Algebra

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Practice Questions for CAT Algebra with Solutions

1. CAT - Functions

If x is a positive real number then the minimum value of (x + 7) (x + 19)/(x + 3) is:

Video Explanation

2. CAT - Functions

If f(x) = x²– 7x and g(x) = x + 3, then the minimum value of f(g(x)) – 3x is

(CAT 2021)

1. -16

2. -15

3. -12

4. -20

Video Explanation

3. CAT - Functions

If 3x + 2 |y| + y = 7 and x + |x| + 3y = 1, then x + 2y is:

(CAT 2021)

1. 8/3

2. -4/3

3. 1

4. 0

Video Explanation

4. CAT - Functions

Find the range of (x²+ 4x + 8)/(x²+ 4x + 5)

Video Explanation

5. CAT - Inequalities

Find the sum of all negative integral values of x where |x – | x – 2 | + 3 | – 4 < 3

Video Explanation

6. CAT - Inequalities

Consider the function f(x) = (x + 4) (x + 6) (x + 8) ………….. (x + 98). The number of integers x for which f(x) is less than 0 are:

Video Explanation

7. CAT - Inequalities

How many integer values of x will satisfy this (x² – 16|x| + 60) / (x² – 14x + 49) < 0

Video Explanation

8. CAT - Inequalities

For how many positive integral values of k is (k-11)(k-15)(k − 19)… (k – 99) < 0?

Video Explanation

9. CAT - Inequalities

The number of integers n that satisfy the inequalities |n -60| < |n-100| < |n-20| is:

(CAT 2021)

Video Explanation

10. CAT - Inequalities

Consider the pair of equations: x² – xy – x = 22 and y² – xy + y = 34. If x > y, then x – y equals (CAT 2021)

1. 8

2. 6

3. 4

4. 7

Video Explanation

11. CAT - Inequalities

f(x) = (x² + 2x -15) /(x² – 7x – 18) is negative if and only if (CAT 2021)

1. x < -5 or 3 < x < 9

2. -5 < x < -2 or 3 < x < 9

3. x < -5 or -2 < x < 3

4. -2 < x < 3 or x > 9

Video Explanation

12. CAT - Inequalities

Let k be a constant. The equations kx + y = 3 and 4x + ky = 4 have a unique solution if and only if (CAT 2020)

1. k ≠ 2
2. |k| = 2
3. k = 2
4. |k| ≠ 2

Video Explanation

13. CAT - Series (AP & GP)

Consider 2 APs 2,6,10……. and 5,12,19………….If S is a set containing the first 100 members of each progression then how many distinct elements are there in S?

Video Explanation

14. CAT - Series (AP & GP)

Three positive integers x, y and z are in arithmetic progression. If y − x greater than 2 and xyz = 5 (x + y + z), then z − x equals: (CAT 2021)

1. 14

2. 10

3. 8

4. 12

Video Explanation

15. CAT - Series (AP & GP)

a + b + c + d = 4 All a,b,c,d are integers Find minimum possible value of 1/a + 1/b + 1/c + 1/d

Video Explanation

16. CAT - Series (AP & GP)

If x, y and z are non – zero real numbers and 9^x = 16^y = 36^z Find the value of 5[xz/(xy -yz) ]

Video Explanation

17. CAT - Series (AP & GP)

If a, b, c are non-zero and 14^a = 36^b = 84^c, then 6b(1/c – 1/a) is equal to:

Video Explanation

18. CAT - Series (AP & GP)

How many 3 term geometric progressions can be made from the series 1, 3, 3², 3³, …… 3⁴⁸?

Video Explanation

19. CAT - Logs

If log_a30 = A, log_a(5/3) = – B and log₂ a = 1 /3, then log₃ a equals: (CAT 2020)

1. 2/(A + B) – 3

2. 2/(A + B – 3)

3. (A + B)/2 – 3

4. (A + B – 3)/2

Video Explanation

20. CAT - Logs

For a real number a , if (log₁₅ a + log₃₂ a) / [(log₁₅ a)(log₃₂ a)] = 4 then a must lie in range

1. 3 < a < 4

2. 2 < a < 3

3. 4 < a < 5

4. a > 5

Video Explanation

21. CAT - Logs

If log₂ [3 + log₂ {4 + log₄ (x – 1)}] – 2 = 0, then 4x equals: (CAT 2021)

Video Explanation

22. CAT - Logs

If 5 – log₁₀√(1 + x) + 4 log₁₀ √(1 – x) = log₁₀(1/√(1 – x²), then 100x equals

Video Explanation

23. CAT - Logs

Log _(x+3) (x² – x ) < 1

Video Explanation

24. CAT - Equations

x² + 9x + |K| = 0, roots of this quadratic are integers. How many integral values of K are possible?

Video Explanation

25. CAT - Equations

The equation ax²+bx+c = 0 and 5x²+12x+13 = 0 have a common root where a,b,c are the side lengths of triangles ABC then find Angle C

Video Explanation

26. CAT - Equations

If r a constant such that |x² – 4x – 13| = r has exactly three distinct real roots, then the value of r is
(CAT 2021)

1. 17

2. 18

3. 15

4. 21

Video Explanation

27. CAT - Equations

Suppose one of the roots of the equation ax² – bx + c = 0 is 2 + root13, where a, b and c are rational numbers and a is not equal to zero. If b = c³, then |a| equals to: (CAT 2021)

1. 4

2. 2

3. 1

4. 3

Video Explanation

28. CAT - Equations

Let m and n be positive integers, If x² + mx + 2n = 0 and x² + 2nx + m = 0 have real roots, then the smallest possible value of m+ n is: (CAT 2020)

1. 5

2. 8

3. 7

4. 6

Video Explanation

29. CAT - Equations

For natural numbers x, y and z, if xy + yz = 19 and yz + xz = 51, then how many such solutions are possible?

Video Explanation

30. CAT - Equations

Given that three roots of f(x) = x⁴ + ax² + bx + c are 2, – 3 and 5, what is the value of a + b + c?

Video Explanation

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