Solve the following
Question: Write $\sum_{r=0}^{m}{ }^{n+r} C_{r}$ in the simplified form. Solution: We know: ${ }^{n} C_{r}+{ }^{n} C_{r-1}={ }^{n+1} C_{r}$ $\sum_{r=0}^{m}{ }^{n+r} C_{r}={ }^{n} C_{0}+{ }^{n+1} C_{1}+{ }^{n+2} C_{2}+{ }^{n+3} C_{3}+\ldots+{ }^{n+m} C_{m}$ $\because{ }^{n} C_{0}={ }^{n+1} C_{0}$ $\therefore \sum_{r=0}^{m}{ }^{n+r} C_{r}={ }^{n+1} C_{0}+{ }^{n+1} C_{1}+{ }^{n+2} C_{2}+{ }^{n+3} C_{3}+\ldots+{ }^{n+m} C_{m}$ Using ${ }^{n} C_{r-1}+{ }^{n} C_{r}={ }^{n+1} C_{r}:$ $\Rightarrow \sum_{...
Read More →Find the 8th term from the end of the A.P. 7, 10, 13, ..., 184.
Question: Find the 8thterm from the end of the A.P. 7, 10, 13, ..., 184. Solution: In the given problem, we need to find the 8thterm from the end for the given A.P. We have the A.P as 7, 10, 13 184 Here, to find the 8thterm from the end let us first find the total number of terms. Let us take the total number of terms asn. So, First term (a) = 7 Last term (an) = 184 Common difference $(d)=10-7=3$ Now, as we know, $a_{n}=a+(n-1) d$ So, for the last term, $184=7+(n-1) 3$ $184=7+3 n-3$ $184=4+3 n$ ...
Read More →Out of 10 persons
Question: Out of 10 personsP1,P2,...,P10, 5 persons are to be arranged in a line such that in each arrangementP1must occur whereasP4andP5do not occur. The number of such arrangements is ___________. Solution: Since out of 5 to be selected,P1is fix i.eP1must occur. ⇒ we need to select 4 person out of 9 remaining. Now, out of remaining 9,P4andP5do not occur ⇒ Available options are 7 ⇒ Number of such arrangement =7C45! (since 5 persons inlinecan be arranged in 5! ways.)...
Read More →If two straight lines intersect each other, then prove that the ray opposite the bisector of one
Question: If two straight lines intersect each other, then prove that the ray opposite the bisector of one of the angles so formed bisects the vertically-opposite angle. Solution: Let $A B$ and $C D$ be the two lines intersecting at a point $O$ and let ray $O E$ bisect $\angle A O C$. Now, draw a ray $O F$ in the opposite direction of $O E$, such that $E O F$ is a straight line. Let $\angle C O E=1, \angle A O E=2, \angle B O F=3$ and $\angle D O F=4$. We know that vertically-opposite angles are...
Read More →The number of permutations of n distinct objects taken r at a time in which three particular objects occurs together
Question: The number of permutations ofndistinct objects takenrat a time in which three particular objects occurs together is ___________. Solution: Total number of objects (distinct) =n Since, 3 objects are taken together (i.e 3 objects are together) The number of ways of selectingrdistinct objects fromnisnCr If three objects occur together, then number of ways =n3Cr3 also number of arrangements of these three things = 3! and number of arrangements of (r 3 + 1) objects = (r 2)! Total possible w...
Read More →Find the term of the arithmetic progression 9, 12, 15, 18, ... which is
Question: Find the term of the arithmetic progression 9, 12, 15, 18, ... which is 39 more than its 36thterm. Solution: In the given problem, let us first find the 36stterm of the given A.P. A.P. is 9, 12, 15, 18 Here, First term (a) = 9 Common difference of the A.P. $(d)=12-9=3$ Now, as we know, $a_{n}=a+(n-1) d$ So, for 36thterm (n= 36), $a_{36}=9+(36-1)(3)$ $=9+35(3)$ $=9+105$ $=114$ Let us take the term which is 39 more than the 36thterm asan. So, $a_{n}=39+a_{36}$ $=39+114$ $=153$ Also, $a_{...
Read More →Let f be the greatest integer function defined as f(x) = [x] and g be the modules function defined as
Question: Let $f$ be the greatest integer function defined as $f(x)=[x]$ and $g$ be the modules function defined as $g(x)=|x|$, then the value of $g o f\left(-\frac{5}{4}\right)$ is___________. Solution: Given:f(x) = [x]andg(x) = |x| $g \circ f\left(-\frac{5}{4}\right)=g\left(f\left(-\frac{5}{4}\right)\right)$ $=g\left(\left[-\frac{5}{4}\right]\right)$ $=g(-2)$ $=|-2|$ $=2$ Hence, the value of $g \circ f\left(-\frac{5}{4}\right)$ is 2 ....
Read More →The number of automobile license plates that can be made if each plate contains two different letters of English alphabet followed by three distinct digits,
Question: The number of automobile license plates that can be made if each plate contains two different letters of English alphabet followed by three distinct digits, is_______. Solution: Each automobile license plates has 5 places to fill 2 places of alphabet (out of 26) can be filled in 26 25 ways Then 3 places of digits can be filled in 10 48 ways Hence number of different automobile license plates = 26 25 10 9 8 = 468000...
Read More →In the given figure, the two lines AB and CD intersect at a point O such that ∠BOC = 125°.
Question: In the given figure, the two lines AB and CD intersect at a point O such that BOC = 125. Find the values ofx, yandz. Solution: Here,AOC and BOC form a linear pair.AOC + BOC = 180⇒x+ 125= 180⇒x = 180125 =55Now,AOD =BOC =125 (Vertically opposite angles)y = 125BOD =AOC=55 (Vertically opposite angles)z = 55Thus, the respective values ofx,yandzare 55, 125 and 55....
Read More →Which term of the arithmetic progression 8, 14, 20, 26, ... will
Question: Which term of the arithmetic progression 8, 14, 20, 26, ... will be 72 more than its 41stterm. Solution: In the given problem, let us first find the 41stterm of the given A.P. A.P. is 8, 14, 20, 26 Here, First term (a) = 8 Common difference of the A.P. $(d)=14-8=6$ Now, as we know, $a_{v}=a+(n-1) d$ So, for 41stterm (n= 41), $a_{41}=8+(41-1)(6)$ $=8+40(6)$ $=8+240$ $=248$ Let us take the term which is 72 more than the 41stterm asan. So, $a_{v}=72+a_{41}$ $=72+248$ $=320$ Also, $a_{n}=a...
Read More →The number of committees of five persons with a chair person can be selected from 12 persons,
Question: The number of committees of five persons with a chair person can be selected from 12 persons, is ___________. Solution: A chair person can be selected in 12 ways Now, we need to select 4 person out of remaining 11. Number of committees of five person with a chair person can be selected in 1211C4ways i.e $12 \times \frac{11 !}{4 ! 7 !}$ $=\frac{12 \times 11 \times 10 \times 9 \times 8}{4 \times 3 \times 2}$ $=3960$...
Read More →In the given figure, three lines AB, CD and EF intersect at a point O such that ∠AOE = 35° and ∠BOD = 40°.
Question: In the given figure, three lines AB, CD and EF intersect at a point O such that AOE = 35 and BOD = 40. Find the measure of AOC, BOF, COF and DOE. Solution: In the given figure,AOC =BOD =40 (Vertically opposite angles)BOF =AOE=35 (Vertically opposite angles)Now,EOC and COF form a linear pair.EOC + COF = 180⇒ (AOE + AOC)+ COF= 180⇒ 35 +40 +COF= 180⇒75 +COF= 180⇒ COF= 18075=105Also, DOE = COF = 105 (Vertically opposite angles)...
Read More →A committee of 6 is to be chosen from 10 men and 7 women so as to contain at least 3 men and 2 women.
Question: A committee of 6 is to be chosen from 10 men and 7 women so as to contain at least 3 men and 2 women. The number of different ways this can be done, if two particular women refuse to serve on the same committee is ___________. Solution: Here, Number of men = 10 Number of women = 7 6 committee numbers can be selected containing at-least 3 men and 2 women in following 2 ways. 4 men and 2 women 3 men and 3 women number of ways of selecting at-least 3 men and 2 women in committee of 6 $={ ...
Read More →Let f : R → R and g : R → R be functions defined by
Question: Let $f: R \rightarrow R$ and $g: R \rightarrow R$ be functions defined by $f(x)=5-x^{2}$ and $g(x)=3 x-4$. Then the value of fog $(-1)$ is___________. Solution: Given: $f(x)=5-x^{2}$ and $g(x)=3 x-4$ $f o g(-1)=f(g(-1))$ $=f(3(-1)-4)$ $=f(-3-4)$ $=f(-7)$ $=5-(-7)^{2}$ $=5-49$ $=-44$ Hence,the value offog(1) is44....
Read More →Two lines AB and CD intersect each other at a point O such that ∠AOC : ∠AOD = 5 : 7.
Question: Two lines AB and CD intersect each other at a point O such that AOC : AOD = 5 : 7. Find all the angles. Solution: Let $\angle \mathrm{AOC}=5 k$ and $\angle \mathrm{AOD}=7 k$, where $k$ is some constant. Here, $\angle A O C$ and $\angle A O D$ form a linear pair. $\therefore \angle \mathrm{AOC}+\angle \mathrm{AOD}=180^{\circ}$ $\Rightarrow 5 k+7 k=180^{\circ}$ $\Rightarrow 12 k=180^{\circ}$ $\Rightarrow k=15^{\circ}$ $\therefore \angle \mathrm{AOC}=5 \mathrm{k}=5 \times 15^{\circ}=75^{\...
Read More →How many three digit numbers are divisible by 7?
Question: How many three digit numbers are divisible by 7? Solution: In this problem, we need to find out how many numbers of three digits are divisible by 7. So, we know that the first three digit number that is divisible by 7 is 105 and the last three digit number divisible by 7 is 994. Also, all the terms which are divisible by 7 will form an A.P. with the common difference of 7. So here, First term (a) = 105 Last term (an) = 994 Common difference (d) = 7 So, let us take the number of terms a...
Read More →The total number of ways in which six '+' and four '–' signs can be arranged in aline such that no two '–' signs occur together
Question: The total number of ways in which six '+' and four '' signs can be arranged in aline such that no two '' signs occur together is ___________. Solution: Number of '+' sign = 6 and Number of '' sign = 4 After placing 6 '+' sign, there are 7 places for '' sign. Now, we need to select 4 sign out of 7 places The total number of ways of arranging signs Such that no two '' are together =7C4 $=\frac{7 !}{4 ! 3 !}=\frac{7 \times 6 \times 5}{2 \times 3}=35$...
Read More →Let f = {(0, –1), (–1, 3), (2, 3), (3, 5)} be a function from Z to Z defined by
Question: Let $f=\{(0,-1),(-1,3),(2,3),(3,5)\}$ be a function from $Z$ to $Z$ defined by $f(x)=a x+b$. Then, $(a, b)=$___________. Solution: Given: $f=\{(0,-1),(-1,-3),(2,3),(3,5)\}$ is a function from $Z$ to $Z$ defined by $f(x)=a x+b$ $f=\{(0,-1),(-1,-3),(2,3),(3,5)\}$ defined by $f(x)=a x+b$ $f(0)=-1$ $\Rightarrow a(0)+b=-1$ $\Rightarrow 0+b=-1$ $\Rightarrow b=-1$ ...(1) $f(2)=3$ $\Rightarrow a(2)+b=3$ $\Rightarrow 2 a+b=3$ $\Rightarrow 2 a-1=3$ (From (1)) $\Rightarrow 2 a=3+1$ $\Rightarrow 2...
Read More →How many multiples of 4 lie between 10 and 250?
Question: How many multiples of 4 lie between 10 and 250? Solution: In this problem, we need to find out how many multiples of 4 lie between 10 and 250. So, we know that the first multiple of 4 after 10 is 12 and the last multiple of 4 before 250 is 248. Also, all the terms which are divisible by 4 will form an A.P. with the common difference of 4. So here, First term (a) = 12 Last term (an) = 248 Common difference (d) = 4 So, let us take the number of terms asn Now, as we know, $a_{n}=a+(n-1) d...
Read More →Three balls are drawn from a bag containing 5 red, 4 white and 3 black balls.
Question: Three balls are drawn from a bag containing 5 red, 4 white and 3 black balls. The number of ways in which this can be done if at least 2 are red is ___________. Solution: Number of red balls = 5 Number of white balls = 4 Number of black balls = 3 Number of ball drawn = 3 Note, at-least 2 red balls can be drawn in following ways 2 red and 1 non red. all 3 reds balls. Number of ways of drawing at-least two red balls isall red5C3+5C27C1 $=\frac{4 \times 5}{2}+\frac{4 \times 5}{2} \times 7...
Read More →Three balls are drawn from a bag containing 5 red, 4 white and 3 black balls.
Question: Three balls are drawn from a bag containing 5 red, 4 white and 3 black balls. The number of ways in which this can be done if at least 2 are red is ___________. Solution: Number of red balls = 5 Number of white balls = 4 Number of black balls = 3 Number of ball drawn = 3 Note, at-least 2 red balls can be drawn in following ways 2 red and 1 non red. all 3 reds balls. $\therefore$ Number of ways of drawing at-least two red balls is all red $\underline{5} \underline{C}_{3}+{ }^{5} C_{2} \...
Read More →For what value of n, the nth terms of the arithmetic progressions
Question: For what value ofn, the nth terms of the arithmetic progressions 63, 65, 67, ... and 3, 10, 17, ... are equal? Solution: Here, we are given two A.P. sequences. We need to find the value ofnfor which thenthterms of both the sequences are equal. We need to findn So let us first find thenthterm for both the A.P. First A.P. is 63, 65, 67 Here, First term (a) = 63 Common difference of the A.P. $(d)=65-63=2$ Now, as we know, $a_{n}=a+(n-1) d$ So, fornthterm, $a_{n}=63+(n-1) 2$ $=63+2 n-2$ $=...
Read More →Two lines AB and CD intersect at a point O, such that ∠BOC + ∠AOD = 280°, as shown in the figure.
Question: Two linesABandCDintersect at a pointO,such that BOC+ AOD= 280, as shown in the figure. Find all the four angles. Solution: We know that if two lines intersect, then the vertically-opposite angles are equal. Let $\angle B O C=\angle A O D=x^{\circ}$ Then, $x+x=280$ $\Rightarrow 2 x=280$ $\Rightarrow x=140^{\circ}$ $\therefore \angle B O C=\angle A O D=140^{\circ}$ Also, let $\angle A O C=\angle B O D=y^{\circ}$ We know that the sum of all angles around a point is $360^{\circ}$. $\theref...
Read More →A box contain 2 white balls, 3 black balls and 4 red balls.
Question: A box contain 2 white balls, 3 black balls and 4 red balls. The number of ways three balls be drawn from the box if at least one black ball is to be included in the draw is ___________. Solution: At-least 1 black ball can be selected in following ways 1 black ball and two non - black or 2 black ball and one non - black or all 3 black balls Total number of ways of selecting is $=\frac{{ }^{3} C_{3}}{\text { all black }}+\frac{{ }^{3} C_{2} \times{ }^{6} C_{1}}{2 \text { black balls }}+\...
Read More →Let A = {1, 2, 3, 4, 5, 6) and B = (2, 4, 6, 8, 10, 12). If f : A → B is given by
Question: Let $A=\{1,2,3,4,5,6)$ and $B=(2,4,6,8,10,12)$. If $f: A \rightarrow B$ is given by $f(x)=2 x$, then $f^{-1}$ as set of ordered pairs, is___________. Solution: Given: A functionf:ABdefined asf(x) = 2x,whereA= {1, 2, 3, 4, 5, 6} andB = {2, 4, 6, 8, 10, 12} Since, $f(x)=2 x$ Therefore, $f=\{(1,2),(2,4),(3,6),(4,8),(5,10),(6,12)\}$ Hence, $f^{-1}=\{(2,1),(4,2),(6,3),(8,4),(10,5),(12,6)\}$ Hence, if $f: A \rightarrow B$ is given by $f(x)=2 x$, then $f^{-1}$ as set of ordered pairs, is{(2,1...
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