What is the maximum value
Question: What is the maximum value of $\frac{1}{\operatorname{cosec} \theta} ?$ Solution: The maximum value of $\frac{1}{\operatorname{cosec} \theta}$ is 1 because the maximum value of $\sin \theta$ is 1 that is $\frac{1}{\operatorname{cosec} \theta}=\sin \theta$ $\frac{1}{\operatorname{cosec} \theta}=1$...
Read More →If in ∆ABC, ∠C = 105°,
Question: If in ∆ABC, C= 105, B= 45 anda= 2, then findb. Solution: We know, $A+B+C=\pi$ $\therefore A=\pi-(\mathrm{B}+\mathrm{C})$ $\Rightarrow A=180^{\circ}-\left(45^{\circ}+105^{\circ}\right)=30^{\circ}$ Now, According to sine rule, $\frac{a}{\sin \mathrm{A}}=\frac{b}{\sin \mathrm{B}}$. $\Rightarrow \frac{2}{\sin 30^{\circ}}=\frac{b}{\sin 45^{\circ}} \quad\left(\because a=2, \angle B=45^{\circ}\right)$ $\Rightarrow \frac{2}{\frac{1}{2}}=\frac{b}{\frac{1}{\sqrt{2}}}$ $\Rightarrow 4 \times \frac...
Read More →What is the maximum value of 1sec θ?
Question: What is the maximum value of $\frac{1}{\sec \theta}$ ? Solution: The maximum value of $\frac{1}{\sec \theta}$ is 1 because the maximum value of $\cos \theta$ is 1 that is $\frac{1}{\sec \theta}=\cos \theta$ $\frac{1}{\sec \theta}=1$...
Read More →Write the maximum and minimum values of cos θ.
Question: Write the maximum and minimum values of $\cos \theta$. Solution: The maximum value of $\cos \theta$ is 1 and the minimum value of $\cos \theta$ is $-1$ because value of $\cos \theta$ lies between $-1$ and 1...
Read More →If in ∆ABC, ∠A = 45°,
Question: If in ∆ABC, A= 45, B= 60 and C= 75, find the ratio of its sides. Solution: Let $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=k$ Then, $\frac{a}{\sin 45^{\circ}}=\frac{b}{\sin 60^{\circ}}=\frac{c}{\sin 75^{\circ}}=k$ $\Rightarrow \frac{a}{\frac{1}{\sqrt{2}}}=\frac{b}{\frac{\sqrt{3}}{2}}=\frac{c}{\frac{1}{2 \sqrt{2}}(1+\sqrt{3})} \quad\left[\because \sin 75^{\circ}=\sin \left(30^{\circ}+45^{\circ}\right)=\sin 30^{\circ} \cos 45^{\circ}+\sin 45^{\circ} \cos 30^{\circ}\right]$ On mul...
Read More →Write the maximum and minimum values of sin θ.
Question: Write the maximum and minimum values of $\sin \theta$ Solution: The maximum value of $\sin \theta$ is 1 and the minimum value of $\sin \theta$ is $-1$ because value of $\sin \theta$ lies between $-1$ and 1...
Read More →Refer to Example 9. How many packets of each food should be used to maximize the amount of vitamin A in the diet?
Question: Refer to Example 9. How many packets of each food should be used to maximize the amount of vitamin A in the diet? What is the maximum amount of vitamin A in the diet? Solution: Let the diet containxandypackets of foods P and Q respectively. Therefore, $x \geq 0$ and $y \geq 0$ The mathematical formulation of the given problem is as follows. Maximize $z=6 x+3 y$ subject to the constraints, $4 x+y \geq 80$ (1) $x+5 y \geq 115$ (2) $3 x+2 y \leq 150$ (3) $x, y \geq 0$ (4) The feasible reg...
Read More →Refer to Example 9. How many packets of each food should be used to maximize the amount of vitamin A in the diet?
Question: Refer to Example 9. How many packets of each food should be used to maximize the amount of vitamin A in the diet? What is the maximum amount of vitamin A in the diet? Solution: Let the diet containxandypackets of foods P and Q respectively. Therefore, $x \geq 0$ and $y \geq 0$ The mathematical formulation of the given problem is as follows. Maximize $z=6 x+3 y$ subject to the constraints, $4 x+y \geq 80$ (1) $x+5 y \geq 115$ (2) $3 x+2 y \leq 150$ (3) $x, y \geq 0$ (4) The feasible reg...
Read More →If sec 2A = cosec (A − 42°),
Question: If $\sec 2 A=\operatorname{cosec}\left(A-42^{\circ}\right)$, where $2 A$ is an acute angles, find the value of $A$. Solution: Given: $\sec 2 A=\operatorname{cosec}\left(A-42^{\circ}\right)$ and $2 A$ is an acute angle We have to find $\theta$ So we proceed as follows to calculate $\theta$ $\sec 2 A=\operatorname{cosec}\left(A-42^{\circ}\right)$ $\Rightarrow \sec 2 A=\sec \left\{90^{\circ}-\left(A-42^{\circ}\right)\right\}$ $\Rightarrow \sec 2 A=\sec \left(90^{\circ}-A+42^{\circ}\right)...
Read More →If sec 4A = cosec (A − 20°),
Question: If $\sec 4 A=\operatorname{cosec}\left(A-20^{\circ}\right)$, where $4 A$ is an acute angles, find the value of $A$. Solution: Given: $\sec 4 A=\operatorname{cosec}\left(A-20^{\circ}\right)$ and $4 A$ is an acute angle We have to find $\theta$ Now $\sec 4 A=\operatorname{cosec}\left(A-20^{\circ}\right)$ $\sec 4 A=\sec \left\{90^{\circ}-\left(A-20^{\circ}\right)\right\}$ $\sec 4 A=\sec \left(90^{\circ}-A+20^{\circ}\right)$ $\sec 4 A=\sec \left(110^{\circ}-A\right)$ $5 A=110^{\circ}$ $A=2...
Read More →If sin 3 θ = cos (θ − 6°),
Question: If $\sin 3 \theta=\cos \left(\theta-6^{\circ}\right)$, where $3 \theta$ and $\theta-6^{\circ}$ are acute angles, find the value of $\theta$. Solution: We have: $\sin 3 \theta=\cos \left(\theta-6^{\circ}\right)$ where $3 \theta$ and $\left(\theta-6^{\circ}\right)$ are acute angles We have to find $\theta$ Now we proceed as to find $\theta$ $\sin 3 \theta=\cos \left(\theta-6^{\circ}\right)$ $\Rightarrow \sin 3 \theta=\sin \left[90^{\circ}-\left(\theta-6^{\circ}\right)\right]$ $\Rightarro...
Read More →Write each of the following in decimal form and say what kind of decimal expansion each has.
Question: Write each of the following in decimal form and say what kind of decimal expansion each has. (i) $\frac{5}{8}$ (ii) $\frac{7}{25}$ (iii) $\frac{3}{11}$ (iv) $\frac{5}{13}$ (v) $\frac{11}{24}$ (vi) $\frac{261}{400}$ (vii) $\frac{231}{625}$ (viii) $2 \frac{5}{12}$ Solution: (i) $\frac{5}{8}=0.625$ By actual division, we have It is a terminating decimal expansion. (ii) $\frac{7}{25}$ $\frac{7}{25}=0.28$ By actual division, we have: It is a terminating decimal expansion. (iii) $\frac{3}{11...
Read More →If cos 2θ = sin 4θ, where 2θ and 4θ are acute angles
Question: If $\cos 2 \theta=\sin 4 \theta$, where $2 \theta$ and $4 \theta$ are acute angles, find the value of $\theta$. Solution: We have: $\cos 2 \theta=\sin 4 \theta$ Given in question $2 \theta$ and $4 \theta$ are acute angles. We have to find $\theta$ Now we have $\cos 2 \theta=\sin 4 \theta$ $\Rightarrow \sin \left(90^{\circ}-2 \theta\right)=\sin 4 \theta$ $\Rightarrow 90^{\circ}-2 \theta=4 \theta$ $\Rightarrow 6 \theta=90^{\circ}$ Therefore $\theta=15^{\circ}$...
Read More →The corner points of the feasible region determined by the following system of linear inequalities:
Question: The corner points of the feasible region determined by the following system of linear inequalities: $2 x+y \leq 10, x+3 y \leq 15, x y \geq 0$ are $(0,0),(5,0),(3,4)$ and $(0,5)$ Let $Z=p x+q y$, where $p, q0$. Condition on $p$ and $q$ so that the maximum of $Z$ occurs at both $(3,4)$ and $(0,5)$ is (A) $p=q$ (B) $p=2 q$ (C) $p=3 q$ (D) $q=3 p$ Solution: The maximum value of Z is unique. It is given that the maximum value of Z occurs at two points, (3, 4) and (0, 5). $\therefore$ Value...
Read More →If θ is a positive acute such that sec θ = cosec 60°
Question: If $\theta$ is a positive acute such that $\sec \theta=\operatorname{cosec} 60^{\circ}$, find the value of $2 \cos ^{2} \theta-1$. Solution: We have: $\sec \theta=\operatorname{cosec} 60^{\circ}$ where $\theta$ is positive acute angle $\Rightarrow \operatorname{cosec}\left(90^{\circ}-\theta\right)=\operatorname{cosec} 60^{\circ}$ $\Rightarrow 90^{\circ}-\theta=60^{\circ}$ $\Rightarrow \theta=30^{\circ}$ Now we have to find $2 \cos ^{2} \theta-1$ Put $\theta=30^{\circ}$ $=2 \times \cos ...
Read More →There are two types of fertilizers F1 and F2.
Question: There are two types of fertilizers F1and F2. F1consists of 10% nitrogen and 6% phosphoric acid and F2consists of 5% nitrogen and 10% phosphoric acid. After testing the soil conditions, a farmer finds that she needs at least 14 kg of nitrogen and 14 kg of phosphoric acid for her crop. If F1cost Rs 6/kg and F2costs Rs 5/kg, determine how much of each type of fertilizer should be used so that nutrient requirements are met at a minimum cost. What is the minimum cost? Solution: Let the farm...
Read More →If 2θ + 45° and 30° − θ are acute angles,
Question: If $2 \theta+45^{\circ}$ and $30^{\circ}-\theta$ are acute angles, find the degree measure of $\theta$ satisfying $\sin \left(2 \theta+45^{\circ}\right)=\cos \left(30^{\circ}-\theta\right)$. Solution: Given that: $\sin \left(2 \theta+45^{\circ}\right)=\cos \left(30^{\circ}-\theta\right)$ where $\left(2 \theta+45^{\circ}\right)$ and $\left(30^{\circ}-\theta\right)$ are acute angles We have to find $\theta$ So we have $\sin \left(2 \theta+45^{\circ}\right)=\cos \left(30^{\circ}-\theta\ri...
Read More →If A, B, C are the interior angles of a ∆ABC, show that :
Question: If A, B, C are the interior angles of a ∆ABC, show that : (i) $\sin \frac{B+C}{2}=\cos \frac{A}{2}$ (ii) $\cos \frac{B+C}{2}=\sin \frac{A}{2}$ Solution: (i) We have to prove: $\sin \left(\frac{B+C}{2}\right)=\cos \frac{A}{2}$ Since we know that in triangle $A B C$ $A+B+C=180^{\circ}$ $\Rightarrow B+C=180^{\circ}-A$ Dividing by 2 on both sides, we get $\Rightarrow \frac{B+C}{2}=90^{\circ}-\frac{A}{2}$ $\Rightarrow \sin \frac{B+C}{2}=\sin \left(90^{\circ}-\frac{A}{2}\right)$ $\Rightarrow...
Read More →If sin θ = cos (θ − 45°), where θ and θ − 45° are acute angles
Question: If $\sin \theta=\cos \left(\theta-45^{\circ}\right)$, where $\theta$ and $\theta-45^{\circ}$ are acute angles, find the degree measure of $\theta .$ Solution: Given that: $\sin \theta=\cos \left(\theta-45^{\circ}\right)$ where $\theta$ and $\left(\theta-45^{\circ}\right)$ are acute angles We have to find $\theta$ $\sin \theta=\cos \left(\theta-45^{\circ}\right)$ $\Rightarrow \cos \left(90^{\circ}-\theta\right)=\cos \left(\theta-45^{\circ}\right)$ $\Rightarrow 90^{\circ}-\theta=\theta-4...
Read More →Evaluate :
Question: Evaluate : (i) $\frac{2}{3}\left(\cos ^{4} 30-\sin ^{4} 45^{\circ}\right)-3\left(\sin ^{2} 60^{\circ}-\sec ^{2} 45^{\circ}\right)+\frac{1}{4} \cot ^{2} 30^{\circ}$ (ii) $4\left(\sin ^{4} 30^{\circ}+\cos ^{4} 60^{\circ}\right)-\frac{2}{3}\left(\sin ^{2} 60^{\circ}-\cos ^{2} 45^{\circ}\right)+\frac{1}{2} \tan ^{2} 60^{\circ}$ (iii) $\frac{\sin 50^{\circ}}{\cos 40^{\circ}}+\frac{\operatorname{cosec} 40^{\circ}}{\sec 50^{\circ}}-4 \cos 50^{\circ} \operatorname{cosec} 40^{\circ}$ (iv) $\tan...
Read More →Write actual division, find which of the following rational numbers are terminating decimals.
Question: Write actual division, find which of the following rational numbers are terminating decimals. (i) $\frac{13}{80}$ (ii) $\frac{7}{24}$ (iii) $\frac{5}{12}$ (iv) $\frac{31}{375}$ (v) $\frac{16}{125}$ Solution: (i) $\frac{13}{80}$ Denominator of $\frac{13}{80}$ is 80 . And, $80=2^{4} \times 5$ Therefore, 80 has no other factors than 2 and 5. Thus, $\frac{13}{80}$ is a terminating decimal. (ii) $\frac{7}{24}$ Denominator of $\frac{7}{24}$ is 24 . And, $24=2^{3} \times 3$ So, 24 has a prime...
Read More →A diet is to contain at least 80 units of vitamin A and 100 units of minerals.
Question: A diet is to contain at least 80 units of vitamin A and 100 units of minerals. Two foods F1and F2are available. Food F1costs Rs 4 per unit food and F2costs Rs 6 per unit. One unit of food F1contains 3 units of vitamin A and 4 units of minerals. One unit of food F2contains 6 units of vitamin A and 3 units of minerals. Formulate this as a linear programming problem. Find the minimum cost for diet that consists of mixture of these two foods and also meets the minimal nutritional requireme...
Read More →If sinx+cosx=a,
Question: If $\sin x+\cos x=a$, find the value of $|\sin x-\cos x|$. Solution: Given: $\sin x+\cos x=a$ Now, $(\sin x+\cos x)^{2}+(\sin x-\cos x)^{2}=\sin ^{2} x+\cos ^{2} x+2 \sin x \cos x+\sin ^{2} x+\cos ^{2} x-2 \sin x \cos x$ $\Rightarrow(\sin x+\cos x)^{2}+(\sin x-\cos x)^{2}=2\left(\sin ^{2} x+\cos ^{2} x\right)$ $\Rightarrow(\sin x+\cos x)^{2}+(\sin x-\cos x)^{2}=2$ $\therefore a^{2}+(\sin x-\cos x)^{2}=2$ $\Rightarrow(\sin x-\cos x)^{2}=2-a^{2}$ $\Rightarrow \sqrt{(\sin x-\cos x)^{2}}=\...
Read More →If sinx+cosx=a,
Question: If $\sin x+\cos x=a$, then find the value of $\sin ^{6} x+\cos ^{6} x$. Solution: Given: $\sin x+\cos x=a$ Squaring on both sides, we get $\sin ^{2} x+\cos ^{2} x+2 \sin x \cos x=a^{2}$ $\Rightarrow 1+2 \sin x \cos x=a^{2}$ $\Rightarrow \sin x \cos x=\frac{a^{2}-1}{2}$ ...(1) Now, $\sin ^{6} x+\cos ^{6} x$ $=\left(\sin ^{2} x+\cos ^{2} x\right)^{3}-3 \sin ^{2} x \cos ^{2} x\left(\sin ^{2} x+\cos ^{2} x\right)$ $=1-3\left(\frac{a^{2}-1}{2}\right)^{2} \quad[\operatorname{Using}(1)]$ $=\f...
Read More →If tanA=
Question: If $\tan A=\frac{1-\cos B}{\sin B}$, then find the value of $\tan 2 A$. Solution: Given, $\tan A=\frac{1-\cos B}{\sin B}$ $\Rightarrow \tan A=\frac{2 \sin ^{2} \frac{B}{2}}{2 \sin \frac{B}{2} \cos \frac{B}{2}} \quad\left(1-\cos 2 \theta=2 \sin ^{2} \theta\right.$ and $\left.\sin 2 \theta=2 \sin \theta \cos \theta\right)$ $\Rightarrow \tan A=\frac{\sin \frac{B}{2}}{\cos \frac{B}{2}}=\tan \frac{B}{2}$ $\Rightarrow A=\frac{B}{2}$ $\Rightarrow 2 A=B$ $\therefore \tan 2 A=\tan B$ Hence, the...
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