CAT 2017 Algebra - Q1) If \(a\) and \(b\) are integers of opposite signs such that \((a + 3)^2 : b^2 = 9 : 1\) and \((a - 1)^2 : (b - 1)^2 = 4 : 1\), then the ratio \(a^2 : b^2\) is:

• (a) 9 : 4
• (b) 81 : 4
• (c) 1 : 4
• (d) 25 : 4

CAT 2017 Algebra - Q2) The area of the closed region bounded by the equation \(|x| + |y| = 2\) in the two-dimensional plane is:

• (a) \(4\pi\) sq. units
• (b) 4 sq. units
• (c) 8 sq. units
• (d) \(2\pi\) sq. units

CAT 2017 Algebra - Q3) Suppose, \(\log_3 x = \log_{12} y = a\), where \(x, y\) are positive numbers. If \(G\) is the geometric mean of \(x\) and \(y\), then \(\log_6 G\) is equal to:

• (a) \(\sqrt{a}\)
• (b) \(2a\)
• (c) \(a/2\)
• (d) \(a\)

CAT 2017 Algebra - Q4) If \(x + 1 = x^2\) and \(x > 0\), then the value of \(2x^4\) is:

• (a) \(6 + 4\sqrt{5}\)
• (b) \(3 + 3\sqrt{5}\)
• (c) \(5 + 3\sqrt{5}\)
• (d) \(7 + 3\sqrt{5}\)

CAT 2017 Algebra - Q5) The value of \[ \log_{0.008} \sqrt{5} + \log_{\sqrt{3}} 81 - 7 \] is equal to:

• (a) \( \frac{1}{3} \)
• (b) \( \frac{2}{3} \)
• (c) \( \frac{5}{6} \)
• (d) \( \frac{7}{6} \)

CAT 2017 Algebra - Q6) If \[ 9^{2x - 1} - 81^{x - 1} = 1944, \] then \(x\) is:

• (a) 3
• (b) \( \frac{9}{4} \)
• (c) \( \frac{4}{9} \)
• (d) \( \frac{1}{3} \)

CAT 2017 Algebra - Q7) The number of solutions \((x, y, z)\) to the equation \[ x - y - z = 25, \] where \(x, y, z\) are positive integers such that \(x \leq 40\), \(y \leq 12\), and \(z \leq 12\), is:

• (a) 101
• (b) 99
• (c) 87
• (d) 105

CAT 2017 Algebra - Q8) For how many integers \(n\), will the inequality \[ (n - 5)(n - 10) - 3(n - 2) \leq 0 \] be satisfied?

CAT 2017 Algebra - Q9) If \(f_1(x) = x^2 + 11x + n\) and \(f_2(x) = x\), then the largest positive integer \(n\) for which the equation \[ f_1(x) = f_2(x) \] has two distinct real roots is:

CAT 2017 Algebra - Q10) If \(a, b, c,\) and \(d\) are integers such that \(a + b + c + d = 30\), then the minimum possible value of \[ (a - b)^2 + (a - c)^2 + (a - d)^2 \] is:

CAT 2017 Algebra - Q11) The shortest distance of the point \[ \left(\frac{1}{2}, 1\right) \] from the curve \[ y = |x - 1| + |x + 1| \] is:

• (a) 1
• (b) 0
• (c) \(\sqrt{2}\)
• (d) \(\sqrt{\frac{3}{2}}\)

CAT 2017 Algebra - Q12) If \[ f(x) = \frac{5x + 2}{3x - 5} \quad \text{and} \quad g(x) = x^2 - 2x - 1, \] then the value of \[ g(f(f(3))) \] is:

• (a) 2
• (b) \(\frac{1}{3}\)
• (c) 6
• (d) \(\frac{2}{3}\)

CAT 2017 Algebra - Q13) Let \(a_1, a_2, \ldots, a_{3n}\) be an arithmetic progression with \(a_1 = 3\) and \(a_2 = 7\). If \[ a_1 + a_2 + \ldots + a_{3n} = 1830, \] then what is the smallest positive integer \(m\) such that \[ m(a_1 + a_2 + \ldots + a_n) > 1830? \]

• (a) 8
• (b) 9
• (c) 10
• (d) 11

CAT 2017 Algebra - Q14) If \(a, b, c\) are three positive integers such that \(a : b = 3 : 4\) and \(b : c = 2 : 1\), then which one of the following is a possible value of \(a + b + c\)?

• (a) 201
• (b) 205
• (c) 207
• (d) 210

CAT 2017 Algebra - Q15) If the product of three consecutive positive integers is 15600, then the sum of the squares of these integers is:

• (a) 1777
• (b) 1785
• (c) 1875
• (d) 1877

CAT 2017 Algebra - Q16) If \(x\) is a real number such that \[ \log_3 5 = \log_5 (2 + x), \] then which of the following is true?

• (a) \(0 < x < 3\)
• (b) \(23 < x < 30\)
• (c) \(x > 30\)
• (d) \(3 < x < 23\)

CAT 2017 Algebra - Q17) Let \(f(x) = x^2\) and \(g(x) = 2^x\) for all real \(x\). Then the value of \[ f[f(g(x)) + g(f(x))] \text{ at } x = 1 \] is:

• (a) 16
• (b) 18
• (c) 36
• (d) 40

CAT 2017 Algebra - Q18) The minimum possible value of the sum of the squares of the roots of the equation \[ x^2 + (a + 3)x - (a + 5) = 0 \] is:

• (a) 1
• (b) 2
• (c) 3
• (d) 4

CAT 2017 Algebra - Q19) If \[ 9^{x - \frac{1}{2}} - 2^{2x - 2} = 4^x - 3^{2x - 3}, \] then \(x\) is:

• (a) \( \frac{3}{2} \)
• (b) \( \frac{2}{5} \)
• (c) \( \frac{3}{4} \)
• (d) \( \frac{4}{9} \)

CAT 2017 Algebra - Q20) If \[ \log(2^a \cdot 3^b \cdot 5^c) \] is the arithmetic mean of \[ \log(2^2 \cdot 3^3 \cdot 5),\quad \log(2^6 \cdot 3 \cdot 5^7),\quad \text{and} \quad \log(2 \cdot 3^2 \cdot 5^4), \] then \(a\) equals:

CAT 2017 Algebra - Q21) Let \(a_1, a_2, a_3, a_4, a_5\) be a sequence of five consecutive odd numbers. Consider a new sequence of five consecutive even numbers ending with \(2a_3\). If the sum of the numbers in the new sequence is 450, then \(a_5\) is:

CAT 2017 Algebra - Q22) How many different pairs \((a, b)\) of positive integers are there such that \(a \geq b\) and \[ \frac{1}{a} + \frac{1}{b} = \frac{1}{6}? \]

CAT 2017 Algebra - Q23) If \(f(ab) = f(a)f(b)\) for all positive integers \(a\) and \(b\), then the largest possible value of \(f(1)\) is:

CAT 2017 Algebra - Q24) Let \(f(x) = 2x - 5\) and \(g(x) = 7 - 2x\). Then \(|f(x)+g(x)| = |f(x)| + |g(x)|\) if and only if:

• (a) \(\frac{5}{2} < x < \frac{7}{2}\)
• (b) \(x \leq \frac{5}{2}\) or \(x \geq \frac{7}{2}\)
• (c) \(x < \frac{5}{2}\) or \(x > \frac{7}{2}\)
• (d) \(\frac{5}{2} \leq x \leq \frac{7}{2}\)

CAT 2017 Algebra - Q25) An infinite geometric progression \(a_1, a_2, \dots\) has the property that \[ a_n = 3(a_{n+1} + a_{n+2} + \ldots) \] for every \(n \geq 1\). If the sum \(a_1 + a_2 + a_3 + \dots = 32\), then \(a_5\) is:

• (a) \(\frac{1}{32}\)
• (b) \(\frac{2}{32}\)
• (c) \(\frac{3}{32}\)
• (d) \(\frac{4}{32}\)

CAT 2017 Algebra - Q26) If \[ a_1 = \frac{1}{2 \times 5},\quad a_2 = \frac{1}{5 \times 8},\quad a_3 = \frac{1}{8 \times 11}, \ldots, \] then \[ a_1 + a_2 + a_3 + \ldots + a_{100} \] is:

• (a) \(\frac{25}{151}\)
• (b) \(\frac{1}{2}\)
• (c) \(\frac{1}{4}\)
• (d) \(\frac{111}{55}\)