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Test Level 1: Quadratic Equations & Linear Equations - 1 - CAT MCQ


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10 Questions MCQ Test Level-wise Tests for CAT - Test Level 1: Quadratic Equations & Linear Equations - 1

Test Level 1: Quadratic Equations & Linear Equations - 1 for CAT 2024 is part of Level-wise Tests for CAT preparation. The Test Level 1: Quadratic Equations & Linear Equations - 1 questions and answers have been prepared according to the CAT exam syllabus.The Test Level 1: Quadratic Equations & Linear Equations - 1 MCQs are made for CAT 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test Level 1: Quadratic Equations & Linear Equations - 1 below.
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Test Level 1: Quadratic Equations & Linear Equations - 1 - Question 1

Total integer pair(s) (x, y) satisfying the equation x + y = xy is/are  

Detailed Solution for Test Level 1: Quadratic Equations & Linear Equations - 1 - Question 1

The equation is x + y = xy. The only two integer pairs satisfying this are (0, 0) and (2, 2).
Hence, there are 2 pairs.

Test Level 1: Quadratic Equations & Linear Equations - 1 - Question 2

If A = 2 and B + P + F = 24, what are the values of Q and S? Consider whole numbers only.
A + B = Z, Z + P = T, T + A = F, F + S = Q, Q - T = 7

Detailed Solution for Test Level 1: Quadratic Equations & Linear Equations - 1 - Question 2

Since it is known that A + B = Z, it follows that A + B + P = T.
We also know that T + A = F.
So, in the equation B + P + F = 24, we can replace F with T + A.
The equation then becomes B + P + T + A = 24 or B + P + T = 22, since A = 2.
Then, we have
B + P + 2 = T ... (i)
Since B + P + T = 22
-B - P + 22 = T ... (ii)
So, from equations (i) and (ii),
24 = 2T
So, T = 12
Now, 12 + A = 14 = F and Q - 12 = 7
So, Q = 19
Therefore, S = 5

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Test Level 1: Quadratic Equations & Linear Equations - 1 - Question 3

Which of the following is the correct value of x for the equation 

Detailed Solution for Test Level 1: Quadratic Equations & Linear Equations - 1 - Question 3


Step 1:
Multiplying both sides by 3, we get:

x + 368 = 93x
Step 2:
Subtracting 93x from both sides, we get:
x + 368 - 93x = 93x - 93x
x + 368 - 93x = 0
-92x + 368 = 0
Step 3:
Subtracting 368 from both sides, we get:
-92x + 368 - 368 = 0 - 368
-92x = -368
Step 4:
Dividing both sides by -92, we get:

x = 4

Test Level 1: Quadratic Equations & Linear Equations - 1 - Question 4

Let x and y be real numbers, such that (x2 - y2)(x2 - 2xy + y2) = 3 and x - y = 1. What is the value of xy?

Detailed Solution for Test Level 1: Quadratic Equations & Linear Equations - 1 - Question 4

From the given information,
3 = (x2 - y2)(x2 - 2xy + y2)
3 = (x - y)(x + y)(x - y)2 = x + y (Because, it is given that x - y = 1)
Hence, 2x = (x + y) + (x - y) = 3 + 1 = 4 and 2y = (x + y) - (x - y) = 3 - 1 = 2
x = 2 and y = 1, so xy = 2.

Test Level 1: Quadratic Equations & Linear Equations - 1 - Question 5

If p, q and r are the roots of the cubic equation x3 - 3x2 + 5x + k = 2 and pqr = 1, then find the value of k.

Detailed Solution for Test Level 1: Quadratic Equations & Linear Equations - 1 - Question 5

If p, q and r are the roots of the cubic equation x3 - 3x2 + 5x + k - 2 = 0, then the product of the roots are as follows.
pqr = 2 - k (Using the formula of product of roots of a cubic equation)
k = 2 - 1
k = 1

Test Level 1: Quadratic Equations & Linear Equations - 1 - Question 6

Find the value of x3 + y3 + z3 - 3xyz when x + y + z = 9 and xy + yz + zx = 11.

Detailed Solution for Test Level 1: Quadratic Equations & Linear Equations - 1 - Question 6

(x3 + y3 + z3 - 3xyz) = (x + y + z)(x2 + y2 + z2 - xy - yz - zx)
= (x + y + z)[(x + y + z)2 - 3(xy + yz + zx)]
= 9(92 - 3 x 11) = 9(81 - 33) = 432

Test Level 1: Quadratic Equations & Linear Equations - 1 - Question 7

If (a2 + b2)3 = (a3 + b3)2, then find the value of 

Detailed Solution for Test Level 1: Quadratic Equations & Linear Equations - 1 - Question 7


Test Level 1: Quadratic Equations & Linear Equations - 1 - Question 8

1/2 (a + b + c){(a - b)2 + (b - c)2 + (c - a)2} is equal to

Detailed Solution for Test Level 1: Quadratic Equations & Linear Equations - 1 - Question 8

1/2 (a + b + c){(a - b)2 + (b - c)2 + (c - a)2}
1/2 (a + b + c){a2 + b2 – 2ab + b2 + c2 – 2bc + c2 + a2 – 2ca}
= (a + b + c)(a2 + b2 + c2 - ab - bc - ca)
= a3 + b3 + c3 - 3abc

Test Level 1: Quadratic Equations & Linear Equations - 1 - Question 9

What is the value of y if  1/x + 2/y = 3/z ?

Detailed Solution for Test Level 1: Quadratic Equations & Linear Equations - 1 - Question 9

Solving, we get:

Test Level 1: Quadratic Equations & Linear Equations - 1 - Question 10

If x = ay and y = bx, where the values of x and y cannot be zero, find the value of 

Detailed Solution for Test Level 1: Quadratic Equations & Linear Equations - 1 - Question 10

x = ay ... (i)
y = bx ... (ii)
Put the value of x in (ii).
y = b(ay)
⇒ a = 1/b  ...(iii)
According to the question,

Put the value of 'a' from (iii).

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