In triangle ABC, prove the following:
Question: In triangle ABC, prove the following: $b \cos B+c \cos C=a \cos (B-C)$ Solution: Let $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=k$ Then, Consider the LHS of the equation $b \cos B+c \cos C=a \cos (B-C)$. $\mathrm{LHS}=b \cos B+c \cos C$ $=k(\sin B \cos B+\sin C \cos C)$ $=\frac{k}{2}(2 \sin B \cos B+2 \sin C \cos C)$ $=\frac{k}{2}(\sin 2 B+\sin 2 C) \quad \ldots(1)$ $\mathrm{RHS}=a \cos (B-C)$ $=k \sin A \cos (B-C)$ $=\frac{k}{2}[2 \sin A \cos (B-C)]$ $=\frac{k}{2}[\sin (A+B-C...
Read More →An oil company has two depots A and B with capacities of 7000 L and 4000 L respectively.
Question: An oil company has two depots A and B with capacities of 7000 L and 4000 L respectively. The company is to supply oil to three petrol pumps, D, E and F whose requirements are 4500L, 3000L and 3500L respectively. The distance (in km) between the depots and the petrol pumps is given in the following table: Assuming that the transportation cost of 10 litres of oil is Re 1 per km, how should the delivery be scheduled in order that the transportation cost is minimum? What is the minimum cos...
Read More →If tan θ=17√, then cosec2 θ−sec2 θcosec2 θ+sec2 θ=
Question: If $\tan \theta=\frac{1}{\sqrt{7}}$, then $\frac{\operatorname{cosec}^{2} \theta-\sec ^{2} \theta}{\operatorname{cosec}^{2} \theta+\sec ^{2} \theta}=$ (a) $\frac{5}{7}$ (b) $\frac{3}{7}$ (c) $\frac{1}{12}$ (d) $\frac{3}{4}$ Solution: Given that: $\tan \theta=\frac{1}{\sqrt{7}}$ We are asked to find the value of the following expression $\frac{\operatorname{cosec}^{2} \theta-\sec ^{2} \theta}{\operatorname{cosec}^{2} \theta+\sec ^{2} \theta}$ Since $\tan \theta=\frac{\text { Perpendicul...
Read More →Give an example of two irrational numbers whose
Question: Give an example of two irrational numbers whose(i) difference is an irrational number.(ii) difference is a rational number.(iii) sum is an irrational number.(iv) sum is a rational number.(v) product is an irrational number.(vi) product is a rational number.(vii) quotient is an irrational number.(viii) quotient is a rational number. Solution: (i) 2 irrational numbers with difference an irrational number will be $3-\sqrt{5}$ and $3+\sqrt{5}$. (ii) 2 irrational numbers with difference is ...
Read More →In triangle ABC, prove the following:
Question: In triangle ABC, prove the following: $a^{2}\left(\cos ^{2} B-\cos ^{2} C\right)+b^{2}\left(\cos ^{2} C-\cos ^{2} A\right)+c^{2}\left(\cos ^{2} A-\cos ^{2} B\right)=0$ Solution: Let $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=k$ Then, Consider the LHS of the equation $a^{2}\left(\cos ^{2} B-\cos ^{2} C\right)+b^{2}\left(\cos ^{2} C-\cos ^{2} A\right)+c^{2}\left(\cos ^{2} A-\cos ^{2} B\right)=0$. LHS $=a^{2}\left(\cos ^{2} B-\cos ^{2} C\right)+b^{2}\left(\cos ^{2} C-\cos ^{2} A\...
Read More →Is the product of two irrationals always irrational?
Question: Is the product of two irrationals always irrational? Justify your answer. Solution: Product of two irrational numbers is not always an irrational number. Example: $\sqrt{5}$ is irrational number. And $\sqrt{5} \times \sqrt{5}=5$ is a rational number. But the product of another two irrational numbers $\sqrt{2}$ and $\sqrt{3}$ is $\sqrt{6}$ which is also an irrational numbers....
Read More →If 8 tan x = 15, then sin x − cos x is equal to
Question: If $8 \tan x=15$, then $\sin x-\cos x$ is equal to (a) $\frac{8}{17}$ (b) $\frac{17}{7}$ (c) $\frac{1}{17}$ (d) $\frac{7}{17}$ Solution: Given that: $8 \tan x=15$ $\tan x=\frac{15}{8}$ $\Rightarrow$ Perpendicular $=15$ $\Rightarrow$ Base $=8$ $\Rightarrow$ Hypotenuse $=\sqrt{225+64}$ $\Rightarrow$ Hypotenuse $=17$ We know that $\sin x=\frac{\text { Perpendicular }}{\text { Hypotenuse }}$ and $\cos x=\frac{\text { Base }}{\text { Hypotenuse }}$ We find: $\sin x-\cos x$ $\Rightarrow \sin...
Read More →In triangle ABC, prove the following:
Question: In triangle ABC, prove the following: $\frac{a^{2} \sin (B-C)}{\sin A}+\frac{b^{2} \sin (C-A)}{\sin B}+\frac{c^{2} \sin (A-B)}{\sin C}=0$ Solution: Let $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=k$ Then, Consider the LHS of the equation $\frac{a^{2} \sin (B-C)}{\sin A}+\frac{b^{2} \sin (C-A)}{\sin B}+\frac{c^{2} \sin (A-B)}{\sin C}=0$. $\mathrm{LHS}=\frac{a^{2} \sin (B-C)}{\sin A}+\frac{b^{2} \sin (C-A)}{\sin B}+\frac{c^{2} \sin (A-B)}{\sin C}$ $=\frac{k^{2} \sin ^{2} A \sin (...
Read More →Let a be a rational number and b be an irrational number. Is ab necessarily an irrational number?
Question: Letabe a rational number andbbe an irrational number. Isabnecessarily an irrational number? Justify your answer with an example. Solution: abe a rational number andbbe an irrational number thenab necessarily will be an irrational number. Example: 6 is a rational number but $\sqrt{5}$ is irrational. And $6 \sqrt{5}$ is also an irrational number....
Read More →If 16 cot x = 12, then sin x−cos xsin x+cos x equals
Question: If $16 \cot x=12$, then $\frac{\sin x-\cos x}{\sin x+\cos x}$ equals (a) $\frac{1}{7}$ (b) $\frac{3}{7}$ (c) $\frac{2}{7}$ (d) 0 Solution: We are given $16 \cot x=12$. We are asked to find the following $\frac{\sin x-\cos x}{\sin x+\cos x}$ We know that: $\cot x=\frac{\text { Base }}{\text { Perpendicular }}$ $\Rightarrow$ Base $=3$ $\Rightarrow$ Perpendicular $=4$ $\Rightarrow$ Hypotenuse $=\sqrt{(\text { Perpendicular })^{2}+(\text { Base })^{2}}$ $\Rightarrow$ Hypotenuse $=\sqrt{16+...
Read More →In triangle ABC, prove the following:
Question: In triangle ABC, prove the following: $a(\sin B-\sin C)+(\sin C-\sin A)+c(\sin A-\sin B)=0$ Solution: Let $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=k$ Then, Consider the LHS of the equation $a(\sin B-\sin C)+(\sin C-\sin A)+c(\sin A-\sin B)=0$. LHS $=a(\sin B-\sin C)+b(\sin C-\sin A)+c(\sin A-\sin B)$ $=k \sin A(\sin B-\sin C)+k \sin B(\sin C-\sin A)+k \sin C(\sin A-\sin B)$ $=k \sin A \sin B-k \sin A \sin C+k \sin B \sin C-k \sin B \sin A+k \sin C \sin A-k \sin C \sin B$ $=0...
Read More →Let x be a rational number and y be an irrational number.
Question: Letxbe a rational number andybe an irrational number. Isx + ynecessarily an irrational number? Give a example in support of your answer. Solution: xbe a rational number andybe an irrational number thenx + y necessarily will be an irrational number. Example: 5 is a rational number but $\sqrt{2}$ is irrational. So, $5+\sqrt{2}$ will be an irrational number....
Read More →If 5 tan θ − 4 = 0, then the value of 5 sin θ−4 cos θ5 sin θ+4 cos θ is
Question: If $5 \tan \theta-4=0$, then the value of $\frac{5 \sin \theta-4 \cos \theta}{5 \sin \theta+4 \cos \theta}$ is (a) $\frac{5}{3}$ (b) $\frac{5}{6}$ (c) 0 (d) $\frac{1}{6}$ Solution: Given that $5 \tan \theta-4=0 .$ We have to find the value of the following expression $\frac{5 \sin \theta-4 \cos \theta}{5 \sin \theta+4 \cos \theta}$ Since $5 \tan \theta-4=0 \Rightarrow \tan \theta=\frac{4}{5}$ We know that: $\tan \theta=\frac{\text { Perpendicular }}{\text { Base }}$ $\Rightarrow$ Base ...
Read More →In triangle ABC, prove the following:
Question: In triangle ABC, prove the following: $\frac{\sqrt{\sin A}-\sqrt{\sin B}}{\sqrt{\sin A}+\sqrt{\sin B}}=\frac{a+b-2 \sqrt{a b}}{a-b}$ Solution: Consider the LHS of the equation $\frac{\sqrt{\sin A}-\sqrt{\sin B}}{\sqrt{\sin A}+\sqrt{\sin B}}=\frac{a+b-2 \sqrt{a b}}{a-b}$. $\mathrm{LHS}=\frac{\sqrt{\sin \mathrm{A}}-\sqrt{\sin \mathrm{B}}}{\sqrt{\sin \mathrm{A}}+\sqrt{\sin \mathrm{B}}}$ $=\frac{\sqrt{\sin A}-\sqrt{\sin B}}{\sqrt{\sin A}+\sqrt{\sin B}} \times \frac{\sqrt{\sin A}-\sqrt{\sin...
Read More →Two godowns A and B have grain capacity of 100 quintals and 50 quintals respectively.
Question: Two godowns A and B have grain capacity of 100 quintals and 50 quintals respectively. They supply to 3 ration shops, D, E and F whose requirements are 60, 50 and 40 quintals respectively. The cost of transportation per quintal from the godowns to the shops are given in the following table: How should the supplies be transported in order that the transportation cost is minimum? What is the minimum cost? Solution: Let godown A supply $x$ and $y$ quintals of grain to the shops $D$ and $E$...
Read More →Classify the following numbers as rational or irrational.
Question: Classify the following numbers as rational or irrational. give reasons to support your answer. (i) $\sqrt{\frac{3}{81}}$ (ii) $\sqrt{361}$ (iii) $\sqrt{21}$ (iv) $\sqrt{1.44}$ (v) $\frac{2}{3} \sqrt{6}$ (vi) $4.1276$ (vii) $\frac{22}{7}$ (viii) $1.232332333 .$ (ix) $3.040040004$ (x) $2.356565656$ (xi) $6.834834 \ldots$ Solution: (i) $\sqrt{\frac{3}{81}}$ $\sqrt{\frac{3}{81}}=\sqrt{\frac{1}{27}}=\frac{1}{3} \sqrt{\frac{1}{3}}$ It is an irrational number. (ii) $\sqrt{\mathbf{3 6 1}}=19$ ...
Read More →In triangle ABC, prove the following:
Question: In triangle ABC, prove the following: $a^{2} \sin (B-C)=\left(b^{2}-c^{2}\right) \sin A$ Solution: Let $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=k$ Then Consider the RHS of the equation $a^{2} \sin (B-C)=\left(b^{2}-c^{2}\right) \sin A$. $\mathrm{RHS}=k^{2} \sin A\left(\sin ^{2} B-\sin ^{2} C\right)$ $=k^{2} \sin A[\sin (B+C) \sin (B-C)] \quad\left[\because \sin ^{2} B-\sin ^{2} C=\sin (B+C) \sin (B-C)\right]$ $=k^{2} \sin A[\sin (\pi-A) \sin (B-C)] \quad[\because A+B+C=\pi]$...
Read More →If tan θ=ab, then a sin θ+b cos θa sin θ−b cos θis equal to
Question: If $\tan \theta=\frac{a}{b}$, then $\frac{a \sin \theta+b \cos \theta}{a \sin \theta-b \cos \theta}$ is equal to (a) $\frac{a^{2}+b^{2}}{a^{2}-b^{2}}$ (b) $\frac{a^{2}-b^{2}}{a^{2}+b^{2}}$ (C) $\frac{a+b}{a-b}$ (s) $\frac{a-b}{a+b}$ Solution: Given: $\tan \theta=\frac{a}{b}$ We have to find the value of following expression in terms ofaandb We know that: $\tan \theta=\frac{\text { Perpendicular }}{\text { Base }}$ $\Rightarrow$ Base $=b$ $\Rightarrow$ Perpendicular $=a$ $\Rightarrow$ H...
Read More →In triangle ABC, prove the following:
Question: In triangle ABC, prove the following: $b \sin B-c \sin C=a \sin (B-C)$ Solution: Let $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=k$ Then, Consider the LHS of he equation $b \sin B-c \sin C=a \sin (B-C)$. $\mathrm{LHS}=k \sin B \sin B-k \sin C \sin C$ $=k\left(\sin ^{2} B-\sin ^{2} C\right)$ $=k[\sin (B+C) \sin (B-C)]$ $\left[\because \sin ^{2} \mathrm{~B}-\sin ^{2} \mathrm{C}=\sin (\mathrm{B}+\mathrm{C}) \sin (\mathrm{B}-\mathrm{C})\right]$ $=k[\sin (\pi-\mathrm{A}) \sin (B-C)]...
Read More →If θ is an acute angle such that cos θ=35, then sin θ tan θ−12 tan2 θ=
Question: If $\theta$ is an acute angle such that $\cos \theta=\frac{3}{5}$, then $\frac{\sin \theta \tan \theta-1}{2 \tan ^{2} \theta}=$ (a) $\frac{16}{625}$ (b) $\frac{1}{36}$ (C) $\frac{3}{160}$ (d) $\frac{160}{3}$ Solution: Given: $\cos \theta=\frac{3}{5}$ and we need to find the value of the following expression $\frac{\sin \theta \tan \theta-1}{2 \tan ^{2} \theta}$ We know that: $\cos \theta=\frac{\text { Base }}{\text { Hypotenuse }}$ $\Rightarrow$ Base $=3$ $\Rightarrow$ Hypotenuse $=5$ ...
Read More →What are irrationl numbers?
Question: What are irrationl numbers? How do they differ from rational numbers? Give examples. Solution: A number that can neither be expressed as a terminating decimal nor be expressed as a repeating decimal is called an irrational number. A rational number, on the other hand, is always a terminating decimal, and if not, it is a repeating decimal.Examples of irrational numbers:0.101001000...0.232332333.....
Read More →An aeroplane can carry a maximum of 200 passengers.
Question: An aeroplane can carry a maximum of 200 passengers. A profit of Rs 1000 is made on each executive class ticket and a profit of Rs 600 is made on each economy class ticket. The airline reserves at least 20 seats for executive class. However, at least 4 times as many passengers prefer to travel by economy class than by the executive class. Determine how many tickets of each type must be sold in order to maximize the profit for the airline. What is the maximum profit? Solution: Let the ai...
Read More →In triangle ABC, prove the following:
Question: In triangle ABC, prove the following: $\frac{a^{2}-c^{2}}{b^{2}}=\frac{\sin (A-C)}{\sin (A+C)}$ Solution: Let $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=k$ Then, Consider the LHS of the equation $\frac{a^{2}-c^{2}}{b^{2}}=\frac{\sin (A-C)}{\sin (A+C)}$. $\mathrm{LHS}=\frac{[k \sin (\mathrm{A})]^{2}-[\mathrm{k} \sin (\mathrm{C})]^{2}}{(\mathrm{k} \sin (\mathrm{B}))^{2}}$ $=\frac{k^{2}\left(\sin ^{2}(A)-\sin ^{2}(C)\right)}{k^{2} \sin ^{2}(B)}$ $=\frac{\sin (A+C) \sin (A-C)}{\si...
Read More →Express in the form of
Question: Express in the form of $\frac{p}{q}: 0 . \overline{38}+1 . \overline{27}$. Solution: Let $0 . \overline{38}=x$ $1 . \overline{27}=y$ x= 0.3838... ...(i)Multiply with 100 as there are 2 repeating digits after decimals100x= 38.3838... ...(ii)Subtracting (i) from (ii) we get99x= 38 $\Rightarrow x=\frac{38}{99}$ Similarly, we takey= 1.2727... ...(iii)Multiply y with 100 as there are 2 repeating digits after decimal.100y= 127.2727... ...(iv)Subtract (iii) from (iv) we get99y= 126 $\Rightarr...
Read More →If tan A=512, find the value of (sin A + cos A) sec A.
Question: If $\tan A=\frac{5}{12}$, find the value of $(\sin \mathrm{A}+\cos \mathrm{A}) \sec \mathrm{A} .$ Solution: Given: $\tan A=\frac{5}{12}$ $\frac{\text { Perpendicular }}{\text { Base }}=\frac{5}{12}$ Perpendicular $=5$ Base $=12$ Hypotenuse $=\sqrt{(\text { Perpendicular })^{2}+(\text { Base })^{2}}$ We know that: $\tan A=\frac{\text { perpendicular }}{\text { Base }}$ Hypotenuse $=\sqrt{(5)^{2}+(12)^{2}}$ Hypotenuse $=\sqrt{169}$ Hypotenuse $=13$ Now we find, $(\sin A+\cos A) \sec A$ $...
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