A company manufactures two types of novelty souvenirs made of plywood.
Question: A company manufactures two types of novelty souvenirs made of plywood. Souvenirs of type A require 5 minutes each for cutting and 10 minutes each for assembling. Souvenirs of type B require 8 minutes each for cutting and 8 minutes each for assembling. There are 3 hours 20 minutes available for cutting and 4 hours of assembling. The profit is Rs 5 each for type A and Rs 6 each for type B souvenirs. How many souvenirs of each type should the company manufacture in order to maximize the p...
Read More →If
Question: If $\frac{\pi}{2}x\pi$, then $\sqrt{\frac{1-\cos 2 x}{1+\cos 2 x}}=$ Solution: Given for $\frac{\pi}{2}x\pi$ $\sqrt{\frac{1-\cos 2 x}{1+\cos 2 x}}$ $\left[\right.$ using identities; $\left.1-\cos 2 x=2 \sin ^{2} x 1+\cos 2 x=2 \cos ^{2} x\right]$ $=\sqrt{\frac{2 \sin ^{2} x}{2 \cos ^{2} x}}$ $=\sqrt{\tan ^{2} x}$ $=|\tan x|$ Since $\frac{\pi}{2}x\pi$ and $\tan$ is negative in second quadrant i. e. $|\tan x|=-\tan x$ $\therefore \sqrt{\frac{1-\cos 2 x}{1+\cos 2 x}}=-\tan x$...
Read More →Find a rational number between
Question: Find a rational number between (i) $\frac{3}{8}$ and $\frac{2}{5}$ (ii) $1.3$ and $1.4$ (iii) $-1$ and $\frac{1}{2}$ (iv) $-\frac{3}{4}$ and $-\frac{2}{5}$ (v) $\frac{1}{9}$ and $\frac{2}{9}$ Solution: (i) $\frac{3}{8}$ and $\frac{2}{5}$ Let: $x=\frac{3}{8}$ and $y=\frac{2}{5}$ Rational number lying betweenxandy: $\frac{1}{2}(x+y)=\frac{1}{2}\left(\frac{3}{8}+\frac{2}{5}\right)$ $=\frac{1}{2}\left(\frac{15+16}{40}\right)=\frac{31}{80}$ (ii) 1.3 and 1.4Let:x= 1.3 andy= 1.4Rational numbe...
Read More →Evaluate the following :
Question: Evaluate the following : (i) $\left(\frac{\sin 49^{\circ}}{\cos 41^{\circ}}\right)^{2}+\left(\frac{\cos 41^{\circ}}{\sin 49^{\circ}}\right)^{2}$ (ii) $\cos 48^{\circ}-\sin 42^{\circ}$ (iii) $\frac{\cot 40^{\circ}}{\tan 50^{\circ}}-\frac{1}{2}\left(\frac{\cos 35^{\circ}}{\sin 55^{\circ}}\right)$ (iv) $\left(\frac{\sin 27^{*}}{\cos 63^{*}}\right)^{2}-\left(\frac{\cos 63^{*}}{\sin 27^{*}}\right)^{2}$ (v) $\frac{\tan 35^{\circ}}{\cot 55^{\circ}}+\frac{\cot 78^{\circ}}{\tan 12^{\circ}}-1$ (...
Read More →A cottage industry manufactures pedestal lamps and wooden shades,
Question: A cottage industry manufactures pedestal lamps and wooden shades, each requiring the use of a grinding/cutting machine and a sprayer. It takes 2 hours on grinding/cutting machine and 3 hours on the sprayer to manufacture a pedestal lamp. It takes 1 hour on the grinding/cutting machine and 2 hours on the sprayer to manufacture a shade. On any day, the sprayer is available for at the most 20 hours and the grinding/cutting machine for at the most 12 hours. The profit from the sale of a la...
Read More →The value of cos
Question: The value of $\cos ^{2} 6^{\circ}-\cos ^{2} 24^{\circ}$ is Solution: $\cos ^{2} 6^{\circ}-\cos ^{2} 24^{\circ}$ $=\left(\cos 6^{\circ}-\cos 24^{\circ}\right)\left(\cos 6^{\circ}+\cos 24^{\circ}\right)$ $=2 \sin \left(\frac{6^{\circ}+24^{\circ}}{2}\right) \sin \left(\frac{24^{\circ}-6^{\circ}}{2}\right) 2 \cos \left(\frac{6^{\circ}+24^{\circ}}{2}\right) \cos \left(\frac{6^{\circ}-24^{\circ}}{2}\right)$ $=2 \sin 15^{\circ} \sin 9^{\circ} 2 \cos 15^{\circ} \cos \left(-9^{\circ}\right)$ $\...
Read More →Represent each of the following rational number line:
Question: Represent each of the following rational number line: (i) $\frac{5}{7}$ (ii) $\frac{8}{3}$ (iii) $-\frac{23}{6}$ (iv) $1.3$ (v) $-2.4$ Solution: (i) $\frac{5}{7}$ (ii) $\frac{8}{3}$ $\frac{8}{3}=2 \frac{2}{3}$ (iii) $-\frac{23}{6}=-3 \frac{5}{6}$ (iv) $1.3$ 1. $3=\frac{13}{10}=1 \frac{3}{10}$ (v) $-2.4$ $-2.4=\frac{-24}{10}=\frac{-12}{5}=-2 \frac{2}{5}$...
Read More →A factory manufactures two types of screws,
Question: A factory manufactures two types of screws, A and B. Each type of screw requires the use of two machines, an automatic and a hand operated. It takes 4 minutes on the automatic and 6 minutes on hand operated machines to manufacture a package of screws A, while it takes 6 minutes on automatic and 3 minutes on the hand operated machines to manufacture a package of screws B. Each machine is available for at the most 4 hours on any day. The manufacturer can sell a package of screws A at a p...
Read More →A factory manufactures two types of screws,
Question: A factory manufactures two types of screws, A and B. Each type of screw requires the use of two machines, an automatic and a hand operated. It takes 4 minutes on the automatic and 6 minutes on hand operated machines to manufacture a package of screws A, while it takes 6 minutes on automatic and 3 minutes on the hand operated machines to manufacture a package of screws B. Each machine is available for at the most 4 hours on any day. The manufacturer can sell a package of screws A at a p...
Read More →The value of cos
Question: The value of $\cos \frac{\pi}{5} \cos \frac{2 \pi}{5} \cos \frac{4 \pi}{5} \cos \frac{8 \pi}{5}$ is Solution: $\cos \frac{\pi}{5} \cos \frac{2 \pi}{5} \cos \frac{4 \pi}{5} \cos \frac{8 \pi}{5}$ multiply and divide above equation by $2 \sin \frac{\pi}{5}$ $=\frac{1}{2 \sin \frac{\pi}{5}}\left(2 \sin \frac{\pi}{5} \cos \frac{\pi}{5} \cos \frac{2 \pi}{5} \cos \frac{4 \pi}{5} \cos \frac{8 \pi}{5}\right)$ $=\frac{1}{2 \sin \frac{\pi}{5}}\left(\sin \frac{2 \pi}{5} \cos \frac{2 \pi}{5} \cos \...
Read More →Is zero a rational number? Justify.
Question: Is zero a rational number? Justify. Solution: Yes, 0 is a rational number. 0 can be expressed in the form of the fraction $\frac{p}{q}$, where $p=0$ and $q$ can be any integer except 0 ....
Read More →If tan x=
Question: If $\tan x=\frac{1}{7}$, tan $y=\frac{1}{3}$ and $\cos 2 x=\sin k y$, then $k=$ _______________________ Solution: If $\tan x=\frac{1}{7}, \tan y=\frac{1}{3}$ $\cos 2 x=\frac{1-\tan ^{2} x}{1+\tan ^{2} x}$ $\cos 2 x=\frac{1-\frac{1}{49}}{1+\frac{1}{49}}=\frac{\frac{48}{49}}{\frac{50}{49}}=\frac{24}{25}$ since, $\sin 2 y=\frac{2 \tan y}{1+\tan ^{2} y}=\frac{2\left(\frac{1}{3}\right)}{1+\frac{1}{4}}=\frac{\frac{2}{3}}{\frac{10}{4}}=\frac{2}{10} \times \frac{9}{3}=\frac{3}{5} \quad \ldots ...
Read More →A manufacturer produces nuts ad bolts.
Question: A manufacturer produces nuts ad bolts. It takes 1 hour of work on machine A and 3 hours on machine B to produce a package of nuts. It takes 3 hours on machine A and 1 hour on machine B to produce a package of bolts. He earns a profit, of Rs 17.50 per package on nuts and Rs. 7.00 per package on bolts. How many packages of each should be produced each day so as to maximize his profit, if he operates his machines for at the most 12 hours a day? Solution: Let the manufacturer producexpacka...
Read More →Evaluate the following :
Question: Evaluate the following : (i) $\frac{\sin 20^{\circ}}{\cos 70^{\circ}}$ (ii) $\frac{\cos 19^{\circ}}{\sin 71^{\circ}}$ (iii) $\frac{\sin 21^{\circ}}{\cos 69^{\circ}}$ (iv) $\frac{\tan 10^{\circ}}{\cot 80^{\circ}}$ (V) $\frac{\sec 11^{\circ}}{\operatorname{cosec} 79^{\circ}}$ Solution: (i) Given that $\frac{\sin 20}{\cos 70}$ Since $\sin (90-\theta)=\cos \theta$ $\Rightarrow \frac{\sin 20}{\cos 70}=\frac{\sin (90-70)}{\cos 70}$ $\Rightarrow \frac{\sin 20}{\cos 70}=\frac{\cos 70}{\cos 70}...
Read More →If tan θ=
Question: If $\tan \theta=\frac{a}{b}$, then $a \sin 2 \theta+b \cos 2 \theta$ is equal to __________________ Solution: Given $\tan \theta=\frac{a}{b}$ $a \sin 2 \theta+b \cos 2 \theta$ $=a\left(\frac{2 \tan \theta}{1+\tan ^{2} \theta}\right)+b\left(\frac{1-\tan ^{2} \theta}{1+\tan ^{2} \theta}\right)$ $\left[\right.$ using identities : $\sin 2 \theta=\frac{2 \tan \theta}{1+\tan ^{2} \theta}$ and $\left.\cos 2 \theta=\frac{1-\tan ^{2} \theta}{1+\tan ^{2} \theta}\right]$ $=\frac{2 a \tan \theta+b...
Read More →In ∆PQR, right-angled at Q, PQ = 3 cm and PR = 6 cm.
Question: In $\triangle P Q R$, right-angled at $Q, P Q=3 \mathrm{~cm}$ and $P R=6 \mathrm{~cm}$. Determine $\angle P$ and $\angle R$. Solution: We are given the following information in the form of triangle To find: $\angle P$ and $\angle R$ Now, in $\triangle P Q R$ $\cos P=\frac{P Q}{P R}$ $\cos P=\frac{3}{6}$ (1) $=\frac{1}{2}$ Now we know that $\cos 60^{\circ}=\frac{1}{2}$...(2) Now by comparing equation (1) and (2) We get, $\angle P=60^{\circ} \ldots \ldots(3)$ Now we have $\sin P=\frac{Q ...
Read More →If tan θ = t,
Question: If tan=t, then tan 2+ sec 2= ___________. Solution: Given $\tan \theta=t$ $\tan 2 \theta+\sec 2 \theta$ $=\frac{2 \tan \theta}{1-\tan ^{2} \theta}+\frac{1+\tan ^{2} \theta}{1-\tan ^{2} \theta} \quad\left[\right.$ using identity $\left.\tan 2 \theta=\frac{2 \tan \theta}{1-\tan ^{2} \theta} \sec 2 \theta=\frac{1+\tan ^{2} \theta}{1-\tan ^{2} \theta}\right]$ $=\frac{1+\tan ^{2} \theta+2 \tan \theta}{\left(1-\tan ^{2} \theta\right)}$ $=\frac{(1+\tan \theta)^{2}}{1-\tan ^{2} \theta}$ $=\fra...
Read More →A factory makes tennis rackets and cricket bats.
Question: A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftsmans time in its making while a cricket bat takes 3 hour of machine time and 1 hour of craftsmans time. In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftsmans time. (ii) What number of rackets and bats must be made if the factory is to work at full capacity? (ii) If the profit on a racket and on a bat is Rs 20 and...
Read More →The value of
Question: The value of $\frac{\cot x-\tan x}{\cot 2 x}$ is__________ Solution: $\frac{\cot x-\tan x}{\cot 2 x}$ $=\frac{\frac{1}{\tan x}-\tan x}{\frac{1}{\tan 2 x}}$ $=\frac{\frac{1-\tan ^{2} x}{\tan x}}{\frac{1}{\tan 2 x}}$ $=\frac{1-\tan ^{2} x}{\tan x} \times \tan ^{2} x$ $=\frac{1-\tan ^{2} x}{\tan x} \times \frac{2 \tan x}{1-\tan ^{2} x}$ Therefore, $\frac{\cot x-\tan x}{\cot 2 x}=2$...
Read More →The value of
Question: The value of $\left(1+\cos \frac{\pi}{8}\right)\left(1+\cos \frac{3 \pi}{8}\right)\left(1+\cos \frac{5 \pi}{8}\right)\left(1+\cos \frac{7 \pi}{8}\right)$ is ___________________ Solution: $\left(1+\cos \frac{\pi}{8}\right)\left(1+\cos \frac{3 \pi}{8}\right)\left(1+\cos \frac{5 \pi}{8}\right)\left(1+\cos \frac{7 \pi}{8}\right)$ $=\left(1+\cos \frac{\pi}{8}\right)\left(1+\cos \frac{3 \pi}{8}\right)\left(1+\cos \left(\pi-\frac{3 \pi}{8}\right)\right)\left(1+\cos \left(\pi-\frac{\pi}{8}\rig...
Read More →If cos
Question: If cos6x+ sin6x+ksin22x= 1, thenk= ___________. Solution: $\cos ^{6} x+\sin ^{6} x+k \sin ^{2} 2 x=1 \quad$ (Given) L . H . S . $=\left(\cos ^{2} x\right)^{3}+\left(\sin ^{2} x\right)^{3}+k \sin ^{2} 2 x$ $=\left(\cos ^{2} x+\sin ^{2} x\right)^{3}+k \sin ^{2} 2 x+3 \sin ^{2} x \cos ^{4} x+3 \sin ^{4} x \cos ^{2} x$ $\left[\right.$ Since $\left.(a+b)^{3}=a^{3}+b^{3}-3 a b(a+b)\right]$ $=(1)^{3}+3 \sin ^{2} x \cos ^{4} x+3 \sin ^{4} x \cos ^{2} x+k \sin ^{2} 2 x$ $=1+3 \sin ^{2} x \cos ^...
Read More →If A and B are acute angles such that tan A=12,
Question: If $\mathrm{A}$ and $\mathrm{B}$ are acute angles such that $\tan A=\frac{1}{2}, \tan B=\frac{1}{3}$ and $\tan (A+B)=\frac{\tan A+\tan B}{1-\tan A \tan B}$, find $A+B$. Solution: Given: $\tan A=\frac{1}{2}$...(1) $\tan B=\frac{1}{3}$....(2) $\tan (A+B)=\frac{\tan A+\tan B}{1-\tan A \tan B} \ldots \ldots$(3) Now by substituting the value of $\tan A$ and $\tan B$ from equation (1) and (2) in equation (3) We get, $\tan (A+B)=\frac{\left(\frac{1}{2}\right)+\left(\frac{1}{3}\right)}{1-\left...
Read More →One kind of cake requires 200g flour and 25g of fat,
Question: One kind of cake requires 200g flour and 25g of fat, and another kind of cake requires 100g of flour and 50g of fat. Find the maximum number of cakes which can be made from 5 kg of flour and 1 kg of fat assuming that there is no shortage of the other ingredients used in making the cakes? Solution: Let there bexcakes of first kind andycakes of second kind. Therefore, $x \geq 0$ and $y \geq 0$ The given information can be complied in a table as follows. $\therefore 200 x+100 y \leq 5000$...
Read More →The least value of 2 sin
Question: The least value of 2 sin2+ 3cos2is ___________. Solution: $2 \sin ^{2} \theta+3 \cos ^{2} \theta$ $=2\left(\sin ^{2} \theta+\cos ^{2} \theta\right)+\cos ^{2} \theta$ $=2+\cos ^{2} \theta$ Since $-1 \leq \cos \theta \leq 1$ $\Rightarrow 0 \leq \cos ^{2} \theta \leq 1$ $\therefore 2 \sin ^{2} \theta+3 \cos ^{2} \theta \geq 2+0$ $2 \sin ^{2} \theta+3 \cos ^{2} \theta \geq 2$ i.e. least value of $2 \sin ^{2} \theta+3 \cos ^{2}$ is $2 .$...
Read More →Find acute angles A and B, if sin (A+2B)=3√2 and cos (A+4B)=0, A>B.
Question: Find acute angles $\mathrm{A}$ and $\mathrm{B}$, if $\sin (A+2 B)=\frac{\sqrt{3}}{2}$ and $\cos (A+4 B)=0, AB$ Solution: Given: $\sin (A+2 B)=\frac{\sqrt{3}}{2}$....(1) $\cos (A+4 B)=0$....(2) We know that, $\sin 60^{\circ}=\frac{\sqrt{3}}{2}$....(3) $\cos 90^{\circ}=0$....(4) Now by comparing equation (1) and (3) We get, $A+2 B=60 \ldots \ldots(5)$ Now by comparing equation (2) and (4) We get, $A+4 B=90 \ldots \ldots(6)$ Now to get the values ofAandB, let us solve equation (5) and (6)...
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