UGAT Bangalore 2025 · Sample Paper

The value of an item depreciates $x\%$ per year. The present value of the item is $₹ 109350$. Its value after 3 years will be $₹ 79716.15$. Two years ago from now, its value, in rupees, was

The slope of the tangent to the curve $y = x^2 - 4\cos x$, $x \in [0, \pi]$ is maximum when $x$ is equal to

Let $f(x) = \tan^{-1}\left(\sqrt{\frac{1-\cos x}{1+\cos x}}\right)$; $x \in (0,\pi)$. A normal to $y = f(x)$ at $x = \frac{\pi}{3}$ passes through the point:

Let $A \times B$ denote the Cartesian product of sets $A$ and $B$. If the cardinality of $A \cap B$ is $10$, then the cardinality of $(A \times B) \cap (B \times A)$ is:

Let $f(x) = \left|\frac{x}{2}\right|$ and $g(x) = \sqrt{3}\cos x + \sin x - 1$, for all real numbers $x$. If $[ \ \ ]$ denotes the greatest integer function, then the range of the function $[f \circ g]$ is:

The perimeters of a regular hexagon and a regular octagon are equal. Find the ratio of the area of the octagon to that of the hexagon. [Use $\tan 67.5° = (\sqrt{2} + 1)$]

The number of girls in XI in the school S3 is approximately ______% of the number of boys in IX in the school S2.

In class XII, the number of girls in the school ______ is more than the combined number of girls in the school ______ and ______.

For all schools, the correct arrangement of classes in ascending order of number of boys is ______.

On arranging schools in descending order of total number of girls in all classes, the correct order is ______.

A shopkeeper marks the price of an item $50\%$ above its cost. During a festival, he offers a sequence of discounts — first a $20\%$ discount, then an additional $10\%$ off on the reduced price, and finally gives a further rebate of $₹30$ on the total bill for 10 identical items. If after all these reductions, he still makes a profit of $5\%$ on selling the 10 items, what is the cost price of one item?

A can complete a piece of work alone in 120 hours, B can complete it alone in 180 hours, while C can complete it alone in 216 hours. In every cycle of three days, only A works on Day 1, only B works on Day 2, and only C works on Day 3. In the $n^{th}$ cycle, each of A, B, and C work for $n$ hours. An exception is made on the last day, if the work is completed mid cycle. What is the total number of days needed for the work to be completed? (Note: if on the final day of work, the person working is supposed to work for p hours, but the work gets over after just q hours, that should be considered as only $\frac{q}{p}$ day.)

Which of the following relations is an equivalence relation?

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The total number of students in the college is 43200 and 15% of all female students study in the 4th year. How many female students study in the 4th year?

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The total number of students in the college is 39600 and 5940 of the students in the 1st year are females. What percentage of all female students study in the 1st year?

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There are 12960 students who are studying vocational courses in the college. A quarter of all female students study in the 2nd year of college. How many male students study in the 2nd year in the college?

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The total number of students in the 3rd year of college is 11520. If 6% of all female students in the college are 3rd year science students, how many male students in the college study science in the 3rd year?

There are 50 distinct lines $L_1, L_2, ..., L_{50}$. All lines $L_{4n}$, where n a positive integer, are parallel to each other. All lines $L_{4n-3}$, where n is a positive integer, pass through a point P. The maximum possible number of points of intersection of pairs of lines in the set $\{L_1, L_2, ...\}$ is

We have a bag of tokens. Each token has a label $m$ chosen from the squares of the first $n$ natural numbers, with frequency $\sqrt{m}$. What is the arithmetic mean of the labels?

P, Q, and R enter into a partnership. P contributes one-fifth of the whole capital while Q contributes twice as much as P and R together contribute. If the profit at the end of the year is ₹555540, what is the difference (in ₹) of Q's share and the combined share of P and R in the profit?

A ten-digit number is selected at random from the set of all possible ten-digit numbers. The probability that the sum of the ten digits in this selected number is odd, is nearest to

Solve the inequality:

$|2x + 5| \leq 3|2x - 1|$

Yaseen was travelling from City A to City B. He travelled the first one-fifth of the distance at 10 km/h, and the next three-twentieth of the distance at 15 km/h. Of the remaining distance, he travelled one part at 20 km/h and the rest of the distance at 12 km/h. If the average speed of Yaseen for the entire trip was 16 km/h, what part of the entire trip did Yaseen cover at 20 km/h?

The integer $n$ for which $\lim_{x \to 0} \frac{(\sin x - x)\tan x}{x^n}$ is a finite non zero number, is

Arun scored 25, 13, 12 and 22 points in the $6^{\text{th}}$, $7^{\text{th}}$, $8^{\text{th}}$ and $9^{\text{th}}$ basketball games, respectively. His points-per game average was higher after nine games than it was after the first five games. If his average after ten games was greater than 19, the least number of points that he could have scored in the $10^{\text{th}}$ game, was

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

Three decorative lights are hung equidistantly on the edge of a circular swimming pool of radius 15 m. These lights are connected with each other by light wires forming a triangle. What is the perpendicular distance of the wire connecting any two lights from the center of the swimming pool?

An article costing ₹80 was marked 25% above the cost price. After offering a discount of X%, the customer now pays ₹81. If the marked price is increased twice successively by X% and the article is sold at the new marked price, then the profit would be what percent more than the profit when the selling price was ₹81?

In a geometric progression, the first term is $a$ (with $a \ne 0$), the common ratio is $r$ (with $r < 1$), and the sum to infinity is $S$. If the second term of this geometric progression is equal to the negative of the sum to infinity, what is the value of $[r]$, where $[r]$ denotes the greatest integer function?

Let $A = [a_{ij}]$ be a 3 × 3 matrix such that $a_{ij} = i + j$, for all $i, j$. If $A^3 + kA^2 + lA + mI = 0$, then $k + l + m$ is equal to