List all the elements of the following sets:
Question: List all the elements of the following sets: (i) $A=\left\{x: x^{2} \leq 10, x \in Z\right\}$ (ii) $B=\left\{x: x=\frac{1}{2 n-1}, 1 \leq n \leq 5\right\}$ (iii) $C=\left\{x: x\right.$ is an integer, $\left.-\frac{1}{2}x\frac{9}{2}\right\}$ (iv)D= {x:xis a vowel in the word "EQUATION"}(v)E= {x:xis a month of a year not having 31 days}(vi)F= {x:xis a letter of the word "MISSISSIPPI"} Solution: (i) $A=\{0, \pm 1, \pm 2, \pm 3\}$ (ii) $B=\left\{1, \frac{1}{3}, \frac{1}{5}, \frac{1}{7}, \f...
Read More →List all the elements of the following sets:
Question: List all the elements of the following sets: (i) $A=\left\{x: x^{2} \leq 10, x \in Z\right\}$ (ii) $B=\left\{x: x=\frac{1}{2 n-1}, 1 \leq n \leq 5\right\}$ (iii) $C=\left\{x: x\right.$ is an integer, $\left.-\frac{1}{2}x\frac{9}{2}\right\}$ (iv)D= {x:xis a vowel in the word "EQUATION"}(v)E= {x:xis a month of a year not having 31 days}(vi)F= {x:xis a letter of the word "MISSISSIPPI"} Solution: (i) $A=\{0, \pm 1, \pm 2, \pm 3\}$ (ii) $B=\left\{1, \frac{1}{3}, \frac{1}{5}, \frac{1}{7}, \f...
Read More →An angle is equal to 8 times its complement. Determine its measure?
Question: An angle is equal to 8 times its complement. Determine its measure? Solution: It is given that required angle = 8 times its complement Let 'x' be the measured angle angle = 8 times complement angle = 8 (90 - x) x = 8(90 - x) x = 720 - 8x x + 8x = 720 9x = 720 x = 80 Therefore measured angle is 80....
Read More →Two supplementary angles differ by 48°.Find the angles?
Question: Two supplementary angles differ by 48.Find the angles? Solution: Given that two supplementary angles differ by48 Let the angle measured bex Therefore, Its supplementary angle will be(180 - x) It is given that: (180 - x) - x = 48 (180 - 48) = 2x 2x = 132 x = 132/2 x = 66 Hence, 180 - x =114 Therefore, the angles are 66 and 114....
Read More →Describe the following sets in set-builder form:
Question: Describe the following sets in set-builder form: (i)A= {1, 2, 3, 4, 5, 6}; (ii) $B=\{1,1 / 2,1 / 3,1 / 4,1 / 5, \ldots\}$; (iii)C= {0, 3, 6, 9, 12, ...}; (iv)D= {10, 11, 12, 13, 14, 15}; (v)E= {0}; (vi) {1, 4, 9, 16, ..., 100} (vii) {2, 4, 6, 8 .....} (viii) {5, 25, 125 625} Solution: Set-builder form: To describe a set, a variablex(each element of the set) is written inside braces. Then, after putting a colon, the common property P(x) possessed by each element of the set is written wi...
Read More →Two supplementary angles are in the ratio 4: 5. Find the angles?
Question: Two supplementary angles are in the ratio 4: 5. Find the angles? Solution: Supplementary angles are in the ratio 4: 5 Let the angles be 4x and 5x It is given that they are supplementary angles Hence 4x + 5x = 180 9x = 180 x = 20 Hence, 4x = 4 (20) = 80 5(x) = 5 (20) = 100 Hence, angles are 80 and 100...
Read More →Describe the following sets in set-builder form:
Question: Describe the following sets in set-builder form: (i)A= {1, 2, 3, 4, 5, 6}; (ii) $B=\{1,1 / 2,1 / 3,1 / 4,1 / 5, \ldots\}$; (iii)C= {0, 3, 6, 9, 12, ...}; (iv)D= {10, 11, 12, 13, 14, 15}; (v)E= {0}; (vi) {1, 4, 9, 16, ..., 100} (vii) {2, 4, 6, 8 .....} (viii) {5, 25, 125 625} Solution: Set-builder form: To describe a set, a variablex(each element of the set) is written inside braces. Then, after putting a colon, the common property P(x) possessed by each element of the set is written wi...
Read More →Solve the following systems of equations:
Question: Solve the following systems of equations: $\frac{5}{x+y}-\frac{2}{x-y}=-1$ $\frac{15}{x+y}+\frac{7}{x-y}=10$ Solution: The given equations are: $\frac{5}{x+y}-\frac{2}{x-y}=-1$ $\frac{15}{x+y}+\frac{7}{x-y}=10$ Let $\frac{1}{x+y}=u$ and $\frac{1}{x-y}=v$ then equations are $5 u-2 v=-1 \ldots(i)$ $15 u+7 v=10 \ldots(i i)$ Multiply equation $(i)$ by 7 and equation $(i i)$ by 2 and add both equations, we get Put the value of $u$ in equation $(i)$, we get $5 \times \frac{1}{5}-2 v=-1$ $\Ri...
Read More →If an angle is 30° more than half of its complement, find the measure of the angle?
Question: If an angle is 30 more than half of its complement, find the measure of the angle? Solution: Let the measured angle be 'x' Hence its complement will be (90 - x) It is given that, Angle = 30 + complement/2 x = 30 + (90 - x) /2 $3 \frac{x}{2}=30+45$ 3x = 150 x = 50 Therefore the angle is 50...
Read More →Prove
Question: $\frac{3 x^{2}}{x^{6}+1}$ Solution: Let $x^{3}=t$ $\therefore 3 x^{2} d x=d t$ $\Rightarrow \int \frac{3 x^{2}}{x^{6}+1} d x=\int \frac{d t}{t^{2}+1}$ $=\tan ^{1} t+C$ $=\tan ^{-1}\left(x^{3}\right)+C$...
Read More →If an angle is 28° less than its complement, find its measure?
Question: If an angle is 28 less than its complement, find its measure? Solution: Let the angle measured be 'x' in degrees Hence, Its complement will be90 x Angle = Complement - 28 x = (90 - x) - 28 2x = 62 x = 31 Therefore, angle measured is 31...
Read More →Describe the following sets in Roster form:
Question: Describe the following sets in Roster form: (i) {x:xis a letter beforeein the English alphabet}; (ii) {xN:x2 25}; (iii) {xN:xis a prime number, 10 x 20}; (iv) {xN:x= 2n,nN}; (v) {xR:xx}. (vi) {x:xis a prime number which is a divisor of 60} (vii) {x:xis a two digit number such that the sum of its digits is 8} (viii) The set of all letters in the word 'Trigonometry' (ix) The set of all letters in the word 'Better'. Solution: Roster form:In this form, a set is defined by listing elements,...
Read More →Write the supplement of each of the following angles:
Question: Write the supplement of each of the following angles: (i) 54 (ii) 132 (iii) 138 Solution: (i) The given angle is 54, Since the sum of an angle and its supplement is 180, Hence, Its supplement will be (180 - 54 = 126) (ii)The given angle is 132, Since the sum of an angle and its supplement is 180, Hence, its supplement will be 180 - 132 = 48 (iii)The given angle is 138, Since the sum of an angle and its supplement is 180, Hence, Its supplement will be 180 - 138 = 42...
Read More →Prove
Question: $\frac{3 x^{2}}{x^{6}+1}$ Solution: Let $x^{3}=t$ $\therefore 3 x^{2} d x=d t$ $\Rightarrow \int \frac{3 x^{2}}{x^{6}+1} d x=\int \frac{d t}{t^{2}+1}$ $=\tan ^{1} t+C$ $=\tan ^{-1}\left(x^{3}\right)+C$...
Read More →Solve the following systems of equations:
Question: Solve the following systems of equations: $\frac{22}{x+y}+\frac{15}{x-y}=5$ $\frac{55}{x+y}+\frac{45}{x-y}=14$ Solution: The given equations are: $\frac{22}{x+y}+\frac{15}{x-y}=5$ $\frac{55}{x+y}+\frac{45}{x-y}=14$ Let $\frac{1}{x+y}=u$ and $\frac{1}{x-y}=v$ then equations are $22 u+15 v=5 \ldots(i)$ $55 u+45 v=14 \ldots(i i)$ Multiply equation $(i)$ by 3 and subtracting (ii) from (i), we get $\Rightarrow u=\frac{1}{11}$ Put the value of $u$ in equation $(i)$, we get $22 \times-\frac{1...
Read More →Write the complement of each of the following angles:
Question: Write the complement of each of the following angles: (i) 20 (ii) 35 (iii) 90 (iv) 77 (v) 30 Solution: (i)Given angle is 20 Since, the sum of an angle and its compliment is 90 Hence, its compliment will be (90 - 20 = 70) (ii)Given angle is 35 Since, the sum of an angle and its compliment is 90 Hence, its compliment will be (90 - 35 = 55) (iii)Given angle is 90 Since, the sum of an angle and its compliment is 90 Hence, its compliment will be (90 - 90 = 0) (iv) Given angle is 77 Since, t...
Read More →If A and B are two sets such that n
Question: If $A$ and $B$ are two sets such that $n(A)=115, n(B)=326, n(A-B)=47$, then write $n(A \cup B)$. Solution: $n(A)=115, n(B)=326$ and $n(A-B)=47$ Now, $n(A)-n(A \cap B)=n(A-B)$ $\Rightarrow 115-n(A \cap B)=47$ $\Rightarrow n(A \cap B)=68$ Thus, we get: $n(A \cup B)=n(A)+n(B)-n(A \cap B)$ $=115+326-68$ $=373$...
Read More →Prove
Question: $\int \frac{e^{x}(1+x)}{\cos ^{2}\left(e^{x} x\right)} d x$ equals A. $-\cot \left(e^{x} x\right)+C$ B. $\tan \left(x e^{x}\right)+C$ C. $\tan \left(e^{x}\right)+C$ D. $\cot \left(e^{x}\right)+C$ Solution: $\int \frac{e^{x}(1+x)}{\cos ^{2}\left(e^{x} x\right)} d x$ Let $e^{x} x=t$ $\Rightarrow\left(e^{x} \cdot x+e^{x} \cdot 1\right) d x=d t$ $e^{x}(x+1) d x=d t$ $\therefore \int \frac{e^{x}(1+x)}{\sec ^{2}\left(x^{x}\right)} d x=\int \frac{d t}{\cos ^{2} t}$ $=\int \sec ^{2} t d t$ $=\...
Read More →If A and B are two sets such that n (A) =20,
Question: If $A$ and $B$ are two sets such that $n(A)=20, n(B)=25$ and $n(A \cup B)=40$, then write $n(A \cap B)$. Solution: We have: $n(A)=20, n(B)=25$ and $n(A \cup B)=40$ We know: $n(A \cup B)=n(A)+n(B)-n(A \cap B)$ $\Rightarrow n(A \cap B)=n(A)+n(B)-n(A \cup B)$ $=20+25-40$ $=5$...
Read More →Fill in the blanks so as to make the following statements true:
Question: Fill in the blanks so as to make the following statements true: (i) Two distinct points in a plane determine a _____________ line. (ii) Two distinct ___________ in a plane cannot have more than one point in common. (iii) Given a line and a point, not on the line, there is one and only _____________ line which passes through the given point and is _______________ to the given line. (iv) A line separates a plane into _________ parts namely the __________ and the _____ itself. Solution: (...
Read More →Solve the following systems of equations:
Question: Solve the following systems of equations: $\frac{x y}{x+y}=\frac{6}{5}$ $\frac{x y}{y-x}=6$ where $x+y \neq 0, y-x \neq 0$ Solution: The given equations are: $\frac{x y}{x+y}=\frac{6}{5}$ $6 x+6 y=5 x y \quad \ldots(i)$ $\frac{x y}{y-x}=6$ $6 y-6 x=x y \quad \ldots(i i)$ Add both equations, we get $6 x+6 y=5 x y$ Put the value of $x$ in equation $(i)$, we get $22 \times-\frac{1}{11}+15 v=5$ $\Rightarrow 15 v=3$ $\Rightarrow v=\frac{1}{5}$ Hence the value of $x=2$ and $y=3$...
Read More →Prove
Question: $\int \frac{\sin ^{2} x-\cos ^{2} x}{\sin ^{2} x \cos ^{2} x} d x$ is equal to A. $\tan x+\cot x+C$ B. $\tan x+\operatorname{cosec} x+C$ C. $-\tan x+\cot x+C$ D. $\tan x+\sec x+C$ Solution: $\int \frac{\sin ^{2} x-\cos ^{2} x}{\sin ^{2} x \cos ^{2} x} d x=\int\left(\frac{\sin ^{2} x}{\sin ^{2} x \cos ^{2} x}-\frac{\cos ^{2} x}{\sin ^{2} x \cos ^{2} x}\right) d x$ $=\int\left(\sec ^{2} x-\operatorname{cosec}^{2} x\right) d x$ $=\tan x+\cot x+\mathrm{C}$ Hence, the correct answer is A....
Read More →IF =
Question: If $A=\left\{(x, y): y=e^{x}, x \in R\right\}$ and $B=\left\{(x, y): y=e^{-x}, x \in R\right\}$, then write $A \cap B$. Solution: We have: $A=\left\{(0,1),(1, e),\left(2, e^{2}\right),\left(3, e^{3}\right), \ldots\right\}$ $B=\left\{(0,1),\left(1, e^{-1}\right),\left(2, e^{-2}\right),\left(3, e^{-3}\right), \ldots\right\}$ Thus, we get: $\mathrm{A} \cap \mathrm{B}=\{(0,1)\}$...
Read More →In the below figure. Name the following:
Question: In the below figure. Name the following: Solution: (i) Five line segments AB, CD, AC, PQ. DS (ii) Five rays (iii) Four collinear points. C, D, Q, S (iv) Two pairs of non--intersecting line segments AB and CD, AB and LS....
Read More →If A =
Question: If $A=\left\{(x, y): y=\frac{1}{x}, 0 \neq x \in R\right\}$ and $B=\{(x, y): y=-x, x \in R\}$, then write $A \cap B$. Solution: We have: $A=\left\{(x, y): y=\frac{1}{x}, 0 \neq x \in R\right\}$ $=\left\{(1,1),\left(2, \frac{1}{2}\right),\left(3, \frac{1}{3}\right),\left(4, \frac{1}{4}\right), \ldots\right\}$ And, $B=\{(x, y): y=-x, x \in R\}$ $=\{(1,-1),(2,-2),(3,-3),(4,-4), \ldots\}$ Thus, we get: $A \cap B=\emptyset$...
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