Linear Equations

The difference between a mother's age and the sum of her two daughter's age is $8$. The average age of two daughters is $24$. What is the age of the mother?

In each of the following number series, a wrong number is given. Find out the wrong number.

$5, 14, 65, 99, 197, 254, 401$

If $5^{5x+5}= 1,$ then x is equal to -

If $a+b+c=0$ then $\frac{a^2+b^2-c^2}{ab}$ is equal to _____.

Study the following information and answer the question given below:

Twelve friends A, B, C, D, E, F, G, H, I, J, K and L were born in different months of the same year. A was born in the month of April and G was born in the month of August. J was born the month immediately preceding the month in which K was born and immediately succeeding the month in which C was born. J was not born in the month of October nor in February. There is a gap of two months between the birthdays of L and B. There were $30$ Days in the month in which L was born. D was born in the month immediately after the month in which I was born. There were $31$ Days in the month in which D was born. There is a gap of one month between the birthdays of B and F. E and H were born in that months which had $31$ Days each.

In which of the following months B was born ?

Directions: In each of these questions, two equations $\left(\mathrm{I}\right)$ and $\left(\mathrm{II}\right)$ are given. You have to solve both the equations and give the answer
A. if $x<y$
B. if $x\le y$
C. if $x=y$, or no relation can be established between $x$ and $y$.
D. if $x>y$
E. if $x\ge y$

$\mathrm{I}. 7x–4y=29$
$\mathrm{II}. 5x+3y–50=0$

In each of the following number series, a wrong number is given. Find out the wrong number.
 

$24, 56, 110, 192, 309, 464, 666$

When a number is divided by another number, the remainder is $23$. When twice of the number is divided by the same divisor the remainder is $11.$ The divisor is _____.

What will come in place of the question mark $\left(?\right)$ in the following number series?

$8, 10, 18, 44, 124, \left(?\right)$

What approximate value will come in place of the question mark ($?$) in the following questions? (You are not expected to calculate the exact value.)
 

$\frac{901}{29}\times \frac{91}{301}÷\frac{51}{599}=?$

Directions: In the following questions two equations numbered $\mathrm{I}$ and $\mathrm{II}$ are given. You have to solve both the equations and give the answer if.
(A) $x>y$
(B) $x\ge y$
(C) $x<y$
(D) $x\le y$
(E) $x=y$ or the relationship cannot be established

$\mathrm{I}$. $5\sqrt{169}{x}^2-585=0$

$\mathrm{II}$.$6\sqrt{324}y+324=0$

$$The sum${11}^2+{12}^2+.................. + {21}^2=?$

If value of $x=1+\sqrt{2}+\sqrt{3}$, then value of $\left(\frac{x+1}{x-1}\right)=?$

Directions: In the following questions two equations numbered $\mathrm{I}$ and $\mathrm{II}$ are given. You have to solve both the equations and give the answer if.
(A) $x>y$
(B) $x\ge y$
(C) $x<y$
(D) $x\le y$
(E) $x=y$ or the relationship cannot be established

$\mathrm{I}$. $\frac{5^2}{\sqrt{x}}-\frac{19}{\sqrt{x}}=x^{\frac{3}{2}}$

$\mathrm{II}$. $y^4-\frac{2^{{\displaystyle \frac{9}{2}}}}{\sqrt{y}}=0$

Father's age was $5$ times his son's age $5$ years ago and will be $3$ times the son's age after $2$ years. The ratio of the present age of the father to that of the son is -

Lara had a certain sum of money. She bought books with $\frac{1}{3}$ of her money, and cosmetics with $\frac{1}{4}$ of the remainder. Finally, she purchased a toy by spending $\frac{1}{5}$ of what remained. She was then left with $₹ 50$. How much money did she have at the beginning?

Directions: In each of these questions, two equations $\left(\mathrm{I}\right)$ and $\left(\mathrm{II}\right)$ are given. You have to solve both the equations and give the answer
A. if $x<y$
B. if $x\le y$
C. if $x=y$, or no relation can be established between $x$ and $y$.
D. if $x>y$
E. if $x\ge y$

$$\begin{gathered} \mathrm{I}. 3x^2+5x -2 = 0 \\ \mathrm{II}. 2y^2 - 7y + 5 = 0 \end{gathered}$$

Ramesh is$60 \mathrm{years}$ old. Suresh is $5 \mathrm{years}$ junior to Ramesh and $4 \mathrm{years}$ senior to Raju. The youngest brother of Raju is Saket and he is $6 \mathrm{years}$ junior to him. What is the age difference between Ramesh and Saket?

Sum of two different positive integers is $11$and the sum of their squares is $61$. What is the ratio of larger integer to the smaller integer?

Directions: In the following questions two equations numbered $\mathrm{I}$ and $\mathrm{II}$ are given. You have to solve both the equations and give the answer if.

(A) $x>y$
(B) $x\ge y$
(C) $x<y$
(D) $x\le y$
(E) $x=y$ or the relationship cannot be established

$\mathrm{I}$. $\frac{7}{17}-\frac{13}{34}=\frac{\sqrt{x}}{102}$
$\mathrm{II}$. $\frac{\sqrt{y}}{12}+\frac{\sqrt{y}}{6}=\frac{1}{2\sqrt{y}}$