Algebra Questions

If $2x+4(x-3)<-2-x<2x-1$, then find the value of $x$ from the given integer options.

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:

From $6$ different novels and $3$ different dictionaries, $4$ novels and $1$ dictionary are to be selected and arranged in a row on the shelf so that the dictionary is always in the middle. Then the number of such arrangements is:

The possible value of $x$ satisfying the equation $\frac{(143÷13\times x-3\times 3)}{(6^2-7\times 5+x^2)}=1$ is:

Let $A$ and $B$ be two sets containing $2$ elements and $4$ elements respectively. The number of subsets of $A\times B$ having $3$ or more elements is:

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

The remainder left out when $8^{2n}-(62{)}^{2n+1}$ is divided by $9$ is:

Coefficient of x in (x+9)(8−5x) is:

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

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

If the coefficients of $r^{\text{th}}$, $(r+1{)}^{\text{th}}$ and $(r+2{)}^{\text{th}}$ terms in the binomial expansion of $(1+y{)}^m$ are in A.P., then $m$ and $r$ satisfy the equation:

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

If roots of the equation $x^2-bx+c=0$ are two consecutive integers, then $b^2-4c$ equals:

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

Find the value of (a−2x)
3
+(b−2x)
3
+(c−2x)
3
−3(a−2x)(b−2x)(c−2x), given that a+b+c=6x.

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

[(4a−5b) 3 +(5b−3c) 3 −(4a−3c) 3 ] is equal to:

If the roots of the equation $b{x}^2+cx+a=0$ be imaginary, then for all real values of $x$, the expression $3b^2{x}^2+6bcx+2c^2$ is:

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

​

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

If
x+2
2x
​
+
2x+4
1
​
−
2x
2
+4x
12−x
​
=a+
x
b
​
, where a and b are integers, then the value of ab=_____.

Simplify (x−2y)
3
−6xy(x−2y).

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

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:

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

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

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:

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

If $xy=-30$ and $x^2+y^2=61$, then find the value of $(x+y)$.

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:

A number is greater than five times its reciprocal by $\frac{19}{2}$. Find the number.

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

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:

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

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

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
​

The sum of two numbers is 19, while the difference between the two numbers is 3. Find the smaller number out of these two numbers.

Find the value of 1006×994.

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

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

If x
3
=184+y
3
and x=4+y, then the value of (x+y) is (given that x>0 and y>0):

If
2
a
​
=
3
b
​
=
4
c
​
=
k
2a−3b+5c
​
, then find the value of k.

Simplify the following expression: (x−3y)
3
+27(y
3
−xy
2
)

The vectors $\vec{a}$ and $\vec{b}$ are not perpendicular and $\vec{c}$ and $\vec{d}$ are two vectors satisfying: $\vec{b}\times \vec{c}=\vec{b}\times \vec{d}$ and $\vec{a}\cdot \vec{d}=0$. Then the vector $\vec{d}$ is equal to:

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

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

The total emoluments of two persons are the same, but one gets allowances to the extent of $65\%$ of his basic pay and the other gets allowances to the extent of $80\%$ of his basic pay. The ratio of the basic pay of the former to the basic pay of the latter is:

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

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:

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

Let $A$ and $B$ be two symmetric matrices of order $3$.Statement-1 : $ABA$ and $A{B}^{2}A$ are symmetric matrices.Statement-2 : $AB$ is a symmetric matrix if matrix multiplication of $A$ and $B$ is commutative.

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

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

Find the value of 13
3
+(−8)
3
+(−5)
3
.

If $\alpha$ and $\beta$ are the roots of the equation $x^2-x+1=0$, then ${\alpha }^{2009}+{\beta }^{2009}=$

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 $14x+6y-32=0$?

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

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?

The value of ${}^{50}{C}_4+{\sum }_{r=1}^6{}^{56-r}{C}_3$ is:

If $\vec{a}=\frac{1}{\sqrt{10}}(3\hat{i}+\hat{k})$ and $\vec{b}=\frac{1}{7}(2\hat{i}+3\hat{j}-6\hat{k})$, then the value of $(\overrightarrow{2a}-\vec{b})\cdot [(\vec{a}\times \vec{b})\times (\vec{a}+\overrightarrow{2b})]$ is:

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

Simplify the following expression:
$$25\times 25+9\times 9+6\times 6+2\times 25\times 9+2\times 9\times 6+2\times 25\times 6$$

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

The coefficient of x in (x+8)(6−3x) is:

In three containers - yellow, white and black, $120$ marbles are placed. There are twice as many marbles in the white and yellow containers combined as there are in the black container. Also, there are twice as many marbles in the black container as there are in the yellow container. How many marbles are there in the white container?

If $\alpha +\beta +\gamma =0$, then find the value of $\frac{3{\beta }^2+{\alpha }^2+{\gamma }^2}{2{\beta }^2-\alpha \gamma }$.

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

If a−b=−5 and a
2
+b
2
=73, then find ab.

If 2x+2(4+3x)<2+3x>2x+x/2; then x can take which of the following values?

Find the simplest value of 5
3
2
​
×2
6
1
​
−2
3
1
​

Find the value of 27a
3
+
a
3

1
​
, if 9a
2
+
a
2

1
​
=43.

If $(2x+3y{)}^3-(2x-3y{)}^3=3y[A{x}^2+B{y}^2]$, then what is the value of $(A-B)$?

If y=
3

​
−
2

​
, then find the value of (
y
3

1
​
−y
3
).

The number of values of $k$ for which the linear equations $4x+ky+2z=0$, $kx+4y+z=0$, and $2x+2y+z=0$ possess a non-zero solution is:

If x
3
−y
3
=40 and x−y=4, then find the value of xy.

Find the sum of 3+3
2
+3
3
+⋯+3
8
.

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

If $x=3y+4$, then what is the value of $(x^3-27y^3-36xy)$?

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

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 2x+3y=0 and 3x−4y=34, then x−y=____.

If $\sin x+\sin y=a$ and $\cos x+\cos y=b$, then find the value of $(\sin x\sin y+\cos x\cos y)$.

If x+y=10 and x
2
+y
2
=68, then find xy.

Which of the following is correct?

If 23% of 'x' is 46, then 'x' is ________

A man saves Rs. $200$ in each of the first three months of his service. In each of the subsequent months his saving increases by Rs. $40$ more than the saving of immediately previous month. His total saving from the start of service will be Rs. $11040$ after:

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
.

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

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