algebra Questions

Simplify the given expression: $\frac{12.43\times 12.43\times 12.43+19.57\times 19.57\times 19.57}{12.43\times 12.43-12.43\times 19.57+19.57\times 19.57}$

If $\left(x-\frac{1}{x}\right)=3$, then the value of $(x-1{)}^2+\frac{1}{(x-1{)}^2}$ is:

If $x+y=12$ and $xy=6$, then the value of $\frac{x}{y}+\frac{y}{x}$ is equal to:

Which of the following quadratic equations has equal roots?

If a 9-digit number 1039m837n is divisible by 72, then find the value of
n
2
−m
4


​
.

Find the value of
0.09
0.23
2
−0.04
2

​
.

If x% of 190 is 15 less than (x+10)% of 180, then (x+30)% of 90 is what per cent more than x% of 150?

If x=3−4y, then (x
3
+64y
3
+36xy) is equal to:

If
1+
39
x
​


​
=5, then the value of x is equal to:

Four distinct positive numbers, 'a', 'b', 'c' and 'd', in the order given, are in proportion. 'b' is 35 more than 'a' and 'd' is 60 more than 'c'. The product of 'a' and 'c' is 5376. What is the sum of 'a', 'b', 'c' and 'd'?

Simplify the following expression:
(x+y)(x−y)(x
2
+y
2
)(x
4
+y
4
)(x
8
+y
8
)(x
1
6+y
1
6)

If x+
x
1
​
=15, then the value of (x
2
+
x
2

1
​
) is:

Name the person who had a major influence in the field of Algebra in India.

The ratio of the sums of money Arun and Ahaan had is $9:5$. If Arun gives ₹$12$ from his share to Ahaan, then the ratio will change to $4:3$. How much money did Arun have initially?

The sum of two numbers is $15$. If the sum of their reciprocals is $\frac{3}{10}$, then find the two numbers?

If the average of $x$ and $\frac{1}{x}$ ($x\ne 0$) is $m$, then the average of $x^2$ and $\frac{1}{x^2}$ is :

A number $X$ lies between $0$ and $1$. Which of the following is true?

Two numbers are in the ratio of $7:6$. If the difference between their squares is $637$, then the numbers are:

For what value of $p$ does the system of equations $18x+36y+45=0$ and $px-54y+67=0$ have no solution?

Two numbers are in the ratio $5:8$. If $4$ is added to each, the ratio becomes $2:3$. Find the smaller number.

The LCM of two numbers is $12$ times their HCF. The sum of HCF and LCM is $1560$. If one of the numbers is $120$, then find the other number.

S varies directly as (R+7), and S=42 when R=17. What is the value of S when R=29?

In an enclosure, there are crows as well as cows. If there are 30 heads and 100 legs, what is the ratio of the cows to crows?

Given the simultaneous linear equations $2x-3y=-12$ and $\frac{x}{y}=\frac{3}{4}$, what are the values of $x$ and $y$?

If $\alpha$ and $\beta$ are the roots of the quadratic equation $a{x}^2+2bx+c=0$ and $\alpha +\gamma$, $\beta +\gamma$ are the roots of $A{x}^2+2Bx+C=0$, then which one of the following is correct?

If $(a,b)\ast (c,d)=(a+d)-\frac{c}{b}$, then $\left(\frac{5}{6},\frac{2}{7}\right)\ast \left(\frac{3}{7},\frac{2}{3}\right)$ will be:

If $f(x)=\frac{x}{c}$ where $c\ne 0$, then the value of $\frac{cf(x)+1}{xf(x)-1}$ will be:

If ${\left(x+\frac{1}{x}\right)}^2=3$, then the value of $x^{2020}+x^{2021}+x^{2022}+x^{2014}+x^{2015}+x^{2016}+1$ is

If $a=2021$, $b=2022$, and $c=2023$, then the value of $a^3+b^3+c^3-3abc$ is:

If $x=5+2\sqrt{6}$, then the value of $\frac{x^6+3x^3+1}{x^3}$ is equal to:

Ram and Shyam solve a quadratic equation. Ram makes a mistake in the constant term and finds the roots as $5,-3$. Shyam makes a mistake in the co-efficient of $x$ and gets the roots as $1,-3$. The correct roots are:

If $a+b+c=0$, then $\frac{1}{x^b+x^{-c}+1}+\frac{1}{x^c+x^{-a}+1}+\frac{1}{x^a+x^{-b}+1}$ is equal to:

If $a^2+b^2+c^2=ab+bc+ca$, then which one is correct?

If

is equal to:

If $2x^2+5y^2+2z^2-2xy-6yz=0$, then $x:y:z$ is:

The age of a father is twice that of his elder son. Ten years hence, the age of the father will be three times that of his younger son. If the difference between the ages of the two sons is $15\text{years}$, the current age of the father is:

The sum of the digits of a two-digit number is $12$. When the digits are interchanged, the resulting number is $36$ more than the original number. What is the original two-digit number?

Free books were distributed equally among children of a class. The number of books each child got was one-eighth of the number of children. Had the number of children been half, each child would have got $16$ books. How many books were distributed in total?

When the numerator of a fraction increases by $4$, the fraction increases by $\frac{2}{3}$. The denominator of the fraction is

Total expenses of a hostel are partly constant and partly vary directly as the number of boarders. When the number of boarders is $120$ and $100$, the total expenses are $\text{Rs. }2000$ and $\text{Rs. }1700$ respectively. Find the total number of boarders if the total expense is $\text{Rs. }1880$.

A cricket team won $3$ matches more than they lost. If a Win gives them $2$ points and Loss $(-1)$ point, how many matches, in all, have they played if their score is $23$?

Simplify the given algebraic expression:
$$\frac{437bxaz}{23ab}$$

The LCM of two numbers is $495$ and their HCF is $5$. If the sum of the numbers is $100$, then find their difference.

If $a+\frac{1}{b}=1$ and $b+\frac{1}{c}=1$, then find the value of $c+\frac{1}{a}$.

If $\frac{x}{2}+\frac{y}{3}=4$ and $\frac{2}{x}+\frac{3}{y}=1$, then what is $x+y$ equal to?

If $a=999$, $b=998$, $c=997$, then the value of $a^3+b^3+c^3-3abc$ will be:

If $\frac{a^2}{b+c}=\frac{b^2}{c+a}=\frac{c^2}{a+b}=1$, then find the value of $\frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c}$.

The sum of the first 16 terms of the A.P.: 10, 6, 2, ... is

If $x=7-4\sqrt{3}$, then $\sqrt{x}+\frac{1}{\sqrt{x}}=\text{?}$

If $4+3x\le 6+x$ and $3x+5>2+2x$, then $x$ can take which of the following values?

Simplify the following expression:
33×31×(2
10
+1)
(3
2
)
4
−1
​

A number $X$ lies between $0$ and $1$. Which of the following is true?

How many ways can a captain and a vice-captain be selected from a team of $11$ players?

The age of a son today is $\frac{1}{4}\text{th}$ of the age of his father. After $20$ years, the age of the father will be twice the age of his son. Find the age of the father today.

Find the roots of the equation $\frac{1}{p+q+x}=\frac{1}{p}+\frac{1}{q}+\frac{1}{x}$.

If $\frac{x}{y}=\frac{a+2}{a-2}$, then find the value of $\frac{x^2-y^2}{x^2+y^2}$.

If x
2
−3x+1=0, then (x
3
+x
−3
) is equal to:

If a+b+c=3, ab+bc+ca=−10 and abc=−24, then the value of
bc
a
2

​
+
ac
b
2

​
+
ab
c
2

​
is:

When x is subtracted from each of the numbers 64,81,85 and 109, the numbers so obtained in this order are in proportion. What is the mean proportional between (x+3) and (2x−1)?

What is the simplified form of the following expression?
(x−
y
1
​
)
3
+(x+
y
1
​
)
3

Simplify $\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}-\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}$.

If $a+b+c=1$, $ab+bc+ca=-8$ and $a^3+b^3+c^3=-11$, then what is the value of $\frac{a}{bc}+\frac{b}{ac}+\frac{c}{ab}$?

If x−x
−1
=6, then the value of x
4
+x
−4
is:

If a+b+c=7, and a
2
+b
2
+c
2
=33, then what is the value of a
3
+b
3
+c
3
−3abc?

The average of 7 consecutive numbers is 20. The largest of these numbers is:

If a+b=2 and ab=4, then find the value of a
4
+b
4
+ab
3
+ba
3
.

If x+y+z=3 and xy+yz+zx=−11, then what is the value of x
3
+y
3
+z
3
−3xyz?

Simplify the following expression:
0.7+0.4
0.7
2
+(0.4+0.2)(0.7)+(0.4×0.2)
​

If the semi-perimeter and area of a rectangular field whose length and breadth are 'x' and 'y' is 12cm and 28cm
2
, respectively, then find the value of x
4
+x
2
y
2
+y
4
.

Find the value of $\frac{m(m^2+3m+3)+1}{m(m+2)+1}$ when $m=87$.

Simplify $\frac{0.32\times 0.72\times 0.56}{(0.32{)}^3+(0.24{)}^3-(0.56{)}^3}$:

Simplify the following expression:
4×4×4
(8.25)
3
−3×(8.25)
2
×0.25+24.75×(0.25)
2
−(0.25)
3

​

If x+y=10 and xy=12, find the value of x
2
−xy+y
2
.

Simplify the given expression.
(2.3)
2
+(15.9)
2
+(3.7)
2
−2.3×15.9−15.9×3.7−3.7×2.3
(2.3)
3
+(15.9)
3
+(3.7)
3
−3×2.3×15.9×3.7
​

Evaluate the expression
$$\frac{(1024{)}^{\frac{n}{5}}\times 4^{2n+1}}{16^n\times 4^{n-1}}$$

The cost of 21 pencils and 9 pens is ₹276, and the cost of 6 pencils and 3 pens is ₹84. The cost of 1 pencil and 1 pen is:

The addition of a rational number and its reciprocal is $\frac{74}{35}$. Calculate the number.

How many solutions will a pair of linear equations have, if the equations are $7x+4y-16=0$ and $14x+6y-32=0$?

If $x=\frac{\sqrt{5}-1}{\sqrt{5}+1}$ and $y=\frac{\sqrt{5}+1}{\sqrt{5}-1}$, then what is the value of $x-y$?

The value of x for which the expressions 15−7x and 15x+7 become equal is ________.

If a:b=3:8, find the value of
2a+b
5a−3b
​
.

If 5−3x<4−x and 5(2−x)>2−2x, then x can take which of the following values?

If 2x+3y=0 and 3x−4y=34, then x−y=____.

Coefficient of x
2
in (x+5)(3−2x)(4x−2) is

If x+y=14 and x
2
+y
2
=100, then find xy.

If 2x+2(1−x)>3x−2>2x−5, then x can take which of the following values?

If $\frac{x}{y}=\frac{4}{5}$ and $7x+6y=29$, find the value of $y$.

Simplify (x−2y)
3
−6xy(x−2y).

If x=7−4
3

​
, then find the value of (x
2
+
x
2

1
​
).