 and 0.15 respectively. One of the insured persons meets with an accident. What is the probability that he is a scooter driver? Solution: Let E1, E2, and E3be the respective events that the driver is a scooter driver, a car driver, and a truck driver. Let A be the event that the person meets with an accident. There are 2000 scooter drivers, 4000 car dri...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. $\frac{1+\cos A}{\sin A}=\frac{\sin A}{1-\cos A}$ Solution: We need to prove $\frac{1+\cos A}{\sin A}=\frac{\sin A}{1-\cos A}$ Now, multiplying the numerator and denominator of LHS by $1-\cos A$, we get $\frac{1+\cos A}{\sin A}=\frac{1+\cos A}{\sin A} \times \frac{1-\cos A}{1-\cos A}$ Further using the identity, $a^{2}-b^{2}=(a+b)(a-b)$, we get $\frac{1+\cos A}{\sin A} \times \frac{1-\cos A}{1-\cos A}=\frac{1-\cos ^{2} A}{\sin A(1-\cos A)}$...
Read More →Answer each of the following questions in one word or one sentence or as per exact requirement of the question.
Question: Answer each of the following questions in one word or one sentence or as per exact requirement of the question. In a $\triangle \mathrm{ABC}$, if $b=20, c=21$ and $\sin A=\frac{3}{5}$, find $a$. Solution: In $\triangle \mathrm{ABC}, b=20, c=21$ and $\sin A=\frac{3}{5}$. Using cosine rule, we have $\cos A=\frac{b^{2}+c^{2}-a^{2}}{2 b c}$ $\Rightarrow \sqrt{1-\left(\frac{3}{5}\right)^{2}}=\frac{20^{2}+21^{2}-a^{2}}{2 \times 20 \times 21} \quad\left(\cos ^{2} A+\sin ^{2} A=1\right)$ $\Rig...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. $\frac{\sec A-\tan A}{\sec A+\tan A}=\frac{\cos ^{2} A}{(1+\sin A)^{2}}$ Solution: We need to prove $\frac{\sec A-\tan A}{\sec A+\tan A}=\frac{\cos ^{2} A}{(1+\sin A)^{2}}$ Here, we will first solve the LHS. Now, using $\sec \theta=\frac{1}{\cos \theta}$ and $\tan \theta=\frac{\sin \theta}{\cos \theta}$, we get $\frac{\sec A-\tan A}{\sec A+\tan A}=\frac{\frac{1}{\cos A}-\frac{\sin A}{\cos A}}{\frac{1}{\cos A}+\frac{\sin A}{\cos A}}$ $=\frac...
Read More →Answer each of the following questions in one word or one sentence or as per exact requirement of the question.
Question: Answer each of the following questions in one word or one sentence or as per exact requirement of the question. In a $\triangle \mathrm{ABC}$, if $\cos A=\frac{\sin B}{2 \sin C}$, then show that $c=a$. Solution: Given: $\cos A=\frac{\sin B}{2 \sin C}$ $\Rightarrow \frac{b^{2}+c^{2}-a^{2}}{2 b c}=\frac{b}{2 c} \quad$ (Using sine rule and cosine rule) $\Rightarrow b^{2}+c^{2}-a^{2}=b^{2}$ $\Rightarrow c^{2}=a^{2}$ $\Rightarrow c=a$...
Read More →Answer each of the following questions in one word or one sentence or as per exact requirement of the question.
Question: Answer each of the following questions in one word or one sentence or as per exact requirement of the question. In a $\triangle \mathrm{ABC}$, if $\cos A=\frac{\sin B}{2 \sin C}$, then show that $c=a$. Solution: Given: $\cos A=\frac{\sin B}{2 \sin C}$ $\Rightarrow \frac{b^{2}+c^{2}-a^{2}}{2 b c}=\frac{b}{2 c} \quad$ (Using sine rule and cosine rule) $\Rightarrow b^{2}+c^{2}-a^{2}=b^{2}$ $\Rightarrow c^{2}=a^{2}$ $\Rightarrow c=a$...
Read More →