Top 50 IPMAT Algebra Questions PDF with Video Solutions

Let $$\alpha$$, $$\beta$$ be the roots of $$x^{2} - x + p = 0$$ and $$\gamma$$, $$\delta$$ be the roots of $$x^{2}- 4x + q = 0$$ where p and q are integers. If $$\alpha, \beta, \gamma, \delta$$ are in geometric progression then p + q is

Let $$f(x) = a^2x^2 + 2bx + c$$ where, $$a \neq 0, b, c$$ are real numbers and x is a real variable then

If the harmonic mean of the roots of the equation $$(5 + \sqrt{2})x^{2} − bx + 8 + 2\sqrt{5} = 0$$ is 4 then the value of b is

If the polynomial $$ax^2 + bx + 5$$ leaves a remainder 3 when divided by $$x - 1$$, and a remainder 2 when divided by $$x + 1$$, then $$2b - 4a$$ equals

Let $$a_{1} a_{2}, a_{3}$$ be three distinct real numbers in geometric progression. If the equations $$a_{1}x^{2} + 2a_{2} x + a_{3} = 0$$ and $$b_{1}x^{2} + 2b_{2}x + b_{3} = 0$$ has a common root,then which of the following is necessarily true?

If $$8x^2 - 2kx + k = 0$$ is a quadratic equation in x, such that one of its roots is p times the other, and p, k are positive real numbers, then k equals

Consider the polynomials $$f(x) = ax^{2} + bx + c$$, where a > 0, b, c are real, and g(x) = -2x. If f(x) cuts the x-axis at (-2,0) and g(x) passes through (a, b), then the minimum value of f(x) + 9a + 1 is

The number of real solutions of the equation $$x^2 - 10 \mid x \mid - 56 = 0$$ is

The length of the line segment joining the two intersection points of the curves $$y = 4970 - |x|$$ and $$y = x^{2}$$ is_________.

Given that
$$f(x)=|x|+2|x−1|+|x−2|+|x−4|+|x−6|+2|x−10|$$, $$x \epsilon (-\infty, \infty)$$
the minimum value of f(x) is _________.

The set of all possible values of f(x) for which $$(81)^{x} + (81)^{(f(x)} = 3$$ is

The sum of the squares of all the roots of the equation $$x^{2} + |x + 4| + |x − 4| − 35 = 0$$ is

For a > b > c > 0, the minimum value of the function f(x) = |x - a| + |x - b| + |x - c| is

The set of all real values of x satisfying the inequality $$\frac{x^{2}(x + 1)}{(x - 1)(2x + 1)^{3}}> 0$$ is

Statements: $$M = V; R \geq S; V < S; M > A; R \leq U$$
Conclusions:
I. $$U > S$$
II. $$R = S$$
III. $$R > S$$

If $$\mid x + 1 \mid + (y + 2)^2 = 0$$ and $$ax - 3ay = 1$$, Then the value of a is

The number of solutions of the equation $$x_1 + x_2 + x_3 + x_4 = 50$$, where $$x_1 , x_2 , x_3 , x_4$$ are integers with $$x_1 \geq1, x_2 \geq 2, x_3 \geq 0, x_4 \geq 0$$ is

The set of values of x which satisfy the inequality $$0.7^{2x^{2}-3x+4} < 0.343$$ is

The number of pairs (x, y) of integers satisfying the inequality $$\mid x - 5 \mid + \mid y - 5 \mid \leq 6$$ is:

If |x|<100 and |y|<100, then the number of integer solutions of (x, y) satisfying the equation 4x + 7y = 3 is

If a, b, c are real numbers $$a^{2} + b^{2} + c^{2} = 1$$, then the set of values $$ab+bc+ca$$ can take is:

The minimum value of $$f(x)=|3-x|+|2+x|+|5-x|$$ is equal to  _____________.

Let [x] denote the greatest integer not exceeding x and {x} = x -[x].
If n is a natural number, then the sum of all values of x satisfying the equation 2[x] = x + n{x} is

For all real values of x, $$\dfrac{3x^{2} - 6x + 12}{x^{2} + 2x + 4}$$ lies between 1 and k, and does not take any value above k. Then k equals...........

If a, b, c are three distinct natural numbers, all less than 100, such that $$\mid a - b \mid + \mid b - c \mid = \mid c - a \mid$$, then the maximum possible value of b is ______

If minimum value of $$f(x) = x^{2} + 2bx + 2c^{2}$$ is greater than the maximum value of $$g(x) = - x^{2} - 2cx + b^{2}$$, then for real value of x.

The set of all real numbers x for which $$x^{2} - |x + 2 |+ x > 0$$, is

Given $$A =2^{65}$$ and $$B = (2^{64} + 2^{63} + 2^{62} + ... + 2^{0})$$, which of the following is true?

Consider the following statements:
(i) When 0 < x < 1, then $$\dfrac{1}{1+x} < 1 - x + x^{2}$$.
(ii) When 0 < x < 1, then $$\dfrac{1}{1+x} > 1 - x + x^{2}$$.
(iii) When -1 < x < 0, then $$\dfrac{1}{1+x} < 1 - x + x^{2}$$.
(iv) When -1 < x < 0, then $$\dfrac{1}{1+x} > 1 - x + x^{2}$$.

Statements: $$A < P = C \geq D; S > U \leq B < A; C < Q > S \geq V$$
Conclusions:
I. $$U > V$$
II. $$B < C$$
III. $$Q > D$$

The total number of positive integer solutions of $$21 \leq a + b + c \leq 25$$ is __________.

If x ∈ (a, b) satisfies the inequality, $$\dfrac{x-3}{x^2+3x+2}\ge1$$ then the largest possible value of b - a is

For some non-zero real values of $$a, b$$ and $$c$$, it is given that $$\mid \frac{c}{a} \mid = 4, \mid \frac{a}{b} \mid = \frac{1}{3}$$ and $$\frac{b}{c} = -\frac{3}{4}$$. If $$ac > 0$$, then $$\left(\frac{b + c}{a}\right)$$

The area enclosed by $$2|x| + 3|y| \leq 6$$ is____________ sq. units

The inequality $$\log_{2} \frac{3x - 1}{2 - x} < 1$$ holds true for

Statements:
Lady's Finger is tastier than cabbage
Cauliflower is tastier than Lady's Finger
Cabbage is not tastier than peas

The numbers $$2^{2024}$$ and $$5^{2024}$$ are expanded and their digits are written out consecutively on one page. The total number of digits written on the page is

If x, y, z are positive real numbers such that $$x^{12} = y^{16} = z^{24}$$, and the three quantities $$3\log_{y}x, 4 \log_{z}y, n\log_{x}z$$ are in arithmetic progression, then the value of n is

If $$\log_{25}\left[5 \log_3 (1 + \log_3(1 + 2 \log_2 x))\right] = \dfrac{1}{2}$$ then x is:

Let a, b, c be real numbers greater than 1, and n be a positive real number not equal to 1. If $$\log_{n}(\log_{2}a) = 1, \log_{n} (log_{2}b) = 2$$ and $$\log_{n}(\log_{2}c) = 3$$, then which of the following is true?

The set of all values of x satisfying the inequality $$\log_{\left(x+\frac{1}{x}\right)}\left[\log_2\left(\frac{x-1}{x+2}\right)\right]>0$$ is

The number of real solutions of the equation $$(x^2 -15x + 55)^{x^2 - 5x + 6} = 1$$ is:

The product of the roots of the equation $$\log_{2}2^{(\log_{2}x)^{2} }− 5\log_{2}x + 6 = 0$$ is ________

If $$4^{\log_2 x} - 4x + 9^{\log_3 y} - 16y + 68 = 0$$, then $$y - x$$ equals:

If $$\log_{(x^{2})}y + \log_{(y^{2})} x= 1$$ and $$y = x^{2} - 30$$, then the value of $$x^{2} + y^{2}$$ is _________

If $$y = a + b \log_e x$$, which of the following is true?

The value of $$0.04^{\log_{\sqrt{5}}(\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+....)}$$ is _____________.

The Set of real values of x for which the inequality $$\log_{27}8\le\log_3x<9^{\ \frac{\ 1}{\log_23}}$$ holds is

If $$\log_{5}\log_{8}(x^2 - 1) = 0$$, then a possible value of x is

Let $$a = \dfrac{(\log_7 4)(\log_7 5 - \log_7 2)}{\log_7 25(\log_7 8 - \log_7 4)}$$. Then the value of $$5^a$$ is