Tick (✓) the correct answer
Question: Tick (✓) the correct answer The value of $\left(3^{-1}+4^{-1}\right)^{-1} \div 5^{-1}$ is (a) $\frac{7}{10}$ (b) $\frac{60}{7}$ (c) $\frac{7}{5}$ (d) $\frac{7}{15}$ Solution: (b) $\frac{60}{7}$ $\left(3^{-1}+4^{-1}\right)^{-1} \div 5^{-1}=\left(\frac{1}{3}+\frac{1}{4}\right)^{-1} \div \frac{1}{5}=\left(\frac{4+3}{12}\right)^{-1} \div \frac{1}{5}=\left(\frac{7}{12}\right)^{-1} \div \frac{1}{5}=\left(\frac{12}{7}\right) \div \frac{1}{5}=\frac{12}{7} \times 5=\frac{60}{7}$...
Read More →Let A be the set of first five natural numbers and let R be a relation on A,
Question: Let A be the set of first five natural numbers and let R be a relation on A, defined by $(x, y) \in R \leftrightarrow x \leq y$ Express $\mathbf{R}$ and $\mathbf{R}^{-1}$ as sets of ordered pairs. Find: dom $\left(R^{-1}\right)$ and range $(R)$. Solution: A = {1, 2, 3, 4, 5} Since, $x \leq y$ $\mathrm{R}=\{(1,1),(1,2),(1,3),(1,4),(1,5),(2,2),(2,3),(2,4),(2,5),(3,3),(3,4),(3,5),(4,$, 4), $(4,5),(5,5)\}$ The domain of R is the set of first co-ordinates of R Dom(R) = {1, 2, 3, 4, 5} The r...
Read More →Tick (✓) the correct answer
Question: Tick (✓) the correct answer $\left(2^{-5} \div 2^{-2}\right)=?$ (a) $\frac{1}{128}$ (b) $\frac{-1}{128}$ (c) $-\frac{1}{8}$ (d) $\frac{1}{8}$ Solution: (d) $\frac{1}{8}$ $\left(2^{-5} \div 2^{-2}\right)=\left(\frac{1}{2^{5}} \div \frac{1}{2^{2}}\right)=\left(\frac{1}{32} \div \frac{1}{4}\right)=\left(\frac{1}{32} \times 4\right)=\frac{4}{32}=\frac{1}{8}$...
Read More →At one end A of a diameter AB
Question: At one end A of a diameter AB of a circle of radius 5 cm, tangent XAY is drawn to the circle. The length of the chord CD parallel to XY and at a distance 8 cm from A, is (a) 4 cm (b) 5 cm (c) 6 cm (d) 8 cm Solution: (d)First, draw a circle of radius 5 cm having centre 0. A tangent XY is drawn at point A. A chord $C D$ is drawn which is parallel to $X Y$ and at a distance of $8 \mathrm{~cm}$ from $A$. Now, $\angle O A Y=90^{\circ}$ [Tangent and any point of a circle is perpendicular to ...
Read More →Tick (✓) the correct answer
Question: Tick (✓) the correct answer The value of $(-2)^{-5}$ is (a) $-32$ (b) $\frac{-1}{32}$ (c) 32 (d) $\frac{1}{32}$ Solution: (b) $\frac{-1}{32}$ $(-2)^{-5}=\frac{1}{(-2)^{5}}=\frac{1}{-32}=\frac{1 \times(-1)}{-32 \times(-1)}=\frac{-1}{32}$...
Read More →Tick (✓) the correct answer
Question: Tick (✓) the correct answer The value of $(-3)^{-4}$ is (a) 12 (b) 81 (c) $-\frac{1}{12}$ (d) $\frac{1}{81}$ Solution: (d) $\frac{1}{81}$ $(-3)^{-4}=\frac{1}{(-3)^{4}}=\frac{1}{(-1)^{4} \times(3)^{4}}=\frac{1}{(3)^{4}}=\frac{1}{81}$...
Read More →Tick (✓) the correct answer
Question: Tick (✓) the correct answer The value of $\left(\frac{2}{5}\right)^{-3}$ is (a) $-\frac{8}{125}$ (b) $\frac{25}{4}$ (c) $\frac{125}{8}$ (d) $-\frac{2}{5}$ Solution: (c) $\frac{125}{8}$ $\left(\frac{2}{5}\right)^{-3}=\left(\frac{5}{2}\right)^{3}=\frac{5^{3}}{2^{3}}=\frac{125}{8}$...
Read More →Solve this
Question: Let $R$ be a relation on $Z$, defined by $(x, y) \in R \leftrightarrow x^{2}+y^{2}=9$. Then, write $R$ as a set of ordered pairs. What is its domain? Solution: $x^{2}+y^{2}=9$ We can have only integral values of x and y. Put $x=0, y=3,0^{2}+3^{2}=9$ Put $x=3, y=0,3^{2}+0^{2}=9$ $R=\{(0,3),(3,0),(0,-3),(-3,0)\}$ The domain of R is the set of first co-ordinates of R $\operatorname{Dom}(R)=\{-3,0,3\}$ The range of R is the set of second co-ordinates of R Range $(R)=\{-3,0,3\}$...
Read More →Write each of the following numbers in usual form:
Question: Write each of the following numbers in usual form: (i) 2.06 105 (ii) 5 107 (iii) 6.82 106 (iv) 5.673 104 (v) 1.8 102 (vi) 4.129 103 Solution: (i) $2.06 \times 10^{-5}=\frac{206}{100} \times \frac{1}{10^{5}}=\frac{206}{10^{2} \times 10^{5}}=\frac{206}{10^{(5+2)}}=\frac{206}{10^{7}}=\frac{206}{10000000}=0.0000206$ (ii) $5 \times 10^{-7}=\frac{5}{10^{7}}=\frac{5}{10000000}=0.0000005$ (iii) $6.82 \times 10^{-6}=\frac{682}{100} \times \frac{1}{10^{6}}=\frac{682}{10^{2} \times 10^{6}}=\frac{...
Read More →Solve this
Question: If $f(x)=\left|\log _{e}\right| x||$, then (a)f(x) is continuous and differentiable for allxin its domain(b)f(x) is continuous for all for allin its domain but not differentiable atx= 1(c)f(x) is neither continuous nor differentiable atx= 1(d) none of these Solution: (b)f(x) is continuous for allxin its domain but not differentiable atx= 1 We have, $f(x)=\left|\log _{e}\right| x||$ We know that $\log$ function is defined for positive value. Here, $|x|$ is positive for all non zero $x$....
Read More →From a point P which is at a distance of 13 cm
Question: From a point P which is at a distance of 13 cm from the centre 0 of a circle of radius 5 cm, the pair of tangents PQ and PR to the circle is drawn. Then, the area of the quadrilateral PQOR is (a) 60 cm2 (b) 65 cm2 (c) 30 cm2 (d) 32.5 cm2 Solution: (a)Firstly, draw a circle of radius 5 cm having centre O. P is a point at a distance of 13 cm from O. A pair of tangents PQ and PR are drawn. Thus, quadrilateral POOR is formed. $\because \quad \quad O Q \perp Q P$[since, AP is a tangent line...
Read More →Solve this
Question: If $f(x)=\left|\log _{e}\right| x||$, then (a)f(x) is continuous and differentiable for allxin its domain(b)f(x) is continuous for all for allin its domain but not differentiable atx= 1(c)f(x) is neither continuous nor differentiable atx= 1(d) none of these Solution: (b)f(x) is continuous for allxin its domain but not differentiable atx= 1 We have, $f(x)=\left|\log _{e}\right| x||$ We know that $\log$ function is defined for positive value. Here, $|x|$ is positive for all non zero $x$....
Read More →Solve the following
Question: (i) 1 micron $=\frac{1}{1000000} \mathrm{~m}$. Express it in standard form. (ii) Size of a bacteria $=0.0000004 \mathrm{~m}$. Express it in standard form. (iii) Thickness of a paper $=0.03 \mathrm{~mm}$. Express it in standard form. Solution: (i) 1 micron $=\frac{1}{1000000} \mathrm{~m}=1 \times 10^{-6} \mathrm{~m}$ (ii) $0.0000004 \mathrm{~m}=\frac{4}{10^{7}} \mathrm{~m}=\left(4 \times 10^{-7}\right) \mathrm{m}$ (iii) Thickness of paper $=0.03 \mathrm{~mm}=\frac{3}{10^{2}} \mathrm{~mm...
Read More →Solve this
Question: Let $R=\{(a, b): a, b \in Z$ and $b=2 a-4\}$. If $(a,-2\} \in R$ and $\left(4, b^{2}\right) \in R$. Then, write the values of a and b. Solution: $b=2 a-4$ $a=\frac{b+4}{2}$ Put $b=-2, a=1$ Put $a=4, b=4$ a = 1 , b = 4...
Read More →In figure, AB is a chord of the circle and AOC
Question: In figure, AB is a chord of the circle and AOC is its diameter such that ACB = 50. If AT is the tangent to the circle at the point A, then BAT is equal to (a) 45 (b) 60 (c) 50 (d) 55 Solution: (c)In figure, AOC is a diameter of the circle. We know that, diameter subtends an angle 90 at the circle. So, $\angle A B C=90^{\circ}$ In $\triangle A C B, \quad \angle A+\angle B+\angle C=180^{\circ}$[since, sum of all angles of a triangle is $180^{\circ}$ ] $\Rightarrow \quad \angle A+90^{\cir...
Read More →Solve this
Question: If $f(x)=\left|\log _{e} x\right|$, then (a) $f^{\prime}\left(1^{+}\right)=1$ (b) $f^{\prime}(1)=-1$ (c) $f^{\prime}(1)=1$ (d) $f^{\prime}(1)=-1$ Solution: (a) $f^{\prime}\left(1^{+}\right)=1$ and (b) $f^{\prime}(1)=-1$ $f(x)=\left|\log _{e} x\right|,=\left\{\begin{array}{l}-\log _{e} x, \text { for } 0x1 \\ \log _{e} x, \text { for } x \geq 1\end{array}\right.$ Differentiablity at $x=1$, we have, $(L H D$ at $x=1)=\lim _{x \rightarrow 1^{-}} \frac{f(x)-f(1)}{x-1}$ $=\lim _{x \rightarr...
Read More →Write each of the following numbers in standard form:
Question: Write each of the following numbers in standard form: (i) 0.0006 (ii) 0.00000083 (iii) 0.0000000534 (iv) 0.0027 (v) 0.00000165 (vi) 0.00000000689 Solution: (i) $0.0006=\frac{6}{10^{4}}=6 \times 10^{-4}$ (ii) $0.00000083=\frac{83}{10^{8}}=\frac{8.3 \times 10}{10^{8}}=8.3 \times 10^{(1-8)}=8.3 \times 10^{-7}$ (iii) $0.0000000534=\frac{534}{10^{10}}=\frac{5.34 \times 10^{2}}{10^{10}}=5.34 \times 10^{(2-10)}=5.34 \times 10^{-8}$ (iv) $0.0027=\frac{27}{10^{4}}=\frac{2.7 \times 10}{10^{4}}=2...
Read More →Solve this
Question: Let $R=\{(a, b): a, b \in Z$ and $b=2 a-4\}$. If $(a,-2\} \in R$ and $\left(4, b^{2}\right) \in R$. Then, write the values of a and b. Solution: $b=2 a-4$ $a=\frac{b+4}{2}$ Put $b=-2, a=1$ Put $a=4, b=4$ a = 1 , b = 4...
Read More →Mass of earth is
Question: Mass of earth is (5.97 1024) kg and mass of moon is (7.35 1022) kg. What is the total mass of the two? Solution: Mass of the Earth $=5.97 \times 10^{24} \mathrm{~kg}$ Now, $5.97 \times 10^{24}=5.97 \times 10^{(2+22)}=5.97 \times 10^{2} \times 10^{22}=597 \times 10^{22}$ So, the mass of the Earth can also be written as $597 \times 10^{22} \mathrm{~kg}$. Mass of the Moon $=7.35 \times 10^{22} \mathrm{~kg}$ Sum of the masses of the Earth and the Moon: $=\left(597 \times 10^{22}\right)+\le...
Read More →In figure, if ∠AOB = 125°, then ∠COD is equal to
Question: In figure, if AOB = 125, then COD is equal to (a) 62.5 (b) 45 (c) 35 (d) 55 Solution: (d)We know that, the opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle. i.e., $\quad \angle A O B+\angle C O D=180^{\circ}$ $\Rightarrow \quad \angle C O D=180^{\circ}-\angle A O B$ $=180^{\circ}-125^{\circ}=55^{\circ}$...
Read More →The height of Mount Everest is 8848 m. Write it in standard form.
Question: (i) The height of Mount Everest is 8848 m. Write it in standard form. (ii) The speed of light is 300000000 m/sec. Express it in standard form. (iii) The distance from the earth to the sun is 149600000000 m. Write it in standard form. Solution: (i) The height of the Mount Everest is 8848 m. In standard form, we have: $8848=8.848 \times 1000 \mathrm{~m}=8.848 \times 10^{3} \mathrm{~m}$ (ii) The speed of light is 300000000 m/s. In standard form, we have $300000000=3 \times 100000000 \math...
Read More →Prove that
Question: Let $A=\{1,2,3\}$ and $R=\left\{(a, b): a, b \in A\right.$ and $\left|a^{2}-b^{2}\right| \leq 5$ Write R as a set of ordered pairs Mention whether $R$ is (i) reflexive (ii) symmetric (iii) transitive. Give reason in each case. Solution: Put $a=1, b=1\left|1^{2}-1^{2}\right| \leq 5,(1,1)$ is an ordered pair. Put $a=1, b=2\left|1^{2}-2^{2}\right| \leq 5,(1,2)$ is an ordered pair. Put $a=1, b=3\left|1^{2}-3^{2}\right|5,(1,3)$ is not an ordered pair. Put $a=2, b=1\left|2^{2}-1^{2}\right| \...
Read More →Solve this
Question: If $f(x)=x^{2}+\frac{x^{2}}{1+x^{2}}+\frac{x^{2}}{\left(1+x^{2}\right)}+\ldots+\frac{x^{2}}{\left(1+x^{2}\right)}+\ldots$ then at $x=0, f(x)$ (a) has no limit (b) is discontinuous (c) is continuous but not differentiable (d) is differentiable Solution: (b) is discontinuous We have, $f(x)=x^{2}+\frac{x^{2}}{1+x^{2}}+\frac{x^{2}}{\left(1+x^{2}\right)}+\ldots+\frac{x^{2}}{\left(1+x^{2}\right)}+\ldots$ When $x=0$ then $x^{2}=0$ and $\frac{x^{2}}{1+x^{2}}=0$ $\therefore f(0)=0+0+0+0 \ldots ...
Read More →If radii of two concentric circles are 4 cm and 5 cm,
Question: If radii of two concentric circles are 4 cm and 5 cm, then length of each chord of one circle which is tangent to the other circle, is (a) 3 cm (b) 6 cm (c) 9 cm (d) 1 cm Solution: (b)Let 0 be the centre of two concentric circles C1and C2, whose radii are r1= 4 cm and r2= 5 cm. Now, we draw a chord AC of circle C2, which touches the circle C1at B. Also, join OB, which is perpendicular to AC. [Tangent at any point of circle is perpendicular to radius throughly the point of contact] Now,...
Read More →Solve this
Question: If $f(x)=x^{2}+\frac{x^{2}}{1+x^{2}}+\frac{x^{2}}{\left(1+x^{2}\right)}+\ldots+\frac{x^{2}}{\left(1+x^{2}\right)}+\ldots$ then at $x=0, f(x)$ (a) has no limit (b) is discontinuous (c) is continuous but not differentiable (d) is differentiable Solution: (b) is discontinuous We have, $f(x)=x^{2}+\frac{x^{2}}{1+x^{2}}+\frac{x^{2}}{\left(1+x^{2}\right)}+\ldots+\frac{x^{2}}{\left(1+x^{2}\right)}+\ldots$ When $x=0$ then $x^{2}=0$ and $\frac{x^{2}}{1+x^{2}}=0$ $\therefore f(0)=0+0+0+0 \ldots ...
Read More →