Which of the following statements are true?
Question: Which of the following statements are true?(i) A line segment has no definite length.(ii) A ray has no end-point.(iii) A line has a definite length. (iv) A line $\overleftrightarrow{A B}$ is same as line $\overleftrightarrow{B A}$ (v) A ray $\overrightarrow{A B}$ is same as ray $\overrightarrow{B A}$. (vi) Two distinct points always determine a unique line.(vii) Three lines are concurrent if they have a common point.(viii) Two distinct lines cannot have more than one point in common.(i...
Read More →A tea party is arranged for 16 persons along two sides of a long table with 8 chairs on each side.
Question: A tea party is arranged for 16 persons along two sides of a long table with 8 chairs on each side. Four persons wish to sit on one particular side and two on the other side. In how many ways can they be seated? Solution: A tea party is arranged for 16 people along two sides of a long table with 8 chairs on each side. 4 people wish to sit on sideAA(say) and two on sideBB(say). Now, 10 people are left, out of which 4 people can be selected for sideAAin10C4ways. And, from the remaining pe...
Read More →A tea party is arranged for 16 persons along two sides of a long table with 8 chairs on each side.
Question: A tea party is arranged for 16 persons along two sides of a long table with 8 chairs on each side. Four persons wish to sit on one particular side and two on the other side. In how many ways can they be seated? Solution: A tea party is arranged for 16 people along two sides of a long table with 8 chairs on each side. 4 people wish to sit on sideAA(say) and two on sideBB(say). Now, 10 people are left, out of which 4 people can be selected for sideAAin10C4ways. And, from the remaining pe...
Read More →Find the number of combinations and permutations of 4 letters taken from the word 'EXAMINATION'
Question: Find the number of combinations and permutations of 4 letters taken from the word 'EXAMINATION' Solution: There are 11 letters in the word EXAMINATION, namely AA, NN, II, E, X, M, T and O. The four-letter word may consist of (i) 2 alike letters of one kind and 2 alike letters of the second kind (ii) 2 alike letters and 2 distinct letters (iii) all different letters Now, we shall discuss the three cases one by one. (i) 2 alike letters of one kind and 2 alike letters of the second kind: ...
Read More →Find the number of combinations and permutations of 4 letters taken from the word 'EXAMINATION'
Question: Find the number of combinations and permutations of 4 letters taken from the word 'EXAMINATION' Solution: There are 11 letters in the word EXAMINATION, namely AA, NN, II, E, X, M, T and O. The four-letter word may consist of (i) 2 alike letters of one kind and 2 alike letters of the second kind (ii) 2 alike letters and 2 distinct letters (iii) all different letters Now, we shall discuss the three cases one by one. (i) 2 alike letters of one kind and 2 alike letters of the second kind: ...
Read More →The distinct linear functions that map [−1, 1] onto [0, 2] are
Question: The distinct linear functions that map [1, 1] onto [0, 2] are (a) $f(x)=x+1, g(x)=-x+1$ (b) $f(x)=x-1, g(x)=x+1$ (c) $f(x)=-x-1, g(x)=x-1$ (d) None of these Solution: Let us substitute the end-points of the intervals in the given functions. Here, domain = [-1, 1] and range =[0, 2]By substituting -1 or 1 in each option, we get: Option (a): $f(-1)=-1+1=0$ and $f(1)=1+1=2$ $g(-1)=1+1=2$ and $g(1)=-1+1=0$ So, option (a) is correct. Option (b): $f(-1)=-1-1=-2$ and $f(1)=1-1=0$ $g(-1)=-1+1=0...
Read More →The first term of an A.P. is 5,
Question: The first term of an A.P. is 5, the common difference is 3 and the last term is 80; Solution: In the given problem, we are given an A.P whose, First term (a) = 5 Last term $\left(a_{e}\right)=80$ Common difference (d) = 3 We need to find the number of terms present in it (n) So here we will find the value of $n$ using the formula, $a_{n}=a+(n-1) d$ So, substituting the values in the above mentioned formula $80=5+(n-1) 3$ $80-5=3 n-3$ $75+3=3 n$ $n=\frac{78}{3}$ $n=26$ Thus, $n=26$ Ther...
Read More →A business man hosts a dinner to 21 guests.
Question: A business man hosts a dinner to 21 guests. He is having 2 round tables which can accommodate 15 and 6 persons each. In how many ways can he arrange the guests? Solution: A businessman hosts a dinner for 21 guests. 15 people can be accommodated at one table in ${ }^{21} \mathrm{C}_{15}$ ways. They can arrange themselves in $(15-1) !=14 !$ ways. The remaining 6 people can be accommodated at another table in ${ }^{6} \mathrm{C}_{6}$ ways. They can arrange themselves in $(6-1) !=5 !$ ways...
Read More →A business man hosts a dinner to 21 guests.
Question: A business man hosts a dinner to 21 guests. He is having 2 round tables which can accommodate 15 and 6 persons each. In how many ways can he arrange the guests? Solution: A businessman hosts a dinner for 21 guests. 15 people can be accommodated at one table in ${ }^{21} \mathrm{C}_{15}$ ways. They can arrange themselves in $(15-1) !=14 !$ ways. The remaining 6 people can be accommodated at another table in ${ }^{6} \mathrm{C}_{6}$ ways. They can arrange themselves in $(6-1) !=5 !$ ways...
Read More →How many terms are there in the A.P.?
Question: How many terms are there in the A.P.? (i) $7,10,13, \ldots 43$. (ii) $-1, \frac{5}{6}, \frac{2}{3}, \frac{1}{2}, \ldots \frac{10}{3}$. (iii) $7,13,19, \ldots, 205$. (iv) $18,15 \frac{1}{2}, 13, \ldots,-47$. Solution: In the given problem, we are given an A.P. We need to find the number of terms present in it So here we will find the value of $n$ using the formula, $a_{n}=a+(n-1) d$ (i) Here, A.P is $7,10,13, \ldots .43$ The first term $(a)=7$ The last term $\left(a_{n}\right)=43$ Now, ...
Read More →How many words can be formed by taking 4 letters at a time from the letters of the word 'MORADABAD'?
Question: How many words can be formed by taking 4 letters at a time from the letters of the word 'MORADABAD'? Solution: There are 9 letters in the word MORADABAD, namely AAA, DD, M, R, B and O. The four-letter word may consists of (i) 3 alike letters and 1 distinct letter (ii) 2 alike letters of one kind and 2 alike letters of the other kind (iii) 2 alike letters and 2 distinct letters (iv) all different letters (i) 3 alike letters and 1 distinct letter: There is one set of three alike letters,...
Read More →How many words can be formed by taking 4 letters at a time from the letters of the word 'MORADABAD'?
Question: How many words can be formed by taking 4 letters at a time from the letters of the word 'MORADABAD'? Solution: There are 9 letters in the word MORADABAD, namely AAA, DD, M, R, B and O. The four-letter word may consists of (i) 3 alike letters and 1 distinct letter (ii) 2 alike letters of one kind and 2 alike letters of the other kind (iii) 2 alike letters and 2 distinct letters (iv) all different letters (i) 3 alike letters and 1 distinct letter: There is one set of three alike letters,...
Read More →Find the number of ways in which :
Question: Find the number of ways in which : (a) a selection (b) an arrangement, of four letters can be made from the letters of the word 'PROPORTION'. Solution: There are 10 letters in the word PROPORTION, namely OOO, PP, RR, I, T and N.(a) The four-letter word may consists of (i) 3 alike letters and 1 distinct letter (ii) 2 alike letters of one kind and 2 alike letters of the second kind (iii) 2 alike letters and 2 distinct letters (iv) all distinct letters Now, we shall discuss these four cas...
Read More →Find the number of ways in which :
Question: Find the number of ways in which : (a) a selection (b) an arrangement, of four letters can be made from the letters of the word 'PROPORTION'. Solution: There are 10 letters in the word PROPORTION, namely OOO, PP, RR, I, T and N.(a) The four-letter word may consists of (i) 3 alike letters and 1 distinct letter (ii) 2 alike letters of one kind and 2 alike letters of the second kind (iii) 2 alike letters and 2 distinct letters (iv) all distinct letters Now, we shall discuss these four cas...
Read More →How many lines can be drawn through a given point?
Question: (i) How many lines can be drawn through a given point?(ii) How many lines can be drawn through two given points?(iii) At how many points can two lines at the most intersect?(iv) IfA,BandCare three collinear points, name all the line segments determined by them Solution: (i) Infinite lines can be drawn through a given point. (ii) Only one line can be drawn through two given points. (iii) At most two lines can intersect at one point. (iv) The line segments determined by three collinear p...
Read More →From the given figure, name the following:
Question: From the given figure, name the following:(a) Three lines(b) One rectilinear figure(c) Four concurrent points Solution: (a) Line $\overleftrightarrow{P Q}$, Line $\overleftrightarrow{R S}$ and Line $\overleftrightarrow{A B}$ (b) $C E F G$ (c) No point is concurrent....
Read More →Find the number of permutations of n different things taken r at a time such that two specified things occur together?
Question: Find the number of permutations ofndifferent things takenrat a time such that two specified things occur together? Solution: We havendifferent things. We are to selectrthings at a time such that two specified things occur together. Remaining things $=n-2$ Out of the remaining $(n-2)$ things, we can select $(r-2)$ things in ${ }^{n-2} C_{r-2}$ ways. Consider the two things as one and mix them with $(r-2)$ things. Now, we have $(r-1)$ things that can be arranged in $(r-1) !$ ways. But, t...
Read More →Find the number of permutations of n different things taken r at a time such that two specified things occur together?
Question: Find the number of permutations ofndifferent things takenrat a time such that two specified things occur together? Solution: We havendifferent things. We are to selectrthings at a time such that two specified things occur together. Remaining things $=n-2$ Out of the remaining $(n-2)$ things, we can select $(r-2)$ things in ${ }^{n-2} C_{r-2}$ ways. Consider the two things as one and mix them with $(r-2)$ things. Now, we have $(r-1)$ things that can be arranged in $(r-1) !$ ways. But, t...
Read More →In the adjoining figure, name:
Question: In the adjoining figure, name:(i) two pairs of intersecting lines and their corresponding points of intersection(ii) three concurrent lines and their points of intersection(iii) three rays(iv) two line segments Solution: (i) Two pairs of intersecting lines and their point of intersection are $\{\overleftrightarrow{E F}, \overleftrightarrow{G H}$, point $R\},\{\overleftrightarrow{A B}, \overleftrightarrow{C D}$, point $P\}$ (ii) Three concurrent lines are $\{\overleftrightarrow{A B}, \o...
Read More →Let g(x)=1+x−[x] and f(x)
Question: Let $g(x)=1+x-[x]$ and $f(x)=\left\{\begin{array}{ll}-1, x0_{\square} \\ 0, x=0, \\ 1, x0\end{array}\right.$, where $[x]$ denotes the greatest integer less than or equal to $x .$ Then for all $\left.x, f(g)\right)$ is equal to (a) $x$ (b) 1 (c) $f(x)$ (d) $g(x)$ Solution: (b) 1 When, $-1x0$ Then, $g(x)=1+x-[x]$ $\quad=1+x-(-1)=2+x$ $\therefore f(g(x))=1$ When, $x=0$ Then, $g(x)=1+x-[x]$ $\therefore f(g(x))=1$ When, $x1$ Then, $g(x)=1+x-[x]$ $\therefore f(g(x))=1+x-1=x$ Therefore, for e...
Read More →Let g(x)=1+x−[x] and f(x)
Question: Let $g(x)=1+x-[x]$ and $f(x)=\left\{\begin{array}{ll}-1, x0_{\square} \\ 0, x=0, \\ 1, x0\end{array}\right.$, where $[x]$ denotes the greatest integer less than or equal to $x .$ Then for all $\left.x, f(g)\right)$ is equal to (a) $x$ (b) 1 (c) $f(x)$ (d) $g(x)$ Solution: (b) 1 When, $-1x0$ Then, $g(x)=1+x-[x]$ $\quad=1+x-(-1)=2+x$ $\therefore f(g(x))=1$ When, $x=0$ Then, $g(x)=1+x-[x]$ $=1+x-0=1+x$ $\therefore f(g(x))=1$ When, $x1$ Then, $g(x)=1+x-[x]$ $=1+x-1=x$ $\therefore f(g(x))=1...
Read More →If F : [1, ∞)→[2, ∞) is given by
Question: If $F:[1, \infty) \rightarrow[2, \infty)$ is given by $f(x)=x+\frac{1}{x}$, then $f^{-1}(x)$ equals (a) $\frac{x+\sqrt{x^{2}-4}}{2}$ (b) $\frac{x}{1+x^{2}}$ (c) $\frac{x-\sqrt{x^{2}-4}}{2}$ (d) $1+\sqrt{x^{2}-4}$ Solution: Let $f^{-1}(x)=y$ $\Rightarrow f(y)=x$ $\Rightarrow y+\frac{1}{y}=x$ $\Rightarrow y^{2}+1=x y$ $\Rightarrow y^{2}-x y+1=0$ $\Rightarrow y^{2}-2 \times y \times \frac{x}{2}+\left(\frac{x}{2}\right)^{2}-\left(\frac{x}{2}\right)^{2}+1=0$ $\Rightarrow y^{2}-2 \times y \t...
Read More →In the adjoining figure, name
Question: In the adjoining figure, name(i) six points(ii) five lines segments(iii) four rays(iv) four lines(v) four collinear points Solution: (i) Points are A, B, C, D, P and R. (ii) $\overline{E F}, \overline{G H}, \overline{F H}, \overline{E G}, \overline{M N}$ (iii) $\overrightarrow{E P}, \overrightarrow{G R}, \overrightarrow{H S}, \overrightarrow{F Q}$ (iv) $\overleftrightarrow{A B}, \overleftrightarrow{C D}, \overleftrightarrow{P Q}, \overleftrightarrow{R S}$ (v) Collinear points are $\mat...
Read More →Define the following terms:
Question: Define the following terms:(i) Line segment(ii) Ray(iii) Intersecting lines(iv) Parallel lines(v) Half line(vi) Concurrent lines(vii) Collinear points(viii) Plane Solution: (i) Line segment:A line segment is a part of line that is bounded by two distinct end-points. A line segment has a fixed length. (ii) Ray: A line with a start point but no end point and without a definite length is a ray. (iii) Intersecting lines:Two lines with a common point are called intersecting lines. (iv) Para...
Read More →If the function f : R→Rf : R→R be such that f(x)=x−[x]fx=x-x,
Question: If the function $f: R \rightarrow R$ be such that $f(x)=x-[x]$, where $[x]$ denotes the greatest integer less than or equal to $x$, then $f^{-1}(x)$ is (a) $\frac{1}{x-[x]}$ (b) $[x]-x$ (c) not defined (d) none of these Solution: $f(x)=x-[x]$ We know that the range of $f$ is $[0,1)$. Co-domain of $f=R$ As range of $f \neq$ Co-domain of $f, f$ is not onto. $\Rightarrow f$ is not a bijective function. So, $f^{-1}$ does not exist. Thus, the answer is (c)....
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