Algebra Questions

If x
2
+y
2
+z
2
=xy+yz+zx, then the value of (
8x
2
y
2
+6y
2
z
2
+10z
2
x
2

17x
4
+9y
4
+16z
4

​
) is:

If for a non-zero x, 7x
2
+5x+7=0, then x
3
+
x
3

1
​
is ________.

If x−y=−9 and xy=−20, then find x
2
+y
2
.

Coefficient of x in (x+9)(8−5x) is:

If 4x−5(2x−1)>2x+3>2−3x; then x can take which of the following values?

What is the value of xy if x−y=3 and x
2
+y
2
=15?

The coefficient of $x^7$ in the expansion of $(1-x-x^2+x^3{)}^6$ is:

Find the value of 1006×994.

Which of the following values of $x$ satisfies the equation $81^{4x+4}=243^{2x+8}$?

What will be the value of $r^2+p^2+s^2-rp-ps-rs$ when $r=4$, $p=2$, and $s=8$?

The sum of 2xy(3x+4y−5z) and 5yz(2x−3y) is:

If 4x−73
2
​
≥3x+1, then x can take which of the following values?

If 4x
2
+y
2
+z
2
+41=2(4x+y+6z), then the value of 2x+y−z is:

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
.

The sum of the ages of a brother and sister at present is 21. Five years ago the product of their ages was 28. What is the age of the brother and the sister?

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

The solution set of 4x−3y=47 and 3x+y=32 is:

If both the roots of the quadratic equation $x^2-2kx+k^2+k-5=0$ are less than $5$, then $k$ lies in the interval:

If $|z-4/z|=2$, then the maximum value of $|z|$ is equal to:

Let $\alpha ,\beta$ be real and $z$ be a complex number. If $z^2+\alpha z+\beta =0$ has two distinct roots on the line $\text{Re}(z)=1$, then it is necessary that:

Find the value of $0.682^2-0.318^2$.

Determine the Least Common Multiple (LCM) of the algebraic expressions $x^{2}y$ and $x{y}^3$.

If cos45
∘
−sec60
∘
=x, then the value of x is:

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

If (9−3x)−(17x−10)=−1, then the value of x is:

If a−b=−7 and a
2
+b
2
=85, then find ab.

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

The sum of a non-zero number and thrice its reciprocal is
7
52
​
. Find the number.

If $\omega \ne 1$ is a cube root of unity, and $(1+{\omega }^2{)}^7=A+B\omega$. Then $(A,B)$ equals:

If $3x-7y=14$ and $\frac{x}{x+y}=\frac{7}{9}$, then what is the value of $xy$?

If 2x−2(3+4x)<−1−2x<−
3
5
​
−
3
x
​
, then x can take which of the following values?

Find the value of $m$:
$$m-\frac{2}{3}=\frac{5}{6}+\frac{1}{2}$$

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

If x+y+z=4, xy+yz+zx=1, and x
3
+y
3
+z
3
=34, then what is the value of 6xyz?

If 3x−8(2−x)=−19, then the value of x is:

If x−y=6 and xy=40, then find x
2
+y
2
.

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

If 2x+
3x
1
​
+1=0 and x

=0, then find the value of 2x+
3x
1
​
.

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

Simplify 52×52−48×48.

1: $\text{adj}(\text{adj }A)=A$ where $A$ is a $2\times 2$ matrix.
2: $|\text{adj }A|=|A|$.

A multiple choice examination has $5$ questions. Each question has three alternative answers of which exactly one is correct. The probability that a student will get $4$ or more correct answers just by guessing is:

If x=7−4
3

​
, then find the value of (x
2
+
x
2

1
​
).

If decreasing $70$ by $x\%$ yields the same result as increasing $60$ by $x\%$, then $x\%$ of $91$ is:

On selling a television at a $6\%$ profit and a fridge at a $9\%$ profit, a shopkeeper gains $\text{Rs. }5,100$. However, if the shopkeeper sells the television at a $9\%$ profit and the fridge at a $6\%$ loss, he gains $\text{Rs. }1,800$ on the whole transaction. The cost price (in $\text{Rs.}$) of the fridge is:

The average of $n$ observations is $40$. If two observations $50$ and $60$ are added, then the average of all the observations will be $41$. What is the value of $n$?

Factorise x
2
+5x−14.

If $(a+b-3{)}^2+(b+c-2{)}^2+(c+a-5{)}^2=0$, then the value of $\sqrt{(b+c{)}^a+(c+a{)}^b-1}$ is:

If the sum of the roots of a quadratic equation is 1 and the product of the roots is −20, find the quadratic equation.

If the equations $x^2+2x+3=0$ and $a{x}^2+bx+c=0$, $a,b,c\in \mathbb{R}$, have a common root, then $a:b:c$ is:

If 3x−3<3+
2
x
​
and x−2≤6+2x, then x can take which of the following values?

If $x^4-3x^2-1=0$, then the value of $\left(x^6-3x^2+\frac{3}{x^2}-\frac{1}{x^6}+1\right)$ is:

The sum of a non-zero number and twice its reciprocal is
4
33
​
. Find the number.

If $x=3+\sqrt{8}$, then the value of $x^3+\frac{1}{x^3}$ is

The age of father is twice that of elder son. Ten years hence, the age of the father will be three times that of the younger son. If the difference of ages of the two sons is $15$ years, find the age of the father.

If $P=\left[\begin{array}{ccc}1& \alpha & 3\\ 1& 3& 3\\ 2& 4& 4\end{array}\right]$ is the adjoint of a $3\times 3$ matrix $A$ and $|A|=4$, then $\alpha$ is equal to:

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 $\text{Rs. }1880$.

The graphs of the lines $x+2y-3=0$ and $2x+4y+11=0$ are

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?

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:

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 expression $\frac{5}{\sqrt{9+\sqrt{45}}+\sqrt{9-\sqrt{45}}}$ is equal to

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?

Let $a,b,c$ be such that $(b+a)c\ne 0$. If
$$\left|\begin{array}{ccc}a& a+1& a-1\\ b& b+1& b-1\\ c& c-1& c+1\end{array}\right|+\left|\begin{array}{ccc}a+1& b+1& c-1\\ a-1& b-1& c+1\\ (-1{)}^{n}a& (-1{)}^{n}b& (-1{)}^{n+1}c\end{array}\right|=0$$
then the value of '$n$ ' is:

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:

The sum of two natural numbers is $8$. Determine the numbers if the sum of their reciprocals is $\frac{8}{15}$.

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:

The roots of the equation $\frac{1}{p+q+x}=\frac{1}{p}+\frac{1}{q}+\frac{1}{x}$ are:

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

The number of values of $k$, for which the system of equations$(k+1)x+8y=4k$$kx+(k+3)y=3k-1$ has no solution, is:

If $x=\frac{\sqrt{\sqrt{5}+2}+\sqrt{\sqrt{5}-2}}{\sqrt{\sqrt{5}+1}}-\sqrt{3-2\sqrt{2}}$, then $x$ equals:

If $\frac{x}{y+z}=\frac{y}{z+x}=\frac{z}{x+y}=K(K\ne 0)$, then the value of $K$ is:

If $x=\frac{8ab}{a+b}$, then find the value of:
$$\frac{x+4a}{x-4a}+\frac{x+4b}{x-4b}$$

If ${\left(\frac{1+i}{1-i}\right)}^x=1$, then:

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 $x$ is so small that $x^3$ and higher powers of $x$ may be neglected, then $\frac{(1+x{)}^{3/2}-(1+\frac{1}{2}x{)}^3}{(1-x{)}^{1/2}}$ may be approximated as:

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

Let $T_n$ be the number of all possible triangles formed by joining vertices of an $n$ -sided regular polygon. If $T_{n+1}-T_n=10$, then the value of $n$ is:

In an Arithmetic Progression, if $a=28$, $d=-4$, $n=7$, then $a_n$ is:

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

The value of x for which the expressions 11x+7 and 17x−1 become equal is:

If (x+y):(x−y)=5:2, find the value of
x−4y
4x+5y
​
.

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.

If 2+2x<5−
2
x
​
and 5x+3>5−5x, then x can take which of the following values?

The sum of a non-zero number and thrice its reciprocal is
2
13
​
. Find the number.

The value of $a$ for which the sum of the squares of the roots of the equation $x^2-(a-2)x-a-1=0$ assumes the least value is:

A function $f$ from the set of natural numbers to integers defined by$f(n)=\{\begin{array}{ll}\frac{n-1}{2},& \text{when }n\text{ is odd}\\ -\frac{n}{2},& \text{when }n\text{ is even}\end{array}$ is:

The sum to the infinity of the series $1+\frac{2}{3}+\frac{6}{3^2}+\frac{10}{3^3}+\frac{14}{3^4}+\dots$ is:

If $z$ is a complex number of unit modulus and argument $\theta$, then $\text{arg}(\frac{1+z}{1+\bar{z}})$ equals:

If (x
2
+
x
2

1
​
)=18 and x>1, what is the value of (x
3
−
x
3

1
​
)?

If $a\ast b=(a\times b)+b$, then $5\ast 7$ is equal to

The minimum value of $(x-2)(x-9)$ is

Find the square root of $(272{)}^2-(128{)}^2$:

If $\frac{a}{3}=\frac{b}{2}=\frac{c}{1}$, then $\frac{a+b+2c}{b}$ is

Find the value of $K$ for which equation $x-Ky=2$, $3x+2y=5$ has unique solution.

The polynomial $a{x}^3+b{x}^2+x-6$ has $(x+2)$ as a factor and leaves remainder $4$ when divided by $(x-2)$. Find the value of $a$ and $b$.

If $a^{m^n}=(a^m{)}^n$, then the value of $m$ is

If $0<x<1$, then it is true that

If HCF of $(p^2-8p+12)$ and $(p^2+4p-12)$ is $(p-a)$, then the value of a is

The factors of polynomial $x^3-6x^2+11x-6$ are