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

1

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

2

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

Topic: NIMCET 2026

3

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

4

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

5

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

Topic: NIMCET 2026

6

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

7

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

8

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

9

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

10

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}…

Topic: NIMCET 2026