- Equations
- Progressions
- Functions
- Maxima Minima
- Logarithms

All these algebra questions are with detailed explanations.

**Question 1:**

For the given pair (x, y) of positive integers, such that 4x-17y=1 and x<1000 how many integer values of y satisfy the given conditions?

[1] 56

[2] 57

[3] 58

[4] 59

**Question 2:**

One year payment to the servant is Rs. 90 plus one turban. The servant leaves after 9 months and receives Rs. 65 and turban. Then find the price of the turban

[1] Rs.10

[2] Rs.15

[3] Rs.7.5

[4] Cannot be determined

**Question 3:**

In CAT 2007 there were 75 questions. Each correct answer was rewarded by 4 marks and each wrong answer was penalized by 1 mark. In how many different combination of correct and wrong answer is a score of 50 possible?

[1] 14

[2] 15

[3] 16

[4] None of these

**Question 4:**

How many integer solutions exist for the equation 8x – 5y = 221 such that \(x \times y < 0\)

[1] 4

[2] 5

[3] 6

[4] 8

**Question 5:**

How many integer solutions exists for the equation 11x + 15y = -1 such that both x and y are less than 100?

[1] 15

[2] 16

[3] 17

[4] 18

**Question 6:**

The number of ordered pairs of natural numbers (a, b) satisfying the equation 2a + 3b = 100 is:

[1] 13

[2] 14

[3] 15

[4] 16

**Question 7:**

For how many positive integral values of N, less than 40 does the equation 3a – Nb = 5, have no integer solution

[1] 13

[2] 14

[3] 15

[4] 12

**Question 8:**

What are the number of integral solutions of the equation 7x + 3y = 123 for x,y > 0

[1] 3

[2] 5

[3] 12

[4] Infinite

**Question 9:**

The cost of 3 hamburgers, 5 milk shakes, and 1 order of fries at a certain fast food restaurant is $23.50. At the same restaurant, the cost of 5 hamburgers, 9 milk shakes, and 1 order of fries is \(\$ 39.50\). What is the cost of 2 hamburgers, 2 milk shakes, and 2 orders of fries at this restaurant?

[1] 10

[2] 15

[3] 7.5

[4] Cannot be determined

**Question 10:**

How many integer solutions are there for the equation: |x| + |y| =7?

[1] 24

[2] 26

[3] 14

[4] None of these

**Question 11:**

A shop stores

*x*kg of rice. The first customer buys half this amount plus half a kg of rice. The second customer buys half the remaining amount plus half a kg of rice. Then the third customer also buys half the remaining amount plus half a kg of rice. Thereafter, no rice is left in the shop. Which of the following best describes the value of

*x*?

[1] 2 ≤ x ≤ 6

[2] 5 ≤ x ≤ 8

[3] 9 ≤ x ≤ 12

[4] 11 ≤ x ≤ 14

**Question 12:**

If p and Q are integers such that \(\frac{7}{10}<\frac{p}{q}<\frac{11}{15} \) , find the smallest possible value of q.

[1] 13

[2] 60

[3] 30

[4] 7

**Question 13:**

Given the system of equations \(\left\{ {\begin{array}{*{20}{c}}{2x + y + 2z = 4}\\{x + 2y + 3z = - 1}\\{3x + 2y + z = 9}\end{array}} \right. \), find the value of x+y+z.

[1] -1

[2] 3.5

[3] 2

[4] 1

**Question 14:**

If x and y are positive integers and x+y+xy=54, find x+y

[1] 12

[2] 14

[3] 15

[4] 16

**Question 15:**

How many pairs of integers (x, y) exist such that x

^{2}+ 4y

^{2}< 100?

[1] 95

[2] 90

[3] 147

[4] 180

**Question 16:**

A test has 20 questions, with 4 marks for a correct answer, –1 mark for a wrong answer, and no marks for an unattempted question. A group of friends took the test. If all of them scored exactly 15 marks, but each of them attempted a different number of questions, what is the maximum number of people who could be in the group?

[1] 3

[2] 4

[3] 5

[4] more than 5

**Question 17:**

How many integers x with |x|< 100 can be expressed as \(x = \frac{{4 - {y^3}}}{4} \) for some positive integer y?

[1] 0

[2] 3

[3] 6

[4] 4

**Question 18:**

The number of roots common between the two equations x

^{3}+3x

^{2}+4x+5=0 and x

^{3}+2x

^{2}+7x+3=0 is:

[1] 0

[2] 1

[3] 2

[4] 3

**Question 19:**

Let u= \({({\log _2}x)^2} - 6{\log _2}x + 12 \) where x is a real number. Then the equation x

^{u}=256, has:

[1] no solution for x

[2] exactly one solution for x

[3] exactly two distinct solutions for x

[4] exactly three distinct solutions for x

**Question 20:**

Let a, b, and c be positive real numbers. Determine the largest total number of real roots that the following three polynomials may have among them: ax

^{2}+ bx + c, bx

^{2}+ cx + a, and cx

^{2}+ ax + b.

[1] 4

[2] 5

[3] 6

[4] 0

**Question 21:**

Given that three roots of f(x) = x

^{4}+ax

^{2}+bx+c are 2, -3, and 5, what is the value of a+b+c?

[1] -79

[2] 79

[3] -80

[4] 80

**Question 22:**

If both a and b belong to the set (1, 2, 3, 4), then the number of equations of the form ax

^{2}+bx+1=0 having real roots is

[1] 10

[2] 7

[3] 6

[4] 12

**Question 23:**

Rakesh and Manish solve an equation. In solving Rakesh commits a mistake in constant term and finds the root 8 and 2. Manish commits a mistake in the coefficient of x and finds the roots -9 and -1. Find the correct roots.

[1] 9,1

[2] -9,1

[3] -8,-2

[4] None of these

**Question 24:**

The number of quadratic equations which are unchanged by squaring their roots is

[1] 2

[2] 4

[3] 6

[4] None of these.

**Question 25:**

If the roots of px

^{2}+qx+2=0 are reciprocals of each other, then

[1] p = 0

[2] p = -2

[3] p= +2

[4] p = √2

**Question 26:**

If x =2+2

^{2/3}+2

^{1/3}, then the value of x

^{3}-6x

^{2}+6x is:

[1] 2

[2] -2

[3] 0

[4] 4

**Question 27:**

If the roots of the equation x

^{2}-2ax+a

^{2}+a-3=0 are real and less than 3, then

[1] a < 2

[2] 2 < a < 3

[3] 3 < a < 4

[4] a > 4

**Question 28:**

Find the value of \(\sqrt {2 + \sqrt {2 + \sqrt {2 + \sqrt {2 + .....} } } } \)

[1] -1

[2] 1

[3] 2

[4] \(\frac{{\sqrt 2 + 1}}{2} \)

**Question 29:**

If a, b and c are the roots of the equation x

^{3}– 3x

^{2}+ x + 1 = 0 find the value of \(\frac{1}{a} + \frac{1}{b} + \frac{1}{c} \)

[1] 1

[2] -1

[3] 1/3

[4] -1/3

**Question 30:**

If p, q and r are the roots of the equation 2z

^{3}+ 4z

^{2}-3z -1 =0, find the value of (1 - p) × (1 - q) × (1 - r)

[1] -2

[2] 0

[3] 2

[4] None of these

**Question 31:**

If $\alpha, \beta$ and $\gamma$ are the roots of the equation $x^{3}-7 x+3=0$ what is the value of $\alpha^{4}+\beta^{4}+\gamma^{4}$ ?

[1] 0

[2] 199

[3] 49

[4] 98

**Question 32:**

For what values of p does the equation 4x

^{2}+ 4px + 4 –3p = 0 have two distinct real roots?

[1] p < -4 or p > 1

[2] -1 < p < 4

[3] p < -1 or p > 4

[4] –4 < p < 1

**Question 33:**

If x

^{2}+ 4x + n > 13 for all real number x, then which of the following conditions is necessarily true?

[1] n > 17

[2] n = 20

[3] n > -17

[4] n < 11

**Question 34:**

If (x + 1)×(x – 2)×(x + 3)×(x – 4)×(x + 5)…(x – 100) = a

_{0}+ a

_{1}x + a

_{2}x

^{2}… + a

_{100}x

^{100 }then the value of a

_{99}is equal to:

[1] 50

[2] 0

[3] -50

[4] -100

**Question 35:**

If a, b, and c are the solutions of the equation x

^{3}– 3x

^{2}– 4x + 5 = 0, find the value of \(\frac{1}{{ab}} + \frac{1}{{bc}} + \frac{1}{{ca}} \)

[1] 3/5

[2] -3/5

[3] -4/5

[4] 4/5

**Question 36:**

If a, b, and g are the roots of the equation x

^{3}– 4x

^{2}+ 3x + 5 = 0, find (a + 1)(b + 1)(g + 1)

[1] -3

[2] 0

[3] 3

[4] 1

**Question 37:**

Let A = (x – 1)

^{4}+ 3(x – 1)

^{3}+ 6(x – 1)

^{2}+ 5(x – 1) + 1. Then the value of A is:

[1] (x – 2)

^{4}

[2] x

^{4}

[3] (x + 1)

^{4}

[4] None of these

**Question 38:**

Find the remainder when 3x

^{5}+ 2x

^{4}– 3x

^{3}– x

^{2}+ 2x + 2 is divided by x

^{2}– 1.

[1] 3

[2] 2x – 2

[3] 2x + 3

[4] 2x – 1

**Question 39:**

A quadratic function f(x) attains a maximum of 3 at x = 1. The value of the function at x = 0 is 1. What is the value of f (x) at x = 10?

[1] -105

[2] -119

[3] -159

[4] -110

**Question 40:**

\(x + \frac{1}{x} = 3\) then, what is the value of \({x^5} + \frac{1}{{x{}^5}}. \)

[1] 123

[2] 144

[3] 159

[4] 186

**Question 41:**

If \(\sqrt {x + \sqrt {x + \sqrt {x + ....} } } = 10. \)What is the value of x?

[1] 80

[2] 90

[3] 100

[4] 110

**Question 42:**

If $\alpha$ and $\beta$ are the roots of the quadratic equation $x^{2}-x-6,$ then find the value of $\alpha^{4}+\beta^{4} ?$

[1] 1

[2] 55

[3] 97

[4] none of these

**Question 43:**

Find the value of \(\sqrt {4 - \sqrt {4 + \sqrt {4 - \sqrt {4 + ...} } } } \)

[1] \(\frac{{\sqrt {13} - 1}}{2} \)

[2] \(\frac{{\sqrt {13} + 1}}{2} \)

[3] \(\frac{{\sqrt {11} + 1}}{2} \)

[4] \(\frac{{\sqrt {15} - 1}}{2} \)

**Question 44:**

If the roots of the equation

*x*

^{3}–

*ax*

^{2}+

*bx*–

*c =*0 are three consecutive integers, then what is the smallest possible value of

*b*?

[1] -1/√3

[2] -1

[3] 0

[4] 1/√3

**Question 45:**

Three consecutive positive integers are raised to the first, second and third powers respectively and then added. The sum so obtained is a perfect square whose square root equals the total of the three original integers. Which of the following best describes the minimum, say

*m*, of these three integers?

[1] 1 ≤ m ≤ 3

[2] 4 ≤ m ≤ 6

[3] 7 ≤ m ≤ 9

[4] 10 ≤ m ≤ 12

**Question 46:**

The price of Darjeeling tea (in rupees per kilogram) is 100 + 0.10

*n*, on the

*n*th day of 2007 (

*n*= 1, 2, ..., 100), and then remains constant. On the other hand, the price of Ooty tea (in rupees per kilogram) is 89 + 0.15

*n*, on the

*n*th day of 2007 (

*n*= 1, 2, ..., 365). On which date in 2007 will the prices of these two varieties of tea be equal?

[1] May 21

[2] April 11

[3] May 20

[4] April 10

**Question 47:**

The polynomial f(x)=x

^{2}-12x+c has two real roots, one of which is the square of the other. Find the sum of all possible value of c.

[1] -37

[2] -12

[3] 25

[4] 91

**Question 48:**

Two sides of a triangle have lengths 10 and 20. How many integers can take the value of the third side length:

[1] 18

[2] 19

[3] 20

[4] 21

**Question 49:**

Which of the following is a solution to: \(6{\left( {x + \frac{1}{x}} \right)^2} - 35\left( {x + \frac{1}{x}} \right) + 50 = 0 \)

[1] 1

[2] 1/3

[3] 4

[4] 6

**Question 50:**

Find x if \(\frac{5}{{3 + \frac{5}{{3 + \frac{5}{{3 + ...}}}}}} = x. \) \( \)

[1] \(\frac{{ - 3 + \sqrt {29} }}{2} \)

[2] \(\frac{{3 + \sqrt {29} }}{2} \)

[3] \(\frac{{ - 1 + \sqrt 5 }}{2} \)

[4] \(\frac{{1 + \sqrt 5 }}{2} \)

**Question 51:**

If $a, b, c$ are the roots of $x^{3}-x^{2}-1=0,$ what's the value of $\frac{a}{b c}+\frac{b}{c a}+\frac{c}{a b}$ ?

[1] -1

[2] 1

[3] 2

[4] -2

**Question 52:**

The sum of the integers in the solution set of |x

^{2}-5x|<6 is:

[1] 10

[2] 15

[3] 20

[4] 0

**Question 53:**

Find abc if a+b+c = 0 and a

^{3}+ b

^{3}+ c

^{3}=216

[1] 48

[2] 72

[3] 24

[4] 216

**Question 54:**

Solve for x: \(\sqrt {x + \sqrt {x + \sqrt x + ....} } = \frac{3}{2} \)

[1] Empty Set

[2] 3/2

[3] 3/4

[4] 3/16

**Question 55:**

Solve for x \(\sqrt {\frac{3}{2} + \sqrt {\frac{3}{2} + \sqrt {\frac{3}{2}} + ....} } = x \)

[1] \(\frac{{1 \pm \sqrt 7 }}{2} \)

[2] \(\frac{{1 + \sqrt 7 }}{2} \)

[3] \(\frac{{\sqrt 7 }}{2} \)

[4] \(\frac{3}{2} \)

**Question 56:**

What is/are the value(s) of

*x*if \(\sqrt {{x^2} + \sqrt {{x^2} + \sqrt {{x^2} + ...} } } = 9 \)

[1] 6√2

[2] 3√10

[3] ±3√10

[4] ±6√2

**Question 57:**

For x ≠ 1 and x ≠ -1, simplify the following expression: \(\frac{{{\rm{(}}{{\rm{x}}^{\rm{3}}} + 1)({{\rm{x}}^3} - 1)}}{{({{\rm{x}}^2} - 1)}} \)

[1] x

^{4}+ x

^{2}+ 1

[2] x

^{4}+ x

^{3}+ x + 1

[3] x

^{6}– 1

[4] x

^{6}+ 1

**Question 58:**

If √x + √y = 6 and xy = 4 then for: x>0, y>0 give the value of x+y

[1] 2

[2] 28

[3] 32

[4] 34

**Question 59:**

Find a for which a<b and \(\sqrt {1 + \sqrt {21 + 12\sqrt 3 } } = \sqrt a + \sqrt b \)

[1] 1

[2] 3

[3] 4

[4] None of these

**Question 60:**

One root of the following given equation \(2{x^5} - 14{x^4} + 31{x^3} - 64{x^2} + 19x + 130 = 0 \) is

[1] 1

[2] 3

[3] 5

[4] 7

**Question 61:**

The equation \(x + \frac{2}{{1 - x}} = 1 + \frac{2}{{1 - x}}, \) has

[1] No real root

[2] One real root

[3] Two equal roots

[4] Infinite roots

**Question 62:**

If \(x = \sqrt {7 + 4\sqrt 3 } , \) then \(x + \frac{1}{x} = \)

[1] 4

[2] 6

[3] 3

[4] 2

**Question 63:**

If A.M. of the roots of a quadratic equation is 8/5 and A.M. of their reciprocals is 8/7, then the equation is

[1] 5x

^{2}-16x+7=0

[2] 7x

^{2}-16x+5=0

[3] 7x

^{2}-16x+8=0

[4] 3x

^{2}-12x+7=0

**Question 64:**

The equation x

^{2}+ ax + (b + 2) = 0 has real roots. What is the minimum value of a

^{2}+ b

^{2}?

[1] 0

[2] 1

[3] 2

[4] 4