Multiply:
Question: Multiply: (i) $3 \sqrt{5}$ by $2 \sqrt{5}$ (ii) $6 \sqrt{15}$ by $4 \sqrt{3}$ (iii) $2 \sqrt{6}$ by $3 \sqrt{3}$ (iv) $3 \sqrt{8}$ by $3 \sqrt{2}$ (v) $\sqrt{10}$ by $\sqrt{40}$ (vi) $3 \sqrt{28}$ by $2 \sqrt{7}$ Solution: (i) $3 \sqrt{5} \times 2 \sqrt{5}=3 \times 2 \times \sqrt{5} \times \sqrt{5}=6 \times 5=30$ (ii) $6 \sqrt{15} \times 4 \sqrt{3}=6 \times 4 \times \sqrt{5} \times \sqrt{3} \times \sqrt{3}=24 \times 3 \times \sqrt{5}=72 \sqrt{5}$ (iii) $2 \sqrt{6} \times 3 \sqrt{3}=2 \...
Read More →If θ and 2θ − 45° are acute angles such that sin θ = cos (2θ − 45°),
Question: If $\theta$ and $2 \theta-45^{\circ}$ are acute angles such that $\sin \theta=\cos \left(2 \theta-45^{\circ}\right)$, then $\tan \theta$ is equal to (a) 1 (b) $-1$ (C) $\sqrt{3}$ (d) $\frac{1}{\sqrt{3}}$ Solution: Given that: $\sin \theta=\cos \left(2 \theta-45^{\circ}\right)$ and $\theta$ and $2 \theta-45$ are acute angles We have to find $\tan \theta$ $\Rightarrow \sin \theta=\cos \left(2 \theta-45^{\circ}\right)$ $\Rightarrow \cos \left(90^{\circ}-\theta\right)=\cos \left(2 \theta-4...
Read More →In ∆ABC,
Question: In ∆ABC, ifa2,b2andc2are in A.P., prove that cotA, cotBand cotCare also in A.P. Solution: Let $\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}=k$ Then, $\sin A=k a, \sin B=k b, \sin C=k c$ $a^{2}, b^{2}$ and $c^{2}$ are in A.P. $\Rightarrow 2 b^{2}=a^{2}+c^{2}$ $\Rightarrow 2\left(a^{2}+c^{2}-b^{2}\right)=2\left(2 b^{2}-b^{2}\right)=2 b^{2}=b^{2}+b^{2}+c^{2}-a^{2}-c^{2}+a^{2}$ $\Rightarrow 2\left(a^{2}+c^{2}-b^{2}\right)=b^{2}+c^{2}-a^{2}+a^{2}+b^{2}-c^{2}$ $\Rightarrow \frac{2\left...
Read More →The value of cos (90°−θ) sec (90°−θ) tan θcosec (90°−θ) sin (90°−θ) cot (90°−θ)+tan (90°−θ)cot θ is
Question: The value of $\frac{\cos \left(90^{\circ}-\theta\right) \sec \left(90^{\circ}-\theta\right) \tan \theta}{\operatorname{cosec}\left(90^{\circ}-\theta\right) \sin \left(90^{\circ}-\theta\right) \cot \left(90^{\circ}-\theta\right)}+\frac{\tan \left(90^{\circ}-\theta\right)}{\cot \theta}$ is (a) 1(b) 1(c) 2(d) 2 Solution: We have to find: $\frac{\cos \left(90^{\circ}-\theta\right) \sec \left(90^{\circ}-\theta\right) \tan \theta}{\operatorname{cosec}\left(90^{\circ}-\theta\right) \sin \left...
Read More →In ∆ABC,
Question: In ∆ABC, if sin2A+ sin2B= sin2C. show that the triangle is right-angled Solution: In ∆ABC, Given, $\sin ^{2} A+\sin ^{2} B=\sin ^{2} C \ldots \ldots$ (1) Suppose $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=k$. $\Rightarrow \sin A=\frac{a}{k}, \quad \sin B=\frac{b}{k}, \quad \sin C=\frac{c}{k}$ On putting these values in equation (1), we get: $\frac{a^{2}}{k^{2}}+\frac{b^{2}}{k^{2}}=\frac{c^{2}}{k^{2}} \Rightarrow a^{2}+b^{2}=c^{2}$ Thus, ∆ABCis right-angled....
Read More →The value of tan 10° tan 15° tan 75° tan 80° is
Question: The value of $\tan 10^{\circ} \tan 15^{\circ} \tan 75^{\circ} \tan 80^{\circ}$ is (a) 1(b) 0(c) 1(d) None of these Solution: Here we have to find: $\tan 10^{\circ} \tan 15^{\circ} \tan 75^{\circ} \tan 80^{\circ}$ Now $\tan 10^{\circ} \tan 15^{\circ} \tan 75^{\circ} \tan 80^{\circ}$ $=\tan \left(90^{\circ}-80^{\circ}\right) \tan \left(90^{\circ}-75^{\circ}\right) \tan 75^{\circ} \tan 80^{\circ}$ $=\cot 80^{\circ} \cot 75^{\circ} \tan 75^{\circ} \tan 80^{\circ}$ $=\left(\cot 80^{\circ} \...
Read More →A coin is tossed three times, where
Question: A coin is tossed three times, where (i) E: head on third toss, F: heads on first two tosses (ii) E: at least two heads, F: at most two heads (iii) E: at most two tails, F: at least one tail Solution: If a coin is tossed three times, then the sample space S is S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} It can be seen that the sample space has 8 elements. (i) E = {HHH, HTH, THH, TTH} F = {HHH, HHT} $\therefore \mathrm{E} \cap \mathrm{F}=\{\mathrm{HHH}\}$ $P(F)=\frac{2}{8}=\frac{1}{4}$ ...
Read More →In ∆ABC, prove that if θ be any angle,
Question: In ∆ABC,prove that if be any angle, thenbcos =ccos (A ) +acos (C+ ). Solution: Suppose $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=k$. ....(1) Consider the RHS of the equationbcos =ccos (A ) +acos (C+ ). RHS $=c \cos (A-\theta)+a \cos (C+\theta)$ $=k \sin C \cos (A-\theta)+k \sin A \cos (C+\theta) \quad$ (from (1)) $=\frac{k}{2}[2 \sin C \cos (A-\theta)+2 \sin A \cos (C+\theta)]$ $=\frac{k}{2}[\sin (A+C-\theta)+\sin (C+\theta-A)+\sin (A+C+\theta)+\sin (A-C-\theta)]$ $=\frac{k}{...
Read More →In ∆ABC, prove that if θ be any angle,
Question: In ∆ABC,prove that if be any angle, thenbcos =ccos (A ) +acos (C+ ). Solution: Suppose $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=k$. ....(1) Consider the RHS of the equationbcos =ccos (A ) +acos (C+ ). RHS $=c \cos (A-\theta)+a \cos (C+\theta)$ $=k \sin C \cos (A-\theta)+k \sin A \cos (C+\theta) \quad$ (from (1)) $=\frac{k}{2}[2 \sin C \cos (A-\theta)+2 \sin A \cos (C+\theta)]$ $=\frac{k}{2}[\sin (A+C-\theta)+\sin (C+\theta-A)+\sin (A+C+\theta)+\sin (A-C-\theta)]$ $=\frac{k}{...
Read More →The value of cos 1° cos 2° cos 3° ..... cos 180° is
Question: The value of $\cos 1^{\circ} \cos 2^{\circ} \cos 3^{\circ}$ $\cos 180^{\circ}$ is (a) 1 (b) 0 (c) $-1$ (d) None of these Solution: Here we have to find: $\cos 1^{\circ} \cos 2^{\circ} \cos 3^{\circ} \ldots \cos 180^{\circ}$ $\cos 1^{\circ} \cos 2^{\prime \prime} \cos 3^{\circ} \ldots \cos 180^{\prime \prime}$ $=0$ Hence the correct option is (b)...
Read More →if the
Question: If $P(A)=\frac{6}{11}, P(B)=\frac{5}{11}$ and $P(A \cup B)=\frac{7}{11}$, find (i) $P(A \cap B)$ (ii) $P(A \mid B)$ (iii) $P(B \mid A)$ Solution: It is given that $\mathrm{P}(\mathrm{A})=\frac{6}{11}, \mathrm{P}(\mathrm{B})=\frac{5}{11}$, and $\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\frac{7}{11}$ (i) $\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\frac{7}{11}$ $\therefore \mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})=\frac{7}{11}$ $\Rightarrow \frac{6}{11...
Read More →The value of tan 1° tan 2° tan 3° ...... tan 89° is
Question: The value of $\tan 1^{\circ} \tan 2^{\circ} \tan 3^{\circ}$ $\tan 89^{\circ}$ is (a) 1 (b) $-1$ (c) 0 (d) None of these Solution: Here we have to find: $\tan 1^{\circ} \tan 2^{\circ} \tan 3^{\circ}$ $\tan 89^{\circ}$ $\tan 1^{\circ} \tan 2^{\circ} \tan 3^{\circ}$ $\tan 89^{\circ}$ $=\tan \left(90^{\circ}-89^{\circ}\right) \tan \left(90^{\circ}-88^{\circ}\right) \tan \left(90^{\circ}-87^{\circ}\right) \ldots \tan 87^{\circ} \tan 88^{\circ} \tan 89^{\circ}$ $=\cot 89^{\circ} \cot 88^{\ci...
Read More →In ∆ABC, prove that a
Question: In $\Delta A B C$, prove that $a(\cos C-\cos B)=2(b-c) \cos ^{2} \frac{A}{2}$. Solution: Consider $a(\cos C-\cos B)$ $=k(\sin A \cos C-\sin A \cos B) \quad\left[\because \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=k\right]$ $=\frac{k}{2}(2 \sin A \cos C-2 \sin A \cos B)$ $=\frac{k}{2}[\sin (A+C)+\sin (A-C)-\sin (A+B)-\sin (A-B)]$ $=\frac{k}{2}[\sin (\pi-\mathrm{B})+\sin (\mathrm{A}-\mathrm{C})-\sin (\pi-\mathrm{C})-\sin (\mathrm{A}-\mathrm{B})] \quad(\because \mathrm{A}+\mathrm{...
Read More →If θ is an acute angle such that sec2 θ = 3, then the value of tan2 θ−cosec2 θtan2 θ+cosec2 θis
Question: If $\theta$ is an acute angle such that $\sec ^{2} \theta=3$, then the value of $\frac{\tan ^{2} \theta-\operatorname{cosec}^{2} \theta}{\tan ^{2} \theta+\operatorname{cosec}^{2} \theta}$ is (a) $\frac{4}{7}$ (b) $\frac{3}{7}$ (c) $\frac{2}{7}$ (d) $\frac{1}{7}$ Solution: Given that: $\sec ^{2} \theta=3$ $\sec \theta=\sqrt{3}$ We need to find the value of the expression $\frac{\tan ^{2} \theta-\operatorname{cosec}^{2} \theta}{\tan ^{2} \theta+\operatorname{cosec}^{2} \theta}$ Since $\s...
Read More →solw this
Question: Add: (i) $(2 \sqrt{3}-5 \sqrt{2})$ and $(\sqrt{3}+2 \sqrt{2})$ (ii) $(2 \sqrt{2}+5 \sqrt{3}-7 \sqrt{5})$ and $(3 \sqrt{3}-\sqrt{2}+\sqrt{5})$ (iii) $\left(\frac{2}{3} \sqrt{7}-\frac{1}{2}+6 \sqrt{11}\right)$ and $\left(\frac{1}{3} \sqrt{7}+\frac{3}{2} \sqrt{2}-\sqrt{11}\right)$ Solution: (i) $2 \sqrt{3}-5 \sqrt{2}+\sqrt{3}+2 \sqrt{2}$ $=(2 \sqrt{3}+\sqrt{3})+(2 \sqrt{2}-5 \sqrt{2})$ $=3 \sqrt{3}-3 \sqrt{2}$ (ii) $2 \sqrt{2}+5 \sqrt{3}-7 \sqrt{5}+3 \sqrt{3}-\sqrt{2}+\sqrt{5}$ $=2 \sqrt{...
Read More →Evaluate
Question: Evaluate $\mathrm{P}(\mathrm{A} \cup \mathrm{B})$, if $2 \mathrm{P}(\mathrm{A})=\mathrm{P}(\mathrm{B})=\frac{5}{13}$ and $\mathrm{P}(\mathrm{A} \mid \mathrm{B})=\frac{2}{5}$ Solution: It is given that, $2 P(A)=P(B)=\frac{5}{13}$ $\Rightarrow P(A)=\frac{5}{26}$ and $P(B)=\frac{5}{13}$ $P(A \mid B)=\frac{2}{5}$ $\Rightarrow \frac{\mathrm{P}(\mathrm{A} \cap \mathrm{B})}{\mathrm{P}(\mathrm{B})}=\frac{2}{5}$ $\Rightarrow P(A \cap B)=\frac{2}{5} \times P(B)=\frac{2}{5} \times \frac{5}{13}=\f...
Read More →State in each case, whether the given statement is true of false.
Question: State in each case, whether the given statement is true of false.(i) The sum of two rational numbers is rational.(ii) The sum of two irrational numbers is irrational.(iii) The product of two rational numbers is rational.(iv) The product of two irrational number is irrational.(v) The sum of a rational number and an irrational number is irrational.(vi) The product of a nonzero rational number and an irrational number is a rational number.(vii) Every real number is rational.(viii) Every r...
Read More →If angles A, B, C to a ∆ABC from an increasing AP, then sin B =
Question: If angles $A, B, C$ to a $\triangle A B C$ from an increasing $A P$, then $\sin B=$ (a) $\frac{1}{2}$ (b) $\frac{\sqrt{3}}{2}$ (c) 1 (d) $\frac{1}{\sqrt{2}}$ Solution: Let the angles of a triangle $\triangle A B C$ be $(a-d),(a),(a+d)$ respectively which constitute an A.P.As we know that sum of all the three angles of a triangle is $180^{\circ} .$ So, $(a-d)+a+(a+d)=180^{\circ}$ $\mathrm{So}, a=60^{\circ}$ Therefore, $\angle B=60^{\circ}$ Hence, $\sin \angle B=\frac{\sqrt{3}}{2}$ So an...
Read More →a (cos B cos C+cos A)=b
Question: $a(\cos B \cos C+\cos A)=b(\cos C \cos A+\cos B)=c(\cos A \cos B+\cos C)$ Solution: Suppose $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=k$ Consider: $a(\cos B \cos C+\cos A)$ $=k \sin A(\cos B \cos C+\cos A)$ $=k(\sin A \cos B \cos C+\cos A \sin A)$ $=k\left[\frac{1}{2} \cos C\{\sin (A+B)+\sin (A-B)\}+\sin A \cos A\right]$ $=k\left[\frac{1}{2}\{\sin (A+B) \cos C+\sin (A-B) \cos C\}+\sin A \cos A\right]$ $=k\left[\frac{1}{2}\left\{\frac{1}{2}[\sin (A+B+C)+\sin (A+B-C)+\sin (A-B+...
Read More →a (cos B cos C+cos A)=b
Question: $a(\cos B \cos C+\cos A)=b(\cos C \cos A+\cos B)=c(\cos A \cos B+\cos C)$ Solution: Suppose $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=k$ Consider: $a(\cos B \cos C+\cos A)$ $=k \sin A(\cos B \cos C+\cos A)$ $=k(\sin A \cos B \cos C+\cos A \sin A)$ $=k\left[\frac{1}{2} \cos C\{\sin (A+B)+\sin (A-B)\}+\sin A \cos A\right]$ $=k\left[\frac{1}{2}\{\sin (A+B) \cos C+\sin (A-B) \cos C\}+\sin A \cos A\right]$ $=k\left[\frac{1}{2}\left\{\frac{1}{2}[\sin (A+B+C)+\sin (A+B-C)+\sin (A-B+...
Read More →if the
Question: If P(A) = 0.8, P(B) = 0.5 and P(B|A) = 0.4, find (i) P(A B) (ii) P(A|B) (iii) P(A B) Solution: It is given that P(A) = 0.8, P(B) = 0.5, and P(B|A) = 0.4 (i) $P(B \mid A)=0.4$ $\therefore \frac{\mathrm{P}(\mathrm{A} \cap \mathrm{B})}{\mathrm{P}(\mathrm{A})}=0.4$ $\Rightarrow \frac{\mathrm{P}(\mathrm{A} \cap \mathrm{B})}{0.8}=0.4$ $\Rightarrow \mathrm{P}(\mathrm{A} \cap \mathrm{B})=0.32$ (ii) $\mathrm{P}(\mathrm{A} \mid \mathrm{B})=\frac{\mathrm{P}(\mathrm{A} \cap \mathrm{B})}{\mathrm{P}...
Read More →If x tan 45° cos 60° = sin 60° cot 60°, then x is equal to
Question: If $x \tan 45^{\circ} \cos 60^{\circ}=\sin 60^{\circ} \cot 60^{\circ}$, then $x$ is equal to (a) 1 (b) $\sqrt{3}$ (c) $\frac{1}{2}$ (d) $\frac{1}{\sqrt{2}}$ Solution: Given that: $x \tan 45^{\circ} \cos 60^{\circ}=\sin 60^{\circ} \cot 60^{\circ}$ Here we have to find the value of $x$ We know that $\left[\begin{array}{l}\tan 45^{\circ}=1 \\ \cos 60^{\circ}=\frac{1}{2} \\ \sin 60^{\circ}=\frac{\sqrt{3}}{2} \\ \cot 60^{\circ}=\frac{1}{\sqrt{3}}\end{array}\right]$ $\Rightarrow x \tan 45^{\...
Read More →Find two irrational numbers between 0.16 and 0.17.
Question: Find two irrational numbers between 0.16 and 0.17. Solution: The two irrational numbersbetween 0.16 and 0.17 are 0.161161116... and 0.1606006000...Disclaimer: There are an infinite number of irrational numbers between two rational numbers....
Read More →Compute
Question: Compute $P(A \mid B)$, if $P(B)=0.5$ and $P(A \cap B)=0.32$ Solution: It is given that $P(B)=0.5$ and $P(A \cap B)=0.32$ $\Rightarrow \mathrm{P}\left(\frac{\mathrm{A}}{\mathrm{B}}\right)=\frac{\mathrm{P}(\mathrm{A} \cap \mathrm{B})}{\mathrm{P}(\mathrm{B})}=\frac{0.32}{0.5}=\frac{16}{25}$...
Read More →Find two rational numbers of the form
Question: Find two rational numbers of the form $\frac{p}{q}$ between the numbers $0.2121121112 \ldots$ and $0.2020020002 .$ Solution: The rational numbers between the numbers 0.2121121112... and 0.2020020002... are: $0.21=\frac{21}{100}$ and $0.205=\frac{205}{1000}=\frac{41}{200}$ Disclaimer:There are an infinite number of rational numbers between two irrational numbers....
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