Show that tan4 θ + tan2 θ = sec4 θ – sec2 θ.
Question: Show that tan4 + tan2 = sec4 sec2. Solution: LHS = tan4 + tan2 = tan2 (tan2+1) = tan2.sec2 [ sec2 = tan2+1] = (sec2-1) . sec2 [tan2 = sec2 1] = sec4 sec2 = RHS...
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Question: Tick (✓) the correct answer Which of the following numbers is in standard form? (a) $\frac{-12}{26}$ (b) $\frac{-49}{71}$ (c) $\frac{-9}{16}$ (d) $\frac{28}{-105}$ Solution: (b) $\frac{-49}{71}$ and (c) $\frac{-9}{16}$ The numbers $\frac{-49}{71}$ and $\frac{-9}{16}$ are in the standard form because they have no common divisor other than 1 and their denominators are positive....
Read More →An observer 1.5 m tall is 20.5 m away from
Question: An observer 1.5 m tall is 20.5 m away from a tower 22 m high. Determine the angle of elevation of the top of the tower from the eye of the observer. Solution: Let the angle of elevation of the top of the tower from the eve of the observe is $\theta$ Given that, $\quad A B=22 \mathrm{~m}, P Q=1.5 \mathrm{~m}=M B$ and $\quad Q B=P M=20.5 \mathrm{~m}$ $\Rightarrow \quad A M=A B-M B$ $=22-1.5=20.5 \mathrm{~m}$ Now, in $\triangle A P M, \quad \tan \theta=\frac{A M}{P M}=\frac{20.5}{20.5}=1$...
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Question: Tick (✓) the correct answer The sum of two rational numbers is $-3$. If one of them is $\frac{-10}{3}$ then the other one is (a) $\frac{-13}{3}$ (b) $\frac{-19}{3}$ (c) $\frac{1}{3}$ (d) $\frac{13}{3}$ Solution: (C) $\frac{1}{3}$ Let the other number bex. Now, $x+\left(-\frac{10}{3}\right)=-3$ $\Rightarrow x=-3+\left(\right.$ Additive inverse of $\left.-\frac{10}{3}\right)$ $\Rightarrow x=\left(-3+\frac{10}{3}\right)$ $=\frac{-3}{1}+\frac{10}{3}$ $=\frac{(-9+10)}{3}$ $=\frac{1}{3}$ Thu...
Read More →Let A = {a, b, c, d}, B = {c, d, e} and C = {d, e, f, g}.
Question: Let A = {a, b, c, d}, B = {c, d, e} and C = {d, e, f, g}. Then verify each of the following identities: (i) $A \times(B \cap C)=(A \times B) \cap(A \times C)$ (ii) $A \times(B-C)=(A \times B)-(A \times C)$ (iii) $(A \times B) \cap(B \times A)=(A \cap B) \times(A \cap B)$ Solution: Given: A = {a, b, c, d,}, B = {c, d, e} and C = {d, e, f, g} (i) Need to prove: $A \times(B \cap C)=(A \times B) \cap(A \times C)$ Left hand side $(B \cap C)=\{d, e\}$ $\Rightarrow A \times(B \cap C)=\{(a, d)...
Read More →Discuss the continuity and differentiability of f (x) = |log |x||.
Question: Discuss the continuity and differentiability off(x) = |log |x||. Solution: We have, f(x) = |log |x|| $|x|=\left\{\begin{array}{cl}-x -\inftyx-1 \\ -x -1x0 \\ x 0x1 \\ x 1x\infty\end{array}\right.$ $\log |x|=\left\{\begin{array}{cc}\log (-x) -\inftyx-1 \\ \log (-x) -1x0 \\ \log (x) 0x1 \\ \log (x) 1x\infty\end{array}\right.$ $|\log | x||=\left\{\begin{array}{lc}\log (-x) -\inftyx-1 \\ -\log (-x) -1x0 \\ -\log (x) 0x1 \\ \log (x) 1x\infty\end{array}\right.$ $(\mathrm{LHD}$ at $x=-1)=\lim...
Read More →Discuss the continuity and differentiability of f (x) = |log |x||.
Question: Discuss the continuity and differentiability off(x) = |log |x||. Solution: We have, f(x) = |log |x|| $|x|=\left\{\begin{array}{cl}-x -\inftyx-1 \\ -x -1x0 \\ x 0x1 \\ x 1x\infty\end{array}\right.$ $\log |x|=\left\{\begin{array}{cc}\log (-x) -\inftyx-1 \\ \log (-x) -1x0 \\ \log (x) 0x1 \\ \log (x) 1x\infty\end{array}\right.$ $|\log | x||=\left\{\begin{array}{lc}\log (-x) -\inftyx-1 \\ -\log (-x) -1x0 \\ -\log (x) 0x1 \\ \log (x) 1x\infty\end{array}\right.$ $(\mathrm{LHD}$ at $x=-1)=\lim...
Read More →Discuss the continuity and differentiability of f (x) = |log |x||.
Question: Discuss the continuity and differentiability off(x) = |log |x||. Solution: We have, f(x) = |log |x|| $|x|=\left\{\begin{array}{cl}-x -\inftyx-1 \\ -x -1x0 \\ x 0x1 \\ x 1x\infty\end{array}\right.$ $\log |x|=\left\{\begin{array}{cc}\log (-x) -\inftyx-1 \\ \log (-x) -1x0 \\ \log (x) 0x1 \\ \log (x) 1x\infty\end{array}\right.$ $|\log | x||=\left\{\begin{array}{lc}\log (-x) -\inftyx-1 \\ -\log (-x) -1x0 \\ -\log (x) 0x1 \\ \log (x) 1x\infty\end{array}\right.$ $(\mathrm{LHD}$ at $x=-1)=\lim...
Read More →Solve the following
Question: Show that $\frac{\cos ^{2}\left(45^{\circ}+\theta\right)+\cos ^{2}\left(45^{\circ}-\theta\right)}{\tan \left(60^{\circ}+\theta\right) \tan \left(30^{\circ}-\theta\right)}=1$ Solution: $\mathrm{LHS}=\frac{\cos ^{2}\left(45^{\circ}+\theta\right)+\cos ^{2}\left(45^{\circ}-\theta\right)}{\tan \left(60^{\circ}+\theta\right) \cdot \tan \left(30^{\circ}-\theta\right)}$ $=\frac{\cos ^{2}\left(45^{\circ}+\theta\right)+\left[\sin \left\{90^{\circ}-\left(45^{\circ}-\theta\right)\right\}\right]^{2...
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Question: Tick (✓) the correct answer What should be subtracted from $\frac{-3}{5}$ to get $-2 ?$ (a) $\frac{-7}{5}$ (b) $\frac{-13}{5}$ (c) $\frac{13}{5}$ (d) $\frac{7}{5}$ Solution: (d) $\frac{7}{5}$ Let the required number bex. Now, $-\frac{3}{5}-x=-2$ $\Rightarrow-\frac{3}{5}=-2+x$ $\Rightarrow x=\left(-\frac{3}{5}+2\right)$ $\Rightarrow x=\frac{(-3+10)}{5}$ $\Rightarrow x=\frac{7}{5}$ Thus, the required number is $\frac{7}{5}$...
Read More →Let A = {1, 2} and B = {2, 3}.
Question: Let $A=\{1,2\}$ and $B=\{2,3\} .$ Then, write down all possible subsets of $A \times B$. Solution: A = {1, 2} and B = {2, 3} Need to write: All possible subsets of A B A = {1, 2} and B = {2, 3} So, all the possible subsets of A B are: $(A \times B)=\left\{(x, y): x^{\in}\right.$ A and $\left.y \in_{B}\right\}$ = {(1, 2), (1, 3), (2,2), (2,3)}...
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Question: Tick (✓) the correct answer The product of two numbers is $\frac{-16}{35}$. If one of the numbers is $\frac{-15}{14}$, the other is (a) $\frac{-2}{5}$ (b) $\frac{8}{15}$ (C) $\frac{32}{75}$ (d) $\frac{-8}{3}$ Solution: (C) $\frac{32}{75}$ Let the other number bex. Now, $x \times \frac{-15}{4}=\frac{-16}{35}$ $\Rightarrow x=\frac{-16}{35} \div \frac{-15}{14}$ $=\frac{-16}{35} \times \frac{14}{-15}$ $=\frac{-(16 \times 14)}{-(35 \times 15)}$ $=\frac{16 \times 14}{35 \times 15}=\frac{224}...
Read More →If 2 sin2 θ – cos2 θ = 2,
Question: If 2 sin2 cos2 = 2, then find the value of . Solution: Given, $\quad 2 \sin ^{2} \theta-\cos ^{2} \theta=2$ $\Rightarrow$ $2 \sin ^{2} \theta-\left(1-\sin ^{2} \theta\right)=2$ $\left[\because \sin ^{2} \theta+\cos ^{2} \theta=1\right]$ $\Rightarrow$ $2 \sin ^{2} \theta+\sin ^{2} \theta-1=2$ $\Rightarrow$ $3 \sin ^{2} \theta=3$ $\Rightarrow$ $\sin ^{2} \theta=1$ $\left[\because \sin 90^{\circ}=1\right]$ $\Rightarrow$ $\sin \theta=1=\sin 90^{\circ}$ $\therefore \quad \theta=90^{\circ}$...
Read More →Tick (✓) the correct answer
Question: Tick (✓) the correct answer The product of two numbers is $\frac{-16}{35}$. If one of the numbers is $\frac{-15}{14}$, the other is (a) $\frac{-2}{5}$ (b) $\frac{8}{15}$ (C) $\frac{32}{75}$ (d) $\frac{-8}{3}$ Solution: (C) $\frac{32}{75}$ Let the other number bex. Now, $x \times \frac{-15}{4}=\frac{-16}{35}$ $\Rightarrow x=\frac{-16}{35} \div \frac{-15}{14} $=\frac{-16}{35} \times \frac{14}{-15}$ $=\frac{-(16 \times 14)}{-(35 \times 15)}$ $=\frac{16 \times 14}{35 \times 15}=\frac{224}{...
Read More →For any two sets A and B, show that
Question: For any two sets A and B, show that A B and B A have an element in common if and only if A and B have an element in common. Solution: We know $(A \times B) \cap(B \times A)=(A \cap B) \times(B \cap A)$ Here $A$ and $B$ have an element in common i.e., $n(A \cap B)=1=(B \cap A)$ So, $n((A \times B) \cap(B \times A))=n((A \cap B) \times(B \cap A))=n(A \cap B) \times n(B \cap A)=1 \times 1=1$ That means, $A \times B$ and $B \times A$ have an element in common if and only if $A$ and $B$ hav...
Read More →Write an example of a function which is everywhere continuous but fails to differentiable exactly at five points.
Question: Write an example of a function which is everywhere continuous but fails to differentiable exactly at five points. Solution: $f(x)=|x|+|x+1|+|x+2|+|x+3|+|x+4|$ The above function is continuous everywhere but not differentiable atx= 0, 1, 2, 3 and 4...
Read More →Simplify (1 + tan2 θ) (1 – sinθ) (1 + sinθ)
Question: Simplify (1 + tan2) (1 sin) (1 + sin) Solution: $\left(1+\tan ^{2} \theta\right)(1-\sin \theta)(1+\sin \theta)=\left(1+\tan ^{2} \theta\right)\left(1-\sin ^{2} \theta\right) \quad\left[\because(a-b)(a+b)=a^{2}-b^{2}\right]$ $=\sec ^{2} \theta \cdot \cos ^{2} \theta$ $\left[\because 1+\tan ^{2} \theta=\sec ^{2} \theta\right.$ and $\left.\cos ^{2} \theta+\sin ^{2} \theta=1\right]$ $=\frac{1}{\cos ^{2} \theta} \cdot \cos ^{2} \theta=1$ $\left[\because \sec \theta=\frac{1}{\cos \theta}\rig...
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Question: Tick (✓) the correct answer The product of two rational numbers is $\frac{-28}{81}$. If one of the numbers is $\frac{14}{27}$ then the other one is (a) $\frac{-2}{3}$ (b) $\frac{2}{3}$ (c) $\frac{3}{2}$ (d) $\frac{-3}{2}$ Solution: (a) $\frac{-2}{3}$ Let the other number bex. Now, $x \times \frac{14}{27}=\frac{-28}{81}$ $\Rightarrow x=\frac{-28}{81} \div \frac{14}{27}$ $=\frac{-28}{81} \times \frac{27}{14}$ $=\frac{(-28) \times 27}{81 \times 14}$ $=\frac{-(28 \times 27)}{81 \times 14}$...
Read More →A ladder 15 m long just reaches the top of a vertical wall.
Question: A ladder 15 m long just reaches the top of a vertical wall.If the ladders makes an angle of 60 with the wall,then find the height of the wall. Solution: Given that,the height of the ladder = 15 m Let the height of the vertical wall = h and the ladder makes an angle of elevation 60 with the wall i.e = 60 In $\triangle Q P R$,$\cos 60^{\circ}=\frac{P R}{P Q}=\frac{h}{15}$ $\Rightarrow$ $\frac{1}{2}=\frac{h}{15}$ $\Rightarrow$ $h=\frac{15}{2} \mathrm{~m}$ Hence, the required height of the...
Read More →Examine the differentialibilty of the function f defined by
Question: Examine the differentialibilty of the functionfdefined by $f(x)=\left\{\begin{array}{l}2 x+3 \text { if }-3 \leq x \leq-2 \\ x+1 \quad \text { if }-2 \leq x0 \\ x+2 \quad \text { if } 0 \leq x \leq 1\end{array}\right.$ Solution: $f(x)=\left\{\begin{array}{cc}2 x+3 \text { if }-3 \leq x \leq-2 \\ x+1 \text { if }-2 \leq x0 \\ x+2 \text { if } 0 \leq x \leq 1\end{array}\right.$ $\Rightarrow f^{\prime}(x)=\left\{\begin{array}{cc}2 \text { if }-3 \leq x \leq-2 \\ 1 \text { if }-2 \leq x0 \...
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Question: Tick (✓) the correct answer $\left(\frac{-3}{7}\right)^{-1}=?$ (a) $\frac{7}{3}$ (b) $\frac{-7}{3}$ (c) $\frac{3}{7}$ (d) none of these Solution: (b) $\frac{-7}{3}$ $\left(-\frac{3}{7}\right)^{-1} \Rightarrow$ Reciprocal of $\frac{-3}{7}$ The reciprocal of $\frac{-3}{7}$ is $\frac{7}{-3}$, i. e., $\frac{-7}{3}$...
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Question: Tick (✓) the correct answer What should be subtracted from $\frac{-5}{3}$ to get $\frac{5}{6} ?$ (a) $\frac{5}{2}$ (b) $\frac{3}{2}$ (c) $\frac{5}{4}$ (d) $\frac{-5}{2}$ Solution: (d) $\frac{-5}{2}$ Let the required number bex. Now, $\frac{-5}{3}-x=\frac{5}{6}$ $\Rightarrow x=\left(\frac{-5}{3}-\frac{5}{6}\right)$ $=\frac{-10-5}{6}$ $=\frac{-15}{6}$ $=\frac{-5}{2}$ Thus, the required number is $\frac{-5}{2}$...
Read More →Find the derivative of the function f defined by
Question: Find the derivative of the function $f$ defined by $f(x)=m x+c$ at $x=0$. Solution: Given: $f(x)=m x+c$ Clearly, being a polynomial function, is differentiable everywhere. Therefore the derivative of $f$ at $x$ is given by: $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ $\Rightarrow f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{m(x+h)+c-m x-c}{h}$ $\Rightarrow f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{m x+m h+c-m x-c}{h}$ $\Rightarrow f^{\prime}(x)=\lim _{h \rightarrow 0...
Read More →If √3 tanθ = 1, then find the value of
Question: If 3 tan = 1, then find the value of sin2 cos2 Solution: Given that, $\sqrt{3} \tan \theta=1$ $\Rightarrow$ $\tan \theta=\frac{1}{\sqrt{3}}=\tan 30^{\circ}$ $\Rightarrow$ $\theta=30^{\circ}$ Now, $\sin ^{2} \theta-\cos ^{2} \theta=\sin ^{2} 30^{\circ}-\cos ^{2} 30^{\circ}$ $=\left(\frac{1}{2}\right)^{2}-\left(\frac{\sqrt{3}}{2}\right)^{2}$ $=\frac{1}{4}-\frac{3}{4}=\frac{1-3}{4}=-\frac{2}{4}=-\frac{1}{2}$...
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Question: Tick (✓) the correct answer What should be added to $\frac{-5}{7}$ to get $\frac{-2}{3} ?$ (a) $\frac{-29}{21}$ (b) $\frac{29}{21}$ (c) $\frac{1}{21}$ (d) $\frac{-1}{21}$ Solution: (c) $\frac{1}{21}$ Let the required number bex. Now, $\frac{-5}{7}+x=\frac{-2}{3}$ $\Rightarrow x=\frac{-2}{3}+\left(\right.$ Additive inverse of $\left.\frac{-5}{7}\right)$ $\Rightarrow x=\left(\frac{-2}{3}+\frac{5}{7}\right)$ $=\frac{(-14)+15}{21}$ $=\frac{1}{21}$...
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