If $x,y$ are real numbers such that $2^{x+\frac{1}{2}}\times 4^{y-\frac{5}{6}}=3^{x-\frac{1}{2}}\times 9^{y-\frac{1}{3}}$ then which of the following is true?

1

The eccentricity of an ellipse whose center is at the origin is $\frac{1}{2}$. If one of its directrices is $x=-4$, fin…

Topic: NIMCET 2026

2

A triangle has a vertex at (1, 2) and the mid points of the two sides through it are (-1, 1) and (2, 3). Then the area …

Topic: NIMCET 2026

3

If $f:[0,\infty)\to R$ is defined by $f(x)=\frac{x^2-1}{x^2+1}$, find the value of: $\int_{-1}^{1} f^{-1}(y)dy$

Topic: NIMCET 2026

4

A die is rolled twice independently. What is the probability that either the first die shows a number no less than 4 or…

Topic: NIMCET 2026

5

Evaluate the following definite integral:$\int_{0}^{\frac{\pi}{4}} \frac{dx}{\cos^4 x}$

Topic: NIMCET 2026

6

Find the area of the triangle formed in the right half-plane by the lines $x-y=0$ and $x+y=0$, and a tangent to the hyp…

Topic: NIMCET 2026

7

Given that $\cos 6x=a\cos^6x+b\cos^4x+c\cos^2x+d$ for any real number $x$, find the value of $a+b+c$.

Topic: NIMCET 2026

8

$\lim_{x\to 0}\frac{x^2+2\cos x-2}{x\sin^3 x}$

Topic: NIMCET 2026

9

Let $A_k$ be the arithmetic mean of squares of $k$ natural numbers. If $\sum_{k=1}^{n}(6A_k-3k)=31$, find the value of …

Topic: NIMCET 2026

10

The arithmetic mean of two numbers a and b is 5, and the harmonic mean is 3.2. Find the numbers a and b.

Topic: NIMCET 2026

1

From the top of a viewpoint at a height of 80 m, the angles of depression of the top and bottom of a flag standing on t…

Topic: NIMCET 2026

2

The set $ (A \Delta B) \cap C $ is not equal to:

Topic: NIMCET 2026

3

Define a relation $\sim$ on the set $\{1,2,3,4,5,6,7,8,9,10\}$ by $a \sim b$ if $a - 2b$ is divisible by $3$. Then:

Topic: NIMCET 2026

4

Let $n$ be the number of injective functions $f:{1,2,3,4}\to {1,2,3,4,5,6,7,8}$ sending an even number to an even numbe…

Topic: NIMCET 2026

5

If there are $5$ sets represented via a Venn diagram, the number of regions that arise in the Venn diagram equals:

Topic: NIMCET 2026

6

Let $R\subset \mathbb{N}\times \mathbb{N}$. Which of the following statement is necessarily true?

Topic: NIMCET 2026

7

The number of triples of sets $(A,B,C)$ with $A,B,C\subseteq \{1,\ldots,n\}$ such that $(A\cap B)\subseteq C\subseteq (…

Topic: NIMCET 2026

8

The number of ordered tuples $(p,q,r)$ in the truth table for which the statement $(\neg p\vee q)\Rightarrow r$ is true…

Topic: NIMCET 2026

9

In a tuition batch of two students, the probability that $X$ will pass the exam is $\frac{2}{5}$ and that of $Y$ is $\f…

Topic: NIMCET 2026

10

Let $A_1,A_2,A_3$ are three events in a sample space with the condition $A_1\cap A_2\neq \phi$, then always:

Topic: NIMCET 2026