The least value of a when f f(x) = x² + ax + 1 is increasing on (1, 2) is __________

The curve y = e^x is ________

If y = sin x and x changes from \(\frac{\pi}{2}\) to ㅠ the approximate change in y is ..............

If u = y sin x then \(\frac { { \partial }^{ 2 }u }{ \partial x\partial y } \) = ..........

The value of \(\int _{ -\pi }^{ \pi }{ { sin }^{ 3 }x \ { cos }^{ 3 }x \ } dx\) is __________

\(\int _{ 0 }^{ \infty }{ { e }^{ -mx } } { x }^{ 7 }\) dx is __________

\(\int _{ a }^{ b }{ f(x) } dx=\) ..............

The general solution of \(4\frac{d^2 y}{dx^2}\) + y = 0 is _________

The general solution of x \(\frac{dy}{dx}\) = y is _________.

A pair of dice numbered 1, 2, 3, 4, 5, 6 of a six-sided die and 1, 2, 3, 4 of a four-sided die is rolled and the sum is determined. Let the random variable X denote this sum. Then the number of elements in the inverse image of 7 is

In eight throws of a die, 1 or 3 is considered a success. Then the mean number of success is _____________

A coin is tossed 3 times. The probability of getting exactly 2 heads is________

For a Bernouli distribution

Which one is the inverse of the statement (pVq)➝(pΛq)?

The number whose multiplication universe does not exist in C.

Find the inverse (if it exists) of the following:
\(\left[ \begin{matrix} -2 & 4 \\ 1 & -3 \end{matrix} \right] \)

Represent the complex number −1−i

Simplify the following:
 \(\sum _{ n=1 }^{ 102 }{ { i }^{ n } } \)

Solve: (2x-1) (x+3) (x-2) (2x+3)+20 = 0

State the reason for cos^-1\([cos(-\frac{\pi}{6})]\neq \frac{\pi}{6}.\)

Find the value of
\(tan^{-1}(tan\frac{5\pi}{4})\)

Find the principal value of
cosec^{-1\((-\sqrt{2})\)}

Find the parametric form of vector equation and Cartesian equations of the straight line passing through the point (−2, 3, 4) and parallel to the straight line \(\frac { x-1 }{ -4 } =\frac { y+3 }{ 5 } =\frac { 8-z }{ 6 } \)

Find the acute angle between the following lines
\(\vec { r } =(4\hat { i } -\hat { j } )+t(\hat { i } +2\hat { j } -2\hat { k } )\), \(\hat{r}=(\hat { i } +2\hat { j } -2\hat { k } )+s(\hat {- i } -2\hat { j } +2\hat { k } )\)

Find the distance of a point (2, 5, −3) from the plane \(\vec { r } .(6\hat { i } -3\hat { j } +2\hat { k } )\) = 5

Find the distance between the planes \(\vec { r } .(2\hat { i } -\hat { j } -2\hat { k } )\) = 6 and \(\vec { r } .(6\hat { i } -\hat { 3j } -\hat { 6k } )\) = 27

Express each of the following physical statements in the form of differential equation.
A saving amount pays 8% interest per year, compounded continuously. In addition, the income from another investment is credited to the amount continuously at the rate of Rs. 400 per year.

Suppose X is the number of tails occurred when three fair coins are tossed once simultaneously. Find the values of the random variable X and number of points in its reverse images.

An urn contains 5 mangoes and 4 apples. Three fruits are taken at randaom. If the number of apples taken is a random variable, then find the values of the random variable and number of points in its inverse images.

Decrypt the received encoded message \(\left[ \begin{matrix} 2 & -3 \end{matrix} \right] \left[ \begin{matrix} 20 & 4 \end{matrix} \right] \) with the encryption matrix \(\left[ \begin{matrix} -1 & -1 \\ 2 & 1 \end{matrix} \right] \) and the decryption matrix as its inverse, where the system of codes are described by the numbers 1 - 26 to the letters A - Z respectively, and the number 0 to a blank space.

Find the inverse of the non-singular matrix A = \(\left[ \begin{matrix} 0 & 5 \\ -1 & 6 \end{matrix} \right] \), by Gauss-Jordan method.

Four men and 4 women can finish a piece of work jointly in 3 days while 2 men and 5 women can finish the same work jointly in 4 days. Find the time taken by one man alone and that of one woman alone to finish the same work by using matrix inversion method.

Find the inverse (if it exists) of the following:
\(\left[ \begin{matrix} 5 & 1 & 1 \\ 1 & 5 & 1 \\ 1 & 1 & 5 \end{matrix} \right] \)

Solve the following system of linear equations by matrix inversion method :
2x  −  y  =  8 ,   3x  +  2y  =  −2.

If z_i = 2− i and z₂ = -4+3i , find the inverse of z₁z₂ and \(\frac { { z }_{ 1 } }{ { z }_{ 2 } } \)

Compute the value of 'c' satisfied by Rolle’s theorem for the function \(f(x)=log(\frac{x^{2}+6}{5x})\) in the interval [2, 3]

Find the absolute extrema of the following function on the given closed interval
f(x) = 3x⁴-4x³ ;[-1, 2]

Without actually solving show that the equation x⁴+2x³-2 = 0 has only one real root in the interval (0, 1).

Find the absolute extrema of the following functions on the given closed interval.
\(f(x)=6x^{ \frac { 3 }{ 4 } }-3x^{ \frac { 1 }{ 3 } };\left[ -1,1 \right] \)

Evaluate : \(\underset{x\rightarrow 1^{-}}{lim}(\frac{log(1-x)}{cot(\pi x)})\).

Find a linear approximation for the following functions at the indicated points.
\(h(x)=\frac{x}{x+1}, x_{0}=1\)

Let (x, y) = e^-2y cos(2x) for all (x, y) ∈ R². Prove that u is a harmonic function in R².

Verify the above theorem for F(x, y) = x² - 2y² + 2xy and x(t) = cos t, y(t) = sin t, t ∈ [0, 2\(\pi\)]

Evaluate the following definite integrals:
\(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { e }^{ x }\left( \frac { 1+sinx }{ 1+cosx } \right) dx } \)

If Rolle's theorem holds for f (x) = x³ + bx² + ax + 5 on [1,3] with c = \(\left( 2+\frac { 1 }{ \sqrt { 3 } } \right) \) find the values of a and b.

missle fired from ground level rises x metres vertically upwards in t seconds and \(x=100t-\frac { 25 }{ 2 } { t }^{ 2 }\). Find the 
(i) initial velocity of the missile
(ii) the time when the height of the missile is maximum
(iii) the maximum height reached
(iv) the velocity which the missile strikes the ground.

Using differential find the approximate value of cos 61; if it is given that sin 60° = 0.86603 and 1⁰ = 0.01745 radians.

Find the area bounded by the curves y=|x|-1 and y=-|x|+1

Show that the ratio of the area under the curve y=sinx and y=sin2x between x=0 and \(x=\frac { \pi }{ 3 } \) and x- axis are as 2 : 3.

Solve: (1 + e^2x) dy + (1 + y²)e^x dx = 0 when y(0) = 1

Sovle \(\left( x+2 \right) \frac { dy }{ dx } =x2+4x-9\) .Also find the domain of the function.

Solve : \(\frac { dy }{ dx } =\left( { sin }^{ 2 }x{ cos }^{ 2 }x+{ xe }^{ x } \right) dx\)

Solve :x²dy+y(x+y)dx=0 given that y=1 when x=1.