- 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
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Login with GoogleQuestion 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 x2 + 4y2 < 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 x3+3x2+4x+5=0 and x3+2x2+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 xu=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: ax2 + bx + c, bx2 + cx + a, and cx2 + ax + b.
[1] 4
[2] 5
[3] 6
[4] 0
Question 21:
Given that three roots of f(x) = x4+ax2+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 ax2+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 px2+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+22/3+21/3, then the value of x3-6x2+6x is:
[1] 2
[2] -2
[3] 0
[4] 4
Question 27:
If the roots of the equation x2-2ax+a2+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 x3 – 3x2 + 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 2z3 + 4z2 -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 4x2 + 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 x2 + 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) = a0 + a1x + a2x2… + a100x100 then the value of a99 is equal to:
[1] 50
[2] 0
[3] -50
[4] -100
Question 35:
If a, b, and c are the solutions of the equation x3 – 3x2 – 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 x3 – 4x2 + 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] x4
[3] (x + 1)4
[4] None of these
Question 38:
Find the remainder when 3x5 + 2x4 – 3x3 – x2 + 2x + 2 is divided by x2 – 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 x3 – ax2 + 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 nth 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 + 0.15n, on the nth 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)=x2-12x+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 |x2-5x|<6 is:
[1] 10
[2] 15
[3] 20
[4] 0
Question 53:
Find abc if a+b+c = 0 and a3+ b3+ c3=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] x4 + x2 + 1
[2] x4 + x3 + x + 1
[3] x6 – 1
[4] x6 + 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] 5x2-16x+7=0
[2] 7x2-16x+5=0
[3] 7x2-16x+8=0
[4] 3x2-12x+7=0
Question 64:
The equation x2 + ax + (b + 2) = 0 has real roots. What is the minimum value of a2 + b2?
[1] 0
[2] 1
[3] 2
[4] 4
