If A and B are two sets such that
Question: If $A$ and $B$ are two sets such that $n(A)=23, n(b)=37$ and $n(A-B)=8$ then find $n(A \cup B)$ $\operatorname{Hint} n(A)=n(A-B)+n(A \cap B) n(A \cap B)=(23-8)=15 .$ Solution: Given: $n(A)=23, n(B)=37, n(A-B)=8$ Using the hint $n(A)=n(A-B)+n(A \cap B)$ $\Rightarrow 23=8+n(A \cap B)$ $\Rightarrow n(A \cap B)=23-8$ $\Rightarrow n(A \cap B)=15$ Visualizing the hint given, We know that $n(A \cup B)=n(A)+n(B)-n(A \cap B)$ $\Rightarrow n(A \cup B)=23+37-15$ $\Rightarrow \mathrm{n}(\mathrm{A}...
Read More →Find the area of the triangle
Question: Find the area of the triangle whose vertices are (-8,4), (-6,6) and (- 3, 9). Solution: Given that, the vertices of triangles Let $\quad\left(x_{1}, y_{1}\right) \rightarrow(-8,4)$ $\left(x_{2}, y_{2}\right) \rightarrow(-6,6)$ and $\quad\left(x_{3}, y_{3}\right) \rightarrow(-3,9)$ We know that, the area of triangle with vertices $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)$ and $\left(x_{3}, y_{3}\right)$ $\Delta=\frac{1}{2}\left[x_{1}\left(y_{2}-y_{3}\right)+x_{2}\left(y_{3}-y...
Read More →If the point A(2, – 4) is equidistant from P(3, 8)
Question: If the point A(2, 4) is equidistant from P(3, 8) and Q(- 10, y), then find the value of y. Also, find distance PQ. Solution: According to the question, A (2, 4) is equidistant from P (3, 8) = 0 (-10, y) is equidistant from A (2, 4) i.e. $P A=Q A$ $\Rightarrow \quad \sqrt{(2-3)^{2}+(-4-8)^{2}}=\sqrt{(2+10)^{2}+(-4-y)^{2}}$ $\left[\because\right.$ distance between two points $\left(x_{1}, y_{1}\right)$ and $\left.\left(x_{2}, y_{2}\right), d=\sqrt{\left(x_{2}-x_{1}\right)+\left(y_{2}-y_{...
Read More →The value of a for which the function
Question: The value ofafor which the function $f(x)=\left\{\begin{array}{ll}5 x-4, \text { if } 0x \leq 1 \\ 4 x^{2}+3 a x, \text { if } 1x2\end{array}\right.$ is continuous at every point of its domain, is (a) $\frac{13}{3}$ (b) 1 (c) 0 (d) $-1$ Solution: (d) $-1$ Given: $f(x)=\left\{\begin{array}{l}5 x-4, \text { if } 0\mathrm{x} \leq 1 \\ 4 x^{2}+3 a x, \text { if } 1\mathrm{x}2\end{array}\right.$ If $f(x)$ is continuous in its domain, then it will be continuous at $x=1$. Now, $\lim _{x \righ...
Read More →The value of a for which the function
Question: The value ofafor which the function $f(x)=\left\{\begin{array}{ll}5 x-4, \text { if } 0x \leq 1 \\ 4 x^{2}+3 a x, \text { if } 1x2\end{array}\right.$ is continuous at every point of its domain, is (a) $\frac{13}{3}$ (b) 1 (c) 0 (d) $-1$ Solution: (d) $-1$ Given: $f(x)=\left\{\begin{array}{l}5 x-4, \text { if } 0\mathrm{x} \leq 1 \\ 4 x^{2}+3 a x, \text { if } 1\mathrm{x}2\end{array}\right.$ If $f(x)$ is continuous in its domain, then it will be continuous at $x=1$. Now, $\lim _{x \righ...
Read More →If A and B are two sets such than
Question: If $A$ and $B$ are two sets such than $n(A)=8, n(B)=11$ and $n(A \cup B)=14$ then find $n(A \cap B)$. Solution: Given: $n(A)=8, n(B)=11, n(A \cup B)=14$ We know that $n(A \cup B)=n(A)+n(B)-n(A \cap B)$ $\Rightarrow 14=8+11-n(A \cap B)$ $\Rightarrow 14=19-n(A \cap B)$ $\Rightarrow n(A \cap B)=19-14$ $\Rightarrow n(A \cap B)=5$ Hence $n(A \cap B)=5$...
Read More →Verify the following:
Question: Verify the following: (i) $\frac{-12}{5}+\frac{2}{7}=\frac{2}{7}+\frac{-12}{5}$ (ii) $\frac{-5}{8}+\frac{-9}{13}=\frac{-9}{13}+\frac{-5}{8}$ (iii) $3+\frac{-7}{12}=\frac{-7}{12}+3$ (iv) $\frac{2}{-7}+\frac{12}{-35}=\frac{12}{-35}+\frac{2}{-7}$ Solution: 1. $\mathrm{LHS}=\frac{-12}{5}+\frac{2}{7}$ LCM of 5 and 7 is 35 . $\frac{-12 \times 7}{5 \times 7}+\frac{2 \times 5}{7 \times 5}=\frac{-84}{35}+\frac{10}{35}=\frac{-84+10}{35}=\frac{-74}{35}$ $\mathrm{RHS}=\frac{2}{7}+\frac{-12}{5}$ LC...
Read More →The points of discontinuity of the function
Question: The points of discontinuity of the function $f(x)= \begin{cases}\frac{1}{5}\left(2 x^{2}+3\right), x \leq 1 \\ 6-5 x , \quad 1x3 \text { is (are) } \\ x-3 \quad, \quad x \geq 3\end{cases}$ (a)x= 1(b)x= 3(c)x= 1, 3(d) none of these Solution: (b) $x=3$ If $x \leq 1$, then $f(x)=\frac{1}{5}\left(2 x^{2}+3\right)$. Since $2 x^{2}+3$ is a polynomial function and $\frac{1}{5}$ is a constant function, both of them are continuous. So, their product will also be continuous. Thus, $f(x)$ is cont...
Read More →The points of discontinuity of the function
Question: The points of discontinuity of the function $f(x)= \begin{cases}\frac{1}{5}\left(2 x^{2}+3\right), x \leq 1 \\ 6-5 x , \quad 1x3 \text { is (are) } \\ x-3 \quad, \quad x \geq 3\end{cases}$ (a)x= 1(b)x= 3(c)x= 1, 3(d) none of these Solution: (b) $x=3$ If $x \leq 1$, then $f(x)=\frac{1}{5}\left(2 x^{2}+3\right)$. Since $2 x^{2}+3$ is a polynomial function and $\frac{1}{5}$ is a constant function, both of them are continuous. So, their product will also be continuous. Thus, $f(x)$ is cont...
Read More →Find the value of m, if the points (5,1),
Question: Find the value of m, if the points (5,1), (- 2, 3) and (8, 2m) are collinear. Solution: Let A (x1,y1) s (5,1), B = (x2, y2) = (- 2, 3), C s (x3, y3) = (8,2m) Since, the points A (5,1), B (- 2, 3) and C (8,2m) are collinear. Area of $\triangle A B C=0$ $\Rightarrow \quad \frac{1}{2}\left[x_{1}\left(y_{2}-y_{3}\right)+x_{2}\left(y_{3}-y_{1}\right)+x_{3}\left(y_{1}-y_{2}\right)\right]=0$ $\Rightarrow \quad \frac{1}{2}[5(-3-2 m)+(-2)(2 m-1)+8\{1-(-3)\}]=0$ $\Rightarrow \quad \frac{1}{2}(-1...
Read More →If n(A) = 3 and n(B) = 5, find:
Question: If $n(A)=3$ and $n(B)=5$, find: (i) The maximum number of elements in $A \cup B$, (ii) The minimum number of elements in $A \cup B$. Solution: Number of elements in set A n(A) = 3 and number of elements in set B n(B) = 5 The number of elements in $A \cup B$ is $n(A \cup B)$. i) Now for elements in $\mathrm{A} \cup \mathrm{B}$ to be maximum, there should not be any intersection between both sets that is $A$ and $B$ both sets must be disjoint sets as shown. Hence the number of elements i...
Read More →Find the coordinates of the point Q on the x-axis
Question: Find the coordinates of the point Q on the x-axis which lies on the perpendicular bisector of the line segment joining the points A (- 5, 2) and B (4, 2). Name the type of triangle formed by the point Q, A and B. Solution: Firstly, we plot the points of the line segment on the paper and join them. We know that, the perpendicular bisector of the line segment AB bisect the segment AB, i.e.,perpendicular bisector of the line segment AB passes through the mid-point of AB. $\therefore \quad...
Read More →Add the following rational numbers:
Question: Add the following rational numbers: (i) $\frac{3}{4}$ and $\frac{-3}{5}$ (ii) $\frac{5}{8}$ and $\frac{-7}{12}$ (iii) $\frac{-8}{9}$ and $\frac{11}{6}$ (iv) $\frac{-5}{16}$ and $\frac{7}{24}$ (v) $\frac{7}{-18}$ and $\frac{8}{27}$ (vi) $\frac{1}{-12}$ and $\frac{2}{-15}$ (vii) $-1$ and $\frac{3}{4}$ (viii) 2 and $\frac{-5}{4}$ (ix) 0 and $\frac{-2}{5}$ Solution: 1. The denominators of the given rational numbers are 4 and 5. LCM of 4 and 5 is 20. Now, $\frac{3}{4}=\frac{3 \times 5}{4 \t...
Read More →Add the following rational numbers:
Question: Add the following rational numbers: (i) $\frac{3}{4}$ and $\frac{-3}{5}$ (ii) $\frac{5}{8}$ and $\frac{-7}{12}$ (iii) $\frac{-8}{9}$ and $\frac{11}{6}$ (iv) $\frac{-5}{16}$ and $\frac{7}{24}$ (v) $\frac{7}{-18}$ and $\frac{8}{27}$ (vi) $\frac{1}{-12}$ and $\frac{2}{-15}$ (vii) $-1$ and $\frac{3}{4}$ (viii) 2 and $\frac{-5}{4}$ (ix) 0 and $\frac{-2}{5}$ Solution: 1. The denominators of the given rational numbers are 4 and 5. LCM of 4 and 5 is 20. Now, $\frac{3}{4}=\frac{3 \times 5}{4 \t...
Read More →If A = ϕ then write P(A).
Question: If $A=\phi$ then write $P(A)$. Solution: The power set of set A is a collection of all subsets of A Here $A=\{\phi\}$ Hence the subset of A will only be a : set $\phi$ Hence $P(A)=\{\phi\}$...
Read More →If a set A and n elements then find the number of elements in its power set
Question: If a set A and n elements then find the number of elements in its power set P(A). Solution: The power set of set A is a collection of all subsets of A. For example: if the set A is {1, 2} then all possible subsets of A would be {} (empty set), {1}, {2}, {1, 2} Hence powerset of $A$ that is $P(A)$ will be $\{\phi,\{1\},\{2\},\{1,2\}\}$ Now if the number of elements in set $A$ is $n$ then the number of elements in power set of A $P(A)$ is $2^{n}$...
Read More →Solve this
Question: If $f(x)=\left\{\begin{array}{cl}\frac{1-\sin ^{2} x}{3 \cos ^{2} x}, x\frac{\pi}{2} \\ a , x=\frac{\pi}{2} . \text { Then, } f(x) \text { is continuous at } x=\frac{\pi}{2}, \text { if } \\ \frac{b(1-\sin x)}{(\pi-2 \mathrm{x})^{2}}, x\frac{\pi}{2}\end{array}\right.$ (a) $a=\frac{1}{3}, b=2$ (b) $a=\frac{1}{3}, b=\frac{8}{3}$ (c) $a=\frac{2}{3}, b=\frac{8}{3}$ (d) none of these Solution: (b) $a=\frac{1}{3}, b=\frac{8}{3}$ Given: $f(x)=\left\{\begin{array}{c}\frac{1-\sin ^{2} x}{3 \cos...
Read More →Find a point which is equidistant from
Question: Find a point which is equidistant from the points A (- 5, 4) and B (- 1, 6). How many such points are there? Solution: Let P (h, k) be the point which is equidistant from the points A (- 5, 4) and B (-1, 6). $\therefore \quad P A=P B \quad\left[\because\right.$ by distance formula, distance $=\sqrt{\left.\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}\right]}$ $\Rightarrow \quad(P A)^{2}=(P B)^{2}$ $\Rightarrow \quad(-5-h)^{2}+(4-k)^{2}=(-1-h)^{2}+(6-k)^{2}$ $\Rightarrow \qua...
Read More →Solve this
Question: If $f(x)=\left\{\begin{array}{cl}\frac{1-\sin ^{2} x}{3 \cos ^{2} x}, x\frac{\pi}{2} \\ a , x=\frac{\pi}{2} . \text { Then, } f(x) \text { is continuous at } x=\frac{\pi}{2}, \text { if } \\ \frac{b(1-\sin x)}{(\pi-2 \mathrm{x})^{2}}, x\frac{\pi}{2}\end{array}\right.$ (a) $a=\frac{1}{3}, b=2$ (b) $a=\frac{1}{3}, b=\frac{8}{3}$ (c) $a=\frac{2}{3}, b=\frac{8}{3}$ (d) none of these Solution: (b) $a=\frac{1}{3}, b=\frac{8}{3}$ Given: $f(x)=\left\{\begin{array}{c}\frac{1-\sin ^{2} x}{3 \cos...
Read More →Solve this
Question: If $f(x)=\left\{\begin{array}{cl}\frac{1-\sin ^{2} x}{3 \cos ^{2} x}, x\frac{\pi}{2} \\ a , x=\frac{\pi}{2} . \text { Then, } f(x) \text { is continuous at } x=\frac{\pi}{2}, \text { if } \\ \frac{b(1-\sin x)}{(\pi-2 \mathrm{x})^{2}}, x\frac{\pi}{2}\end{array}\right.$ (a) $a=\frac{1}{3}, b=2$ (b) $a=\frac{1}{3}, b=\frac{8}{3}$ (c) $a=\frac{2}{3}, b=\frac{8}{3}$ (d) none of these Solution: (b) $a=\frac{1}{3}, b=\frac{8}{3}$ Given: $f(x)=\left\{\begin{array}{c}\frac{1-\sin ^{2} x}{3 \cos...
Read More →A class has 175 students. The following description gives the number of
Question: A class has 175 students. The following description gives the number of students one or more of the subjects in this class: mathematics 100, physics 70, chemistry 46, mathematics and physics 30; mathematics and chemistry 28; physics and chemistry 23; mathematics, physics and chemistry 18. Find (i) how many students are enrolled in mathematics alone, physics alone and chemistry alone, (ii) The number of students who have not offered any of these subjects. Solution: Given: - Number of st...
Read More →Find the value of a, if the distance between
Question: Find the value of a, if the distance between the points A(- 3, 14) and B (a, 5) is 9 units. Solution: According to the question, Distance between A (- 3, -14) and 8 (a, 5), AB = 9 $\left[\because\right.$ distance between two points $\left(x_{1}, y_{1}\right)$ and $\left.\left(x_{2}, y_{2}\right), d=\sqrt{\left(x_{2}-x_{1}\right)+\left(y_{2}-y_{1}\right)^{2}}\right]$ $\Rightarrow \quad \sqrt{(a+3)^{2}+(-5+14)^{2}}=9$ $\Rightarrow \quad \sqrt{(a+3)^{2}+(9)^{2}}=9$ On squaring both the si...
Read More →Add the following rational numbers:
Question: Add the following rational numbers: (i) $\frac{-2}{5}$ and $\frac{4}{5}$ (ii) $\frac{-6}{11}$ and $\frac{-4}{11}$ (iii) $\frac{-11}{8}$ and $\frac{5}{8}$ (iv) $\frac{-7}{3}$ and $\frac{1}{3}$ (v) $\frac{5}{6}$ and $\frac{-1}{6}$ (vi) $\frac{-17}{15}$ and $\frac{-1}{15}$ Solution: 1. $\frac{-2}{5}+\frac{4}{5}=\frac{-2+4}{5}=\frac{2}{5}$ 2. $\frac{-6}{11}+\frac{-4}{11}=\frac{-6+(-4)}{11}=\frac{-6-4}{11}=\frac{-10}{11}$ 3. $\frac{-11}{8}+\frac{5}{8}=\frac{-11+5}{8}=\frac{-6}{8}=\frac{-3 \...
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) $\frac{-3}{5}$ lies to the left of 0 on the number line. (ii) $\frac{-12}{7}$ lies to the right of 0 on the number line. (iii) The rational numbers $\frac{1}{3}$ and $\frac{-5}{2}$ are on opposite sides of 0 on the number line. (iv) The rational number $\frac{-18}{-13}$ lies to the left of 0 on the number line. Solution: (i) TrueA negative number always lies to the left of 0 on the number line.(ii) FalseA negative numb...
Read More →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 $0 \leq x \leq 1$...
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