List of top Mathematics Questions

The function \( f(x) \) is given by:

\[ f(x) = \begin{cases} x \sin \left( \frac{1}{x} \right), & \text{for } x \neq 0 \\ 0, & \text{for } x = 0 \end{cases} \]

For the parabola \( y^2 = -12x \), the equation of the directrix is  \( x = a \). The value of  \( a \) is:

The principal value of \(\sin^{-1} \left( \sin \frac{5\pi}{3} \right)\) is:

The function \( f(x) \) is given by:

\[ f(x) = \begin{cases} x \lfloor x \rfloor & \text{if } 0 \leq x < 2 \\ (x - 1)x & \text{if } 2 \leq x < 3 \end{cases} \]

The function is:

If \( \alpha, \beta \) are the roots of the equation \( ax^2 + bx + c = 0 \), then \[ \frac{\alpha}{a\beta + b} + \frac{\beta}{a\alpha + b} = \]

The value of the definite integral

\[ \int_0^{\frac{\pi}{2}} \log(\tan x) \, dx \]

is:

The following determinant is equal to:

\[ \begin{vmatrix} \sin^2 x & \cos^2 x & 1 \\ \cos^2 x & \sin^2 x & 1 \\ -10 & 12 & 2 \end{vmatrix} \]
Let f : \(\R^2 → \R\) be the function defined by
\(f(x,y) = \begin{cases}   x^2\sin\frac{1}{x}+y^2\cos y, & x \ne 0 \\   0, & x=0. \end{cases}\)
Then which one of the following statements is NOT true ?
Let f : [1, 2] → \(\R\) be the function defined by
\(f(t)=\int^t_1\sqrt{x^2e^{x^2}-1}\ dx.\)
Then the arc length of the graph of f over the interval [1, 2] equals
Let F : [0, 2] → \(\R\) be the function defined by
\(F(x)=\int_{x^2}^{x+2}e^{x[t]}dt,\)
where [t] denotes the greatest integer less than or equal to t. Then the value of the derivative of F at x = 1 equals
Let the system of equations
x + ay + z = 1
2x + 4y + z = -b
3x + y + 2z = b + 2
have infinitely many solutions, where a and b are real constants. Then the value of 2a + 8b equals
Let \(A=\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 1 \end{pmatrix}\). Then the sum of all the elements of A100 equals
Let {an}n≥1 be a sequence of real numbers such that \(a_n=\frac{1}{3^n}\) for all n ≥ 1. Then which of the following statements is/are true ?
Let f : \(\R^2 → \R\) be the function defined by
f(x, y) = 8(x2 - y2) - x4 + y4.
Then which of the following statements is/are true ?
If n ≥ 2, then which of the following statements is/are true ?
Let {an}n≥1 be a sequence of real numbers such that a1+5m = 2, a2+5m = 3, a3+5m = 4, a4+5m = 5, a5+5m = 6, m = 0, 1, 2, … . Then limsupn→∞ an + liminfn→∞ an equals __________
Let f : \(\R → \R\) be a function such that
20(x - y) ≤ f(x) - f(y) ≤ 20(x - y) + 2(x - y)2
for all x, y ∈ \(\R\) and f(0) = 2. Then f(101) equals __________
Let A be a 3 × 3 real matrix such that det(A) = 6 and
\(adj\ A=\begin{pmatrix} 1 & -1 & 2 \\ 5 & 7 & 1 \\ -1 & 1 & 1 \end{pmatrix}\)
where adj A denotes the adjoint of A.
Then the trace of A equals __________ (round off to 2 decimal places)
Let f ∶ \(\R^2 → \R\) be the function defined by f(x, y) = x2 - 12y. If M and m be the maximum value and the minimum value, respectively, of the function f on the circle x2 + y2 = 49, then |M| + |m| equals __________
The value of
\(\int^2_0 \int_0^{2-x}(x+y)^2e^{\frac{2y}{x+y}}\ dy\ dx\)
equals __________ (round off to 2 decimal places)
Let \(A=\begin{pmatrix} 1 & -1 & 2 \\ -1 & 0 & 1 \\ 2 & 1 & 1 \end{pmatrix}\) and let \(\begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}\)be an eigenvector corresponding to the smallest eigenvalue of A, satisfying \(x^2_1+x^2_2+x^2_3=1\). Then the value of |x1| + |x2| + |x3| equals __________ (round off to 2 decimal places)

The equation of the hyperbola with vertices at

\[ (0, \pm 6) \quad \text{and} \quad e = \frac{5}{3} \]

is:

For what value of \( k \), does the equation \[ 9x^2 + y^2 = k(x^2 - y^2 - 2x) \] represent the equation of a circle?

The area enclosed between the graph of \( y = x^3 \) { and the lines} \[ x = 0, \, y = 1, \, y = 8 { is:} \]

The value of the integral \[ \int_0^{\frac{\pi}{2}} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} \, dx \] is: