Write the truth value (T/F) of each of the following statements:
Question: Write the truth value (T/F) of each of the following statements: (i) Two lines intersect in a point. (ii) Two lines may intersect in two points (iii) A segment has no length. (iv) Two distinct points always determine a line. (v) Every ray has a finite length. (vi) A ray has one end-point only. (vii) A segment has one end-point only. (viii) The ray AB is same as ray BA. (ix) Only a single line may pass through a given point. (x) Two lines are coincident if they have only one point in co...
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Question: $\frac{1}{\cos (x-a) \cos (x-b)}$ Solution: $\frac{1}{\cos (x-a) \cos (x-b)}=\frac{1}{\sin (a-b)}\left[\frac{\sin (a-b)}{\cos (x-a) \cos (x-b)}\right]$ $=\frac{1}{\sin (a-b)}\left[\frac{\sin [(x-b)-(x-a)]}{\cos (x-a) \cos (x-b)}\right]$ $=\frac{1}{\sin (a-b)} \frac{[\sin (x-b) \cos (x-a)-\cos (x-b) \sin (x-a)]}{\cos (x-a) \cos (x-b)}$ $=\frac{1}{\sin (a-b)}[\tan (x-b)-\tan (x-a)]$ $\Rightarrow \int \frac{1}{\cos (x-a) \cos (x-b)} d x=\frac{1}{\sin (a-b)} \int[\tan (x-b)-\tan (x-a)] d x...
Read More →Solve the following systems of equations:
Question: Solve the following systems of equations: $\frac{6}{x+y}=\frac{7}{x-y}+3$ $\frac{1}{2(x+y)}=\frac{1}{3(x-y)^{\dagger}}$ where $x+y \neq 0$ and $x-y \neq 0$ Solution: The given equations are: $\frac{6}{x+y}=\frac{7}{x-y}+3$ $\frac{1}{2(x+y)}=\frac{1}{3(x-y)}$ Let $\frac{1}{x+y}=u$ and $\frac{1}{x-y}=v$ then equations are $6 u=7 v+3 \ldots(i)$ $\frac{u}{2}=\frac{v}{3}$ ...(ii) Multiply equation (ii) by 12 and subtract (ii) from (i), we get Put the value of $v$ in equation $(i)$, we get $...
Read More →(i) Given two points P and Q. Find how many line segments do they determine.
Question: (i) Given two points P and Q. Find how many line segments do they determine. (ii) Name the line segments determined by the three collinear points P. Q and R. Solution: (i) One (ii) PQ, QR, PR...
Read More →(i) How many lines can pan through a given point?
Question: (i) How many lines can pan through a given point? (ii) In how many points can two distinct lines at the most intersect? Solution: (i) Infinitely many (ii) One...
Read More →Define the following terms.
Question: Define the following terms. (i) Line segment (ii) Collinear points (iii) Parallel lines (iv) Intersecting lines (v) Concurrent lines (vi) Ray (vii) Half-line Solution: (i)Line-segment: Give two points A and B on a line I. the connected part (segment) of the line with end points at A and B is called the line segment AB. (ii)Collinear points: Three or more points are said to be collinear if there is a line which contains all of them. (iii)Parallel lines: Two lines l and m in a plane are ...
Read More →Let A and B be two sets having 4 and 7 elements respectively.
Question: Let $A$ and $B$ be two sets having 4 and 7 elements respectively. Then write the maximum number of elements that $A \cup B$ can have. Solution: We know that $n(A \cup B)=n(A)+n(B)-n(A \cap B)$ $n(A \cup B)$ is maximum when $n(A \cap B)$ is minimum so, $n(A \cap B)=0$ Hence, $n(A \cup B)=n(A)+n(B)-n(A \cap B)$ $=4+7-0$ $=11$...
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Question: $\sin ^{-1}(\cos x)$ Solution: $\sin ^{-1}(\cos x)$ Let $\cos x=t$ Then, $\sin x=\sqrt{1-t^{2}}$ $\Rightarrow(-\sin x) d x=d t$ $d x=\frac{-d t}{\sin x}$ $d x=\frac{-d t}{\sqrt{1-t^{2}}}$ $\therefore \int \sin ^{-1}(\cos x) d x=\int \sin ^{-1} t\left(\frac{-d t}{\sqrt{1-t^{2}}}\right)$ $=-\int \frac{\sin ^{-1} t}{\sqrt{1-t^{2}}} d t$ Let $\sin ^{-1} t=u$ $\Rightarrow \frac{1}{\sqrt{1-r^{2}}} d t=d u$ $\therefore \int \sin ^{-1}(\cos x) d x=\int 4 d u$ $=-\frac{u^{2}}{2}+C$ $=\frac{-\le...
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Question: $\sin ^{-1}(\cos x)$ Solution: $\sin ^{-1}(\cos x)$ Let $\cos x=t$ Then, $\sin x=\sqrt{1-t^{2}}$ $\Rightarrow(-\sin x) d x=d t$ $d x=\frac{-d t}{\sin x}$ $d x=\frac{-d t}{\sqrt{1-t^{2}}}$ $\therefore \int \sin ^{-1}(\cos x) d x=\int \sin ^{-1} t\left(\frac{-d t}{\sqrt{1-t^{2}}}\right)$ $=-\int \frac{\sin ^{-1} t}{\sqrt{1-t^{2}}} d t$ Let $\sin ^{-1} t=u$ $\Rightarrow \frac{1}{\sqrt{1-r^{2}}} d t=d u$ $\therefore \int \sin ^{-1}(\cos x) d x=\int 4 d u$ $=-\frac{u^{2}}{2}+C$ $=\frac{-\le...
Read More →If A and B are two sets such that
Question: If $A$ and $B$ are two sets such that $A \subset B$, then write $B^{\prime}-A^{\prime}$ in terms of $A$ and $B$. Solution: $B^{\prime}-A^{\prime}=$ Not $B-$ Not $A$ $=$ Nothing common in them $=\emptyset$...
Read More →Solve the following systems of equations:
Question: Solve the following systems of equations: $\frac{2}{x}+\frac{3}{y}=\frac{9}{x y}$ $\frac{4}{x}+\frac{9}{y}=\frac{21}{x y}, x \neq 0, y \neq 0$ Solution: The given equations are: $\frac{2}{x}+\frac{3}{y}=\frac{9}{x y} \quad \ldots(i)$ $\frac{4}{x}+\frac{9}{y}=\frac{21}{x y} \ldots($ ii $)$ Multiply equation (i) by 3 and subtract (ii) from (i), we get $\frac{6}{x}+\frac{9}{y}=\frac{27}{x y}$ Put the value of $x$ in equation $(i)$, we get $\Rightarrow \frac{2}{x}+\frac{3}{3}=\frac{9}{3 x}...
Read More →If A
Question: If $A=\left\{x \in C: x^{2}=1\right\}$ and $B=\left\{x \in C: x^{4}=1\right\}$, then write $A-B$ and $B-A$. Solution: We have: $A=\left\{x \in C: x^{2}=1\right\}$ $\Rightarrow A=\{-1,1\}$ And, $B=\left\{x \in C: x^{4}=1\right\}$ $\Rightarrow \mathrm{B}=\left\{x^{4}-1=0\right\}$ $\Rightarrow \mathrm{B}=\left\{\left(x^{2}-1\right)\left(x^{2}+1\right)=0\right\}$ $\Rightarrow \mathrm{B}=\{-1,1,-\mathrm{i}, \mathrm{i}\}$ Thus, we get: $A-B=\emptyset$ And, BA= {-i, i}...
Read More →If A
Question: If $A=\left\{x \in C: x^{2}=1\right\}$ and $B=\left\{x \in C: x^{4}=1\right\}$, then write $A-B$ and $B-A$. Solution: We have: $A=\left\{x \in C: x^{2}=1\right\}$ $\Rightarrow A=\{-1,1\}$ And, $B=\left\{x \in C: x^{4}=1\right\}$ $\Rightarrow \mathrm{B}=\left\{x^{4}-1=0\right\}$ $\Rightarrow \mathrm{B}=\left\{\left(x^{2}-1\right)\left(x^{2}+1\right)=0\right\}$ $\Rightarrow \mathrm{B}=\{-1,1,-\mathrm{i}, \mathrm{i}\}$ Thus, we get: $A-B=\emptyset$ And, BA= {-i, i}...
Read More →Solve the following systems of equations:
Question: Solve the following systems of equations: $\frac{1}{5 x}+\frac{1}{6 y}=12$ $\frac{1}{3 x}-\frac{3}{7 y}=8, x \neq 0, y \neq 0$ Solution: The given equations are: $\frac{1}{5 x}+\frac{1}{6 y}=12$$\ldots(i)$ $\frac{1}{3 x}-\frac{3}{7 y}=8$...(ii) Multiply equation $(i)$ by $\frac{3}{7}$ and equation (ii) by $\frac{1}{6}$, add both equations, we get $\frac{3}{35 x}+\frac{3}{42 y}=\frac{36}{7}$ Put the value of $x$ in equation $(i)$, we get $\frac{1}{\frac{5 \times 89}{4080}}+\frac{1}{6 y}...
Read More →Let A and B be two sets having 3 and 6 elements respectively.
Question: Let $A$ and $B$ be two sets having 3 and 6 elements respectively. Write the minimum number of elements that $A \cup B$ can have. Solution: We know that $n(A \cup B)=n(A)+n(B)-n(A \cap B)$ $n(A \cup B)$ is minimum when $n(A \cap B)$ is maximum so, $n(A \cap B)=3$ Hence, $n(A \cup B)=n(A)+n(B)-n(A \cap B)$ $=3+6-3$ $=6$...
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Question: $\frac{\cos 2 x}{(\cos x+\sin x)^{2}}$ Solution: $\frac{\cos 2 x}{(\cos x+\sin x)^{2}}=\frac{\cos 2 x}{\cos ^{2} x+\sin ^{2} x+2 \sin x \cos x}=\frac{\cos 2 x}{1+\sin 2 x}$ $\therefore \int \frac{\cos 2 x}{(\cos x+\sin x)^{2}} d x=\int \frac{\cos 2 x}{(1+\sin 2 x)} d x$ Let $1+\sin 2 x=t$ $\Rightarrow 2 \cos 2 x d x=d t$ $\therefore \int \frac{\cos 2 x}{(\cos x+\sin x)^{2}} d x$$=\frac{1}{2}$$\int_{t}^{1} \frac{1}{d t}$ $=\frac{1}{2} \log |t|+\mathrm{C}$ $=\frac{1}{2} \log |1+\sin 2 x|...
Read More →Let A = {x : x ∈ N, x is a multiple of 3}
Question: Let $A=\{x: x \in N, x$ is a multiple of 3$\}$ and $B=\{x: x \in N$ and $x$ is a multiple of 5$\}$. Write $A \cap B$. Solution: A= {x:xNandxis a multiple of 3.} = {3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45,...} B= {x:xNandxis a multiple of 5.} ={5, 10, 15, 20, 25, 30, 35, 40, 45,...} Thus, we have: $A \cap B=\{15,30,45, \ldots\}$ $=\{x: x \in N$, where $x$ is a multiple of $15 .\}$...
Read More →Factorize each of the following polynomials:
Question: Factorize each of the following polynomials: 1. $x^{3}+13 x^{2}+31 x-45$ given that $x+9$ is a factor 2. $4 x^{3}+20 x^{2}+33 x+18$ given that $2 x+3$ is a factor Solution: 1. $x^{3}+13 x^{2}+31 x-45$ given that $x+9$ is a factor let, $f(x)=x^{3}+13 x^{2}+31 x-45$ given that (x + 9) is the factor divide f(x) with (x + 9) to get other factors by , long division $x^{2}+4 x-5$ $x+9 x^{3}+13 x^{2}+31 x-45$ $x^{3}+9 x^{2}$ (-) (-) $4 x^{2}+31 x$ $4 x^{2}+36 x$ (-) (-) -5x 45 -5x 45 (+) (+) ...
Read More →Write the number of elements in the power set of null set.
Question: Write the number of elements in the power set of null set. Solution: We know that a set of $n$ elements has $2^{n}$ subsets or elements. A null set has no element(s) in it. $\therefore$ Number of elements in the power set of null set $=2^{0}=1$...
Read More →If a set contains n elements,
Question: If a set containsnelements, then write the number of elements in its power set. Solution: A set having $n$ elements has $2^{n}$ subsets or elements....
Read More →If a set contains n elements,
Question: If a set containsnelements, then write the number of elements in its power set. Solution: A set havingnelements has2n2nsubsets or elements....
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Question: $\frac{1}{\sin x \cos ^{3} x}$ Solution: $\frac{1}{\sin x \cos ^{3} x}=\frac{\sin ^{2} x+\cos ^{2} x}{\sin x \cos ^{3} x}=\frac{\sin x}{\cos ^{3} x}+\frac{1}{\sin x \cos x}$ $\Rightarrow \frac{1}{\sin x \cos ^{3} x}=\tan x \sec ^{2} x+\frac{\frac{1}{\cos ^{2} x}}{\frac{\sin x \cos x}{\cos ^{2} x}}=\tan x \sec ^{2} x+\frac{\sec ^{2} x}{\tan x}$ $\therefore \int \frac{1}{\sin x \cos ^{3} x} d x=\int \tan x \sec ^{2} x d x+\int \frac{\sec ^{2} x}{\tan x} d x$ Let $\tan x=t \Rightarrow \se...
Read More →If A and B are two finite sets such that
Question: IfAandBare two finite sets such thatn(A) n(B) and the difference of the number of elements of the power sets ofAandBis 96, thenn(A) n(B) = ____________. Solution: Ifn(A) n(B) andn(P(A)) n(P(B)) = 96 given where $P(A)$ and $P(B)$ represents power left of $A \neq B$ respectively. Letn(A) =nandn(B) =m i.en(P(A)) =2nandn(P(B)) = 2m i.e2n 2m= 96 2m(2nm 1) = 96= 25 3 i.e 2m= 25 i.em= 5and 2nm 1 = 3 2nm = 4 = 22 i.e.nm= 2 i.en= 2 +m n= 2 + 5 i.e.n= 7 n(A) n(B) =nm= 2...
Read More →For any two sets A and B,
Question: For any two setsAandB, ifn(A) =15,n(B) = 12,AB ϕ andBA, then the maximum and and minimum possible values ofn(A∆B) are _______ and___________respectively. Solution: Ifn(A) =15 n(B) = 12 AB ϕ BA Then maximum and possible values ofn(A∆B) = ? SinceABAandABB ⇒n(AB)n(A) andn(AB) n(B) ⇒n(AB min {n(A),n(B)} = 12 ⇒ n(AB) 12 i.en(AB) 12 alsoAA⋃B, BA⋃B i.en(A) n(A⋃B) andn(B) n(A⋃B) ⇒n(A⋃B) max {n(A),n(B)} = 15 i.e.n(A⋃B) 15 ⇒n(A∆B) =n(A⋃B) n(AB) 15 12 = 3 i.en(A∆B) 3 i.e maximum value ofn(A∆B) = ...
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Question: $\frac{\cos 2 x+2 \sin ^{2} x}{\cos ^{2} x}$ Solution: $\frac{\cos 2 x+2 \sin ^{2} x}{\cos ^{2} x}$ $=\frac{\cos 2 x+(1-\cos 2 x)}{\cos ^{2} x}$ $\left[\cos 2 x=1-2 \sin ^{2} x\right]$ $=\frac{1}{\cos ^{2} x}$ $=\sec ^{2} x$ $\therefore \int \frac{\cos 2 x+2 \sin ^{2} x}{\cos ^{2} x} d x=\int \sec ^{2} x d x=\tan x+\mathrm{C}$...
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