Find all point of discontinuity of the function
Question: Find all point of discontinuity of the function $f(t)=\frac{1}{t^{2}+t-2}$, where $t=\frac{1}{x-1}$ Solution: $f(t)=\frac{1}{t^{2}+t-2}$ Now, let $u=\frac{1}{x-1}$ $\therefore f(u)=\frac{1}{u^{2}+2 u-u-2}=\frac{1}{u^{2}+u-2}=\frac{1}{(u+2)(u-1)}$ So, $f(u)$ is not defined at $u=-2$ and $u=1$ If $u=-2$, then $-2=\frac{1}{x-1}$ $\Rightarrow 2 x=1$ $\Rightarrow x=\frac{1}{2}$ If $u=1$, then $1=\frac{1}{x-1}$ $\Rightarrow x=2$ Hence, the function is discontinuous at $x=\frac{1}{2}, 2$...
Read More →A bag contains 3 red balls, 5 black balls and 4 white balls.
Question: A bag contains 3 red balls, 5 black balls and 4 white balls. A ball is drawn at random from the bag. What is the probability that the ball drawn is: (i) white? (ii) red? (iii) black? (iv) not red? Solution: Number of red balls $=3$ Number of black balls $=5$ Number of white balls $=4$ Total number of balls $=3+5+4=12$ Therefore, the total number of cases is 12 . (i) Since there are 4 white balls, the number of favourable outcomes is 4 . $\mathrm{P}($ a white ball $)=\frac{\text { Numbe...
Read More →Given the function
Question: Given the function $f(x)=\frac{1}{x+2}$. Find the points of discontinuity of the function $f(f(x))$. Solution: $f[f(x)]=\frac{1}{\frac{1}{x+2}+2}=\frac{x+2}{2 x+5}$ So, $f[f(x)]$ is not defined at $x+2=0$ and $2 x+5=0$ If $x+2=$, then $x=-2$ If $2 x+5=0$, then $x=-\frac{5}{2}$ Hence, the function is discontinuous at $x=-\frac{5}{2}$ and $-2$...
Read More →Let A = {a, b, c, e, f} B = {c, d, e, g} and C = {b, c, f, g} be subsets of the set U
Question: Let A = {a, b, c, e, f} B = {c, d, e, g} and C = {b, c, f, g} be subsets of the set U = {a, b, c, d, e, f, g, h}. (i) $\mathbf{A} \cap \mathbf{B}$ (ii) $A \cup(B \cap C)$ (iii) $\mathbf{A}-\mathbf{B}$ (iv) $\mathbf{B}-\mathbf{A}$ (v) $A-(B \cap C)$ (vi) $(B-C) \cup(C-B)$ Solution: (i) $A^{\cap} B$ will contain the common elements of $A$ and $B$ $\mathrm{A}^{\cap} \mathrm{B}=\{\mathrm{c}, \mathrm{e}\}$ (ii) $A U\left(B^{\cap} C\right)$ $B^{\cap} C=\{c, d, g\}$ $\mathrm{AU}\left(\mathrm{...
Read More →An urn contains 10 red and 8 white balls. One ball is drawn at random.
Question: An urn contains 10 red and 8 white balls. One ball is drawn at random. Find the probability that the ball drawn is white. Solution: Number of red balls $=10$ Number of white balls $=8$ Total number of balls in the urn $=10+8=18$ Therefore, the total number of cases is 18 and the number of favourable cases is 8 . $\therefore \mathrm{P}$ (The ball drawn is white) $=\frac{\text { Number of favourable cases }}{\text { Total number of cases }}=\frac{8}{18}=\frac{4}{9}$...
Read More →Solve this
Question: Determine if $f(x)=\left\{\begin{array}{cl}x^{2} \sin \frac{1}{x}, x \neq 0 \\ 0, x=0\end{array}\right.$ is a continuous function? Solution: The given function $f$ is $f(x)= \begin{cases}x^{2} \sin \frac{1}{x}, \text { if } x \neq 0 \\ 0, \text { if } x=0\end{cases}$ It is evident that $f$ is defined at all points of the real line. Let $c$ be a real number. Case I: If $c \neq 0$, then $f(c)=c^{2} \sin \frac{1}{c}$ $\lim _{x \rightarrow c} f(x)=\lim _{x \rightarrow c}\left(x^{2} \sin \f...
Read More →A card is drawn at random from a pack of 52 cards. Find the probability that the card drawn is:
Question: A card is drawn at random from a pack of 52 cards. Find the probability that the card drawn is: (i) a black king (ii) either a black card or a king (iii) black and a king (iv) a jack, queen or a king (v) neither a heart nor a king (vi) spade or an ace (vii) neither an ace nor a king (viii) neither a red card nor a queen. (ix) other than an ace (x) a ten (xi) a spade (xii) a black card (xiii) the seven of clubs (xiv) jack (xv) the ace of spades (xvi) a queen (xvii) a heart (xviii) a red...
Read More →0 is the point of intersection of the diagonals AC and BD
Question: 0 is the point of intersection of the diagonals AC and BD of a trapezium ABCD with AB || DC. Through 0, a line segment PQ is drawn parallel to AB meeting AD in P and BC in Q, prove that PO = QO. Solution: Given ABCD is a trapezium. Diagonals AC and BD are intersect at 0. PQ||AB||DC. To prove $P O=Q O$ Proof in $\triangle A B D$ and $\triangle P O D$, $P O \| A B$ $[\because P Q \| A B]$ $\angle D=\angle D \quad$ [common angle] $\angle A B D=\angle P O D \quad$ [corresponding angles] $\...
Read More →Three coins are tossed together. Find the probability of getting:
Question: Three coins are tossed together. Find the probability of getting: (i) exactly two heads (ii) at least two heads (iii) at least one head and one tail (iv) no tails Solution: When 3 coins are tossed together, the outcomes are as follows: $\mathrm{S}=\{(\mathrm{h}, \mathrm{h}, \mathrm{h}),(\mathrm{h}, \mathrm{h}, \mathrm{t}),(\mathrm{h}, \mathrm{t}, \mathrm{h}),(\mathrm{h}, \mathrm{t}, \mathrm{t}),(\mathrm{t}, \mathrm{h}, \mathrm{h}),(\mathrm{t}, \mathrm{h}, \mathrm{t}),(\mathrm{t}, \math...
Read More →For any sets A and B, prove that:
Question: For any sets A and B, prove that: (i) $A \cap B^{\prime}=\phi \Rightarrow A \subset B$ (ii) $A^{\prime} \cup B^{\prime}=U \Rightarrow A \subset B$ Solution: (i) The Venn Diagram for the given condition is given below As can be seen from the Venn Diagram, A is a proper subset of B $\Rightarrow \mathrm{A} \subset \mathrm{B}$ (ii) Wrong question. If $A$ is a proper subset of $B$ then $A^{, U} B^{\prime} \neq U$...
Read More →Find all the points of discontinuity
Question: Find all the points of discontinuity of $f$ defined by $f(x)=|x|-|x+1|$. Solution: Given: $f(x)=|x|-|x+1|$ The two functions, $g$ and $h$, are defined as $g(x)=|x|$ and $h(x)=|x+1|$ Then, $f=g-h$ The continuity of $g$ and $h$ is examined first. $g(x)=|x|$ can be written as $g(x)= \begin{cases}-x, \text { if } x0 \\ x, \text { if } x \geq 0\end{cases}$ Clearly, $g$ is defined for all real numbers. Let $c$ be a real number. Case I: If $c0$, then $g(c)=-c$ and $\lim _{x \rightarrow c} g(x...
Read More →For any sets A and B, prove that:
Question: For any sets A and B, prove that: (i) $(A-B) \cap B=\phi$ (ii) $A \cup(B-A)=A \cup B$ (iii) $(A-B) \cup(A \cap B)=A$ (iv) $(A \cup B)-B=A-B$ (iv) $A-(A \cup B)=A-B$ Solution: Two sets are shown with the following Venn Diagram The yellow region is denoted by 1. Blue region is denoted by 2. The common region is denoted by 3. (i) A - B denotes region 1 B denotes region (2+3) So their intersection is a : set $\Rightarrow(A-B)^{\cap} B=\varnothing$ (ii) B - A denotes region 2 A denotes regi...
Read More →In a simultaneous throw of a pair of dice, find the probability of getting:
Question: In a simultaneous throw of a pair of dice, find the probability of getting: (i) 8 as the sum (ii) a doublet (iii) a doublet of prime numbers (iv) a doublet of odd numbers (v) a sum greater than 9 (vi) an even number on first (vii) an even number on one and a multiple of 3 on the other (viii) neither 9 nor 11 as the sum of the numbers on the faces (ix) a sum less than 6 (x) a sum less than 7 (xi) a sum more than 7 (xii) at least once (xiii) a number other than 5 on any dice. Solution: W...
Read More →In figure, PA, QB, RC and SD are
Question: In figure, PA, QB, RC and SD are all perpendiculars to a line i, AB = 6 cm, BC = 9 cm, CD = 12 cm and SP = 36 cm. Find PQ, QR and RS. Solution: Given, AS = 6 cm, BC = 9 cm, CD = 12 cm and SP = 36 cm Also, PA, QB, RC and SD are all perpendiculars to line l. PA || QS|| SC || SD By basic proportionality theorem, $P Q: Q R: R S=A B: B C: C D$ $=6: 9: 12$ Let $\quad P Q=6 x, Q R=9 x$ and $R S=12 x$ Since, length of$P S=36 \mathrm{~km}$ $\therefore$ $P Q+Q R+R S=36$ $\Rightarrow \quad 6 x+9 ...
Read More →Show that f (x) = | cos x | is a continuous function.
Question: Show that $f(x)=|\cos x|$ is a continuous function. Solution: The given function is $f(x)=|\cos x|$ This function $f$ is defined for every real number and $f$ can be written as the composition of two functions as, $f=g \circ h$, where $g(x)=|x|$ and $h(x)=\cos x$ $[\because(g o h)(x)=g(h(x))=g(\cos x)=|\cos x|=f(x)]$ It has to be first proved that $g(x)=|x|$ and $h(x)=\cos x$ are continuous functions. $g(x)=|x|$ can be written as $g(x)= \begin{cases}-x, \text { if } x0 \\ x, \text { if...
Read More →Given an example of three sets A, B, C such that
Question: Given an example of three sets $A, B, C$ such that $A \cap C \neq \phi, B \cap C \neq \varnothing, A \cap C$ $\neq \phi$, and $A \cap B \cap C=\phi$ Solution: Let A = {1, 2} $B=\{2,3\}$ $C=\{1,3,4\}$ $A^{\cap} B=\{2\}$ $A^{\cap} C=\{1\}$ $B^{\cap} C=\{3\}$ $\mathrm{A}^{\cap} \mathrm{B} \cap \mathrm{C}=\{2\} \cap\{1,3,4\}=\varnothing$ So the three sets are valid and satisfy the given conditions...
Read More →In given figure,
Question: In given figure, $l \| m$ and line segments $A B, C D$ and $E F$ are concurrent at point $P$. Prove that $\frac{A E}{B F}=\frac{A C}{B D}=\frac{C E}{F D}$. Solution: Given $l \| m$ and line segments $A B, C D$ and $E F$ are concurrent at point $P$. To prove $\frac{A E}{B F}=\frac{A C}{B D}=\frac{C E}{F D}$ Proof in $\triangle \mathrm{APC}$ and $\triangle B P D$.$\angle A P C=\angle B P D$ [vertically opposite angles] $\angle P A C=\angle P B D$ [alternate angles] $\therefore$ $\triangl...
Read More →Let A = {a, b, c}, B = {b, c, d, e} and = {c, d, e, f} be subsets of U = {a, b, c, d,
Question: Let A = {a, b, c}, B = {b, c, d, e} and = {c, d, e, f} be subsets of U = {a, b, c, d, e, f}. Then verify that: (i) $\left(A^{\prime}\right)^{\prime}=A$ (ii) $(A \cup B)^{\prime}=\left(A^{\prime} \cap B^{\prime}\right)$ (iii) $(A \cap B)^{\prime}=\left(A^{\prime} \cup B^{\prime}\right)$ Solution: (i) A = {d, e, f} $\left(A^{\prime}\right)^{\prime}=\{a, b, c\}=A$ Hence proved (ii) $A^{U} B=\{a, b, c, d, e\}$ $\left(\mathrm{A}^{\cup} \mathrm{B}\right)^{\prime}=\{\mathrm{f}\}$ $\mathrm{A}^...
Read More →If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8}, and = {2, 3, 5, 7} verify that:
Question: If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8}, and = {2, 3, 5, 7} verify that: (i) $(A \cup B)^{\prime}=\left(A^{\prime} \cap B^{\prime}\right)$ (ii) $(A \cap C)^{\prime}=\left(A^{\prime} \cup B^{\prime}\right)$ Solution: (i) $A^{U} B=\{2,3,4,5,6,7,8\}$ $\left(\mathrm{A}^{U} \mathrm{~B}\right)^{\prime}=\{1,9\}$ $\mathrm{A}^{\prime}=\{1,3,5,7,9\}$ $\mathrm{B}^{\prime}=\{1,4,6,8,9\}$ $A^{\prime} \cap_{B^{\prime}}=\{1,9\}$ $\Rightarrow\left(\mathrm{A}^{\mathrm{U}} \mathrm{B}\right)...
Read More →Show that
Question: Show that $f(x)=\cos x^{2}$ is a continuous function. Solution: Given: $f(x)=\cos \left(x^{2}\right)$ This function $f$ is defined for every real number and $f$ can be written as the composition of two functions as $f=g \circ h$, where $g(x)=\cos x$ and $h(x)=x^{2}$ $\left[\because(g o h)(x)=g(h(x))=g\left(x^{2}\right)=\cos \left(x^{2}\right)=f(x)\right]$ It has to be first proved that $g(x)=\cos x$ and $h(x)=x^{2}$ are continuous functions. It is evident that $g$ is defined for every ...
Read More →If A = {x : x ϵ N, x ≤ 7}, B = {x : x is prime, x < 8} and C = {x : x ϵ N, x is odd
Question: If A = {x : x ϵ N, x 7}, B = {x : x is prime, x 8} and C = {x : x ϵ N, x is odd and x 10}, verify that (i) $A \cup(B \cap C)=(A \cup B) \cap(A \cup C)$ (ii) $A \cap(B \cup C)=(A \cap B) \cup(A \cap C)$ Solution: Natural numbers start from 1 $A=\{1,2,3,4,5,6,7\}$ $B=\{2,3,5,7\}$ $C=\{1,3,5,7,9\}$ (i) $B^{\cap} C=\{3,5,7\}$ $\mathrm{AU}\left(\mathrm{B}^{\cap} \mathrm{C}\right)=\{1,2,3,4,5,6,7\}$ $\mathrm{A}^{U} \mathrm{~B}=\{1,2,3,4,5,6,7\}$ $\mathrm{A}^{U} \mathrm{C}=\{1,2,3,4,5,6,7,9\}...
Read More →In a quadrilateral ΔBCD, ∠A+ ∠D = 90°.
Question: In a quadrilateral ΔBCD, A+ D = 90. Prove that AC2+ BD2= AD2+ BC2. Solution: Given Quadrilateral ΔBCD, in which A+ D = 90 To prove AC2+ BD2= AD2+ BC2 Construct Produce AB and CD to meet at E. Also, join $A C$ and $B D$. Proof In $\triangle A E D, \quad \angle A+\angle D=90^{\circ}$[given] $\therefore$ $\angle E=180^{\circ}-(\angle A+\angle D)=90^{\circ}$ Then, by Pythagoras theorem, $\quad A D^{2}=A E^{2}+D E^{2}$ In $\triangle B E C$, by Pythagoras theorem, $B C^{2}=B E^{2}+E F^{2}$ O...
Read More →A die is thrown. Find the probability of getting:
Question: A die is thrown. Find the probability of getting: (i) a prime number (ii) 2 or 4 (iii) a multiple of 2 or 3 Solution: When a die is thrown, the possible outcomes are $1,2,3,4,5$ and 6 . Thus, the sample space will be as follows: $\mathrm{S}=\{1,2,3,4,5,6\}$ (i) Let $A$ be the event of getting a prime number. There are 3 prime numbers $(2,3$ and 5$)$ in the sample space. Thus, the number of favourable outcomes is 3 . Hence, the probability of getting a prime number is as follows: $\math...
Read More →Discuss the continuity of the following functions:
Question: Discuss the continuity of the following functions: (i) $f(x)=\sin x+\cos x$ (ii) $f(x)=\sin x-\cos x$ (iii) $f(x)=\sin x \cos x$ Solution: It is known that if $g$ and $h$ are two continuous functions, then $g+h, g-h$ and $g \times h$ are also continuous. It has to proved first that $g(x)=\sin x$ and $h(x)=\cos x$ are continuous functions. Let $g(x)=\sin x$ It is evident that $g(x)=\sin x$ is defined for every real number. Let $c$ be a real number. Put $x=c+h$ If $x \rightarrow c$, then...
Read More →Discuss the continuity of the following functions:
Question: Discuss the continuity of the following functions: (i) $f(x)=\sin x+\cos x$ (ii) $f(x)=\sin x-\cos x$ (iii) $f(x)=\sin x \cos x$ Solution: It is known that if $g$ and $h$ are two continuous functions, then $g+h, g-h$ and $g \times h$ are also continuous. It has to proved first that $g(x)=\sin x$ and $h(x)=\cos x$ are continuous functions. Let $g(x)=\sin x$ It is evident that $g(x)=\sin x$ is defined for every real number. Let $c$ be a real number. Put $x=c+h$ If $x \rightarrow c$, then...
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