List of top Mathematics Questions

Let
A=\(\begin{bmatrix} 2 & -1 \\ 0 & 2 \end{bmatrix}\)
If B = I – 5C1(adjA) + 5C2(adjA)2 – …. – 5C5(adjA)5, then the sum of all elements of the matrix B is

If y = y(x) is the solution of the differential equation

\(2x^2\frac{dy}{dx}-2xy+3y^2=0\) such that \(y(e)=\frac{e}{3},\)

then y(1) is equal to

Let ƒ :R→R be a function defined by 
\(f(x) = \frac{2e^{2x}}{e^{2x} + e^x}\)
Then \(f\left(\frac{1}{100}\right) + f\left(\frac{2}{100}\right) + f\left(\frac{3}{100}\right) + \ldots + f\left(\frac{99}{100}\right)\)
 is equal to ________.

Let the common tangents to the curves 4(x2 + y2) = 9 and y2 = 4x intersect at the point Q. Let an ellipse, centered at the origin O, has lengths of semi-minor and semi-major axes equal to OQ and 6, respectively. If e and I respectively denote the eccentricity and the length of the latus rectum of this ellipse, then 
\(\frac{1}{e^2}\) is equal to

The area of the region 
\(\begin{array}{l} \left\{\left(x,y\right);\left|x-1\right|\leq y \leq \sqrt{5-x^2} \right\}\end{array}\)
is equal to
Let the solution curve of the differential equation \(x\frac{dy}{dx}-y=\sqrt{y^2+16x^2},\)  y(1) = 3 be y = y(x). Then y(2) is equal to 
Let H:  \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1, a > 0, b > 0,\) be a hyperbola such that the sum of lengths of the transverse and the conjugate axes is \(4(2\sqrt 2 + \sqrt {14}).\) If the eccentricity H is \(\frac{\sqrt {11}}{2}\), then the value of \(a^2 + b^2 \) is equal to ______.
Let A be a 3 × 3 invertible matrix. If |adj (24A)| = |adj (3 adj (2A))|, then |A|2 is equal to :

Let 
\(\stackrel{→}{a} = \hat{i} + \hat{j} + \hat{2k}, \stackrel{→}{b} = \hat{2i} - \hat{3j} + \hat{k}\)
and 
\(\stackrel{→}{c}= \hat{i} - \hat{j} + \hat{k}\)
be three given vectors.
Let \(\stackrel{→}{v}\) be a vector in the plane of \(\stackrel{→}{a}\) and \(\stackrel{→}{b}\) whose projection on \(\stackrel{→}{c}\) is \(\frac{2}{\sqrt3}\).
If \(\stackrel{→}{v}.\hat{j}\) = 7 , then \(\stackrel{→}{v}.(\hat{i}+\hat{k})\) is equal to :

If the two lines 
\(l1:\frac{(x−2)}{3}=\frac{(y+1)}{−2},z=2 \)
and
\( l2:\frac{(x−1)}{1}=\frac{(2y+3)}{α}=\frac{(z+5)}{2} \)
are perpendicular, then an angle between the lines l2 and 
\(l3:\frac{(1−x)}{3}=\frac{(2y−1)}{−4}=\frac{z}{4} \)
is

Let the image of the point P(1, 2, 3) in the line 
\(L:\frac{x−6}{3}=\frac{y−1}{2}=\frac{z−2}{3} \)
be Q. Let R (α, β, γ) be a point that divides internally the line segment PQ in the ratio 1 : 3. Then the value of 22(α + β + γ) is equal to ________.

A vector \(\vec{a}\)
is parallel to the line of intersection of the plane determined by the vectors
\(\hat{i},\hat{i}+\hat{j} \)and the plane determined by the vectors 
\(\hat{i}−\hat{j},\hat{i}+\hat{k}\). The obtuse angle between 
\(\vec{a}\) and the vector \(\vec{b}=\hat{i}−2\hat{j}+2\hat{k}\)
is

Numbers are to be formed between 1000 and 3000, which are divisible by 4, using the digits 1, 2, 3, 4, 5 and 6 without repetition of digits. Then the total number of such numbers is ______________.

Let the slope of the tangent to a curve y = f(x) at (x, y) be given by 2 tanx(cosx – y). If the curve passes through the point (π/4, 0) then the value of 
\(\int_{0}^{\frac{\pi}{2}} y \,dx\)
is equal to :

Consider a cuboid of sides 2x, 4x and 5x and a closed hemisphere of radius r. If the sum of their surface areas is a constant k, then the ratio x : r, for which the sum of their volumes is maximum, is

Let
\(M = \begin{bmatrix}   0 & -\alpha \\   \alpha & 0 \\ \end{bmatrix}\)
where α is a non-zero real number an
\(N =   \sum\limits_{k=1}^{49} M^{2k}. \) If \((I - M^2)N = -2I\)
then the positive integral value of α is ____ .

A circle of radius 2 unit passes through the vertex and the focus of the parabola y2 = 2x and touches the parabola \(y=\left (x−\frac{1}{4}\right)^2+α,\) where \(α > 0\). Then \((4α – 8)^2\) is equal to ___________.
The area enclosed by the curves 
\(\begin{array}{l} y=\text{log}_e\left(x+e^2\right),\ x=\text{log}_e\left(\frac{2}{y}\right)\text{ and }x=\text{ log }_e\ 2,\end{array}\)
above the line y = 1 is
Let α and β be the roots of the equation x2 + (2i – 1) = 0. Then, the value of |α2 + β2| is equal to
A water tank has the shape of a right circular cone with axis vertical and vertex downwards. Its semi-vertical angle is tan-1(3/4). Water is poured in it at a constant rate of 6 cubic meter per hour. The rate (in square meter per hour), at which the wet curved surface area of the tank is increasing, when the depth of water in the tank is 4 meters, is

The sum of the infinite series
\(1 + \frac{5}{6} + \frac{12}{6^2} + \frac{22}{6^3} + \frac{35}{6^4} +\frac{51}{6^5} + \frac{70}{6^6}+…..\)
is equal to

If the length of the latus rectum of the ellipse x2 + 4y2 + 2x + 8y – λ = 0 is 4, and l is the length of its major axis, then λ + l is equal to ________.
Let A be a matrix of order 2 × 2, whose entries are from the set {0, 1, 3, 4, 5}. If the sum of all the entries of A is a prime number p, \(2 < p < 8\), then the number of such matrices A is ___________.

Let
\(f(x) = \begin{vmatrix} a & -1 & 0\\ ax & a & -1\\ ax^2 & ax & a \end{vmatrix}\)
a ∈ R. Then the sum of the square of all the values of a, for which 2f′(10) –f′(5) + 100 = 0, is

Bag I contains 3 red, 4 black and 3 white balls and Bag II contains 2 red, 5 black and 2 white balls. One ball is transferred from Bag I to Bag II and then a ball is drawn from Bag II. The ball so drawn is found to be black in colour. Then the probability, that the transferred ball is red, is :