List of top Quantitative Ability Questions on Algebra

What is the value of \( \log_{10} 50 + \frac{\log_{0.5} 5}{1+\log_{2} 5} \)?

Suppose M = (1, 1), N = (-1, 3), S = (2, 7), T = (0, -4). If A and B, respectively, divide MN and ST in the ratio 2:3, what is the equation of line AB?

For p,q,r \(\neq\) 0, let \(f(x) = x^3 - px^2 + qx - r\), \(g(x) = x^3 - \frac{p}{r}x^2 + \frac{q}{r^2}x - \frac{1}{r}\). Which of the following statements is/are true?
I. \(f(a) = 0 \implies g(\frac{1}{a}) = 0\).
II. \(f(a) = 0 \implies g(1 + a) = 0\).
III. \(g(a) = 0 \implies f(\frac{r}{a}) = 0\).}

The area enclosed by |x| + |y| = 2 is:

Denote the origin by O, (2,1) by P and (–1,2) by Q. If R and S are marked on the XY-plane such that OR = 30 units, OS = 40 units, P lies on OR and Q on OS, then what is the distance between R and S, in the given unit?

If we draw the graph of \(f(x) = \log_{10(x + 1)\) on the domain of definition, which quadrants does it pass through?}

If \(\frac{x^3}{z^2}<\frac{x^3+y^3+z^3}{x^2+y^2+z^2}<\frac{z^3}{x^2}\); \(x, y, z\) are positive real numbers, then which of the following options always ensure the given inequality to be true?

If \(x = 7\), then what is the value of
\(\frac{x^{\frac{1}{2}} + x^{-\frac{1}{2}}}{1 - x} + \frac{1 - x^{-\frac{1}{2}}}{1 - \sqrt{x}} = ?\)

If \(x\) and \(y\) are positive real numbers satisfying \(x + y = 52\), then the minimum possible value of \(91\left(1+\frac{1}{x}\right)\left(1+\frac{1}{y}\right)\) is:

Assume that \(f: (-2, -1) \to (1, 2)\) is an onto function and for \(i = 1, 2, 3, 4\), define \(g_1(x) = f(x) - 2\), \(g_2(x) = f(-x)\), \(g_3(x) = -f(-x)\) and \(g_4(x) = f(-x - 2)\). What is the correct arrangement of \(g_1, g_2, g_3, g_4\) such that the graph of the \(k^{\text{th}}\) function lies in the \(k^{\text{th}}\) quadrant for \(k = 1, 2, 3, 4\)?

The longest chord of the circumcircle of the triangle made by x-axis, y-axis and \( 4x+3y=24 \) is:

For real numbers \( \alpha \) and \( \beta \), let \( p(x)=x^{2}-(\alpha+\beta)x+\alpha\beta \) and \( q(x)=x^{2}-(\alpha+\beta+2)x+(\alpha+1)(\beta+1) \). Which of the following statements is true?}

Which of the following inequalities is true for any positive real numbers a, b and c?
I. \( ab(a+b)+bc(b+c)+ca(c+a) \le 6abc \)
II. \( \frac{a^{2}+b^{2}+c^{2}}{abc} \le \frac{1}{a}+\frac{1}{b}+\frac{1}{c} \)

PQRS is a quadrilateral with side \( PS=7 \) cm and \( QR=11 \) cm. \( \angle SPQ \) and \( \angle QRS \) are both right angles. If E and F are points on PQ and RS, respectively, and QE is twice SF; \( SF=n \) cm and n is an integer, then what is the value of n such that the area of the quadrilateral EQFS is \( 225~cm^{2} \)?