### Dictionary Rank of a Word | Permutations & Combinations

PERMUTATIONS & COMBINATIONS Rank of the word or Dictionary order of the English words like COMPUTER, COLLEGE, SUCCESS, SOCCER, RAIN, FATHER, etc. Dictionary Rank of a Word Method of finding the Rank (Dictionary Order) of the word  “R A I N” Given word: R A I N Total letters = 4 Letters in alphabetical order: A, I, N, R No. of words formed starting with A = 3! = 6 No. of words formed starting with I = 3! = 6 No. of words formed starting with N = 3! = 6 After N there is R which is required R ----- Required A ---- Required I ---- Required N ---- Required RAIN ----- 1 word   RANK OF THE WORD “R A I N” A….. = 3! = 6 I……. = 3! = 6 N….. = 3! = 6 R…A…I…N = 1 word 6 6 6 1 TOTAL 19 Rank of “R A I N” is 19 Method of finding the Rank (Dictionary Order) of the word  “F A T H E R” Given word is :  "F A T H E R" In alphabetical order: A, E, F, H, R, T Words beginni

### Math Assignment Class X Ch-3 | Pair of Linear Equations in Two Variables

CHAPTER 3   CLASS  X

PAIR OF LINEAR EQUATIONS IN TWO VARIABLE

Extra questions of chapter 3 class 10 : Pair of Linear Equations in Two Variable with answer and  hints . Useful math assignment for the students of class 10

For better results

LEVEL-1

Q1 By comparing the ratios of the coefficients  find the nature of sol./ graph / consistency or inconsistency

a)   2x - 3y = 1 ;  x + 2y = 2

Ans: Unique solution, intersecting lines, consistent

b)   3x - y = 6 ;  6x - 2y = 4

Ans: No solution, parallel lines, inconsistent

c)   x + 3y = 8 ;  2x - 3y = 12

Ans: Unique solution, intersecting lines, consistent

d)   9x + 3y + 12 = 0 ;  18x + 6y = - 24

Ans: Many solution, coincident lines, consistent

e)   6x - 3y +10 = 0 ;  2x – y + 9 = 0

Ans: No solution, parallel lines, inconsistent

Q2) Find the value of p so that the following equations have coincident lines

a)    2x + y - 5 = 0;  6x + 3y + p = 0      Ans [p = -15]

b)   x + 2y + 7 = 0;   2x + py + 14 = 0    Ans [p = 4]

c)  3x + 4y + 2 = 0;  9x + 12y + p = 0    Ans[p = 6]

d)    x + 5y - 7 = 0 ;  4x + 20y + p = 0    Ans [p = -28]

Q3) Find k if the following equations are inconsistent

a)   8x + 5y = 9;   kx + 10y = 15      Ans [k = 6 ]

b) (3k + 1)x + 3y - 2 = 0;    (k+ 1)x + (k - 2)y  -  5 = 0   Ans[k = -1]

c)   3x + y = 1;   (2k - 1)x + (k - 1)y = 2k + 1     Ans [k = 2]

d) kx + 3y = k  –  3;  12x + ky = k      Ans [k = -6]

Q4) Find a & b or k so that the following equations have infinitely many solutions

a)  (a + b)x  - 2by = 5a + 2b + 1;    3x - y = 14           Ans [a = 5, b =  1]

b)     2x - 3y = 7;  (a + b)x  -  (a + b - 3)y = 4a + b     Ans[-5, -1]

c)  2x + 3y = 7;    (a - b)x - (a + b - 3)y = 3a + b - 2   Ans[a = 5, b = 1]

d) (2a - 1)x + 3y – 5 = 0;   3x + (b - 1)y – 2 = 0          Ans[a = 17/4,   b= 11/5]

e)   kx + 3y = 2k + 1;  2(k + 1)x + 9y = 7k + 1           Ans [k = 2]

f)   2x + 3y = 7;   (k – 1)x + (k + 2)y = 3k                  Ans [k = 7]

Q5) Solve the following equations graphically

a) x + y = 3;   2x + 5y = 12                           Ans [1, 2]

b)  2x + 3y + 5 = 0;   3x - 2y - 12 = 0           Ans [2,-3]

Q6 Solve the following equations graphically. Shade the area between lines and x-axis. Also find the area of the shaded triangle.

a) x + y = 5;      2x - y = - 2                     [Sol. (1, 4) & Area= 12]

b) x - y + 1 = 0;    3x + 2y - 12 = 0          [Area= 7.5]

c)  2x + y = 6;      2x - y + 2 = 0               [Area= 8]

Q7)  Solve 2(x - 1) = y & x + 3y = 15 graphically and find the coordinates where lines meet the y-axis. Also shade the triangle between the lines and y-axis.      Ans [(3, 4), (0, -2), (0, 5)]

Q8)  Solve x – y = 1 & 2x + y = 8 graphically and shade the area between the lines and the y-axis. Also find the area of the triangle.      Ans [Area = 13.5]

LEVEL-2

Q 9) Solve the following pair of linear equations in two variables

a)     x+ y = 7    ;   3x - 2y = 1                 Ans [3, 4]

b)     3x + 2y = 10  ; 4x - y = 6                Ans [2, 2]

c)     $\frac{x}{2}+\frac{2y}{3} =-1;\; \; x-\frac{y}{3}=3$y}    Ans [2,-3]

d) 2x + y – 35 = 0   ;  3x + 4y - 65 = 0     Ans [15, 5]

e)    ax + by = c ;       bx + ay = 1 + c      $Ans\: \left [\frac{bc-ac+b}{b^{2}-a^{2}},\; \frac{bc-ac+a}{b^{2}-a^{2}} \right ]$

f) ax + by = a - b ;  bx - ay = a + b          Ans [1, - 1]

g) ax + by = c  ;  bx - ay = 0                   $Ans\: \left [\frac{ac}{a^{2}+b^{2}},\; \frac{bc}{a^{2}+b^{2}} \right ]$

h)  x/a - y/b = 0 ;  bx + ay = 4ab            Ans[2a, 2b]

i)   6(ax + by) = 3a + 2b ;  6(bx +  ay) = 3b + 2a     Ans [1/2, 1/3]

j) - 6x + 5y = 2 ;    - 5x + 6y = 9             Ans[3, 4]

k)  ax + by = a+ b2 ; bx - ay = 0         Ans [a, b]

l)  23x - 29y = 98  ;   29x - 23y = 110     Ans [3, - 1]

m)   99x + 101y =  499 ; 101x +  99y = 501     Ans [x = 3, y = 2]

n)    a(x + y) + b(x - y) = a– ab + b2 ;  a(x + y) - b(x - y) = a+ ab + b2

$Ans\: \left [\frac{b^{2}}{2a},\frac{2a^{2}+b^{2}}{2a} \right ]$

o) (a - b)x + (a + b)y = a- 2ab - b;  (a + b)(x + y)  = a+ b

$Ans\; \left [ (a+b),\left ( \frac{-2ab}{a+b} \right ) \right ]$

p) 2(ax - by) + (a + 4b) = 0 ;  2(bx + ay) + (b - 4a) = 0     Ans [-1/2, 2]

q)  x +  y = 5xy  ;   3x + 2y = 13xy          Ans [x = 1/2 ,  y = 1/3]

r)  $\frac{b}{a}x+\frac{a}{b}y=a^{2}+b^{2};\; \; x+y=2ab$

LEVEL-3

PROBLEMS BASED ON DAILY LIFE SITUATIONS

Q10) Find two numbers whose sum is 75 and difference is 15

Ans [45 and 30]

Q11) Sum of two numbers is 8. Find the numbers if sum of their reciprocal is 8/15.

Q12) The ratio of incomes of two persons is 9 : 7  and the ratio of their expenditure  is  4 : 3 If each of them  saves Rs 200/ month, find their monthly incomes.

Ans [Rs1800 & Rs 1400]

Q13) A person can row downstream 20km in 2 hours. And upstream 4km in 2 hours. Find the speed of boat in still water and that of the current.

Ans [6,4]

Q14) A boat covers 32 km upstream and 36 km downstream in 7 hours. Also it covers 40 km upstream and 48 km downstream in 9 hours. Find the speed of the boat in still water and that of the stream.

Ans  [10 km/h and  2 km/h]

Q 15) If 2 is subtracted from the numerator and 1 is added to the denominator, a fraction becomes  1/2 , but when 4 is added to the numerator and 3 is subtracted from the denominator , it becomes 3/2. Find the fractions.

Ans [8/11]

Q 16 37 pens and 53 pencils together cost Rs 320, while 53 pens and 37 pencils together cost Rs 400. Find the cost of a pen and that of a pencil.

Ans  [Rs 6.50 and Rs 1.50]

Q 17) In a two digit number , the ten’s digit is three times the unit’s digit. When the number is decreased by 54, the digits are reversed. Find the number.

Ans  [93]

Q 18) The sum of the two digit number and the number obtained by reversing the order of its digits is 121, and the two digits are differ by 3. Find the numbers.

Ans [47 or 74]

Q 19)  The sum of a two digit number and the number formed by interchanging the digits is 110. If 10 is subtracted from the first number , the new number is 4 more than 5 times the sum of the digits in the first number. Find the first number

Ans   [64]

Q 20) A fraction is such that if the numerator is multiplied by 3 and the denominator is reduced by 3, we get 18/11, but if the numerator is increased by 8 and the denominator is doubled , we get 2/5.  Find the fraction.

Ans  [12/25]

Q 21) The denominator of a fraction is 4 more than twice the numerator. When both the numerator and denominator are decreased by 6, then the denominator becomes 12 times the numerator. Find the fraction

Ans  [7/18]

Q 22) If twice the son’s age in years is added to the fathers age,  the sum is 70. But if twice the father’s age is added to the son’s age, the sum is 95. Find the age of father and son.                Ans  [40,  15]

Q 23) Two years ago, a father was 5 times as old as his son. Two years later , his age will be 8 more than 3 times the age of his son. Find the present ages of father and son.

Ans  [42, 10]

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HIGHER ORDER THINKING SKILL (HOTS)

Question 24)
$\frac{1}{2(2x+3y)}+\frac{12}{7(3x-2y)}=\frac{1}{2}$

$\frac{7}{(2x+3y)}+\frac{4}{(3x-2y)}=2$

Ans: x = 2, y = 1]

Question 25)

$\frac{2}{(3x+2y)}+\frac{3}{(3x-2y)}=\frac{17}{5}$

$\frac{5}{(3x+2y)}+\frac{1}{(3x-2y)}=2$

Ans: x = 1, y = 1

Question 26)

$\frac{44}{x+y}+\frac{30}{x-y}=10$

$\frac{55}{x+y}+\frac{40}{x-y}=13$

Ans: x = 8, y = 3

THANKS FOR YOUR VISIT

1. Good collections