List of top Mathematics Questions

If A and B are two events such that \(P(A)=\frac1{3},P(B)=\frac1{5}\)  and \(P(A∪B)=\frac1{2}\) then \(P(\frac{A}{B'}) + P(\frac{B}{A'})\) is equal to

A common tangent T to the curves
\(C_1:\frac{x^2}{4}+\frac{y^2}{9} = 1\)
and
\(C_2:\frac{x^2}{4^2}\frac{-y^2}{143} = 1\)
does not pass through the fourth quadrant. If T touches C1 at (x1, y1) and C2 at (x2, y2), then |2x1 + x2| is equal to ______.

2sin(\(\frac{\pi}{22}\))sin(\(\frac{3\pi}{22}\))sin(\(\frac{5\pi}{22}\))sin(\(\frac{7\pi}{22}\))sin(\(\frac{9\pi}{22}\)) is equal to

Let \(\vec a\)=\(\hat i\)−\(\hat j\)+2\(\hat k\) and \(\vec b\) be a vector such that 
\(\vec a\)×\(\vec b\)=2\(\hat i\)−\(\hat k\) and \(\vec a\)⋅\(\vec b\)=3. 
Then the projection of  \(\vec b\) on the vector \(\vec a\)−\(\vec b\) is :
Let \(\overset{→}a = \hat{i}- \hat{J}+2\hat{K}\) and \(\overset{→}b\) be a vector such that \(\overset{→}a×\overset{→}b=2\hat{i}−\hat{k} \) and \(\overset{→}a⋅\overset{→}b=3\). Then the projection of \(\overset{→}b\) on the vector \(\overset{→}a-\overset{→}b\) is :

For k ∈ R, let the solution of the equation
\(\cos\left(\sin^{-1}\left(x \cot\left(\tan^{-1}\left(\cos\left(\sin^{-1}\right)\right)\right)\right)\right) = k, \quad 0 < |x| < \frac{1}{\sqrt{2}}\)
Inverse trigonometric functions take only principal values. If the solutions of the equation x2 – bx – 5 = 0 are
\(\frac{1}{α^2}+\frac{1}{β^2} \)and \(\frac{α}{β}\)
, then b/k2 is equal to_____.

The mean and variance of 10 observation were calculated as 15 and 15 respectively by a student who took by mistake 25 instead of 15 for one observation. Then, the correct standard deviation is ________.

Let the line
\(\frac{x - 3}{7} = \frac{y - 2}{-1} = \frac{z - 3}{-4}\)
intersect the plane containing the lines
\(\frac{x - 4}{1} = \frac{y + 1}{-2} = \frac{z}{1}\) and \(4ax-y+5z-7a = 0 = 2x-5y-z-3, a∈R\)
at the point P(α, β, γ). Then the value of α + β + γ equals _____.

An ellipse
\(E:\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)
passes through the vertices of the hyperbola
\(H:\frac{x^2}{49} - \frac{y^2}{64} = -1\)
Let the major and minor axes of the ellipse E coincide with the transverse and conjugate axes of the hyperbola H, respectively. Let the product of the eccentricities of E and H be 1/2. If the length of the latus rectum of the ellipse E, then the value of 113l is equal to _____.

Let y = y(x) be the solution curve of the differential equation
\(\sin(2x^2) \log_e(\tan(x^2)) \,dy + (4xy - 4\sqrt{2}x\sin(x^2 - \frac{\pi}{4})) \,dx = 0, \quad 0 < x < \sqrt{\frac{\pi}{2}}\)
which passes through the point \((\sqrt{\frac{π}{6}},1)\). Then \(|y(\sqrt{\frac{π}{3}})|\)
is equal to _______.

Let M and N be the number of points on the curve y5 – 9xy + 2x = 0, where the tangents to the curve are parallel to x-axis and y-axis, respectively. Then the value of M + N equals ________.

\(\begin{array}{l} \frac{2^3-1^3}{1\times7}+\frac{4^3-3^3+2^2-1^3}{2\times 11}+\frac{6^3-5^3+4^3-3^3+2^3-1^3}{3\times 15}+\cdots+\frac{30^3-29^3+28^3-27^3+\cdots+2^3-1^3}{15\times63}\end{array}\)

is equal to _______.

If \(z = 2 + 3i\), then \(z^5 + (\overline{z})^5\) is equal to:
Let A and B be two \(3 × 3\) non-zero real matrices such that AB is a zero matrix. Then
If \(\frac{1}{(20-a)(40-a)} + \frac{1}{(40-a)(60-a)} + \ldots + \frac{1}{(180-a)(200-a)} = \frac{1}{256}\), then the maximum value of a is :

Let
\(f(x) = 2x^2 - x - 1\ and\ S = \{ n \in \mathbb{Z} : |f(n)| \leq 800 \}\)
Then, the value of ∑n∈S f(n) is equal to ________.

For the curve
\(\begin{array}{l}C : \left(x^2 + y^2 – 3\right) + \left(x^2 – y^2 – 1\right)^5 = 0, \end{array}\)
\(\begin{array}{l}\text{the value of}\ 3y’ – y^3y”, \text{at the point}\end{array}\)
\((α, α), α > 0\), on C is equal to ________.
Let S be the set containing all 3 × 3 matrices with entries from {-1, 0, 1}. The total number of matrices A ∈ S such that the sum of all the diagonal elements of AT A is 6 is ________.
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 ________.
Which of the following matrices can NOT be obtained from the matrix \(\begin{bmatrix} -1 &2 \\  1 & -1 \end{bmatrix}\) by a single elementary row operation?

Let
\(S = \left\{z∈C : z^2+\overline{z} = 0 \right\}\). Then \(∑_{z∈S}(Re(z)+Im(z))\)
is equal to____.

If \(\lim_{{x \to 0}} \frac{\alpha e^x + \beta e^{-x} + \gamma \sin x}{x \sin^2 x} = \frac{2}{3}\), where \(\alpha, \beta, \gamma \in \mathbb{R}\) then which of the following is NOT correct?
The integral \(\int_0^{\frac{\pi}{2}} \frac{1}{3 + 2\sin x + \cos x} \, dx\) is equal to

If the system of equations 
$ x + y + z = 6 $, 
$ 2x + 5y + \alpha z = \beta $, 
$ x + 2y + 3z = 14 $ 
has infinitely many solutions, then $ \alpha + \beta $ is equal to:

Let
\(\begin{array}{l}f\left(x\right) = min \{\left[x – 1\right], \left[x – 2\right], …., \left[x – 10\right]\}\end{array}\)
where [t] denotes the greatest integer ≤t. Then
\(\begin{array}{l} \displaystyle\int\limits_{0}^{10}f\left(x\right)dx+\displaystyle\int\limits_{0}^{10}\left(f\left(x\right)\right)^2dx+\displaystyle\int\limits_{0}^{10}\left|f\left(x\right)\right|dx\end{array}\)
is equal to ________.