List of top Mathematics Questions

Let $f(x)= \begin{vmatrix} 1+\sin ^2 x & \cos ^2 x & \sin 2 x \\ \sin ^2 x & 1+\cos ^2 x & \sin 2 x \\ \sin ^2 x & \cos ^2 x & 1+\sin 2 x\end{vmatrix}, x \in\left[\frac{\pi}{6}, \frac{\pi}{3}\right] $ If $\alpha$ and $\beta$ respectively are the maximum and the minimum values of $f$, then

Let $y=y(x)$ be the solution of the differential equation $\frac{d y}{d x}=\frac{4 y^3+2 y x^2}{3 x y^2+x^3}, y(1)=1$ If for some $n \in N , y (2) \in[ n -1, n )$, then $n$ is equal to ______

The value of $\displaystyle\sum_{r=0}^{22}{ }^{22} C_r{ }^{23} C_r$ is

Given the following sets:
𝐴 = {2, 4, 6, 8, 10, 12}
𝐵 = {8, 10, 12, 14, 16, 18}
𝐶 = {7, 8, 9, 10 11, 12, 13}
(𝐴 ∩ 𝐵) ∪ (𝐵 ∩ 𝐶) is

The order of differential equation $\frac{𝑑^3𝑦}{dx^3} + 2 \frac{𝑑^2𝑦}{dx^2} − 3 \frac{𝑑𝑦}{dx} + 6𝑥^4𝑦$ = 0 is _______.

The value of $\lim_{𝑥→−3}\frac{(2𝑥+6)}{(𝑥+3)}$ is

Given data consists of distinct values of $𝑥_𝑖$ occurring with frequencies $𝑓_𝑖.$ The mean value for the data is _____. (rounded off to one decimal place)

$X_i$	5	6	8	10
$F_i$	8	10	10	12

Let {𝑎_𝑛 }_𝑛≥1 be a sequence such that 𝑎₁=1 and 4𝑎_𝑛+1=\(\sqrt{45 + 16𝑎_𝑛} ,𝑛\)=1, 2, 3, … . Then, which one of the following statements is TRUE?

Let the series 𝑆 and 𝑇 be defined by
\(∑^∞_{n-0}\frac{ 2 ⋅ 5 ⋅ 8 ⋯ (3𝑛 + 2) }{1 ⋅ 5 ⋅ 9 ⋯ (4𝑛 + 1)}\) and \(∑^∞ _{n=1}(1 + \frac{1 }{𝑛} ) ^{−𝑛^2}\)
respectively. Then, which one of the following statements is TRUE?

The volume of the region 𝑅={(𝑥, 𝑦, 𝑧) ∈ ℝ × ℝ × ℝ ∶ 𝑥 ² + 𝑦 ² ≤ 4, 0 ≤ 𝑧 ≤ 4 − 𝑦} is

For real constants 𝛼 and 𝛽, suppose that the system of linear equations
𝑥+2𝑦+3𝑧=6; x+ 𝑦 +𝛼𝑧 = 3; 2𝑦+𝑧=𝛽, 
has infinitely many solutions. Then, the value of 4𝛼 + 3𝛽 equals

Let M= \(\begin{pmatrix} 1 & -1 & 0\\ -a & 2 & -1 \\0&-1&1\end{pmatrix}\)If a non-zero vector 𝑋=(𝑥, 𝑦, 𝑧)^T∈ℝ³ satisfies 𝑀⁶𝑋=𝑋, then a subspace of ℝ3 that contains the vector 𝑋 is

Let 𝑀=𝑀₁𝑀₂, where 𝑀₁ and 𝑀₂ are two 3×3 distinct matrices. Consider the following two statements:
(I) The rows of 𝑀 are linear combinations of rows of 𝑀₂. 
(II) The columns of 𝑀 are linear combinations of columns of 𝑀₁. 
Then,

Among all the points on the sphere 𝑥²+𝑦²+𝑧²=24, the point (𝛼, 𝛽, 𝛾) is closest to the point (1, 2,−1). Then, the value of 𝛼+𝛽+𝛾 equals

Let 𝑀 be a 3×3 real matrix. If 𝑃=𝑀+𝑀^𝑇 and 𝑄=𝑀-𝑀^𝑇, then which of the following statements is/are always TRUE?

For the function 𝑓∶ℝ×ℝ→ℝ defined by 𝑓(𝑥, 𝑦)=2𝑥²-𝑥𝑦-3𝑦 ²-3𝑥+7𝑦 , the point (1,1) is

Let 𝑃 be a 3×3 matrix having the eigenvalues 1,1 and 2. Let (1,−1,2) ^𝑇 be the only linearly independent eigenvector corresponding to the eigenvalue 1. If the adjoint of the matrix 2𝑃 is denoted by 𝑄, then which of the following statements is/are TRUE?

Let 𝑓:ℝ×ℝ→ℝ be a function defined by
\(f(x,y)=\begin{cases} \frac{xy(x+y}{x^2+y^2} & \quad \text{if }(x,y)≠(0,0),\\  0, & \quad if\,(x,y)=(0,0). \end{cases}\)
Then, which of the following statements is/are TRUE?

Which of the following is/are TRUE ?

Let 𝑓: ℝ→ ℝ be a function defined by 𝑓(𝑥)=𝑥²−𝑥, 𝑥 ∈ ℝ. Let 𝑔: ℝ→ℝ be a twice differentiable function such that 𝑔(𝑥)=0 has exactly three distinct roots in the open interval (0, 1). Let ℎ(𝑥) = 𝑓(𝑥)𝑔(𝑥), 𝑥 ∈ ℝ, and ℎ′′ be the second order derivative of the function ℎ. If 𝑛 is the number of roots of ℎ′′(𝑥)=0 in (0, 1), then the minimum possible value of 𝑛 equals ________

Let 𝑋1, 𝑋2,be a sequence of 𝑖. 𝑖. 𝑑. random variables, each having the probability density function
\(f(x) =\begin{cases}   \frac{x^2r^{-x}}{2}       & \quad \text{if }x ≥0,\\  0, & \quad Otherwise \end{cases}\)
For some real constants 𝛽, 𝛾 and 𝑘, suppose that 
\(lim_{→∞} ( \frac{1}{ 𝑛} ∑^n_{i=1}𝑋_𝑖≤ 𝑥) \)=\(\begin{cases}   0     & \quad if\,x< β.\\  kx, & \quad if\,β≤x≤y.\\  ky, & \quad if\,x>y.\end{cases}\)
Then, the value of 2𝛽 + 3𝛾 + 6𝑘 equals _______

Let 𝛼 and 𝛽 be real constants such that\(lim_{x-0^+}\frac{∫^x_0(\frac{αt^2}{1+t^4})dt}{βx-sin\,x}\)=1 .
Then, the value of 𝛼 + 𝛽 equals ________

For 𝑥∈ℝ, the curve 𝑦=𝑥² intersects the curve 𝑦=𝑥 sin 𝑥+cos 𝑥 at exactly 𝑛 points. Then, 𝑛 equals

Let
\(𝑆_𝑛 =∑_{k=1}^n\frac{1+k2^k}{4^{k-1}},n=1,2,...\)
Then, lim𝑛_→∞ 𝑆_𝑛 equals (round off to two decimal places)