List of top Mathematics Questions

Let \( a_1, a_2, a_3, \ldots \) be in G.P. If \( a_1 \cdot a_2 \cdot a_3 = 64 \) and \( a_1 \cdot a_2 \cdot a_3 \cdot a_4 \cdot a_5 = 32 \), then common ratio is

In a G.P., the first and third terms are 4 and 8 respectively. Then the \(21^{\text{st}}\) term is

Let \( z = \frac{2 - i}{\alpha + i} \), where \( \alpha \) is a real number. If \( 4\text{Re}(z) = 3\text{Im}(\bar{z}) \), then the value of \( \alpha \) is

If \( z + \bar{z} = 6 \) and \( z - \bar{z} = 4i \), then \( |z|^2 = \)

The modulus of the complex number \( (2\sqrt{2} + i2\sqrt{2})^2 \) is equal to

Let \( z = 1 - i \). Then the value of \( z^4 \) is equal to

Let \( f(x)=\log_5 x \, (x > 0) \) and \( g(x)=\cos^{-1}(x) \, (-1\le x \le 1) \). Then the domain of \( g \circ f \) is

Let \( f(x) = \cos x \). Then the value of \( \frac{1}{2}[f(x+y) + f(y-x)] - f(x)f(y) \) is equal to

Let \( f(x) = x^2 - 10x - 19, \, x \in \mathbb{R} \). Then the inverse image of 5, \( f^{-1}(5) = \)

Let \( A, B, C \) be any three finite sets. If \( n(A \times B) = 160\), \( n(B \times C) = 80 \) and \( n(C \times A) = 200\), then \( n(A) = \)

The number of common tangents that can be drawn to the curves $\frac{x^2}{16}-\frac{y^2}{9}=1$ and $x^2+y^2=16$ is

If \[ \lim_{x \to \infty} \left( \frac{e}{1 - e} \left( \frac{1}{e} - \frac{x}{1 + x} \right) \right)^x = \alpha, \] then the value of \[ \frac{\log_e \alpha}{1 + \log_e \alpha} \] equals:

If \(\alpha, \beta\) are the roots of the equation \(x^2 + 3x + k = 0\) and \(\alpha + 1/\beta\), \(\beta + 1/\alpha\) are the roots of the equation \(4x^2 + px + 18 = 0\) then k satisfies the equation

If \( f(x) = x^{2}+bx+c \) and \( f(1+k) = f(1-k) \) \( \forall K \in \mathbb{R} \), for two real numbers b and c, then

If \( A = \begin{bmatrix} 4 & \lambda & -3 0 & 2 & 5 1 & 1 & 3 \end{bmatrix} \), then \( A^{-1} \) exists if:

1+(1+3)+(1+3+5)+(1+3+5+7)+... to 10 terms =

AB and CD are diameters of a circle with centre O and radius 7 cm. If \(\angle BOD = 30^\circ\), then find the area and perimeter of the shaded region.

Solution of the differential equation \( \frac{dy}{dx} + x = 0 \) represents a family of

The expression for K is given as a sum of terms up to infinity. Find the value of K.

One of the roots of the equation \((x+1)^4 + 81 = 0\) is

An urn contains 7 red, 5 white and 3 black balls. Three balls are drawn randomly one after the other without replacement. If it is known that first ball drawn is red and the second ball drawn is white, then the probability that the third ball drawn is not red is

If \( x = \alpha, y = \beta, z = \gamma \) is the solution of the system of equations \( 2x+3y+z=-1 \), \( 3x+y+z=4 \), \( x-3y-2z=1 \), then the value of \( \beta \) is

The tangents drawn from a point (2,-1) touch the circle \(x^2+y^2+4x-2y+1=0\) at the points A and B. If C is the centre of the circle, then the area (in sq. units) of the triangle ABC is

One card is drawn from a well shuffled deck of 52 cards. Then the probability that the card will be a face card is ________