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
Solution:
We first need to find out a solution for x & y. Once we get a solution, values of x would be in an AP
with a common difference of 17 whereas values of y would be in an AP with a common difference of
4.
Valid Solutions:
x = 13, y = 3
x = 30, y = 7
x = 47, y = 11
.
.
x = 999, y = 235
No. of terms =999−1317+1=999−1317+1= = 58 + 1 = 59. Option D
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
Show Answer & Explanation
Payment for 12 months = 90 + t {Assuming t as the value of a turban}
Payment for 9 months should be ¾(90 + t)
Payment for 9 months is given to us as 65 + t
Equating the two values we get
¾(90 + t) = 65 + t
270 + 3t = 260 + 4t
t = 10 Rs. Option A
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
Show Answer & Explanation
Solution:
Correct (c) + Wrong (w) + Not attempted (n) = 75
4c – w + 0n= 50
Adding the two equations we get
5c + n = 125
Values of both c & n will be whole numbers in the range [0, 50]
c (max) = 25; when n = 0
c (min) = 13; when n = 60 {Smallest value of ‘c’ which will take the marks from correct
questions greater than or equal to 50}
No. of valid combinations will be for all value of ‘c’ from 13 to 25 = 13. Option D
Question 4:
How many integer solutions exist for the equation 8x – 5y = 221 such that x×y<0
[1] 4
[2] 5
[3] 6
[4] 8
Show Answer & Explanation
We first need to find out a solution for x & y. Once we get a solution, values of x would be in an
AP with a common difference of 5 whereas values of y would be in an AP with a common
difference of 8.
x = 32; y = 7
x = 37; y = 15
x = 42; y = 23
But we need the solutions where one variable is negative whereas the other one is positive. so,
we will move in the other direction.
x = 27; y = -1
x = 22; y = -9
x = 17; y = -17
x = 12; y = -25
x = 7; y = -33
x = 2; y = -41
So, number of integer solutions where x×y<0 is 6. Option C
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
Show Answer & Explanation
First integer solution of the equation 11x + 15y = -1 using hit and trial method is
x = 4 and y = -3
RULE: The value of x vary by coefficient of y (i.e. 15 in this case) and value of y changes by
coefficient of x (i.e. 11 in this case i.e. Next solution will be
x = 4+15 = 19 and y = -3-11 = -14
x = 19+15 = 34 and y = -14-11 = -25
i.e. Positive values of x further will be {4, 19, 34, 49, 64, 79, 94} i.e. 7 solutions
Similarly, values of y will change by 11 and positive values of y will be (first positive value of y =
-3+11 = 8}
so all positive values of y will be {8, 19, 30, 41, 52, 63, 74, 85, 96} = 9 Solutions
Total Solutions = 7+9 = 16 : Option B
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
Show Answer & Explanation
Valid solutions:
a = 2; b = 32
a = 5; b = 30
.
a = 47; b = 2
No. of solutions = 16. Option D
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
Show Answer & Explanation
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
Show Answer & Explanation
x = 3; y = 34
x = 6; y = 27
.
.
x = 15; y = 6
Number of integral solutions such that x, y > 0 are 5. Option B
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
Show Answer & Explanation
Question 10:
How many integer solutions are there for the equation: |x| + |y| =7?
[1] 24
[2] 26
[3] 14
[4] None of these
Show Answer & Explanation
x can take any integer value from [-7,7].
So, there are 15 valid values of x.
For each of these values, there are 2 corresponding values of y. eg: For x = 3; y can be 4 or -4.
Except when x = 7 or -7; where the only possible value of y is 0.
Total valid values of x = 13×2 + 1 + 1 = 28. Option D
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
Show Answer & Explanation
After first customer, amount of rice left is 0.5x – 0.5
After second customer, amount of rice left is 0.5(0.5x -0.5) – 0.5
After third customer, amount of rice left is 0.5(0.5(0.5x -0.5) – 0.5) – 0.5 = 0
0.5(0.5(0.5x -0.5) – 0.5) = 0.5
0.5(0.5x -0.5) – 0.5 = 1
0.5x -0.5 = 3
x = 7. Option B
Verification for better understanding:
Originally there were 7 kgs of rice.
First customer purchased 3.5kgs + 0.5kgs = 4 kgs.
After first customers, amount of rice left is 3 kgs.
Second customer purchased 1.5kgs + 0.5 kgs = 2 kgs.
After second customer, amount of rice left is 1 kg.
Third customer purchased 0.5kgs + 0.5kgs = 1 kg.
No rice is left after the third customer.
Question 12:
If p and Q are integers such that 7/10 < p/q < 11/15, find the smallest possible value of q.
[1] 13
[2] 60
[3] 30
[4] 7
Show Answer & Explanation
The fraction lies in the range (0.7,0.733333)
We know that 8/1 = 0.727272.. is a valid value.
The smallest value of q has to be less than or equal to 11. Only 7 fits in the range.
With a little hit and trial, we get a valid value of p/q as 5/7
The smallest value of q = 7. Option D
Question 13:
Given the system of equations 2x+y+2z=4, x+2y+3z=−1, 3x+2y+z=9, find the value of x+y+z.
[1] -1
[2] 3.5
[3] 2
[4] 1
Show Answer & Explanation
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
Show Answer & Explanation
Question 15:
How many pairs of integers (x, y) exist such that x2 + 4y2 < 100?
[1] 95
[2] 90
[3] 147
[4] 180
Show Answer & Explanation
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
Show Answer & Explanation
Question 17:
How many integers x with |x|< 100 can be expressed as x=4−y34�=4−�34 for some positive
integer y?
[1] 0
[2] 3
[3] 6
[4] 4
Show Answer & Explanation
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
Show Answer & Explanation
Question 19:
Let u= (log2x)2−6log2x+12(log2�)2−6log2�+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
Show Answer & Explanation
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
Show Answer & Explanation
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
Show Answer & Explanation
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
Show Answer & Explanation
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
Show Answer & Explanation
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.
Show Answer & Explanation
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
Show Answer & Explanation
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
Show Answer & Explanation
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
Show Answer & Explanation
Question 28:
Find the value of √ 2+√2+√ 2+√ 2+..... 2+2+2+2+.....
[1] -1
[2] 1
[3] 2
[4] √ 2 +122+12
Show Answer & Explanation
Question 29:
If a, b and c are the roots of the equation x3 – 3x2 + x + 1 = 0 find the value of 1a+1b+1c1�+1�+1�
[1] 1
[2] -1
[3] 1/3
[4] -1/3
Show Answer & Explanation
Question 30:
If p, q and r are the roots of the equation 2z 3 + 4z2 -3z -1 =0, find the value of (1 - p) × (1 - q) × (1 - r)
[1] -2
[2] 0
[3] 2
[4] None of these
Show Answer & Explanation
Question 31:
If α,β�,� and γ� are the roots of the equation x3−7x+3=0�3−7�+3=0 what is the value
of α4+β4+γ4�4+�4+�4 ?
[1] 0
[2] 199
[3] 49
[4] 98
Show Answer & Explanation
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
Show Answer & Explanation
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
Show Answer & Explanation
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
Show Answer & Explanation
Question 35:
If a, b, and c are the solutions of the equation x3 – 3x2 – 4x + 5 = 0, find the value
of 1ab+1bc+1ca1��+1��+1��
[1] 3/5
[2] -3/5
[3] -4/5
[4] 4/5
Show Answer & Explanation
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
Show Answer & Explanation
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
Show Answer & Explanation
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
Show Answer & Explanation
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
Show Answer & Explanation
Question 40:
x+1x=3�+1�=3 then, what is the value of x5+1x5.�5+1�5.
[1] 123
[2] 144
[3] 159
[4] 186
Show Answer & Explanation
Question 41:
If √ x+√ x+√ x+.... =10.�+�+�+....=10.What is the value of x?
[1] 80
[2] 90
[3] 100
[4] 110
Show Answer & Explanation
Question 42:
If α� and β� are the roots of the quadratic equation x2−x−6,�2−�−6, then find the value
of α4+β4?�4+�4?
[1] 1
[2] 55
[3] 97
[4] none of these
Show Answer & Explanation
Question 43:
Find the value of √ 4−√4+√ 4−√ 4+... 4−4+4−4+...
[1] √ 13 −1213−12
[2] √ 13 +1213+12
[3] √ 11 +1211+12
[4] √ 15 −1215−12
Show Answer & Explanation
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
Show Answer & Explanation
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
Show Answer & Explanation
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
Show Answer & Explanation
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
Show Answer & Explanation
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
Show Answer & Explanation
Question 49:
Which of the following is a solution to: 6(x+1x)2−35(x+1x)+50=06(�+1�)2−35(�+1�)+50=0
[1] 1
[2] 1/3
[3] 4
[4] 6
Show Answer & Explanation
Question 50:
Find x if 53+53+53+...=x.53+53+53+...=�.
[1] −3+√ 29 2−3+292
[2] 3+√ 29 23+292
[3] −1+√ 5 2−1+52
[4] 1+√ 5 21+52
Show Answer & Explanation
Question 51:
If a,b,c�,�,� are the roots of x3−x2−1=0,�3−�2−1=0, what's the value
of abc+bca+cab���+���+��� ?
[1] -1
[2] 1
[3] 2
[4] -2
Show Answer & Explanation
Question 52:
The sum of the integers in the solution set of |x2-5x|<6 is:
[1] 10
[2] 15
[3] 20
[4] 0
Show Answer & Explanation
Question 53:
Find abc if a+b+c = 0 and a3+ b3+ c3=216
[1] 48
[2] 72
[3] 24
[4] 216
Show Answer & Explanation
Question 54:
Solve for x: √ x+√ x+√ x +....=32�+�+�+....=32
[1] Empty Set
[2] 3/2
[3] 3/4
[4] 3/16
Show Answer & Explanation
Question 55:
Solve for x √ 32+√32+√32+.... =x32+32+32+....=�
[1] 1±√ 7 21±72
[2] 1+√ 7 21+72
[3] √ 7 272
[4] 3232
Show Answer & Explanation
Question 56:
What is/are the value(s) of x if √ x2+√ x2+√ x2+... =9�2+�2+�2+...=9
[1] 6√2
[2] 3√10
[3] ±3√10
[4] ±6√2
Show Answer & Explanation
Question 57:
For x ≠ 1 and x ≠ -1, simplify the following expression: (x3+1)(x3−1)(x2−1)(x3+1)(x3−1)(x2−1)
[1] x4 + x2 + 1
[2] x4 + x3 + x + 1
[3] x6 – 1
[4] x6 + 1
Show Answer & Explanation
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
Show Answer & Explanation
Question 59:
Find a for which a<b and √ 1+√ 21+12√ 3 =√ a +√ b 1+21+123=�+�
[1] 1
[2] 3
[3] 4
[4] None of these
Show Answer & Explanation
Question 60:
One root of the following given
equation 2x5−14x4+31x3−64x2+19x+130=02�5−14�4+31�3−64�2+19�+130=0 is
[1] 1
[2] 3
[3] 5
[4] 7
Show Answer & Explanation
Question 61:
The equation x+21−x=1+21−x,�+21−�=1+21−�, has
[1] No real root
[2] One real root
[3] Two equal roots
[4] Infinite roots
Show Answer & Explanation
Question 62:
If x=√7+4√ 3 ,�=7+43, then x+1x=�+1�=
[1] 4
[2] 6
[3] 3
[4] 2
Show Answer & Explanation
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
Show Answer & Explanation
Question 64:
The equation x2 + ax + (b + 2) = 0 has real roots. What is the minimum value of a 2 + b2?
[1] 0
[2] 1
[3] 2
[4] 4