UGAT Bangalore 2025 · Actual Paper

Let $f: \mathbb{R} \to \mathbb{R}$ be a function such that

$f'(x) = (x - 2024)^3(x - 2025)(x - 2026)^2$ for all $x \in \mathbb{R}$.

Let $g: \mathbb{R} \to (0, \infty)$ be a function such that $g(x) = \sqrt{f(x)}$ for all $x \in \mathbb{R}$.

Then the number of points at which $g(x)$ has a local maximum is:

The centroid of a triangle with vertices $(2a,b),(-b,a)$ and $(b,3a)$ is $(4,11)$. The area of the triangle is ___ square units.

In how many ways can three letters (not necessarily distinct from one another) be selected from the letters of the word PARALLELOGRAMS?

25000 blood donors were registered with Red Cross in a city. The number of donors increased at the rate of 10% every 6 months. Find the total time (in years) at the end of which the total number of blood donors becomes 33275.

A sportsperson could be an expert in badminton and / or squash. Given that someone is a squash expert, the probability that they are expert at badminton as well is 0.8. There are twice as many badminton experts as there are squash experts. Given that Saina is a badminton expert, what is the likelihood that she is an expert in squash?

In 2010, the total spending per tourist in Region E was ₹6,000, and it increased by 10% in each successive year, while arrivals decreased as per the graph. What was the total spending (in ₹ crore) from Region E across 2010–2012?

Foreign visitors to Region B form 15%, 20% and 25% of the tourist visits during 2010, 2011 and 2012 respectively. Assume that each foreign tourist spends ₹ 7,000, while each domestic tourist spends ₹ 3,000. What is the total revenue (in ₹ crores) from tourist spends during 2010–2012?

In which region was the highest average tourist arrival recorded during 2010–2012?

In 2012, if 10% of visitors in Region E were foreign tourists, then how many domestic tourists (in thousands) visited Region E in that year?

Let $b$ and $c$ be two real numbers such that the numbers $3, b, c$ are in a geometric progression. For the matrix $A = \begin{bmatrix} 1 & 1 & 1 \\ 3 & b & c \\ 9 & b^2 & c^2 \end{bmatrix}$, if $\det[A] = 162$, then the value of $c$ is

Which of the following functions is a one-to-one and onto function from $\mathbb{R}$ to $\mathbb{R}$?

Let A, B, and C be the three mutually independent events with their probabilities equal to $x, y$ and $z$ respectively. Then the probability that exactly two of these events occur is

Let $f: (0, \frac{6}{5}) \to \mathbb{R}$ & $g: (0, \frac{6}{5}) \to \mathbb{R}$ be functions defined by

$f(x) = [x^2]$ and $g(x) = (|x - 1| + |x - 2|)f(x)$

Here $[a] =$ the highest integer $\leq a$. Then

Two positive integers m and n satisfy HCF (m, n) = 14 and LCM (m, n) = 168. If m < n and m divides 84, what is the maximum possible value of m?

Assume the population of city R was 96000 in 2016. Then, in percentage terms, by how much was it more than the male population of city T?

Assume the population of Panchagarh in 2016 is 450000. Then the combined non-male population of cities P, Q and T is approximately what percent more/less than the combined male populations of cities R, S and T?

Assume the total population of Panchagarh state in 2016 is 350000. Then find the average female population of cities P, Q, and S in 2016.

Assume the population of Panchagarh in 2016 was 1.5 lakh. In 2017, the population of city P increased by 15%, while that of city S increased by 20%. What is the ratio of the male population of city P to that of city S in 2017?

If $2y + z > 0$, $2z > y$, and $z < 3$, find the range of possible values of $(y + z)$.

Two rectangles are drawn on line segments of fixed widths. When the heights are 7 m and 8 m respectively, their areas sum up to 156 m². But if these heights are 5 m and 7 m respectively, then their areas sum up to 123 m². Find the sum of the areas of the squares (in m²) drawn on the line segments.

In Saroja's sari shop, the marked price of each sari is x% above the cost of its procurement. She sells each sari after a discount of 25%. In addition, if someone buys a dozen saris, she offers an extra sari for free, and still makes a profit of 8% over their cost of procurement. If Saroja had sold the saris without discounting, but with the offer of an extra saree for every dozen sold, then her profit percentage would have been:

You are given three positive numbers such that

i) A is the sum of the first two numbers.

ii) B is the sum of the first two numbers taken away from the third number.

iii) C is the sum of all these numbers.

iv) $\frac{A}{B} = \frac{B}{C}$

Select the correct option from below:

Alice can complete a certain work in 24 days. Bob is twice as efficient as Alice. Both of them worked together for x days and stopped. The remaining work was completed by Carol working alone for (x+1) days. If Carol is 25 % less efficient than Bob, then the total number of days it took to complete the whole work was

A bag consists of tokens whose labels are selected from all of the first $n$ even natural numbers. The frequency of each label equals half the label itself. Then the variance of all the labels the bag is

Let $A = \{a_1, a_2, ..., a_i, ...\}$ be an arithmetic progression, and let $B = \{b_1, b_2, ..., b_i, ...\}$ be a geometric progression.

The common difference for A is 2.

The common ratio for B is 0.2, and $b_1 = 0.8$

The infinite sum of the products $a_i b_i$ is 1, where $i = 1, 2, 3, ...$

What is $a_i$?

Let A, B and C be three sets. Then $A \cup B - C$ is not equal to

The roots of the equation $\sqrt{2}x^2 - \frac{3}{\sqrt{2}}x + c = 0$ are $p$ and $2p$.

Let $a > 0$, and one root of the equation $a^2x^2 + 12a - 7 = 0$ is $64(p^6 + c^{12})$.

What is the value of $a$?

Let $h(x) = min[\sin x], [\cos x]]$, for all real numbers $x$. Let S be the set of points in $(0, \frac{\pi}{2})$ where $h(x)$ is not differentiable. Then the cardinality of S is:

Let $A = \begin{bmatrix} x & 1 \\ 0 & 1 \end{bmatrix}$ where $x$ is a real number, and $B = \begin{bmatrix} 2\sqrt{2} & 3 + \sqrt{2} \\ 0 & 1 \end{bmatrix}$. If $A^3 = B$, then $x$ is equal to

The proportions of gold in three alloys are 40%, 50% and 80% respectively. These alloys are mixed in certain proportions to obtain 30 kg of a new alloy, which has 55% gold. If the amount of the first alloy is 9 kg, what is the amount in kg of the third alloy used?