Mathematics GATE previous year questions with answer
Ques 40 Gate 2016 Set-2
Suppose that a shop has an equal number of LED bulbs of two different types. The probability of an LED bulb lasting more than 100 hours given that it is of Type 1 is 0.7, and given that it is of Type 2 is 0.4. The probability that an LED bulb chosen uniformly at random lasts more than 100 hours is_________.
Ques 41 Gate 2016 Set-2
Let f (x) be a polynomial and g(x) = f^^(x) be its derivative. If the degree of (f(x) + f(−x)) is 10, then the degree of (g(x) − g(−x)) is ___________ .
Ques 42 Gate 2016 Set-2
The value of the expression 1399(mod 17), in the range 0 to 16, is
PROCESSED
Ques 43 Gate 2016 Set-2
Consider the systems, each consisting of m linear equations in n variables.
I. If m < n, then all such systems have a solution
II. If m > n, then none of these systems has a solution
III. If m = n, then there exists a system which has a solution
Which one of the following is CORRECT?
Given:
A system of m linear equations in n variables.
Statement I: If m < n, then all such systems have a solution.
Statement II: If m > n, then none of these systems has a solution.
Statement III: If m = n, then there exists a system which has a solution.
To find:
The correct statement(s) among I, II, and III.
Solution:
Let me analyze each statement one by one.
First, checking statement I: "If m < n, then all such systems have a solution."
Let me take a counterexample with m = 2 equations and n = 3 variables:
x + y + z = 1
x + y + z = 2
Here, the two parallel planes never intersect, so there is no solution.
Since an inconsistent underdetermined system can exist, not all such systems have a solution.
So statement I is FALSE.
Now, checking statement II: "If m > n, then none of these systems has a solution."
Let me test an overdetermined system where m = 3 and n = 2:
x + y = 2
2x + 2y = 4
3x + 3y = 6
Clearly, this has infinitely many solutions. Even a system like x = 1, y = 1, x + y = 2 has a unique solution.
So overdetermined systems can have solutions.
Thus, statement II is FALSE.
Now, looking at statement III: "If m = n, then there exists a system which has a solution."
This statement just says "there exists", so I only need to show one example.
Let m = n and choose the identity matrix for coefficients:
In X = B
Since det(In) = 1 ≠ 0, this system always has a unique solution.
So statement III is definitely TRUE.
Therefore, only statement III is correct.
Answer:
(C) Only III is true
Ques 44 Gate 2016 Set-1
Consider the following experiment.
Step 1. Flip a fair coin twice.
Step 2. If the outcomes are (TAILS, HEADS) then output Y and stop.
Step 3. If the outcomes are either (HEADS, HEAD) or (HEADS, TAILS), then output N and stop.
Step 4. If the outcomes are (TAILS, TAILS), then go to Step 1.
The probability that the output of the experiment is Y is (up to two decimal places).
Ques 45 Gate 2016 Set-1
A function f : N+ → N+, defined on the set of positive integers N+, satisfies the following
properties:
f(n) = f(n/2) if nis even
f(n) = f(n+5) if nis odd
Let R = {i|∃ j : f(j) = i} be the set of distinct values that f takes. The maximum possible size
of R is .
Ques 46 Gate 2016 Set-1
The coefficient of x12 in (x3 + x4 + x5 + x6 + ...)3 is_________
Ques 47 Gate 2016 Set-1
Two eigenvalues of a 3 x 3 real matrix P are (2 + √ -1) and 3. The determinant of P is _____
Ques 48 Gate 2016 Set-1
A probability density function on the interval [a, 1] is given by 1 / x2 and outside this interval the value of the function is zero. The value of a is :
Ques 49 Gate 2016 Set-1
If f(x) = 2x7 + 3x - 5 Which of the following is a factor of f(x)?
Given:
f(x) = 2x7 + 3x - 5
To find:
Which of the given options is a factor of f(x).
Solution:
According to the factor theorem, (x - a) is a factor of f(x) if f(a) = 0.
Let me quickly test the simplest linear factors from the options.
Let's check option (B) first, which gives x = 1.
Now substituting x = 1 into f(x):
f(1) = 2(1)7 + 3(1) - 5
So we get:
f(1) = 2 + 3 - 5 = 0
Since f(1) = 0, (x - 1) is definitely a factor of f(x).
No need to check the rest, option B works directly.
Answer: (B) (x - 1)
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