For the matrix A, if A³ = I, then find A^-1.

Show that the system of equations is inconsistent. 2x + 5y= 7, 6x + 15y = 13.

Find the rank of the matrix A =\(\left[ \begin{matrix} 4 \\ 7 \end{matrix}\begin{matrix} 5 \\ -3 \end{matrix}\begin{matrix} -6 \\ 0 \end{matrix}\begin{matrix} 1 \\ 8 \end{matrix} \right] \).

Find k if the equations x + 2y + 2z = 0, x - 3y - 3z = 0, 2x + y + kz = 0 have only the trivial solution.

Find Re (z) and im (z) if z = 5i¹¹ + 7i³

If z =\(\left( \frac { \sqrt { 3 } }{ 2 } +\frac { i }{ 2 } \right) ^{ 107 }+\left( \frac { \sqrt { 3 } }{ 2 } -\frac { i }{ 2 } \right) ^{ 107 }\), then show that Im (z) = 0

Find the argument of -2

Find value of a for which the sum of the squares of the equation x² - (a- 2) x - a -1 = 0 assumes the least value.

Find the number of positive and negative roots of the equation x⁷ - 6x⁶ + 7x⁵ + 5x²+2x+2

If \({ cot }^{ -1 }\left( \frac { 1 }{ 7 } \right) =\theta \) find the value of cos \(\theta \)

Evaluate \(sin\left( { cos }^{ -1 }\left( \frac { 3 }{ 5 } \right) \right) \)

Find the equation of tangent to the circle x² +y² + 2x - 3y - 8 = 0 at (2, 3).

If the line y = 3x + 1, touches the parabola y² = 4ax, find the length of the latus rectum?

For the ellipse x² + 3y² = a², find the length of major and minor axis.

Find the Cartesian equation of a line passing through the points A(2, -1, 3) and B(4, 2, 1)

A PHP Error was encountered

Severity: Warning

Message: A non-numeric value encountered

Filename: material/details.php

Line Number: 1002

()

-1

Find the parametric form of vector equation of the plane passing through the point (1, -1, 2) having 2, 3, 3 as direction ratios of normal to the plane.

A PHP Error was encountered

Severity: Warning

Message: A non-numeric value encountered

Filename: material/details.php

Line Number: 1002

()

2

Find the equation of the plane containing the line of intersection of the planes x + y + Z - 6 = 0 and 2x + 3y + 4z + 5 = 0 and passing through the point (1, 1, 1)

A PHP Error was encountered

Severity: Warning

Message: A non-numeric value encountered

Filename: material/details.php

Line Number: 1002

()

x = -1 is one root

Find the equation of the tangent to the curve y²=4x+5 and which is parallel to y=2x+7

Verify Rolle ’s Theorem for \(f(x)=\left| x-1 \right| ,O\le x\le 2\) 

Obtain Maclaurin’s Series expansion for e^2x.

Evaluate the following limits, if necessary using L’Hopitalrule
(i) \(\underset { x\rightarrow 2 }{ lim } \cfrac { sin\pi x }{ 2-x } \) 
(ii) \(\cfrac { lim }{ x\rightarrow 2 } \cfrac { { x }^{ n }-{ a }^{ n } }{ x-2 } \) 
(iii) \(\underset { x\rightarrow \infty }{ lim } \cfrac { sin\frac { 2 }{ x } }{ \frac { 1 }{ x } } \)
(iv) \(\underset { x\rightarrow \infty }{ lim } \cfrac { { x }^{ 2 } }{ { e }^{ x } } \)

Find the absolute extreme of the function f(x) = x²-2x+2 on the closed interval [0, 3]

Prove that the function f(x)=2x²+3x is strictly increasing on \(\left[ -\frac { 1 }{ 2 } ,\frac { 1 }{ 2 } \right] \)

Determine the domain of convexity of the function y=e^x

IF u(x, y) = x² + 3xy + y², x, y, ∈ R, find tha linear appraoximation for u at (2, 1) 

If w=log(x²+y²),x=cosθ,y=sinθ, find \(\frac { dw }{ d\theta } \)

Without using any kind of computational aid use linear approximation to find the value of e^0.1

Find the area of the region enclosed by the curve y = \(\sqrt x\) + 1, the axis of x and the lines x = 0, x = 4.

Find the volume of the solid obtained by revolving the area of the triangle whose sides are x = 4, y = 0 and 3x - 4y = 0 about x - axis

If \(\int _{ 0 }^{ \infty }{ \frac { { x }^{ 2 }dx }{ \left( { x }^{ 2 }+{ a }^{ 2 } \right) \left( { x }^{ 2 }+{ b }^{ 2 } \right) \left( { x }^{ 2 }+{ c }^{ 2 } \right) } } =\frac { \pi }{ 2(a+b)(b+c)(c+a) } \) then find \(\int _{ 0 }^{ \infty }{ \frac { dx }{ \left( { x }^{ 2 }+4 \right) \left( { x }^{ 2 }+9 \right) } } \)

Find the area bounded by y=x²+2, x-axis, x=1 and x=2.

Find the area of the region bounded by the curve y=sin x and the ordinate x=0.\(x=\frac { \pi }{ 3 } \)

Find the area bounded by y=cosx,y=x+1,y=0.

Determine the order and degree of \(\frac { \left[ 1+\left( \frac { dy }{ dx } \right) ^{ 2 } \right] ^{ \frac { 3 }{ 2 } } }{ \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } } =k\)

Find the order and degree of \(\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) ^{ 2 }+cos\left( \frac { dy }{ dx } \right) =0\)

Form the Differential Equation representing the family of curves y = A cos(x + B) where A and B are parameters.

Solve : \(\frac { dy }{ dx } =\frac { { e }^{ x }-{ e }^{ -x } }{ { e }^{ x }+{ e }^{ -x } } \)

Consider the binary operation ∗ defined on the set A = {a, b, c, d} by the following table:

*	a	b	c	d
a	a	c	b	d
b	d	a	b	c
c	c	d	a	a
d	d	b	a	c

Is it commutative and associative?

In the set of integers under the operation * defined by a * b = a + b - 1. Find the identity element.