How many different selections of 4 books can be made from 10 different books,
Question: How many different selections of 4 books can be made from 10 different books, if (i) there is no restriction; (ii) two particular books are always selected; (iii) two particular books are never selected? Solution: (i) Required ways of selecting 4 books from 10 books without any restriction $={ }^{10} C_{4}=\frac{10}{4} \times \frac{9}{3} \times \frac{8}{2} \times 7=210$ (ii) Two particular books are selected from 10 books. So, 2 books need to be selected from 8 books. Required number o...
Read More →Plot the points A(2, 5), B(–2, 2) and C(4, 2) on a graph paper. Join AB, BC and AC. Calculate the area of ∆ABC.
Question: Plot the pointsA(2, 5),B(2, 2) andC(4, 2) on a graph paper. JoinAB, BCandAC. Calculate the area of ∆ABC. Solution: Abscissa ofD= Abscissa ofA= 2Ordinate ofD= Ordinate ofB= 2Now,BC = (2 + 4) units = 6 unitsAD = (5 2) units = 3 units Area of $\Delta A B C=\frac{1}{2} \times$ Base $\times$ Height $=\frac{1}{2} \times B C \times A D$ $=\frac{1}{2} \times 6 \times 3$ $=9$ Hence, area of ∆ABCis 9 square units....
Read More →A function f from the set of natural numbers to the set of integers defined by
Question: A functionffrom the set of natural numbers to the set of integers defined by $f(n) \begin{cases}\frac{n-1}{2}, \text { when } n \text { is odd } \\ -\frac{n}{2}, \text { when } n \text { is even }\end{cases}$ (a) neither one-one nor onto(b) one-one but not onto(c) onto but not one-one(d) one-one and onto Solution: Injectivity:Letxandybe any two elements in the domain (N). Case-1: Both $x$ and $y$ are even. Let $f(x)=f(y)$ $\Rightarrow \frac{-x}{2}=\frac{-y}{2}$ $\Rightarrow-x=-y$ $\Rig...
Read More →From a class of 12 boys and 10 girls, 10 students are to be chosen for a competition; at least including 4 boys and 4 girls.
Question: From a class of 12 boys and 10 girls, 10 students are to be chosen for a competition; at least including 4 boys and 4 girls. The 2 girls who won the prizes last year should be included. In how many ways can the selection be made? Solution: Two girls who won the prizes last year are to be included in every selection. So, we have to select 8 students out of 12 boys and 8 girls, choosing at least 4 boys and 2 girls. Number of ways in which it can be done $={ }^{12} C_{6} \times{ }^{8} C_{...
Read More →How many different products can be obtained by multiplying two or more of the numbers 3, 5, 7, 11 (without repetition)?
Question: How many different products can be obtained by multiplying two or more of the numbers 3, 5, 7, 11 (without repetition)? Solution: Required number of ways of getting different products $={ }^{4} C_{2}+{ }^{4} C_{3}+{ }^{4} C_{4}=6+4+1=11$...
Read More →Which of the following points lie on the x-axis?
Question: Which of the following points lie on thex-axis?(i)A(0, 8)(ii)B(4, 0)(iii)C(0, 3)(iv)D(6, 0)(v)E(2, 1)(vi)F(2, 1)(vii)G(1, 0)(viii)H(0, 2) Solution: (i)A(0, 8)The given point does not lies on thex-axis.(ii)B(4, 0)The ordinate of the point (4, 0) is zero.Hence, the (4, 0) lies on thex-axis.(iii)C(0, 3)The given point does not lies on thex-axis.(iv)D(6, 0)The ordinate of the point (6, 0) is zero.Hence, the (6, 0) lies on thex-axis.(v)E(2, 1)The given point does not lies on thex-axis.(vi)F...
Read More →Write the axis on which the given point lies.
Question: Write the axis on which the given point lies.(i)(2, 0)(ii) (0, 5)(iii) (4, 0)(iv) (0, 1) Solution: (i)(2, 0)The ordinate of the point (2, 0) is zero.Hence, the (2, 0) lies on thex-axis.(ii) (0, 5)The abscissa of the point (0, 5) is zero.Hence, the (0, 5) lies on they-axis.(iii) (4, 0)The ordinate of the point (4, 0) is zero.Hence, the (4, 0) lies on thex-axis.(iv) (0, 1)The abscissa of the point (0, 1) is zero.Hence, the (0, 1) lies on they-axis....
Read More →Which of the following sequences are arithmetic progressions.
Question: Which of the following sequences are arithmetic progressions. For those which are arithmetic progressions, find out the common difference.(i) 3, 6, 12, 24, ...(ii) 0, 4, 8, 12, ... (iii) $\frac{1}{2}, \frac{1}{4}, \frac{1}{6}, \frac{1}{8}, \ldots$ (iv) 12, 2, 8, 18, ...(v) 3, 3, 3, 3, ... (vi) $p, p+90, p+180 p+270, \ldots$ where $p=(999)^{989}$ (vii) $1.0,1.7,2.4,3.1, \ldots$ (viii) $-225,-425,-625,-825, \ldots$ (ix) $10,10+2^{5}, 10+2^{6}, 10+2^{7}, \ldots$ $(x) a+b,(a+1)+b,(a+1)+(b+...
Read More →For each of the following points, write the quadrant in which it lies
Question: For each of the following points, write the quadrant in which it lies(i) (6, 3)(ii) (5, 3)(iii)(11, 6)(iv) (1, 4)(v) (7, 4)(vi)(4, 1)(vii)(3, 8)(viii) (3, 8) Solution: (i) (6, 3)Points of the type (, +) lie in the II quadrant.Hence, the point lies (6, 3) in the II quadrant.(ii) (5, 3)Points of the type (, ) lie in the III quadrant.Hence, the point lies (5, 3) in the III quadrant.(iii)(11, 6)Points of the type (+, +) lie in the I quadrant.Hence, the point lies (11, 6) in the I quadrant....
Read More →There are 10 professors and 20 students out of whom a committee of 2 professors and 3 students is to be formed.
Question: There are 10 professors and 20 students out of whom a committee of 2 professors and 3 students is to be formed. Find the number of ways in which this can be done. Further find in how many of these committees: (i) a particular professor is included. (ii) a particular student is included. (iii) a particular student is excluded. Solution: Clearly, 2 professors and 3 students are selected out of 10 professors and 20 students, respectively. Required number of ways $={ }^{10} C_{2} \times{ }...
Read More →Write down the coordinates of each of the points A, B, C, D, E shown below:
Question: Write down the coordinates of each of the pointsA,B,C,D,Eshown below: Solution: Draw perpendicularAL, BM, CN, DP and EQon theX-axis. (i) Distance ofAfrom theY-axis =OL=-6 unitsDistance ofAfrom theX-axis =AL= 5 unitsHence, the coordinates ofAare (-6,5).(ii) Distance ofBfrom theY-axis =OM=5 unitsDistance ofBfrom theX-axis =BM= 4 unitsHence, the coordinates ofBare (5,4).(iii) Distance ofCfrom theY-axis =ON=-3 units Distance ofCfrom theX-axis =CN= 2 units Hence, the coordinates ofCare (-3,...
Read More →The function f : R→R,
Question: The function $f: R \rightarrow R, f(x)=x^{2}$ is (a) injective but not surjective(b) surjective but not injective(c) injective as well as surjective(d) neither injective nor surjective Solution: Injectivity:Letxandybe any two elements in the domain (R), such thatf(x) = f(y). Then, $x^{2}=y^{2}$ $\Rightarrow x=\pm y$ So,fis not one-one. Surjectivity: As $f(-1)=(-1)^{2}=1$ and $f(1)=1^{2}=1$, $f(-1)=f(1)$ So, both $-1$ and 1 have the same images. $\Rightarrow f$ is not onto. So, the answ...
Read More →In how many ways can a football team of 11 players be selected from 16 players?
Question: In how many ways can a football team of 11 players be selected from 16 players? How many of these will (i) include 2 particular players? (ii) exclude 2 particular players? Solution: Number of ways in which 11 players can be selected out of $16={ }^{16} C_{11}=\frac{16 !}{11 ! 5 !}=\frac{16 \times 15 \times 14 \times 13 \times 12}{5 \times 4 \times 3 \times 2 \times 1}=4368$ (i) If 2 particular players are included, it would mean that out of 14 players, 9 players are selected. Required ...
Read More →On the plane of a graph paper draw X'OX and YOY' as coordinate axes and plot each of the following points.
Question: On the plane of a graph paper drawX'OXandYOY'as coordinate axes and plot each of the following points.(i)A(5, 3)(ii)B(6, 2)(iii)C(5, 3)(iv)D(4, 6)(v)E(3, 2)(vi)F(4, 4)(vii)G(3, 4)(viii)H(5, 0)(ix)I(0, 6)(x)J(3, 0)(xi)K(0, 2)(xii)O(0, 0) Solution: (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x) (xi) (xii)...
Read More →In how many ways can a student choose 5 courses out of 9 courses if 2 courses are compulsory for every student?
Question: In how many ways can a student choose 5 courses out of 9 courses if 2 courses are compulsory for every student? Solution: We are given that 2 courses are compulsory out of the 9 available courses, Thus, a student can choose 3 courses out of the remaining 7 courses. Number of ways $={ }^{7} C_{3}=\frac{7 !}{3 ! 4 !}=\frac{7 \times 6 \times 5 \times 4 !}{3 \times 2 \times 1 \times 4 !}=35$...
Read More →How many different boat parties of 8, consisting of 5 boys and 3 girls,
Question: How many different boat parties of 8, consisting of 5 boys and 3 girls, can be made from 25 boys and 10 girls? Solution: Clearly, out of the 25 boys and 10 girls, 5 boys and 3 girls will be chosen. Then, different boat parties of $8={ }^{25} C_{5} \times{ }^{10} C_{3}$ $=\frac{25 !}{5 ! 20 !} \times \frac{10 !}{3 ! 7 !}$ $=\frac{25 \times 24 \times 23 \times 22 \times 21}{5 \times 4 \times 3 \times 2 \times 1} \times \frac{10 \times 9 \times 8}{3 \times 2 \times 1}$ =6375600...
Read More →How many different boat parties of 8, consisting of 5 boys and 3 girls,
Question: How many different boat parties of 8, consisting of 5 boys and 3 girls, can be made from 25 boys and 10 girls? Solution: Clearly, out of the 25 boys and 10 girls, 5 boys and 3 girls will be chosen. Then, different boat parties of $8={ }^{25} C_{5} \times{ }^{10} C_{3}$ $=\frac{25 !}{5 ! 20 !} \times \frac{10 !}{3 ! 7 !}$ $=\frac{25 \times 24 \times 23 \times 22 \times 21}{5 \times 4 \times 3 \times 2 \times 1} \times \frac{10 \times 9 \times 8}{3 \times 2 \times 1}$ =6375600...
Read More →From a group of 15 cricket players, a team of 11 players is to be chosen.
Question: From a group of 15 cricket players, a team of 11 players is to be chosen. In how many ways can this be done? Solution: Required number of ways $={ }^{15} C_{11}$ Now, ${ }^{15} C_{11}={ }^{15} C_{4}$ $=\frac{15}{4} \times \frac{14}{3} \times \frac{13}{2} \times \frac{12}{1} \times 11 C_{0}$ = 1365...
Read More →Prove that:
Question: Prove that: $\frac{(2 n+1) !}{n !}=2^{n}[1 \cdot 3 \cdot 5 \ldots(2 n-1)(2 n+1)]$ Solution: $\mathrm{LHS}=\frac{(2 n+1) !}{n !}$ $=\frac{(2 n+1)(2 n)(2 n-1) \ldots .(4)(3)(2)(1)}{n !}$ $=\frac{[(1)(3)(5) \ldots \ldots \ldots .(2 n-1)(2 n+1)][(2)(4)(6) \ldots \ldots \ldots(2 n)]}{n !}$ $=\frac{2^{n}[(1)(3)(5) \ldots \ldots \ldots(2 n-1)(2 n+1)][(1)(2)(3) \ldots \ldots \ldots(n)]}{n !}$ $=\frac{2^{n}[(1)(3)(5) \ldots \ldots \ldots(2 n-1)(2 n+1)][n !]}{n !}$ $=2^{n}[(1)(3)(5) \ldots \ldot...
Read More →f : R→R is defined by f(x)
Question: $f: R \rightarrow R$ is defined by $f(x)=\frac{e^{x^{2}}-e^{-x^{2}}}{e^{x^{2}+e^{-x^{2}}}}$ is (a) one-one but not onto(b) many-one but onto(c) one-one and onto(d) neither one-one nor onto Solution: (d) neither one-one nor onto We have, $f(x)=\frac{e^{x^{2}}-e^{-x^{2}}}{e^{x^{2}+e^{-x^{2}}}}$ Here, $-2,2 \in R$ Now, $2 \neq-2$ But, $f(2)=f(-2)$ But, $f(2)=f(-2)$ Therefore, function is not one - one. And, The minimum value of the function is 0 and maximum value is 1 That is range of the...
Read More →Prove that:
Question: Prove that: (i) $\frac{n !}{(n-r) !}=n(n-1)(n-2) \ldots(n-(r-1))$ (ii) $\frac{n !}{(n-r) ! r !}+\frac{n !}{(n-r+1) !(r-1) !}=\frac{(n+1) !}{r !(n-r+1) !}$ Solution: (i) $\mathrm{LHS}=\frac{n !}{(n-r) !}$ $=\frac{n(n-1)(n-2)(n-3)(n-4) \ldots(n-r+1)[(n-r) !]}{(n-r) !}$ $=n(n-1)(n-2)(n-3)(n-4) \ldots(n-r+1)$ $=n(n-1)(n-2)(n-3)(n-4) \ldots[n-(r-1)]=$ RHS (ii) $\mathrm{LHS}=\frac{n !}{(n-\mathrm{r}) ! \mathrm{r} !}+\frac{n !}{(n-\mathrm{r}+1) !}$ $=\frac{n !}{(n-r) ! r !}+\frac{n !}{(n-r+1)...
Read More →If the point (3, 4) lies on the graph of 3y = ax + 7 then the value of a is
Question: If the point (3, 4) lies on the graph of 3y=ax+ 7 then the value ofais (a) $\frac{2}{5}$ (b) $\frac{5}{3}$ (c) $\frac{3}{5}$ (d) $\frac{2}{7}$ Solution: Given equation: $3 y=a x+7$. Also, $(3,4)$ lies on the graph of the equation. Putting $x=3, y=4$ in the equation, we get: $3 \times 4=3 a+7$ $\Rightarrow 12=3 a+7$ $\Rightarrow 3 a=12-7=5$ $\Rightarrow a=\frac{5}{3}$ Hence, the correct answer is option (b)....
Read More →Solve the following
Question: If $\frac{(2 n) !}{3 !(2 n-3) !}$ and $\frac{n !}{2 !(n-2) !}$ are in the ratio $44: 3$, find $n$. Solution: $\frac{(2 n) !}{3 !(2 n-3) !}: \frac{n !}{2 !(n-2) !}=44: 3$ $\Rightarrow \frac{(2 n) !}{3 !(2 n-3) !} \times \frac{2 !(n-2) !}{n !}=\frac{44}{3}$ $\Rightarrow \frac{(2 n)(2 n-1)(2 n-2)[(2 n-3) !]}{3(2 !)(2 n-3) !} \times \frac{2 !(n-2) !}{n(n-1)[(n-2) !]}=\frac{44}{3}$ $\Rightarrow \frac{(2 n)(2 n-1)(2 n-2)}{3} \times \frac{1}{n(n-1)}=\frac{44}{3}$ $\Rightarrow \frac{(2 n)(2 n-...
Read More →Let f : R→R be a function defined by f(x)
Question: Let $f: R \rightarrow R$ be a function defined by $f(x)=\frac{x^{2}-8}{x^{2}+2}$. Then, $f$ is (a) one-one but not onto(b) one-one and onto(c) onto but not one-one(d) neither one-one nor onto Solution: Injectivity:Letxandybe two elements in the domain (R), such that $f(x)=f(y)$ $\Rightarrow \frac{x^{2}-8}{x^{2}+2}=\frac{y^{2}-8}{y^{2}+2}$ $\Rightarrow\left(x^{2}-8\right)\left(y^{2}+2\right)=\left(x^{2}+2\right)\left(y^{2}-8\right)$ $\Rightarrow x^{2} y^{2}+2 x^{2}-8 y^{2}-16=x^{2} y^{2...
Read More →If (n + 3)! = 56 [(n + 1)!], find n.
Question: If (n+ 3)! = 56 [(n+ 1)!], findn. Solution: (n+ 3)! = 56 [(n+ 1)!] $\Rightarrow(n+3) \times(n+2) \times(n+1) !=56[(n+1) !]$ $\Rightarrow(n+3) \times(n+2)=56$ $\Rightarrow(n+3) \times(n+2)=8 \times 7$ $\Rightarrow n+3=8$ $\therefore n=5$...
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