Given two independent events A and B such that
Question: Given two independent events A and B such that P (A) = 0.3, P (B) = 0.6. Find (i) $P(A$ and $B)$ ] (ii) $P$ ( $A$ and not $B$ ) (iii) $P$ ( $A$ or $B$ ) (iv) $P$ (neither $A$ nor $B$ ) Solution: It is given that P (A) = 0.3 and P (B) = 0.6 Also, A and B are independent events. (i) $\therefore \mathrm{P}(\mathrm{A}$ and $\mathrm{B})=\mathrm{P}(\mathrm{A}) \cdot \mathrm{P}(\mathrm{B})$ $\Rightarrow \mathrm{P}(\mathrm{A} \cap \mathrm{B})=0.3 \times 0.6=0.18$ (ii) $P(A$ and not $B)=P\left(...
Read More →Mark the correct alternative in each of the following:
Question: Mark the correct alternative in each of the following: In any ∆ABC, 2(bccosA+cacosB+abcosC) = (a) $a b c$ (b) $a+b+c$ (c) $a^{2}+b^{2}+c^{2}$ (d) $\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}$ Solution: Using cosine rule, we have $2(b c \cos A+c a \cos B+a b \cos C)$ $=2 b c\left(\frac{b^{2}+c^{2}-a^{2}}{2 b c}\right)+2 c a\left(\frac{c^{2}+a^{2}-b^{2}}{2 c a}\right)+2 a b\left(\frac{a^{2}+b^{2}-c^{2}}{2 a b}\right)$ $=b^{2}+c^{2}-a^{2}+c^{2}+a^{2}-b^{2}+a^{2}+b^{2}-c^{2}$ $=a^{2}+b...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. $\frac{\left(1+\cot ^{2} \theta\right) \tan \theta}{\sec ^{2} \theta}=\cot \theta$ Solution: We have to prove $\frac{\left(1+\cot ^{2} \theta\right) \tan \theta}{\sec ^{2} \theta}=\cot \theta$ We know that, $\sec ^{2} \theta-\tan ^{2} \theta=1$ So, $\frac{\left(1+\cot ^{2} \theta\right) \tan \theta}{\sec ^{2} \theta}=\frac{\left(1+\cot ^{2} \theta\right) \tan \theta}{\left(1+\tan ^{2} \theta\right)}$ $=\frac{\left(1+\frac{1}{\tan ^{2} \thet...
Read More →Mark the correct alternative in each of the following:
Question: Mark the correct alternative in each of the following: If the sides of a triangle are in the ratio $1: \sqrt{3}: 2$, then the measure of its greatest angle is (a) $\frac{\pi}{6}$ (b) $\frac{\pi}{3}$ (C) $\frac{\pi}{2}$ (d) $\frac{2 \pi}{3}$ Solution: Let $\triangle \mathrm{ABC}$ be the given triangle such that its sides are in the ratio $1: \sqrt{3}: 2$. $\therefore a=k, b=\sqrt{3} k, c=2 k$ Now, $a^{2}+b^{2}=k^{2}+3 k^{2}=4 k^{2}=c^{2}$ So, ∆ABC is a right triangle right angled at C. ...
Read More →Mark the correct alternative in each of the following:
Question: Mark the correct alternative in each of the following: If the sides of a triangle are in the ratio $1: \sqrt{3}: 2$, then the measure of its greatest angle is (a) $\frac{\pi}{6}$ (b) $\frac{\pi}{3}$ (C) $\frac{\pi}{2}$ (d) $\frac{2 \pi}{3}$ Solution: Let $\triangle \mathrm{ABC}$ be the given triangle such that its sides are in the ratio $1: \sqrt{3}: 2$. $\therefore a=k, b=\sqrt{3} k, c=2 k$ Now, $a^{2}+b^{2}=k^{2}+3 k^{2}=4 k^{2}=c^{2}$ So, ∆ABC is a right triangle right angled at C. ...
Read More →Write the rationalising factor of the denominator in
Question: Write the rationalising factor of the denominator in $\frac{1}{\sqrt{2}+\sqrt{3}}$. Solution: $\frac{1}{\sqrt{2}+\sqrt{3}}$ $=\frac{1}{\sqrt{3}+\sqrt{2}} \times \frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}-\sqrt{2}}$ $=\frac{\sqrt{3}-\sqrt{2}}{(\sqrt{3})^{2}-(\sqrt{2})^{2}}$ $=\frac{\sqrt{3}-\sqrt{2}}{3-2}$ $=\frac{\sqrt{3}-\sqrt{2}}{1}$ Here, the denominator i.e. 1 is a rational number. Thus, the rationalising factor of the denominator in $\frac{1}{\sqrt{2}+\sqrt{3}}$ is $\sqrt{3}-\sqrt{2}$....
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. $(\operatorname{cosec} \theta+\sin \theta)(\operatorname{cosec} \theta-\sin \theta)=\cot ^{2} \theta+\cos ^{2} \theta$ Solution: We have to prove $(\operatorname{cosec} \theta+\sin \theta)(\operatorname{cosec} \theta-\sin \theta)=\cot ^{2} \theta+\cos ^{2} \theta$ We know that, $\sin ^{2} \theta+\cos ^{2} \theta=1$ $\operatorname{cosec}^{2} \theta-\cot ^{2} \theta=1$ So, $(\operatorname{cosec} \theta+\sin \theta)(\operatorname{cosec} \theta...
Read More →Events A and B are such that
Question: Events $A$ and $B$ are such that $P(A)=\frac{1}{2}, P(B)=\frac{7}{12}$ and $P($ not $A$ or not $B)=\frac{1}{4} .$ State whether $A$ and $B$ are independent? Solution: It is given that $\mathrm{P}(\mathrm{A})=\frac{1}{2}, \mathrm{P}(\mathrm{B})=\frac{7}{12}$, and $\mathrm{P}($ not $\mathrm{A}$ or not $\mathrm{B})=\frac{1}{4}$ $\Rightarrow P\left(A^{\prime} \cup B^{\prime}\right)=\frac{1}{4}$ $\Rightarrow P\left((A \cap B)^{\prime}\right)=\frac{1}{4}$ $\left[A^{\prime} \cup B^{\prime}=(A...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. $\frac{1-\sin \theta}{1+\sin \theta}=(\sec \theta-\tan \theta)^{2}$ Solution: We have to prove $\frac{1-\sin \theta}{1+\sin \theta}=(\sec \theta-\tan \theta)^{2}$ We know that, $\sin ^{2} \theta+\cos ^{2} \theta=1$. Multiplying both numerator and denominator by $(1-\sin \theta)$, we have $\frac{1-\sin \theta}{1+\sin \theta}=\frac{(1-\sin \theta)(1-\sin \theta)}{(1+\sin \theta)(1-\sin \theta)}$ $=\frac{(1-\sin \theta)^{2}}{1-\sin ^{2} \theta...
Read More →Mark the correct alternative in each of the following:
Question: Mark the correct alternative in each of the following: In a $\triangle \mathrm{ABC}$, if $a=2, \angle B=60^{\circ}$ and $\angle C=75^{\circ}$, then $b=$ (a) $\sqrt{3}$ (b) $\sqrt{6}$ (c) $\sqrt{9}$ (d) $1+\sqrt{2}$ Solution: It is given that $a=2, \angle B=60^{\circ}$ and $\angle C=75^{\circ}$. In ∆ABC, $\angle A+\angle B+\angle C=180^{\circ} \quad$ (Angle sum property) $\Rightarrow \angle A+60^{\circ}+75^{\circ}=180^{\circ}$ $\Rightarrow \angle A=180^{\circ}-135^{\circ}=45^{\circ}$ Us...
Read More →Mark the correct alternative in each of the following:
Question: Mark the correct alternative in each of the following: In a $\triangle \mathrm{ABC}$, if $a=2, \angle B=60^{\circ}$ and $\angle C=75^{\circ}$, then $b=$ (a) $\sqrt{3}$ (b) $\sqrt{6}$ (c) $\sqrt{9}$ (d) $1+\sqrt{2}$ Solution: It is given that $a=2, \angle B=60^{\circ}$ and $\angle C=75^{\circ}$. In ∆ABC, $\angle A+\angle B+\angle C=180^{\circ} \quad$ (Angle sum property) $\Rightarrow \angle A+60^{\circ}+75^{\circ}=180^{\circ}$ $\Rightarrow \angle A=180^{\circ}-135^{\circ}=45^{\circ}$ Us...
Read More →Visualize the representation of
Question: Visualize the representation of $4 . \overline{67}$ on the number line up to 4 decimal places. Solution: $4 . \overline{67}=4.6767$ (Upto 4 decimal places) 4 4.6767 5Divide the gap between 4 and 5 on the number line into 10 equal parts.Now, 4.6 4.6767 4.7In order to locate the point 4.6767 on the number line, divide the gap between 4.6 and 4.7 into 10 equal parts.Further, 4.67 4.6767 4.68To locate the point 4.6767 on the number line, again divide the gap between 4.67 and 4.68 into 10 e...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. $\frac{\sin \theta}{1-\cos \theta}=\operatorname{cosec} \theta+\cot \theta$ Solution: We have to prove $\frac{\sin \theta}{1-\cos \theta}=\operatorname{cosec} \theta+\cot \theta$. We know that, $\sin ^{2} \theta+\cos ^{2} \theta=1$ Multiplying both numerator and denominator by $(1+\cos \theta)$, we have $\frac{\sin \theta}{1-\cos \theta}=\frac{\sin \theta(1+\cos \theta)}{(1-\cos \theta)(1+\cos \theta)}$ $=\frac{\sin \theta(1+\cos \theta)}{1...
Read More →Mark the correct alternative in each of the following:
Question: Mark the correct alternative in each of the following: In any $\triangle \mathrm{ABC}, \sum a^{2}(\sin B-\sin C)=$ (a) $a^{2}+b^{2}+c^{2}$ (b) $a^{2}$ (c) $b^{2}$ (d) 0 Solution: Using sine rule, we have $\sum a^{2}(\sin B-\sin C)$ $=a^{2}\left(\frac{b}{k}-\frac{c}{k}\right)+b^{2}\left(\frac{c}{k}-\frac{a}{k}\right)+c^{2}\left(\frac{a}{k}-\frac{b}{k}\right)$ $=\frac{1}{k}\left(a^{2} b-a^{2} c+b^{2} c-b^{2} a+c^{2} a-c^{2} b\right)$ This expression cannot be simplified to match with any...
Read More →Visualize the representation of 3.765 on
Question: Visualize the representation of 3.765 on the number line using successive magnification. Solution: 3 3.765 4Divide the gap between 3 and 4 on the number line into 10 equal parts.Now, 3.7 3.765 3.8In order to locate the point 3.765 on the number line, divide the gap between 3.7 and 3.8 into 10 equal parts.Further, 3.76 3.765 3.77So, to locate the point 3.765 on the number line, again divide the gap between 3.76 and 3.77 into 10 equal parts.Now, the number 3.765 can be located on the num...
Read More →Mark the correct alternative in each of the following:
Question: Mark the correct alternative in each of the following: In any $\triangle \mathrm{ABC}, \sum a^{2}(\sin B-\sin C)=$ (a) $a^{2}+b^{2}+c^{2}$ (b) $a^{2}$ (c) $b^{2}$ (d) 0 Solution: Using sine rule, we have $\sum a^{2}(\sin B-\sin C)$ $=a^{2}\left(\frac{b}{k}-\frac{c}{k}\right)+b^{2}\left(\frac{c}{k}-\frac{a}{k}\right)+c^{2}\left(\frac{a}{k}-\frac{b}{k}\right)$ $=\frac{1}{k}\left(a^{2} b-a^{2} c+b^{2} c-b^{2} a+c^{2} a-c^{2} b\right)$ This expression cannot be simplified to match with any...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. $\frac{1-\cos \theta}{\sin \theta}=\frac{\sin \theta}{1+\cos \theta}$ Solution: We have to prove $\frac{1-\cos \theta}{\sin \theta}=\frac{\sin \theta}{1+\cos \theta}$. We know that, $\sin ^{2} \theta+\cos ^{2} \theta=1$ Multiplying both numerator and denominator by $(1+\cos \theta)$, we have $\frac{1-\cos \theta}{\sin \theta}=\frac{(1-\cos \theta)(1+\cos \theta)}{\sin \theta(1+\cos \theta)}$ $=\frac{1-\cos ^{2} \theta}{\sin \theta(1+\cos \t...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. $\sqrt{\frac{1-\cos \theta}{1+\cos \theta}}=\operatorname{cosec} \theta-\cot \theta$ Solution: We know that, $\sin ^{2} \theta+\cos ^{2} \theta=1$ Multiplying numerator and denominator under the square root by $(1-\cos \theta)$, we have $\sqrt{\frac{1-\cos \theta}{1+\cos \theta}}=\sqrt{\frac{(1-\cos \theta)(1-\cos \theta)}{(1+\cos \theta)(1-\cos \theta)}}$ $=\sqrt{\frac{(1-\cos \theta)^{2}}{1-\cos ^{2} \theta}}$ $=\sqrt{\frac{(1-\cos \theta...
Read More →Two ships leave a port at the same time.
Question: Two ships leave a port at the same time. One goes 24 km/hr in the directionN38Eand other travels 32 km/hr in the directionS52 E. Find the distance between the ships at the end of 3 hrs. Solution: After three hours, let the ships be at $P$ and $Q$ respectively. Then, $O P=24 \times 3=72 \mathrm{~km}$ and $O Q=32 \times 3=96 \mathrm{~km}$ From figure, we have $\angle P O Q=180^{\circ}-\angle N O P-\angle S O Q$ $=180^{\circ}-38^{\circ}-52^{\circ}$ $=90^{\circ}$ Now, Since $O P O$ is a ri...
Read More →If A and B are two events such that
Question: If $A$ and $B$ are two events such that $P(A)=\frac{1}{4}, P(B)=\frac{1}{2}$ and $P(A \cap B)=\frac{1}{8}$, find $P($ not $A$ and not $B)$. Solution: It is given that, $\mathrm{P}(\mathrm{A})=\frac{1}{2}$ and $\mathrm{P}(\mathrm{A} \cap \mathrm{B})=\frac{1}{8}$ $P($ not on $A$ and not on $B)=P\left(A^{\prime} \cap B^{\prime}\right)$ $\mathrm{P}($ not on $\mathrm{A}$ and not on $\mathrm{B})=\mathrm{P}((\mathrm{A} \cup \mathrm{B}))^{\prime}\left[\mathrm{A}^{\prime} \cap \mathrm{B}^{\prim...
Read More →Represent
Question: Represent $(1+\sqrt{9.5})$ on the number line. Solution: To represent $(1+\sqrt{9.5})$ on the number line, follow the following steps of construction: (i)Mark two points A and B on a given line such that AB = 9.5 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 inter...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. $\sin ^{2} A+\frac{1}{1+\tan ^{2} A}=1$ Solution: We know that, $\sin ^{2} A+\cos ^{2} A=1$ $\sec ^{2} A-\tan ^{2} A=1$ So, $\sin ^{2} A+\frac{1}{1+\tan ^{2} A}=\sin ^{2} A+\frac{1}{\sec ^{2} A}$ $=\sin ^{2} A+\left(\frac{1}{\sec A}\right)^{2}$ $=\sin ^{2} A+(\cos A)^{2}$ $=\sin ^{2} A+\cos ^{2} A$ $=1$...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. $\cos ^{2} A+\frac{1}{1+\cot ^{2} A}=1$ Solution: We know that, $\sin ^{2} A+\cos ^{2} A=1$ $\operatorname{cosec}^{2} A-\cot ^{2} A=1$ So, $\cos ^{2} A+\frac{1}{1+\cot ^{2} A}=\cos ^{2} A+\frac{1}{\operatorname{cosec}^{2} A}$ $=\cos ^{2} A+\left(\frac{1}{\operatorname{cosec} A}\right)^{2}$ $=\cos ^{2} A+(\sin A)^{2}$ $=\cos ^{2} A+\sin ^{2} A$ $=1$...
Read More →In ΔABC if cos C
Question: In $\Delta A B C$ if $\cos C=\frac{\sin A}{2 \sin B}$, prove that the triangle is isosceles. Solution: Let $\Delta A B C$ be any triangle. Suppose $\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}=k$ If $\cos C=\frac{\sin A}{2 \sin B}$, then $\frac{b^{2}+a^{2}-c^{2}}{2 a b}=\frac{k a}{2 k b} \quad\left(\because \cos C=\frac{b^{2}+a^{2}-c^{2}}{2 a b}\right)$ $\Rightarrow b^{2}+a^{2}-c^{2}=a^{2}$ $\Rightarrow b^{2}-c^{2}=0$ $\Rightarrow(b-c)(b+c)=0$ $\Rightarrow b-c=0$ $\Rightarrow b=c...
Read More →Let A and B be independent events with P (A) = 0.3 and P (B) = 0.4. Find
Question: Let A and B be independent events with P (A) = 0.3 and P (B) = 0.4. Find (i) $P(A \cap B)$ (ii) $P(A \cup B)$ (iii) $P$ (A|B) (iv) $P$ (B|A) Solution: It is given that P (A) = 0.3 and P (B) = 0.4 (i) If A and B are independent events, then $P(A \cap B)=P(A) \cdot P(B)=0.3 \times 0.4=0.12$ (ii) It is known that, $\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})$ $\Rightarrow \mathrm{P}(\mathrm{A} \cup \mathrm{B})...
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