CAT Algebra questions from Linear equations and Quadratic equations that appear in the Quantitative Aptitude section of the CAT Exam consists of concepts from Equations and Algebra. Get as much practice as you can in these two topics because the benefits of being good at framing equations can be enormous and useful in other CAT topics as well. In CAT Exam, one can generally expect to get 1~2 questions from Linear Equations and Quadratic Equations. Make use of 2IIMs Free CAT Questions, provided with detailed solutions and Video explanations to obtain a wonderful CAT score. If you would like to take these questions as a Quiz, head on here to take these questions in a test format, absolutely free.
3x + 4|y| = 33. How many integer values of (x, y) are possible?
(|x| - 3) (|y| + 4) = 12. How many pairs of integers (x, y) satisfy this equation?
x + |y| = 8, |x| + y = 6. How many pairs of x, y satisfy these two equations?
What is the number of real solutions of the equation x^{2} - 7|x| - 18 = 0?
x^{2} - 9x + |k| = 0 has real roots. How many integer values can 'k' take?
x^{2} - 11x + |p| = 0 has integer roots. How many integer values can 'p' take?
2x + 5y = 103. Find the number of pairs of positive integers x and y that satisfy this equation.
Consider three numbers a, b and c. Max (a,b,c) + Min (a,b,c) = 13. Median (a,b,c) - Mean (a,b,c) = 2. Find the median of a, b, and c.
a_{1}x + b_{1}y + c_{1}z = d_{1}, a_{2}x + b_{2}y + c_{2}z = d_{2}, a_{3}x + b_{3}y + c_{3}z = d_{3}.
Which of the following statements if true would imply that the above system of equations does not have a unique solution?
i. \\frac{a_{1}}{a_{2}}\\) = \\frac{b_{1}}{b_{2}}\\) = \\frac{c_{1}}{c_{2}}\\) ≠ \\frac{d_{1}}{d_{2}}\\)
ii. \\frac{a_{1}}{a_{2}}\\) = \\frac{ a_{2} }{ a_{3} }\\) ; \\frac{ b_{1} }{ b_{2} }\\) = \\frac{ b_{2} }{ b_{3} }\\)
iii. a_{1}, a_{2}, a_{3} are integers; b_{1}, b_{2}, b_{3} are rational numbers, c_{1}, c_{2}, c_{3} are irrational numbers
Equation x^{2} + 5x – 7 = 0 has roots a and b. Equation 2x^{2} + px + q = 0 has roots a + 1 and b + 1. Find p + q.
Sum of the roots of a quadratic equation is 5 less than the product of the roots. If one root is 1 more than the other root, find the product of the roots?
How many real solutions are there for the equation x^{2} – 7|x| - 30 = 0?
If (3x+2y-22)^{2} + (4x-5y+9)^{2} = 0 and 5x-4y = 0. Find the value of x+y.
Let x^{3}- x^{2} + bx + c = 0 has 3 real roots which are in A.P. which of the following could be true
(3 + 2√2)^{(x2 - 3)} + (3 - 2√2)^{(x2 - 3)} = b which of the following can be the value of b?
If f(y) = x^{2} + (2p + 1)x + p^{2} - 1 and x is a real number, for what values of ‘p' the function becomes 0?
A merchant decides to sell off 100 articles a week at a selling price of Rs. 150 each. For each 4% rise in the selling price he sells 3 less articles a week. If the selling price of each article is Rs x, then which of the below expression represents the number of articles sold by the merchant in that week?
The Questions that follow, are from actual CAT papers. If you wish to take them separately or plan to solve actual CAT papers at a later point in time, It would be a good idea to stop here.
For some real numbers \(a\) and \(b\), the system of equations \(x+y=4\) and \((a+5) x+\left(b^2-15\right) y=8 b\) has infinitely many solutions for \(x\) and \(y\). Then, the maximum possible value of \(a b\) is
A quadratic equation \(x^2+b x+c=0\) has two real roots. If the difference between the reciprocals of the roots is \(\frac{1}{3}\), and the sum of the reciprocals of the squares of the roots is \(\frac{5}{9}\), then the largest possible value of \((b+c)\) is
The sum of all possible values of \(x\) satisfying the equation \(2^{4 x^2}-2^{2 x^2+x+16}+2^{2 x+30}=0\), is
Let \(k\) be the largest integer such that the equation \((x-1)^2+2 k x+11=0\) has no real roots. If \(y\) is a positive real number, then the least possible value of \(\frac{k}{4 y}+9 y\) is
If \(\sqrt{5 x+9}+\sqrt{5 x-9}=3(2+\sqrt{2})\), then \(\sqrt{10 x+9}\) is equal to
Let \(\alpha\) and \(\beta\) be the two distinct roots of the equation \(2 x^2-6 x+k=0\), such that \((\alpha+\beta)\) and \(\alpha \beta\) are the distinct roots of the equation \(x^2+p x+p=0\). Then, the value of \(8(k-p)\) is
The equation \(x^3+(2 r+1) x^2+(4 r-1) x+2=0\) has -2 as one of the roots. If the other two roots are real, then the minimum possible non-negative integer value of \(r\) is
Suppose \(k\) is any integer such that the equation \(2 x^2+k x+5=0\) has no real roots and the equation \(x^2+(k-5) x+1=0\) has two distinct real roots for \(x\). Then, the number of possible values of \(k\) is
If \((3+2 \sqrt{2})\) is a root of the equation \(a x^2+b x+c=0\), and \((4+2 \sqrt{3})\) is a root of the equation \(a y^2+m y+n=0\), where \(a, b, c, m\) and \(n\) are integers, then the value of \(\left(\frac{b}{m}+\frac{c-2 b}{n}\right)\) is
A donation box can receive only cheques of ₹100, ₹250, and ₹500. On one good day, the donation box was found to contain exactly 100 cheques amounting to a total sum of ₹15250. Then, the maximum possible number of cheques of ₹500 that the donation box may have contained, is
Let \(r\) and \(c\) be real numbers. If \(r\) and \(-r\) are roots of \(5 x^3+c x^2-10 x+9=0\), then \(c\) equals
The number of integer solutions of the equation \(\left(x^2-10\right)^{\left(x^2-3 x-10\right)}=1\) is
Let \(a, b, c\) be non-zero real numbers such that \(b^2 \lt 4 a c\), and \(f(x)=a x^2+b x+c\). If the set \(S\) consists of al integers \(m\) such that \(f(m)\lt0\), then the set \(S\) must necessarily be
If \(3 x+2|y|+y=7\) and \(x+|x|+3 y=1\), then \(x+2 y\) is
Suppose one of the roots of the equation a x^{2} - b x + c = 0 is 2 + √3, where a, b and c are rational numbers and a ≠ 0. If b = c^{3} then |a| equals
Let k be a constant. The equations kx + y = 3 and 4x + ky = 4 have a unique solution if and only if
Let m and n be positive integers, If x^{2 }+ mx + 2n = 0 and x^{2 }+ 2nx + m = 0 have real roots, then the smallest possible value of m + n is
The number of pairs of integers(x,y) satisfying x ≥ y ≥ -20 and 2x + 5y = 99 is
The number of integers that satisfy the equality (x^{2} - 5x + 7)^{x + 1} = 1 is
In how many ways can a pair of integers (x , a) be chosen such that x^{2} − 2 | x | + | a - 2 | = 0 ?
Aron bought some pencils and sharpeners. Spending the same amount of money as Aron, Aditya bought twice as many pencils and 10 less sharpeners. If the cost of one sharpener is 2 more than the cost of a pencil, then the minimum possible number of pencils bought by Aron and Aditya together is
The number of distinct real roots of the equation
(x + \\frac{1}{x}))^{2} - 3(x + \\frac{1}{x}))
+ 2 = 0 equals
How many distinct positive integer-valued solutions exist to the equation (x^{2} - 7x + 11)^{(x2 - 13x + 42)} = 1?
Let a, b, x, y be real numbers such that a^{2} + b^{2} = 25 , x^{2} + y^{2} = 169 and ax + by = 65. If k = ay - bx, then
What is the largest positive integer such that \\frac{n^2+7n+12}{n^2-n-12}) is also positive integer?
Let A be a real number. Then the roots of the equation x^{2} - 4x - log_{2}A = 0 are real and distinct if and only if
The quadratic equation x^{2} + bx + c = 0 has two roots 4a and 3a, where a is an integer. Which of the following is a possible value of b^{2} + c?
The product of two positive numbers is 616. If the ratio of the difference of their cubes to the cube of their difference is 157 : 3, then the sum of the two numbers is
The number of solutions of the equation |x|(6x^{2} + 1) = 5x^{2} is [TITA]
The product of the distinct roots of |x^{2} - x - 6| = x + 2 is
If u^{2} + (u−2v−1)^{2} = −4v(u + v), then what is the value of u + 3v?
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
If x + 1 = x^{2} and x > 0, then 2x^{4} is:
The Questions that follow, are from actual XAT papers. If you wish to take them separately or plan to solve actual XAT papers at a later point in time, It would be a good idea to stop here.
Two different quadratic equations have a common root. Let the three unique roots of the two equations be A, B and C - all of them are positive integers. If (A + B + C) = 41 and the product of the roots of one of the equations is 35, which of the following options is definitely correct?
The Questions that follow, are from actual IPMAT papers. If you wish to take them separately or plan to solve actual IPMAT papers at a later point in time, It would be a good idea to stop here.
If \|x| < 100\\) and \|y| < 100\\), then the number of integer solutions of (x, y) satisfying the equation 4x + 7y = 3 is
If a, b, c are real numbers a^{2} + b^{2} + c^{2} = 1, then the set of values ab + bc + ca can take is:
Let \\alpha, \beta\\) be the roots of x^{2} - x + p = 0 and \\gamma, \delta\\) be the roots of x^{2} - 4x + q = 0 where p and q are integers. If \\alpha, \beta, \gamma, \delta\\) are in geometric progression then p + q is
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