Mathematics Moc...

Which of the following is NOT a property of definite integral?

The solution of the differential equation \(\frac{dx}{dy} + Px =Q\), where P and Q are constants or functions of y, is given by

If A and B are two mutually exclusive events such that P(A) = 0.4 and P(A ∪ B) = 0.6 then P(B) = ? 

Let \(A = \begin{bmatrix} \cos^2 x & \sin^2 x \\\ \sin^2 x & \cos^2 x \end{bmatrix}\) and \(B = \begin{bmatrix} \sin^2 x & \cos^2 x \\\ \cos^2 x & \sin^2 x \end{bmatrix}\). Then the determinant of the matrix A + B is

What is the value of cos (2tan-1 x + 2cot-1 x) ?

Which of the following statements is FALSE for a differentiable function f(x)?

If \(A = \left[ {\begin{array}{*{20}{c}} 3&4&9\\ {11}&6&7\\ 8&9&5 \end{array}} \right]\) and |2A| = k then find the value of k ?

The following plot shows a function y which varies linearly with x. The value of \(I = \mathop \smallint \nolimits_1^2 ydx\) is ________.

The component of the vector \(\hat i - \hat j\) along the vector \(\vec i + \vec j\)  will be a -

Consider the following statements for f(x) = e^-|x| ;

1. The function is continuous at x = 0.

2. The function is differentiable at x = 0.

Which of the above statements is / are correct?

The value (round off to one decimal place) of  \(\mathop \smallint \nolimits_{ - 1}^1 x\;{e^{\left| x \right|}}dx\) ______

The equation of the tangent to the curve given by x = a sin³ t, y = b cos³ t at a point where t = \(\frac{\pi}{2}\) will be -

According to the Mean Value Theorem, for a continuous function f(x) in the interval [a, b], there exists a value ξ in this interval such that \(\mathop \smallint \limits_a^b f\left( x \right)dx =\)

If f(x) = (x + 1) and g(x) = (x² - 7) and t(x) = f(x) + g(x) then find the minimum value of t(x)

Two digits out of 1, 2, 3, 4, 5 are chosen at random and multiplied together. What is the probability that the last digit in the product appears as 0?

If tan^-1 x + tan^-1 y + tan^-1 z = \(\frac{\pi}{2}\), then

The rate of change of a variable is proportional to that variable. How will the variation of that variable with respect to time?

Let X be a random variable with probability density function,

\(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {0.}6\\ {\begin{array}{*{20}{c}} {0.2}\\ 0 \end{array}} \end{array}}&{\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {for\left| x \right| \le 1}\\ {for1 < \left| x\right| \le 4} \end{array}}\\ {otherwise} \end{array}} \end{array}} \right.\)

The probability P(0.5 < × < 5)  is _________

In binomial probability distribution, the mean is 3 and the standard deviation is \( \frac{3}{{2}}.\) Then the probability distribution is

The orthogonal curve to y = mx is

Let E₁ and E₂ be two independent events. Let P(E) denotes the probability of the occurrence of the event E. Further, let \(\rm E_1^{'}\) and \(\rm E_2^{'}\) denote the complements of E₁ and E₂ respectively. If \(P(E_1'{\cap}E_2) = \frac{2}{15}\) and \(\rm P(E_1\cap E_2^{'})=\frac{1}{6}\), then P(E₁) is

If * is a binary operation on Q where Q is set of rational numbers such that a * b = ab/2 ∀ a, b ∈ Q then find the identity element of Q with respect to operation * ? 

Direction Cosines of the line (6x - 8) = 3y + 2 = 2z + 4 are:

If A = {0, 1, 2, 3}, B = {1, - 1} and R is a relation from A to B such that {(x, y) : x ∈ A, y ∈ B and y = i^x where i is the imaginary number} then find the domain of R ?

The shaded region in the graph is shown by the inequalities (Constraints):

The maximum value of Z = 3x + 4y subjected to the constraints 2x + y ≤ 4, x + 2y ≥ 12, x ≥ 0, y ≥ 0

If a₁, a₂, a₃, ..., a₉ are in G.P., then what is the value of the determinant \(\begin{vmatrix} \rm \ln a_1 & \rm \ln a_2 & \rm \ln a_3 \\ \rm \ln a_4 & \rm \ln a_5 & \rm \ln a_6 \\ \rm \ln a_7 & \rm \ln a_8 & \rm \ln a_9 \end{vmatrix}\)?

If A is square matrix such that A² = A, then (I - A)³ + A is equal to

A square non-singular matrix A satisfies the equation x² - 4x + 3 = 0, then A^-1 is equal to

\(\rm\int{dx\over x^5 +x^3}\) = ?

The area of the triangle formed by the tangents to the circle x² + y² = 25 at (3, 4) and co-ordinate axes is

A linear programming problem is shown below:

Maximize        3x + 7y

                        3x + 7y ≤ 10

Subject to        4x + 6y ≤ 8

                        x, y ≥ 0

The area of the region satisfying the inequalities |x| - y ≤ 1, y ≥ 0 and y ≤ 1 is

The line y = 0 divides the line joining the points (3, -5) and (-4, 7) in the ratio:

If \(\Delta=\rm \begin{vmatrix}a_1&b_1&c_1\\\ a_2 &b_2&c_2\\\ a_3&b_3&c_3\end{vmatrix}\) and A₁, B₁, C₁ denote the cofactors of a₁, b₁, c₁ respectively, then the value of the determinant \(\rm \begin{vmatrix}A_1&B_1&C_1\\\ A_2 &B_2&C_2\\ A_3&B_3&C_3\end{vmatrix}\) is

Find the degree and order of differential equation y''' - sin(y') + y = 0 is

The number of solutions of the equations x + 4y - z = 0, 3x - 4y - z = 0, x-3y + z = 0 is 

Let â, b̂ and ĉ  be three unit vectors such that  \(\hat a \times (\hat b \times \hat c)=\frac{\sqrt3}{2}(\hat b+ \hat c)\). If b̂ is not parallel to ĉ, then the angle between â and ĉ is

Find the distance between the point  P (1, 6, 3) and foot of the perpendicular drawn from the point P (1, 6, 3) on the line \(\frac{x}{1} = \frac{{y - 1}}{2} = \frac{{z - 2}}{3}\) ?

An equation of a plane parallel to x + 2y + 2z = 5 and at a unit distance from the origin is given by 

Find distance between the lines L₁ and L₂ whose vector equations are \(\vec r = \;\hat i + \;2\hat j + \lambda \times \left( {2\hat i + \;3\hat j + \;6\hat k} \right)\;and\;\vec r = 3\hat i + \;3\hat j - 5\hat k + \mu \times \left( {2\hat i + 3\hat j + 6\hat k} \right)\)

If points (a, 0), (0, b) and (1, 1) are collinear, then \(\left( \frac{a + b}{ab} \right)\) =

If A is a skew symmetric matrix and n is a positive integer, then Aⁿ is

The probability distribution of a random variable X is given as

The value of p is -

Find x²y₂ + xy₁, if y = sin (log x) ?

The plane x - 3y + 5z - 8 = 0 makes an angle sin^-1 (∝) with Z -axis . the value of ∝ is equal to

If P = {1, 2, 3, 4} and Q = {a₁, a₂}, then the number of onto functions from P to Q is?

Consider the function \(y = x^2 + \dfrac{250}{x}\) At x = 5, the function attains.

If  \(A = \,\left[ {\begin{array}{*{20}{c}} {\log x}&{ - 1}\\ { - \log x}&2 \end{array}} \right]\,\)and if det (A) = 2, then the value of x is equal to

In a Binomial Distribution (BD) the mean is 15 and variance is 10, then parameter n (number of trials) is