IIM Bangalore UG 13th Dec 2025

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

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)$$\dfrac{A}{B} = \dfrac{B}{C}$$
Select the correct option from below:

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 equation $$a^{2}x^{2} + 12a - 7 = 0$$ is $$64\left(p^{6}+c^{12}\right)$$.
What is the value of a ?

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

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

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

Let $$f : \left(0, \frac{6}{5}\right) \rightarrow R$$ & $$g : \left(0, \frac{6}{5}\right) \rightarrow R$$ be functions defined by $$f\left(x\right) = \left[x^{2}\right] \text{and} g\left(x\right) = \left(|x-1| + |x-2|\right)f\left(x\right)$$ Here $$[a] = \text{the highest integer} \leq a$$. Then

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

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

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: