List of top Mathematics Questions on Algebra asked in JEE Main

Let $ S = \{x^3 + ax^2 + bx + c : a, b, c \in \mathbb{N} \text{ and } a, b, c \le 20\} $ be a set of polynomials. Then the number of polynomials in $ S $, which are divisible by $ x^2 + 2 $, is:

Let the direction cosines of two lines satisfy the equations : $ 4l + m - n = 0 $ and $ 2mn + 5nl + 3lm = 0 $. Then the cosine of the acute angle between these lines is :

Find the value of \[ \frac{6}{3^{26}} + 10\cdot\frac{1}{3^{25}} + 10\cdot\frac{2}{3^{24}} + \cdots + 10\cdot\frac{2^{24}}{3^{1}} : \]

Given $\frac{1}{\alpha} - \frac{1}{\beta} = \frac{1}{3}$ such that roots of the quadratic equation $\lambda x^2 + (\lambda+1)x + 3 = 0$ are $\alpha$ & $\beta$, then sum of values of $\lambda$ is equal to :

If \[ \sum_{r=1}^{25}\frac{r}{r^4+r^2+1}=\frac{p}{q}, \] where $p$ and $q$ are coprime positive integers, then $p+q$ is equal to:

Consider the 10 observations 2, 3, 5, 10, 11, 13, 15, 21, a and b such that mean of observation is a and variance is 34.2. Then the mean deviation about median, is :

The sum of coefficients of $x^{499}$ and $x^{500}$ in the expression: \[ (1+x)^{1000} + x(1+x)^{999} + x^2(1+x)^{998} + \cdots + x^{1000} \] is:

If $g(x) = 3x^2 + 2x - 3, f(0) = -3, 4g(f(x)) = 3x^2 - 32x + 72$ then find $f(g(2))$ where $f(x)>0$ for all valid x:

Product of first 3 terms of a G.P. is 27 and sum is $R - \{a,b\}$, then $a^2 + b^2$ is equal to :

A complex number 'z' satisfy both $|z-6|=5$ & $|z+2-6i|=5$ simultaneously. Find the value of $z^3 + 3z^2 - 15z + 141$.

Find possible no. of triplets (b, c, d), such that $x^2 + 2$ is divisor of $x^3 + bx^2 + cx + d$ & b, c, d $ \le $ 20 & b, c, d $ \in $ N :

Let x be the number of 9 digit numbers formed by taking digits from first 9 natural numbers, where only one digit is repeated twice & y be the number of 9 digit numbers formed from first 9 natural numbers, such that exactly 2 digits repeated twice then

If $\vec{a}, \vec{b} \& \vec{c}$ are unit vectors such that $(\vec{a}-\vec{b})^2 + (\vec{b}-\vec{c})^2 + (\vec{c}-\vec{a})^2 = 9$. Find positive k if $|2\vec{a} + k\vec{b} + k\vec{c}| = 3$ :

In the binomial expansion of \[ (ax^2 + bx + c)(1 - 2x)^{26}, \] the coefficients of $x, x^2, x^3$ are $-56, 0$ and $0$ respectively. Then the value of $ (a + b + c) $ is:

If $ x^2 + x + 1 = 0 $, then evaluate \[ \left(x + \frac{1}{x}\right)^4 + \left(x^2 + \frac{1}{x^2}\right)^4 + \left(x^3 + \frac{1}{x^3}\right)^4 + \cdots + \left(x^{25} + \frac{1}{x^{25}}\right)^4. \]

If the range of the function $ f(x) = \sqrt{3 - x} + \sqrt{5 + x} $ is $[\alpha, \beta]$, then $ \alpha^2 + \beta^2 $ is equal to:

If all the words, with or without meaning, made using all the letters of the word ``RANCHI'' are arranged as in a dictionary, then the word at $560^{\text{th}}$ position is:

The number of integral terms in the binomial expansion of \[ \left(11^{\frac{1}{2}} + 17^{\frac{1}{8}}\right)^{1024} \] is:

Let $ f : \mathbb{R} \setminus \{0\} \to (-\infty, 1) $ be a polynomial of degree 2, satisfying $ f(x)f\left( \frac{1}{x} \right) = f(x) + f\left( \frac{1}{x} \right) $. If $ f(K) = -2K $, then the sum of squares of all possible values of $ K $ is:

The sum of the squares of the roots of $ |x - 2|^2 + |x - 2| - 2 = 0 $ and the squares of the roots of $ x^2 |x - 3| - 5 = 0 $, is:

Let $ b_1 = 3, b_2, b_3, \ldots $ be a geometric progression of increasing positive numbers. Let \[ \sum_{n=1}^{20} b_{3n} = 4\sum_{n=1}^{20} b_{3n-2}. \] Then, the sum of the first ten terms of the G.P. is:

If the system of linear equations : \[ x+y+z=4,\quad x+2y+3z=6,\quad 4x+5y+\lambda z=\mu \] has more than one solution, then the value of $ \lambda+\mu $ is equal to:

Two persons $A$ and $B$ alternately throw a pair of dice. $A$ wins if he throws a sum of $4$ before $B$ throws a sum of $9$, and $B$ wins if he throws a sum of $9$ before $A$ throws a sum of $4$. The probability that $A$ wins if $B$ makes the first throw is:

Let the value of $p$, such that the sum of the squares of the roots of the quadratic equation \[ x^2+(7-p)x+4=p \] has the least value, be $\alpha$, and the corresponding roots be $\beta$ and $\gamma$. Then $\alpha^3+\beta^3+\gamma^3$ equals \underline{\hspace{2cm}.}

Let $25^x+25^{-x},\ \dfrac{\alpha}{3},\ 20^{1+x}+20^{1-x}$, where $x,\alpha\in\mathbb{R}$, be the first three terms of an A.P. of increasing terms. For the least value of $\alpha$, the sum of its first $10$ terms is _____