The points of discontinuity of the function
Question: The points of discontinuity of the function $f(x)=\left\{\begin{array}{cl}2 \sqrt{x}, 0 \leq x \leq 1 \\ 4-2 x, 1x\frac{5}{2} \text { is }(\text { are }) \\ 2 x-7, \frac{5}{2} \leq x \leq 4\end{array}\right.$ (a) $x=1, x=\frac{5}{2}$ (b) $x=\frac{5}{2}$ (c) $x=1, \frac{5}{2}, 4$ (d) $x=0,4$ Solution: (b) $x=\frac{5}{2}$ If $0 \leq x \leq 1$, then $f(x)=2 \sqrt{x}$. Since $f(x)=2 \sqrt{x}$ is a polynomial function, it is continuous. Thus, $f(x)$ is continuous for every $0 \leq x \leq 1$...
Read More →What type of quadrilateral do the points A (2, -2),
Question: What type of quadrilateral do the points A (2, -2), B (7, 3) C (11, 1) and D (6, 6) taken in that order form? Solution: To find the type of quadrilateral, we find the length of all four sides as well as two diagonals and see whatever condition of quadrilateral is satisfy by these sides as well as diagonals. Now, using distance formula between two points, sides, $A B=\sqrt{(7-2)^{2}+(3+2)^{2}}$ $=\sqrt{(5)^{2}+(5)^{2}}=\sqrt{25+25}$ $=\sqrt{50}=5 \sqrt{2}$ $\left[\right.$ since, distanc...
Read More →Represent each of the following numbers on the number line:
Question: Represent each of the following numbers on the number line: (i) $\frac{-1}{3}$ (ii) $\frac{-3}{4}$ (iii) $-1 \frac{2}{3}$ (iv) $-3 \frac{1}{7}$ (v) $-4 \frac{3}{5}$ (vi) $-2 \frac{5}{6}$ (vii) $-3$ (viii) $-2 \frac{7}{8}$ Solution: (i)(ii)(iii)(iv)(v)(vi)(vii)(viii)...
Read More →Represent each of the following numbers on the number line:
Question: Represent each of the following numbers on the number line: (i) $\frac{1}{3}$ (ii) $\frac{2}{7}$ (iii) $1 \frac{3}{4}$ (iv) $2 \frac{2}{5}$ (v) $3 \frac{1}{2}$ (vi) $5 \frac{5}{7}$ (vii) $4 \frac{2}{3}$ (viii) 8 Solution: (i)(ii)(iii)(iv)(v)(vi)(vii)(viii)...
Read More →In a town of 10,000 families, it was found that 40% of the families buy
Question: In a town of 10,000 families, it was found that 40% of the families buy newspaper A, 20% buy newspaper B, 10% buy newspaper C, 5% buy A and B; 3% buy B and C, and 4% buy A and C. IF 2% buy all the three newspapers, find the number of families which buy (i) A only, (ii) B only, (iii) none of $A, B$, and $C$. Solution: Given: Total number of families $=10000$ Percentage of families that buy newspaper $A=40$ Percentage of families that buy newspaper $B=20$ Percentage of families that buy ...
Read More →Which of the following statements are true and which are false?
Question: Which of the following statements are true and which are false? (i) Every whole number is a rational number. (ii) Every integer is a rational number. (iii) 0 is a whole number but it is not a rational number. Solution: 1. True A whole number can be expressed as $\frac{a}{b}$, with $b=1$ and $a \geq 0 .$ Thus, every whole number is rational. 2. True Every integer is a rational number because any integer can be expressed as $\frac{a}{b}$, with $b=1$ and $0a \geq 0$. Thus, every integer i...
Read More →The values of the constants a, b and c for which the function
Question: The values of the constantsa,bandcfor which the function $f(x)=\left\{\begin{array}{ll}(1+a x)^{1 / x} , x0 \\ b , \quad x=0 \\ \frac{(x+c)^{1 / 3}-1}{(x+1)^{1 / 2}-1}, x0\end{array}\right.$ may be continuous at $x=0$, are (a) $a=\log _{e}\left(\frac{2}{3}\right), b=-\frac{2}{3}, c=1$ (b) $a=\log _{e}\left(\frac{2}{3}\right), b=\frac{2}{3}, c=-1$ (c) $a=\log _{e}\left(\frac{2}{3}\right), b=\frac{2}{3}, c=1$ (d) none of these Solution: (c) $a=\log \frac{2}{3}, b=\frac{2}{3}, c=1$ Given:...
Read More →Find the points on the X-axis which are at a
Question: Find the points on the X-axis which are at a distance of 25 from the point (7, -4). How many such points are there? Solution: We know that, every point on the X-axis in the form (x, 0). Let P(x, 0) the point on the X-axis have 25 distance from the point Q (7, 4). By given condition, $\quad P Q=2 \sqrt{5} \quad\left[\because\right.$ distance formula $\left.=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}\right]$ $\Rightarrow \quad(P Q)^{2}=4 \times 5$ $\Rightarrow \quad...
Read More →The values of the constants a, b and c for which the function
Question: The values of the constantsa,bandcfor which the function $f(x)=\left\{\begin{array}{ll}(1+a x)^{1 / x} , x0 \\ b , \quad x=0 \\ \frac{(x+c)^{1 / 3}-1}{(x+1)^{1 / 2}-1}, x0\end{array}\right.$ may be continuous at $x=0$, are (a) $a=\log _{e}\left(\frac{2}{3}\right), b=-\frac{2}{3}, c=1$ (b) $a=\log _{e}\left(\frac{2}{3}\right), b=\frac{2}{3}, c=-1$ (c) $a=\log _{e}\left(\frac{2}{3}\right), b=\frac{2}{3}, c=1$ (d) none of these Solution: (c) $a=\log \frac{2}{3}, b=\frac{2}{3}, c=1$ Given:...
Read More →Arrange the following rational numbers in descending order:
Question: Arrange the following rational numbers in descending order: (i) $-2, \frac{-13}{6}, \frac{8}{-3}, \frac{1}{3}$ (ii) $\frac{-3}{10}, \frac{7}{-15}, \frac{-11}{20}, \frac{14}{-30}$ (iii) $\frac{-5}{6}, \frac{-7}{12}, \frac{-13}{18}, \frac{23}{-24}$ (iv) $\frac{-10}{11}, \frac{-19}{22}, \frac{-23}{33}, \frac{-39}{44}$ Solution: (i) We will first write each of the given numbers with positive denominators. We have: $\frac{8}{-3}=\frac{8 \times(-1)}{-3 \times(-1)}=\frac{-8}{3}$ Thus, the giv...
Read More →Name the type of triangle formed
Question: Name the type of triangle formed by the points A (-5, 6), B(- 4, 2) and C(7, 5). Solution: To find the type of triangle, first we determine the length of all three sides and see whatever condition of triangle is satisfy by these sides. Now, using distance formula between two points, $A B=\sqrt{(-4+5)^{2}+(-2-6)^{2}}$ $=\sqrt{(1)^{2}+(-8)^{2}}$ $=\sqrt{1+64}=\sqrt{65} \quad\left[\because d=\sqrt{\left(x_{2}-x_{1}\right)+\left(y_{2}-y_{1}\right)^{2}}\right]$ $B C=\sqrt{(7+4)^{2}+(5+2)^{2...
Read More →The points A(- 1, – 2), B (4, 3),
Question: The points A(- 1, 2), B (4, 3), C (2, 5) and D (- 3, 0) in that order form a rectangle. Solution: True Distance between $A(-1,-2)$ and $B(4,3)$, $A B=\sqrt{(4+1)^{2}+(3+2)^{2}}$ $=\sqrt{5^{2}+5^{2}}=\sqrt{25+25}=5 \sqrt{2}$ Distance between $C(2,5)$ and $D(-3,0)$, $C D=\sqrt{(-3-2)^{2}+(0-5)^{2}}$ $=\sqrt{(-5)^{2}+(-5)^{2}}$ $=\sqrt{25+25}=5 \sqrt{2}$ $\left[\because\right.$ distance between the points $\left(x_{1}, y_{1}\right)$ and $\left.\left(x_{2}, y_{2}\right), d=\sqrt{\left(x_{2...
Read More →The value of k which makes
Question: The value ofkwhich makes $f(x)=\left\{\begin{array}{cc}\sin \frac{1}{x}, x \neq 0 \\ k, x=0\end{array}\right.$ continuous at $x=0$, is (a) 8(b) 1(c) 1(d) none of these Solution: (d) none of these If $f(x)$ is continuous at $x=0$, then $\lim _{x \rightarrow 0} f(x)=f(0)$ $\Rightarrow \lim _{x \rightarrow 0}\left(\sin \frac{1}{x}\right)=k$ But $\lim _{x \rightarrow 0}\left(\sin \frac{1}{x}\right)$ does not exist. Thus, there does not exist any $k$ that makes $f(x)$ a continuous function....
Read More →Arrange the following rational numbers in ascending order:
Question: Arrange the following rational numbers in ascending order: (i) $\frac{4}{-9}, \frac{-5}{12}, \frac{7}{-18}, \frac{-2}{3}$ (ii) $\frac{-3}{4}, \frac{5}{-12}, \frac{-7}{16}, \frac{9}{-24}$ (iii) $\frac{3}{-5}, \frac{-7}{10}, \frac{-11}{15}, \frac{-13}{20}$ (iv) $\frac{-4}{7}, \frac{-9}{14}, \frac{13}{-28}, \frac{-23}{42}$ Solution: (i) We will write each of the given numbers with positive denominators. We have: $\frac{4}{-9}=\frac{4 \times(-1)}{-9 \times(-1)}=\frac{-4}{9}$ and $\frac{7}{...
Read More →Solve this
Question: If $f(x)=x \sin \frac{1}{x}, x \neq 0$, then the value of the function at $x=0$, so that the function is continuous at $x=0$, is (a) 0(b) 1(c) 1(d) indeterminate Solution: (a) 0 Given: $f(x)=x \sin \frac{1}{x}, \quad x \neq 0$ Here, $\lim _{x \rightarrow 0} x \sin \left(\frac{1}{x}\right)=\lim _{x \rightarrow 0} x \lim _{x \rightarrow 0} \sin \left(\frac{1}{x}\right)=0 \times \lim _{x \rightarrow 0} \sin \left(\frac{1}{x}\right)=0$ If $f(x)$ is continuous at $x=0$, then $\lim _{x \righ...
Read More →The point P(- 2, 4) lies on a circle
Question: The point P(- 2, 4) lies on a circle of radius 6 and centre (3, 5). Solution: False If the distance between the centre and any point is equal to the radius, then we say that point lie on the circle. Now, distance between P (-2,4) and centre (3, 5) $=\sqrt{(3+2)^{2}+(5-4)^{2}}$ $=\sqrt{5^{2}+1^{2}}$ $=\sqrt{25+1}=\sqrt{26}$ $\left[\because\right.$ distance between the points $\left(x_{1}, y_{1}\right)$ and $\left.\left(x_{2}, y_{2}\right), d=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2...
Read More →The points A(-6,10), B(-4,6) and
Question: The points $A(-6,10), B(-4,6)$ and $C(3,-8)$ are collinear such that $\mathrm{AB}=\frac{2}{9} \mathrm{AC}$. Solution: True If the area of triangle formed by the points (x1,y2), (x2, y2) and (x3, y3) is zero, then the points are collinear, $\because \quad$ Area of triangle $=\frac{1}{2}\left[x_{1}\left(y_{2}-y_{3}\right)+x_{2}\left(y_{3}-y_{1}\right)+x_{3}\left(y_{1}-y_{2}\right)\right]$ Here, $x_{1}=-6, x_{2}=-4, x_{3}=3$ and $y_{1}=10, y_{2}=6, y_{3}=-8$ $\therefore \quad$ Area of $\t...
Read More →Solve this
Question: If $f(x)=\left\{\begin{array}{cc}\frac{1-\cos 10 x}{x^{2}} , \quad x0 \\ a , \quad x=0 \\ \frac{\sqrt{x}}{\sqrt{625+\sqrt{x}-25}}, x0\end{array}\right.$ then the value of a so thatf(x) may be continuous atx= 0, is(a) 25(b) 50(c) 25(d) none of these Solution: (b) 50 If $f(x)$ is continuous at $x=0$, then $\lim _{\mathrm{x} \rightarrow 0^{-}} f(x)=f(0)$ $\Rightarrow \lim _{h \rightarrow 0} f(-h)=f(0)$ $\Rightarrow \lim _{h \rightarrow 0} \frac{(1-\cos (-10 h))}{(-h)^{2}}=f(0)$ $\Rightarr...
Read More →Solve this
Question: If $f(x)=\left\{\begin{array}{cc}\frac{1-\cos 10 x}{x^{2}} , \quad x0 \\ a , \quad x=0 \\ \frac{\sqrt{x}}{\sqrt{625+\sqrt{x}-25}}, x0\end{array}\right.$ then the value of a so thatf(x) may be continuous atx= 0, is(a) 25(b) 50(c) 25(d) none of these Solution: (b) 50 If $f(x)$ is continuous at $x=0$, then $\lim _{\mathrm{x} \rightarrow 0^{-}} f(x)=f(0)$ $\Rightarrow \lim _{h \rightarrow 0} f(-h)=f(0)$ $\Rightarrow \lim _{h \rightarrow 0} \frac{(1-\cos (-10 h))}{(-h)^{2}}=f(0)$ $\Rightarr...
Read More →If the function f (x) defined by
Question: If the functionf(x) defined by $f(x)=\left\{\begin{array}{cl}\frac{\log (1+3 x)-\log (1-2 x)}{x}, x \neq 0 \\ k , x=0\end{array}\right.$ is continuous at $x=0$, then $k=$ (a) 1 (b) 5 (c) $-1$ (d) none of these Solution: (b) 5 Given: $f(x)=\left\{\begin{array}{l}\frac{\log (1+3 x)-\log (1-2 x)}{x}, x \neq 0 \\ k, x=0\end{array}\right.$ If $f(x)$ is continuous at $x=0$, then $\lim _{x \rightarrow 0} f(x)=f(0)$. $\Rightarrow \lim _{x \rightarrow 0}\left(\frac{\log (1+3 x)-\log (1-2 x)}{x}...
Read More →The point P(5, – 3) is one of the two points
Question: The point P(5, 3) is one of the two points of trisection of line segment joining the points A(7, 2) and B(1, 5). Solution: True Let P (5,-3) divides the line segment joining the points A (7,-2) and B (1 ,-5) in the ratio k: 1 internally. By section formula, the coordinate of point P will be $\left(\frac{k(1)+(1)(7)}{k+1}, \frac{k(-5)+1(-2)}{k+1}\right)$ i.e.,$\left(\frac{k+7}{k+1}, \frac{-5 k-2}{k+1}\right)$ Now, $(5,-3)=\left(\frac{k+7}{k+1}, \frac{-5 k-2}{k+1}\right)$ $\Rightarrow$ $...
Read More →If the function f (x) defined by
Question: If the functionf(x) defined by $f(x)=\left\{\begin{array}{cl}\frac{\log (1+3 x)-\log (1-2 x)}{x}, x \neq 0 \\ k , x=0\end{array}\right.$ is continuous at $x=0$, then $k=$ (a) 1 (b) 5 (c) $-1$ (d) none of these Solution: (b) 5 Given: $f(x)=\left\{\begin{array}{l}\frac{\log (1+3 x)-\log (1-2 x)}{x}, x \neq 0 \\ k, x=0\end{array}\right.$ If $f(x)$ is continuous at $x=0$, then $\lim _{x \rightarrow 0} f(x)=f(0)$. $\Rightarrow \lim _{x \rightarrow 0}\left(\frac{\log (1+3 x)-\log (1-2 x)}{x}...
Read More →Fill in the blanks with the correct symbol out of >, = and <:
Question: Fill in the blanks with the correct symbol out of , = and : (i) $\frac{-3}{7} \ldots . \frac{6}{-13}$ (ii) $\frac{5}{-13} \ldots . \frac{-35}{91}$ (iii) $-2 \ldots \frac{-13}{5}$ (iv) $\frac{-2}{3} \ldots \frac{5}{-8}$ (v) $0 \ldots \frac{-3}{-5}$ (vi) $\frac{-8}{9} \ldots \frac{-9}{10}$ Solution: (i)We will write each of the given numbers with positive denominators. One number $=\frac{-3}{7}$ Other number $=\frac{6}{-13}=\frac{6 \times(-1)}{-13 \times(-1)}=\frac{-6}{13}$ LCM of 7 and ...
Read More →Solve this
Question: If $f(x)=\left\{\begin{array}{cc}a x^{2}+b, 0 \leq x1 \\ 4, x=1 \\ x+3, 1x \leq 2\end{array}\right.$ then the value of (a,b) for whichf(x) cannot be continuous atx= 1, is(a) (2, 2)(b) (3, 1)(c) (4, 0)(d) (5, 2) Solution: (d) $(5,2)$ If $f(x)$ is continuous at $x=1$, then $\lim _{x \rightarrow 1^{-}} f(x)=f(1)$ $\Rightarrow \lim _{h \rightarrow 0} f(1-h)=4 \quad[\because f(1)=4]$ $\Rightarrow \lim _{h \rightarrow 0} a(1-h)^{2}+b=4$ $\Rightarrow(a+b)=4$ Thus, the possible values of $(a, ...
Read More →Solve this
Question: If $f(x)=\left\{\begin{array}{ll}a \sin \frac{\pi}{2}(x+1), x \leq 0 \\ \frac{\tan x-\sin x}{x^{3}}, x0\end{array}\right.$ is continuous at $x=0$, then a equals (a) $\frac{1}{2}$ (b) $\frac{1}{3}$ (C) $\frac{1}{4}$ (d) $\frac{1}{6}$ Solution: (a) $\frac{1}{2}$ Given: $f(x)=\left\{\begin{array}{l}a \sin \frac{\pi}{2}(x+1), x \leq 0 \\ \frac{\tan x-\sin x}{x^{3}}, x0\end{array}\right.$ We have (LHL at $x=0)=\lim _{x \rightarrow 0^{-}} f(x)=\lim _{h \rightarrow 0} f(0-h)=\lim _{h \rightar...
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