List of top Mathematics Questions

Let \((\alpha,\beta,\gamma)\) be the foot of the perpendicular from the point \((25,2,41)\) on the line \[ \frac{x-4}{3}=\frac{y+1}{7}=\frac{z-2}{3}. \] Then \(\alpha+\beta+\gamma\) is equal to:
Let the distance between the foci of an ellipse \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \quad (a>b) \] be \(4\) and the distance between its directrices be \(10\). Then the length of its latus rectum is:

Let \(\vec a = 2\hat i + 3\hat j + 5\hat k\), \(\vec b = \hat i - \hat j + 3\hat k\) and \(\vec c\) be a vector such that \[ \vec a \cdot \vec c = 104 \quad \text{and} \quad \vec a \times \vec c = \vec c \times \vec b. \] Then \(\vec b \cdot \vec c\) is equal to ______

Let the points \(A(a,-1,2)\), \(B(1,b,-4)\), \(C(-1,1,c)\) and \(D(1,-2,8)\) be the vertices of a parallelogram \(ABCD\). Then its area is equal to:
The slope of the curve \( y = -x^3 + 3x^2 + 8x - 20 \) is maximum at:
Which one of the following plots represents \( f(x) = -\frac{|x|}{x} \), where \( x \) is a non-zero real number?
Note: The figures shown are representative.
Let \( A = \begin{bmatrix} 1 & -2 & -1 \\ 0 & 4 & -1 \\ -3 & 2 & 1 \end{bmatrix}, B = \begin{bmatrix} -5 \\ -2 \end{bmatrix}, C = [9 \ \ 7], \) which of the following is defined?

For the matrix [A] given below, the transpose is __________. 

\[ A = \begin{bmatrix} 2 & 3 & 4 \\ 1 & 4 & 5 \\ 4 & 3 & 2 \end{bmatrix} \]

Integration of \(\ln(x)\) with \(x\), i.e. \(\int \ln(x)dx =\) __________.
 

If \( E \) and \( F \) are two independent events such that \( P(E) = \frac{2}{3} \), \( P(F) = \frac{3}{7} \), then \( P(E \,|\, F) \) is equal to:
If \( f(x) = |x| + |x - 1| \), then which of the following is correct?
Find the domain of the function $f(x) = \sin^{-1}(-x^2)$.
A cylindrical water container has developed a leak at the bottom. The water is leaking at the rate of 5 cm$^3$/s from the leak. If the radius of the container is 15 cm, find the rate at which the height of water is decreasing inside the container, when the height of water is 2 meters.
If \( A \) denotes the set of continuous functions and \( B \) denotes the set of differentiable functions, then which of the following depicts the correct relation between set \( A \) and \( B \)?
If $ f(x) = 2x^2 - 3x + 5 $, find $ f(3) $.

There is a circular park of diameter 65 m as shown in the following figure, where AB is a diameter. An entry gate is to be constructed at a point P on the boundary of the park such that distance of P from A is 35 m more than the distance of P from B. Find distance of point P from A and B respectively.

If \( x = a \left( \cos \theta + \log \tan \frac{\theta}{2} \right) \) and \( y = \sin \theta \), then find \( \frac{d^2y}{dx^2} \) at \( \theta = \frac{\pi}{4} \).
A bank offers loans to its customers on different types of interest rates
A bank offers loans to its customers on different types of interest rates namely, fixed rate, floating rate, and variable rate. From the past data with the bank, it is known that a customer avails loan on fixed rate, floating rate, or variable rate with probabilities 10%, 20%, and 70% respectively. A customer after availing a loan can pay the loan or default on loan repayment. The bank data suggests that the probability that a person defaults on loan after availing it at fixed rate, floating rate, and variable rate is 5%, 3%, and 1% respectively. Based on the above information, answer the following:
(i) What is the probability that a customer after availing the loan will default on the loan repayment?
(ii) A customer after availing the loan, defaults on loan repayment. What is the probability that he availed the loan at a variable rate of interest?
The integrating factor of the differential equation \( (x + 2y^3) \frac{dy}{dx} = 2y \) is:
$ \int \frac{e^{-x}}{16 + 9e^{-2x}} \, dx$ is equal to :

Consider the following statements: Statement I: \( 5 + 8 = 12 \) or 11 is a prime. Statement II: Sun is a planet or 9 is a prime. 
Which of the following is true? 
 

The value of \[ \int \sin(\log x) \, dx + \int \cos(\log x) \, dx \] is equal to
 

The value of \[ \lim_{x \to \infty} \left( e^x + e^{-x} - e^x \right) \] is equal to
 

Consider the function
\[ f(x) = \begin{cases} \frac{\log(1+3x) - \log(1-2x)}{x}, & x \neq 0 \\ k, & x = 0 \end{cases} \] If \( f \) is continuous at \( x = 0 \), then the value of \( k \) must be
The solution of the differential equation \( \frac{d^2 y}{dx^2} + y = 0 \) with \( y(0) = 1, y\left( \frac{\pi}{2} \right) = 2 \) is