List of top Mathematics Questions

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 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 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

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

Which of the following is/are TRUE ?

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?

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

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)

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 ________

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

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

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

The order of differential equation $\frac{𝑑^3𝑦}{dx^3} + 2 \frac{𝑑^2𝑦}{dx^2} − 3 \frac{𝑑𝑦}{dx} + 6𝑥^4𝑦$ = 0 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 number of solutions of cos4θ – 2cos2θ + 3sin6θ + 1 = 0 in [0,2π] is

The points of intersection of the line $a x+b y=0,(a \neq b)$ and the circle $x^2+y^2-2 x=0$ are $A (\alpha, 0)$ and $B (1, \beta)$ The image of the circle with $AB$ as a diameter in the line $x+y+2=0$ is :

If the circles $x^2+y^2+6 x+8 y+16=0$ and $x^2+y^2+2(3-\sqrt{3}) x+x+2(4-\sqrt{6}) y$ $= k +6 \sqrt{3}+8 \sqrt{6}, k >0$ touch internally at the point $P(\alpha, \beta)$, then $(\alpha+\sqrt{3})^2+(\beta+\sqrt{6})^2$ is equal to _______

Let $S_1$ and $S_2$ be respectively the sets of all $a \in R -\{0\}$ for which the system of linear equations $ a x+2 a y-3 a z=1$ $ (2 a+1) x+(2 a+3) y+(a+1) z=2$ $(3 a+5) x+(a+5) y+(a+2) z=3$ has unique solution and infinitely many solutions Then

Let a smooth curve $y=f(x)$ be such that the slope of the tangent at any point $(x, y)$ on it is directly proportional to $\left(\frac{-y}{x}\right)$ If the curve passes through the point $(1,2)$ and $(8,1)$, then $\left|y\left(\frac{1}{8}\right)\right|$ is equal to

Let $x=2$ be a local minima of the function $f(x)=2 x^4-18 x^2+8 x+12$, $x \in(-4,4)$ If $M$ is local maximum value of the function $f$ in $(-4,4)$, then $M =$

The sum $1^2-2 \cdot 3^2+3 \cdot 5^2-4 \cdot 7^2+5 \cdot 9^2-\ldots+15 \cdot 29^2$ is_____