List of top Mathematics Questions asked in GATE XH-C1

If \( X \) is a continuous random variable whose probability density function is given by

\[ f_X(x) = \begin{cases} c x^3 + 0.25, & \text{for } 0 \leq x \leq 1,\ c \in \mathbb{R} \\ 0, & \text{elsewhere} \end{cases} \]

Then the value of \( c \) is ________________ (in integer).
If \( X \) is a continuous random variable whose probability density function is given by

\[ f_X(x) = \begin{cases} \dfrac{1}{x^2}, & \text{for } 1 < x < \infty \\ 0, & \text{elsewhere} \end{cases} \]

Then the median of \( X \) is ________________ (in integer).

Consider a lottery with three possible outcomes: 

The maximum amount that a risk-neutral person would be willing to pay to play the above lottery is INR ____________.

A coin has a true probability \( \mu \) of turning up Heads. This coin is tossed 100 times and shows up Heads 60 times. The following hypothesis is tested: \[ H_0: \mu = 0.5 ({Null Hypothesis}), H_1: \mu>0.5 ({Alternative Hypothesis}) \] Using the Central Limit Theorem, the \( p \)-value of the above test is ________ (round off to three decimal places). \[ {Hint: If \( Z \) is a random variable that follows a standard normal distribution, then } P(Z \leq 2) = 0.977. \]
Consider a square field ABCD. The diagonal AC is 50 meter. The cost of laying grass in the field is Rs. 5 per square-meter. The total cost for laying grass in the field ABCD is Rs. ________________ (rounded off to two decimal places).
In the following figure, four overlapping shapes (rectangle, triangle, circle, and hexagon) are given. The sum of the numbers which belong to only two overlapping shapes is ________.
\includegraphics[width=0.5\linewidth]{25.png}

In the following figure, four overlapping shapes (rectangle, triangle, circle, and hexagon) are given. The sum of the numbers which belong to only two overlapping shapes is ________

Given a series \( 5, 8, 11, 14, \dots \), if the \( n \)-th term of the given series is 320, then find \( n \) (where \( n \geq 1 \)):
If \( \times \) means \( + \), \( + \) means \( \div \), \( - \) means \( \times \), and \( \div \) means \( - \), then evaluate: 8 \( \times \) 7 \( - \) 8 \( + \) 40 \( \div \) 2.

The table shows the data of 450 candidates who appeared in the examination of three subjects – Social Science, Mathematics, and Science. How many candidates have passed in at least one subject? 

How many candidates have passed in at least one subject?

A coin has a true probability \( \mu \) of turning up Heads. This coin is tossed 100 times and shows up Heads 60 times. The following hypothesis is tested: 
\[ H_0: \mu = 0.5 \quad ({Null Hypothesis}), \quad H_1: \mu>0.5 \quad ({Alternative Hypothesis}) \] 
Using the Central Limit Theorem, the \( p \)-value of the above test is ________ (round off to three decimal places). 
Hint: If Z is a random variable that follows a standard normal distribution, then P (Z ≤ 2) = 0.977.

If \( X \) is a continuous random variable whose probability density function is given by

\[ f_X(x) = \begin{cases} cx^3 + 0.25, & \text{for } 0 \leq x \leq 1, \ c \in \mathbb{R} \\ 0, & \text{elsewhere} \end{cases} \]

Then the value of \( c \) is ________________ (in integer).
If \( X \) is a continuous random variable whose probability density function is given by

\[ f_X(x) = \begin{cases} \frac{1}{x^2}, & \text{for } 1 < x < \infty \\ 0, & \text{elsewhere} \end{cases} \]

Then the median of \( X \) is ________________ (in integer).

Consider a lottery with three possible outcomes: 

The maximum amount that a risk-neutral person would be willing to pay to play the above lottery is INR ____________.

Let \( f(x) = -3x^2(1 - x) - 3x(1 - x)^2 - (1 - x)^3 \). Then, \( \frac{df(x)}{dx} = \)

Consider a square field ABCD. The diagonal AC is 50 meter. The cost of laying grass in the field is Rs. 5 per square-meter. The total cost for laying grass in the field ABCD is Rs. ________________ (rounded off to two decimal places).

In the following figure, four overlapping shapes (rectangle, triangle, circle, and hexagon) are given. The sum of the numbers which belong to only two overlapping shapes is ________.


 

The table shows the data of 450 candidates who appeared in the examination of three subjects – Social Science, Mathematics, and Science. How many candidates have passed in at least one subject?


 

Which one of the following numbers is odd one out?
31541 42651 53791 64871 75981
Let \( f(x, y, z) = x^2 y^3 z \). Then, what is the value of \[ x \frac{\partial f}{\partial x}(x, y, z) + y \frac{\partial f}{\partial y}(x, y, z) + z \frac{\partial f}{\partial z}(x, y, z)? \]
Given a series \( 5, 8, 11, 14, \dots \), if the \( n \)-th term of the given series is 320, then find \( n \) (where \( n \geq 1 \)):
If \( \times \) means \( + \), \( + \) means \( \div \), \( - \) means \( \times \), and \( \div \) means \( - \), then evaluate: 8 \( \times \) 7 \( - \) 8 \( + \) 40 \( \div \) 2.
Which one of the following plots represents \( f(x) = -\frac{|x|}{x} \), where \( x \) is a non-zero real number?
Note: The figures shown are representative.