Mark (✓) against the correct answer
Question: Mark (✓) against the correct answer The product of two numbers is $\frac{-1}{4}$. If one of them is $\frac{-3}{10}$, then the other is (a) $\frac{5}{6}$ (b) $\frac{-5}{6}$ (c) $\frac{4}{3}$ (d) $\frac{-8}{5}$ Solution: (a) $\frac{5}{6}$ Let the required number bex. Now, $\frac{-3}{10} \times x=\frac{-1}{4}$ $\Rightarrow x=\frac{-1}{4} \div\left(\frac{-3}{10}\right)$ $\Rightarrow x=\frac{-1}{4} \times \frac{10}{-3}$ $\Rightarrow x=\frac{10}{12}=\frac{5}{6}$...
Read More →Solve this
Question: If $f(x)=\left\{\begin{array}{ll}\frac{|x+2|}{\tan ^{-1}(x+2)} , x \neq-2 \\ 2 , x=-2\end{array}\right.$, then $f(x)$ is (a) continuous atx= 2(b) not continuous atx= 2(c) differentiable atx= 2(d) continuous but not derivable atx= 2 Solution: (b) not continuous atx= 2 Given: $f(x)=\left\{\begin{array}{cc}\frac{|x+2|}{\tan ^{-1}(x+2)}, x \neq-2 \\ 2, x=-2\end{array}\right.$ $\Rightarrow f(x)= \begin{cases}\frac{-(x+2)}{\tan ^{-1}(x+2)}, x-2 \\ \frac{(x+2)}{\tan ^{-1}(x+2)}, x-2 \\ 2, x=-...
Read More →Mark (✓) against the correct answer
Question: Mark (✓) against the correct answer $\left(\frac{-5}{4}\right)^{-1}=?$ (a) $\frac{4}{5}$ (b) $\frac{-4}{5}$ (C) $\frac{5}{4}$ (d) $\frac{3}{5}$ Solution: (b) $\frac{-4}{5}$ We ahve: $\left(\frac{-5}{4}\right)^{-1}$ $=\frac{1}{\left(\frac{-5}{4}\right)}$ $=1 \times \frac{4}{-5}$ $=\frac{4}{-5}$ $=\frac{4 \times-1}{-5 \times-1}$ $=\frac{-4}{5}$...
Read More →Prove the following
Question: If $\tan \theta+\sec \theta=1$, then prove that $\sec \theta=\frac{l^{2}+1}{2 l}$. Solution: Given, $\tan \theta+\sec \theta=l$ ......(i) [multiply by $(\sec \theta-\tan \theta)$ on numerator and denominator LHS] $\Rightarrow \quad \frac{(\tan \theta+\sec \theta)(\sec \theta-\tan \theta)}{(\sec \theta-\tan \theta)}=l \quad \Rightarrow \frac{\left(\sec ^{2} \theta-\tan ^{2} \theta\right)}{(\sec \theta-\tan \theta)}=l$ $\Rightarrow$ $\frac{1}{\sec \theta-\tan \theta}=l$ $\left[\because \...
Read More →Mark (✓) against the correct answer
Question: Mark (✓) against the correct answer What should be subtracted from $\frac{-2}{3}$ to get $\frac{3}{4} ?$ (a) $\frac{-11}{12}$ (b) $\frac{-13}{12}$ (c) $\frac{-5}{4}$ (d) $\frac{-17}{12}$ Solution: (d) $\frac{-17}{12}$ Let the number bex. Now, $\frac{-2}{3}-x=\frac{3}{4}$ $\Rightarrow-1 \times\left(\frac{2}{3}+x\right)=\frac{3}{4}$ $\Rightarrow \frac{2}{3}+x=\frac{-3}{4}$ $\Rightarrow x=\frac{-3}{4}+\left(\right.$ Additive inverse of $\left.\frac{2}{3}\right)$ $\Rightarrow x=\frac{-3}{4...
Read More →The set of points where the function f (x) = x |x| is differentiable is
Question: The set of points where the functionf(x) =x|x| is differentiable is (a) $(-\infty, \infty)$ (b) $(-\infty, 0) \cup(0, \infty)$ (c) $(0, \infty)$ (d) $[0, \infty]$ Solution: (a) $(-\infty, \infty)$ We have, $f(x)=x|x|$ $\Rightarrow f(x)=\left\{\begin{array}{cc}-x^{2}, x0 \\ 0, x=0 \\ x^{2}, x0\end{array}\right.$ When, $x0$, we have $f(x)=-x^{2}$ which being a polynomial function is continuous and differentiable in $(-\infty, 0)$ When, $x0$, we have $f(x)=x^{2}$ which being a polynomial ...
Read More →The set of points where the function f (x) = x |x| is differentiable is
Question: The set of points where the functionf(x) =x|x| is differentiable is (a) $(-\infty, \infty)$ (b) $(-\infty, 0) \cup(0, \infty)$ (c) $(0, \infty)$ (d) $[0, \infty]$ Solution: (a) $(-\infty, \infty)$ We have, $f(x)=x|x|$ $\Rightarrow f(x)=\left\{\begin{array}{cc}-x^{2}, x0 \\ 0, x=0 \\ x^{2}, x0\end{array}\right.$ When, $x0$, we have $f(x)=-x^{2}$ which being a polynomial function is continuous and differentiable in $(-\infty, 0)$ When, $x0$, we have $f(x)=x^{2}$ which being a polynomial ...
Read More →A vertical tower stands on a horizontal
Question: A vertical tower stands on a horizontal plane and is surmounted by a vertical flag staff of height h. At a point on the plane, the angles of elevation of the bottom and the top of the flag staff are and respectively. Prove that the height of the tower is$\left(\frac{h \quad \tan \alpha}{\tan \beta-\tan \alpha}\right)$ Solution: Let the height of the tower be H and OR = x Given that, height of flag staff = h = FP and PRO = , FRO = Now, in $\triangle P R O$,$\tan \alpha=\frac{P O}{R O}=\...
Read More →Mark (✓) against the correct answer
Question: Mark (✓) against the correct answer What should be added to $\frac{-3}{5}$ to get $\frac{-1}{3} ?$ (a) $\frac{4}{5}$ (b) $\frac{8}{15}$ (c) $\frac{4}{15}$ (d) $\frac{2}{5}$ Solution: (C) $\frac{4}{15}$ Let the number bex. Now, $\frac{-3}{5}+x=\frac{-1}{3}$ $\Rightarrow x=\frac{-1}{3}+\left(\right.$ Additive inverse of $\left.\frac{-3}{5}\right)$ $\Rightarrow x=\frac{-1}{3}+\frac{3}{5}$ $\Rightarrow x=\left(\frac{-1 \times 5}{3 \times 5}\right)+\left(\frac{3 \times 3}{5 \times 3}\right)...
Read More →Find two rational numbers lying between
Question: Find two rational numbers lying between $\frac{-1}{3}$ and $\frac{1}{2}$. Solution: Required number $=\frac{1}{2} \times\left(\frac{-1}{3}+\frac{1}{2}\right)$ $=\frac{1}{2} \times\left(\frac{-2+3}{6}\right)$ $=\frac{1}{2} \times \frac{1}{6}$ $=\frac{1}{12}$ $\frac{-1}{3}\frac{1}{12}\frac{1}{2}$ Rational number between $\frac{-1}{3}$ and $\frac{1}{12}$ : $\frac{1}{2} \times\left(\frac{-1}{3}+\frac{1}{12}\right)$ $=\frac{1}{2} \times\left(\frac{1-4}{12}\right)$ $=\frac{1}{2} \times \frac...
Read More →Name the property of multiplication shown by each of the following statements:
Question: Name the property of multiplication shown by each of the following statements: (i) $\frac{-12}{5} \times \frac{3}{4}=\frac{3}{4} \times \frac{-12}{5}$ (ii) $\frac{-8}{15} \times 1=\frac{-8}{15}$ (iii) $\left(\frac{-2}{3} \times \frac{7}{8}\right) \times \frac{-5}{7}=\frac{-2}{3} \times\left(\frac{7}{8} \times \frac{-5}{7}\right)$ (iv) $\frac{-2}{3} \times 0=0$ (v) $\frac{2}{5} \times\left(\frac{-4}{5}+\frac{-3}{10}\right)=\left(\frac{2}{5} \times \frac{-4}{5}\right)+\left(\frac{2}{5} \...
Read More →The function
Question: The function $f(x)=\sin ^{-1}(\cos x)$ is (a) discontinuous atx= 0(b) continuous atx= 0(c) differentiable atx= 0(d) none of these Solution: (b) continuous atx= 0 Given: $f(x)=\sin ^{-1}(\cos x)$ Continuity atx= 0: We have, $(\mathrm{LHL}$ at $x=0)$ $\lim _{x \rightarrow 0^{-}} f(x)$ $=\lim _{h \rightarrow 0} \sin ^{-1}\{\cos (0-h)\}$ $=\lim _{h \rightarrow 0} \sin ^{-1}(\cos h)$ $=\sin ^{-1}(1)$ $=\frac{\pi}{2}$ (RHL atx= 0) $\lim _{x \rightarrow 0^{+}} f(x)$ $=\lim _{h \rightarrow 0} ...
Read More →Evaluate:
Question: Evaluate: (i) $\frac{-3}{5} \times \frac{10}{7}$ (ii) $\left(\frac{-5}{8}\right)^{-1}$ (iii) $(-6)^{-1}$ Solution: (i) $\frac{-3}{5} \times \frac{10}{7}$ $=\frac{-3 \times 10}{5 \times 7}$ $=\frac{-30}{35}$ $=\frac{-6 \times 5}{7 \times 5}$ $=\frac{-6}{7}$ (ii) $\left(\frac{-5}{8}\right)^{-1}$ $=\frac{1}{\left(\frac{-5}{8}\right)}$ $=1 \times \frac{8}{-5}$ $=\frac{8}{-5}$ $=\frac{8 \times-1}{-5 \times-1}$ $=\frac{-8}{5}$ (iii) $(-6)^{-1}$ $=\frac{1}{-6}$ $=\frac{1 \times-1}{-6 \times-1...
Read More →Let R = {(x, x + 5): x ϵ {9, 1, 2, 3, 4, 5}}.
Question: Let $R=\{(x, x+5): x \in\{9,1,2,3,4,5\}\}$ (i) Write $R$ in roster form. (ii) Find dom (R) and range (R). Solution: Given: $R=\{(x, x+5): x \in\{9,1,2,3,4,5\}\}$ (i) R is Foster Form is, R = {(9, 14), (1, 6), (2, 7), (3, 8), (4, 9), (5, 10)} (ii) Dom(R) = {1, 2, 3, 4, 5, 9} Range(R) = {6, 7, 8, 9, 10, 14}...
Read More →The function
Question: The function $f(x)=\sin ^{-1}(\cos x)$ is (a) discontinuous atx= 0(b) continuous atx= 0(c) differentiable atx= 0(d) none of these Solution: (b) continuous atx= 0 Given: $f(x)=\sin ^{-1}(\cos x)$ Continuity atx= 0: We have, $(\mathrm{LHL}$ at $x=0)$ $\lim _{x \rightarrow 0^{-}} f(x)$ $=\lim _{h \rightarrow 0} \sin ^{-1}\{\cos (0-h)\}$ $=\lim _{h \rightarrow 0} \sin ^{-1}(\cos h)$ $=\sin ^{-1}(1)$ $=\frac{\pi}{2}$ (RHL atx= 0) $\lim _{x \rightarrow 0^{+}} f(x)$ $=\lim _{h \rightarrow 0} ...
Read More →The shadow of a tower standing on a level
Question: The shadow of a tower standing on a level plane is found to be 50 m longer when Suns elevation is 30 than when it is 60. Find the height of the tower. Solution: Let the height of the tower be h and RQ = x m Given that $P R=50 \mathrm{~m}$ and $\angle S P Q=30^{\circ}, \angle S R Q=60^{\circ}$ Now, in $\triangle S R Q$; $\tan 60^{\circ}=\frac{S Q}{R Q}$ $\Rightarrow$ $\sqrt{3}=\frac{h}{x} \Rightarrow x=\frac{h}{\sqrt{3}}$ ......(i) and in $\triangle S P Q, \quad \tan 30^{\circ}=\frac{S ...
Read More →The product of two numbers is -8.
Question: The product of two numbers is $-8$. If one of them is $-12$, find the other. Solution: Let the other number be $x$. Thus, we have: $-12 \times x=-8$ $\Rightarrow x=(-8) \div(-12)$ $\Rightarrow x=-8 \times \frac{1}{-12}$ $\Rightarrow x=\frac{8}{12}$ $\Rightarrow x=\frac{2}{3}$...
Read More →Let A = {1, 2, 3, 4, 5, 6}.
Question: Let A = {1, 2, 3, 4, 5, 6}. Define a relation R from A to A by R = {(x, y): y = x + 1}. (i) Write R in roster form. (ii) Find dom (R) and range (R). (iii) What is its co-domain? (iv) Depict R by using arrow diagram Solution: Given: A = {1, 2, 3, 4, 5, 6} (i) $R=\{(x, y): y=x+1\}$ So, R is Roster Form is, R = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6)} (ii) Dom(R) = {1, 2, 3, 4, 5} Range(R) = {2, 3, 4, 5, 6} (iii) Here, y = x + 1 So, the CoD(R) = {1, 2, 3, 4, 5, 6, } (iv)...
Read More →Find the multiplicative inverse of:
Question: Find the multiplicative inverse of: (i) $\frac{-3}{4}$ (ii) $\frac{11}{4}$. Solution: (i) Multiplicative inverse of $\frac{-3}{4}$ is $\frac{4}{-3}$, i. e., $\frac{-4}{3}$. (ii) Multiplicative inverse of $\frac{11}{4}$ is $\frac{4}{11}$....
Read More →What number should be subtracted from
Question: What number should be subtracted from $\frac{-3}{4}$ to get $\frac{-1}{2} ?$ Solution: Let the required number be $x$. Thus, we have : $\frac{-3}{4}-x=\frac{-1}{2}$ $\Rightarrow \frac{-3}{4}+\frac{1}{2}=x$ $\Rightarrow x=\frac{2-3}{4}$ $\Rightarrow x=\frac{-1}{4}$...
Read More →The angle of elevation of the top of a tower
Question: The angle of elevation of the top of a tower from two points distant s and t from its foot are complementary. Prove that the height of the tower is st. Solution: Let the height of the tower is h. and ABC = Given that, BC = s, PC =t and angle of elevation on both positions are complementary. i.e , APC = 90- [if two angles are complementary to each other, then the sum of both angles is equal to $\left.90^{\circ} .\right]$ Now in $\triangle A B C$.$\tan \theta=\frac{A C}{B C}=\frac{h}{s}$...
Read More →Solve this
Question: Let $A=\{(x, y): x+3 y=12, x \in N$ and $y \in N\}$ (i) Write R in roster form. (ii) Find dom (R) and range (R). Solution: Given: $A=\{(x, y): x+3 y=12, x \in N$ and $y \in N\}$ (i) So, R in Roster Form is, $R=\{(3,3),(6,2),(9,1)\}$ (ii) $\operatorname{Dom}(R)=\{3,6,9\}$ Range(R) $=\{1,2,3\}$...
Read More →Solve this
Question: Let $f(x)=|x|$ and $g(x)=\left|x^{3}\right|$, then (a)f(x) andg(x) both are continuous atx= 0(b)f(x) andg(x) both are differentiable atx= 0(c)f(x) is differentiable butg(x) is not differentiable atx= 0(d)f(x) andg(x) both are not differentiable atx= 0 Solution: Option (a)f(x) andg(x) both are continuous atx= 0 Given: $f(x)=|x|, g(x)=\left|x^{3}\right|$ We know $|x|$ is continuous at $\mathrm{x}=0$ but not differentiable at $\mathrm{x}=0$ as $(\mathrm{LHD}$ at $\mathrm{x}=0) \neq(\mathr...
Read More →What number should be added to
Question: What number should be added to $\frac{-3}{5}$ to get $\frac{2}{3} ?$ Solution: Let the required number be $x$. Thus, we have : $x+\frac{(-3)}{5}=\frac{2}{3}$ $\Rightarrow x-\frac{3}{5}=\frac{2}{3}$ $\Rightarrow x=\frac{2}{3}+\frac{3}{5}=\frac{2 \times 5+3 \times 3}{15}$ $\Rightarrow x=\frac{10+9}{15}$ $\Rightarrow x=\frac{19}{15}$...
Read More →Let A = {1, 2, 3, 5} AND B = {4, 6, 9}.
Question: Let A = {1, 2, 3, 5} AND B = {4, 6, 9}. Let $R=\{(x, y): x \in A, y \in B$ and $(x-y)$ is odd $\} .$ Write R in roster form. Solution: Given: $A=\{1,2,3,5\}$ AND $B=\{4,6,9\}$ $R=\{(x, y): x \in A, y \in B$ and $(x-y)$ is odd $\}$ Therefore, R in Roster Form is, R = {(1, 4), (1, 6), (2, 9), (3, 4), (3, 6), (5, 4), (5, 6)}...
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