Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. $\frac{\cos \theta}{1+\sin \theta}=\frac{1-\sin \theta}{\cos \theta}$ Solution: We know that, $\sin ^{2} \theta+\cos ^{2} \theta=1$ Multiplying the both numerator and the denominator by $(1-\sin \theta)$, we have $\frac{\cos \theta}{1+\sin \theta}=\frac{\cos \theta(1-\sin \theta)}{(1+\sin \theta)(1-\sin \theta)}$ $=\frac{\cos \theta(1-\sin \theta)}{\left(1-\sin ^{2} \theta\right)}$ $=\frac{\cos \theta(1-\sin \theta)}{\cos ^{2} \theta}$ $=\f...
Read More →If in ΔABC,
Question: If in $\Delta A B C, \cos ^{2} A+\cos ^{2} B+\cos ^{2} C=1$, prove that the triangle is right-angled. Solution: LetABCbe any triangle. In $\triangle \mathrm{ABC}$ $\cos ^{2} A+\cos ^{2} B+\cos ^{2} C=1$ $\Rightarrow \cos ^{2} A+\cos ^{2} B+\cos ^{2}[\pi-(B+A)]=1 \quad(\because A+B+C=\pi)$ $\Rightarrow \cos ^{2} A+\cos ^{2} B+\cos ^{2}(B+A)=1$ $\Rightarrow \cos ^{2} A+\cos ^{2} B=1-\cos ^{2}(B+A)$ $\Rightarrow \cos ^{2} A+\cos ^{2} B=\sin ^{2}(B+A)$ $\Rightarrow \cos ^{2} A+\cos ^{2} B=...
Read More →Represent 7.28−−−−√ geometrically on the number line.
Question: Represent $\sqrt{7.128}$ geometrically on the number line. Solution: To represent $\sqrt{7.28}$ on the number line, follow the following steps of construction: (i)Mark two points A and B on a given line such that AB = 7.28 units.(ii) From B, mark a point C on the same given line such that BC = 1 unit.(iii) Find the mid point of AC and mark it as O.(iv)With O as centre and radius OC, draw a semi-circle touching the given line at points A and C.(v) At point B, draw a line perpendicular t...
Read More →If in ΔABC,
Question: If in $\Delta A B C, \cos ^{2} A+\cos ^{2} B+\cos ^{2} C=1$, prove that the triangle is right-angled. Solution: LetABCbe any triangle. In $\triangle \mathrm{ABC}$ $\cos ^{2} A+\cos ^{2} B+\cos ^{2} C=1$ $\Rightarrow \cos ^{2} A+\cos ^{2} B+\cos ^{2}[\pi-(B+A)]=1 \quad(\because A+B+C=\pi)$ $\Rightarrow \cos ^{2} A+\cos ^{2} B+\cos ^{2}(B+A)=1$ $\Rightarrow \cos ^{2} A+\cos ^{2} B=1-\cos ^{2}(B+A)$ $\Rightarrow \cos ^{2} A+\cos ^{2} B=\sin ^{2}(B+A)$ $\Rightarrow \cos ^{2} A+\cos ^{2} B=...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. $\frac{\cos \theta}{1-\sin \theta}=\frac{1+\sin \theta}{\cos \theta}$ Solution: We know that, $\sin ^{2} \theta+\cos ^{2} \theta=1$ Multiplying both numerator and the denominator by $(1+\sin \theta)$, we have $\frac{\cos \theta}{1-\sin \theta}=\frac{\cos \theta(1+\sin \theta)}{(1-\sin \theta)(1+\sin \theta)}$ $=\frac{\cos \theta(1+\sin \theta)}{\left(1-\sin ^{2} \theta\right)}$ $=\frac{\cos \theta(1+\sin \theta)}{\cos ^{2} \theta}$ $=\frac{...
Read More →Given that the events A and B are such that P(A)
Question: Given that the events $A$ and $B$ are such that $P(A)=\frac{1}{2}, P(A \cup B)=\frac{3}{5}$ and $P(B)=p$. Find $p$ if they are (i) mutually exclusive (ii) independent. Solution: It is given that $\mathrm{P}(\mathrm{A})=\frac{1}{2}, \mathrm{P}(\mathrm{A} \cap \mathrm{B})=\frac{3}{5}$, and $\mathrm{P}(\mathrm{B})=p$ (i) When $A$ and $B$ are mutually exclusive, $A \cap B=\Phi$ $\therefore P(A \cap B)=0$ It is known that, $\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A})+\math...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. $\tan \theta+\frac{1}{\tan \theta}=\sec \theta \operatorname{cosec} \theta$ Solution: We know that, $\sec ^{2} \theta-\tan ^{2} \theta=1$ So, $\tan \theta+\frac{1}{\tan \theta}=\frac{\tan ^{2} \theta+1}{\tan \theta}$ $=\frac{\sec ^{2} \theta}{\tan \theta}$ $=\sec \theta \frac{\sec \theta}{\tan \theta}$ $=\sec \theta \frac{\frac{1}{\cos \theta}}{\frac{\sin \theta}{\cos \theta}}$ $=\sec \theta \frac{1}{\sin \theta}$ $=\sec \theta \operatornam...
Read More →Let E and F be events with
Question: Let $E$ and $F$ be events with $P(E)=\frac{3}{5}, P(F)=\frac{3}{10}$ and $P(E \cap F)=\frac{1}{5} .$ Are $E$ and $F$ independent? Solution: It is given that $\mathrm{P}(\mathrm{E})=\frac{3}{5}, \mathrm{P}(\mathrm{F})=\frac{3}{10}$, and $\mathrm{P}(\mathrm{EF})=\mathrm{P}(\mathrm{E} \cap \mathrm{F})=\frac{1}{5}$ $\mathrm{P}(\mathrm{E}) \cdot \mathrm{P}(\mathrm{F})=\frac{3}{5} \cdot \frac{3}{10}=\frac{9}{50} \neq \frac{1}{5}$ $\Rightarrow \mathrm{P}(\mathrm{E}) \cdot \mathrm{P}(\mathrm{F...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. $\left(\sec ^{2} \theta-1\right)\left(\operatorname{cosec}^{2} \theta-1\right)=1$ Solution: We know that, $\sec ^{2} \theta-\tan ^{2} \theta=1$ $\operatorname{cosec}^{2} \theta-\cot ^{2} \theta=1$ So, $\left(\sec ^{2} \theta-1\right)\left(\operatorname{cosec}^{2} \theta-1\right)=\tan ^{2} \theta \times \cot ^{2} \theta$ $=(\tan \theta \times \cot \theta)^{2}$ $=\left(\tan \theta \times \frac{1}{\tan \theta}\right)^{2}$ $=(1)^{2}$ $=1$...
Read More →Represent 10.5−−−−√ on the number line.
Question: Represent $\sqrt{10.15}$ on the number line Solution: To represent $\sqrt{10.5}$ on the number line, follow the following steps of construction: (i)Mark two points A and B on a given line such that AB = 10.5 units.(ii) From B, mark a point C on the same given line such that BC = 1 unit.(iii) Find the mid point of AC and mark it as O.(iv)With O as centre and radius OC, draw a semi-circle touching the given line at points A and C.(v) At point B, draw a line perpendicular to AC intersecti...
Read More →In ΔABC,
Question: In $\Delta A B C$, if $\angle B=60^{\circ}$, prove that $(a+b+c)(a-b+c)=3 c a$. Solution: Given, $\angle B=60^{\circ}$ We know that, $\cos B=\frac{a^{2}+c^{2}-b^{2}}{2 a c}$ $\Rightarrow \cos 60^{\circ}=\frac{a^{2}+c^{2}-b^{2}}{2 a c}$ $\Rightarrow \frac{1}{2}=\frac{a^{2}+c^{2}-b^{2}}{2 a c} \quad\left(\because \cos 60^{\circ}=\frac{1}{2}\right)$ $\Rightarrow a c=a^{2}+c^{2}-b^{2}$ $\Rightarrow 3 a c-2 a c=a^{2}+c^{2}-b^{2}$ $\Rightarrow 3 a c=a^{2}+c^{2}+2 a c-b^{2}$ $\Rightarrow 3 a ...
Read More →A die marked 1, 2, 3 in red and 4, 5, 6 in green is tossed. Let A be the event,
Question: A die marked 1, 2, 3 in red and 4, 5, 6 in green is tossed. Let A be the event, the number is even, and B be the event, the number is red. Are A and B independent? Solution: When a die is thrown, the sample space (S) is S = {1, 2, 3, 4, 5, 6} Let A: the number is even = {2, 4, 6} $\Rightarrow \mathrm{P}(\mathrm{A})=\frac{3}{6}=\frac{1}{2}$ B: the number is red $=\{1,2,3\}$ $\Rightarrow \mathrm{P}(\mathrm{B})=\frac{3}{6}=\frac{1}{2}$ $\therefore \mathrm{A} \cap \mathrm{B}=\{2\}$ $P(A B)...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. $\operatorname{cosec} \theta \sqrt{1-\cos ^{2} \theta}=1$ Solution: We know that, $\sin ^{2} \theta+\cos ^{2} \theta=1$ So, $\operatorname{cosec} \theta \sqrt{1-\cos ^{2} \theta}=\operatorname{cosec} \theta \sqrt{\sin ^{2} \theta}$ $=\operatorname{cosec} \theta \sin \theta$ $=\frac{1}{\sin \theta} \times \sin \theta$ $=1$...
Read More →In Δ ABC,
Question: In $\Delta A B C, \frac{b+c}{12}=\frac{c+a}{13}=\frac{a+b}{15}$. Prove that $\frac{\cos A}{2}=\frac{\cos B}{7}=\frac{\cos C}{11}$. Solution: Let $\frac{b+c}{12}=\frac{c+a}{13}=\frac{a+b}{15}=k$ $\Rightarrow b+c=12 k, c+a=13 k, a+b=15 k$ $\Rightarrow b+c+c+a+a+b=12 k+13 k+15 k$ $\Rightarrow 2(a+b+c)=40 k$ $\Rightarrow a+c+b=20 k$ $\Rightarrow a+12 k=20 k \quad(\because b+c=12 k)$ $\Rightarrow a=8 k$ Also, $c+a=13 k$ $\Rightarrow c=13 k-a=13 k-8 k=5 k$ and, $a+b=15 k$ $\Rightarrow b=15 k...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. $\tan ^{2} \theta \cos ^{2} \theta=1-\cos ^{2} \theta$ Solution: We know that, $\sin ^{2} \theta+\cos ^{2} \theta=1$. So, $\tan ^{2} \theta \cos ^{2} \theta=(\tan \theta \times \cos \theta)^{2}$ $=\left(\frac{\sin \theta}{\cos \theta} \times \cos \theta\right)^{2}$ $=(\sin \theta)^{2}$ $=\sin ^{2} \theta$ $=1-\cos ^{2} \theta$...
Read More →Represent 4.7−−−√ geometrically on the number line.
Question: Represent $\sqrt{4 . \overline{4}}$ geometrically on the number line. Solution: To represent $\sqrt{4.7}$ on the number line, follow the following steps of construction: (i)Mark two points A and B on a given line such that AB = 4.7 units.(ii) From B, mark a point C on the same given line such that BC = 1 unit.(iii) Find the mid point of AC and mark it as O.(iv)With O as centre and radius OC, draw a semi-circle touching the given line at points A and C.(v) At point B, draw a line perpen...
Read More →A fair coin and an unbiased die are tossed.
Question: A fair coin and an unbiased die are tossed. Let A be the event head appears on the coin and B be the event 3 on the die. Check whether A and B are independent events or not. Solution: If a fair coin and an unbiased die are tossed, then the sample space S is given by, $\mathrm{S}=\left\{\begin{array}{l}(\mathrm{H}, \mathrm{l}),(\mathrm{H}, 2),(\mathrm{H}, 3),(\mathrm{H}, 4),(\mathrm{H}, 5),(\mathrm{H}, 6), \\ (\mathrm{T}, 1),(\mathrm{T}, 2),(\mathrm{T}, 3),(\mathrm{T}, 4),(\mathrm{T}, 5...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. $\left(1+\cot ^{2} A\right) \sin ^{2} A=1$ Solution: We know that, $\operatorname{cosec}^{2} A-\cot ^{2} A=1$ So, $\left(1+\cot ^{2} A\right) \sin ^{2} A=\operatorname{cosec}^{2} A \sin ^{2} A$ $=(\operatorname{cosec} A \sin A)^{2}$ $=\left(\frac{1}{\sin A} \times \sin A\right)^{2}$ $=(1)^{2}$ $=1$...
Read More →In ∆ABC, prove the following:
Question: In ∆ABC, prove the following: $\sin ^{3} A \cos (B-C)+\sin ^{3} B \cos (C-A)+\sin ^{3} C \cos (A-B)=3 \sin A \sin B \sin C$ Solution: Let $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=k \quad \ldots(1)$ $\mathrm{LHS}=\sin ^{3} A \cos (B-C)+\sin ^{3} B \cos (C-A)+\sin ^{3} C \cos (A-B)$ $=\sin ^{2} A\{\sin A \cos (B-C)\}+\sin ^{2} B\{\sin B \cos (C-A)\}+\sin ^{2} A\{\sin A \cos (A-B)\}$ $=\frac{a^{2}}{k^{2}}\{\sin A \cos (B-C)\}+\frac{b^{2}}{k^{2}}\{\sin B \cos (C-A)\}+\frac{c^{2}...
Read More →A box of oranges is inspected by examining three randomly selected oranges drawn without replacement.
Question: A box of oranges is inspected by examining three randomly selected oranges drawn without replacement. If all the three oranges are good, the box is approved for sale, otherwise, it is rejected. Find the probability that a box containing 15 oranges out of which 12 are good and 3 are bad ones will be approved for sale. Solution: Let A, B, and C be the respective events that the first, second, and third drawn orange is good. Therefore, probability that first drawn orange is good, $P(A)=\f...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. $\left(1-\cos ^{2} A\right) \operatorname{cosec}^{2} A=1$ Solution: We know that, $\sin ^{2} A+\cos ^{2} A=1$ so $\left(1-\cos ^{2} A\right) \operatorname{cosec}^{2} A=\sin ^{2} A \operatorname{cosec}^{2} A$ $=(\sin A \operatorname{cosec} A)^{2}$ $=\left(\sin A \times \frac{1}{\sin A}\right)^{2}$ $=(1)^{2}$ $=1$...
Read More →Locate 8–√ on the number line.
Question: Locate $\sqrt{8}$ on the number line. Solution: To represent $\sqrt{8}$ on the number line, follow the following steps of construction: (i) Mark points 0 and 2 as $O$ and $B$, respectively. (ii) At point $B$, draw $A B \perp O A$ such that $A B=2$ units. (iii) Join OA. (iv) With $O$ as centre and radius $O A$, draw an arc intersecting the number line at point $P$. Thus, point P represents $\sqrt{8}$ on the number line. Justification: In right $\triangle \mathrm{OAB}$, Using Pythagoras ...
Read More →In ∆ABC, prove the following:
Question: In ∆ABC, prove the following: $4\left(b c \cos ^{2} \frac{A}{2}+c a \cos ^{2} \frac{B}{2}+a b \cos ^{2} \frac{C}{2}\right)=(a+b+c)^{2}$ Solution: LHS $=4\left(b c \cos ^{2} \frac{A}{2}+c a \cos ^{2} \frac{B}{2}+a b \cos ^{2} \frac{C}{2}\right)$ $=4\left[b c\left(\frac{1+\cos A}{2}\right)+c a\left(\frac{1+\cos B}{2}\right)+a b\left(\frac{1+\cos C}{2}\right)\right]$ $=2 b c+2 b c \cos A+2 c a+2 c a \cos B+2 a b+2 a b \cos C$ $=2(a b+b c+c a)+2 b c\left(\frac{b^{2}+c^{2}-a^{2}}{2 b c}\rig...
Read More →Two cards are drawn at random and without replacement from a pack of 52 playing cards.
Question: Two cards are drawn at random and without replacement from a pack of 52 playing cards. Find the probability that both the cards are black. Solution: There are 26 black cards in a deck of 52 cards. Let P (A) be the probability of getting a black card in the first draw. $\therefore P(A)=\frac{26}{52}=\frac{1}{2}$ Let P (B) be the probability of getting a black card on the second draw. Since the card is not replaced, $\therefore \mathrm{P}(\mathrm{B})=\frac{25}{51}$ Thus, probability of g...
Read More →In ∆ABC, prove the following:
Question: In ∆ABC, prove the following: $a^{2}=(b+c)^{2}-4 b c \cos ^{2} \frac{A}{2}$ Solution: RHS $=(b+c)^{2}-4 b c \cos ^{2} \frac{A}{2}$ $=b^{2}+c^{2}+2 b c-4 b c\left(\frac{1+\cos A}{2}\right)$ $=b^{2}+c^{2}+2 b c-2 b c(1+\cos A)$ $=b^{2}+c^{2}+2 b c(1-1-\cos A)$ $=b^{2}+c^{2}-2 b c \cos A$ $=b^{2}+c^{2}-2 b c\left(\frac{b^{2}+c^{2}-a^{2}}{2 b c}\right) \quad\left(\because \cos A=\frac{b^{2}+c^{2}-a^{2}}{2 b c}\right)$ $=b^{2}+c^{2}-b^{2}-c^{2}+a^{2}$ $=a^{2}=\mathrm{LHS}$ Hence proved....
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