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

Reduce the matrix \(\left[ \begin{matrix} 3 & -1 & 2 \\ -6 & 2 & 4 \\ -3 & 1 & 2 \end{matrix} \right] \) to a row-echelon form.

Show that the equations 3x + y + 9z = 0, 3x + 2y + 12z = 0 and 2x + y + 7z = 0 have nontrivial solutions also.

Solve 6x - 7y = 16, 9x - 5y = 35 using (Cramer's rule).

Find z⁻¹, if z = (2 + 3i) (1− i).

Simplify the following:
i ^-1924+ i²⁰¹⁸

If (cosθ + i sinθ)² = x + iy, then show that x²+y² =1

Find the argument of -2

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

Show that the polynomial 9x⁹+ 2x⁵- x⁴- 7x²+ 2 has at least six imaginary roots.

If sin ∝, cos ∝ are the roots of the equation ax² + bx + c-0 (c ≠ 0), then prove that (n + c)² - b² + c²

Find x If \(x=\sqrt { 2+\sqrt { 2+\sqrt { 2+....+upto\infty } } } \)

Find the principal value of sin^-1(2), if it exists.

Is cos^-1(-x) = \(\pi\)-cos⁻¹(x) true? Justify your answer.

Find the principal value of \({ tan }^{ -1 }\left( \frac { -1 }{ \sqrt { 3 } } \right) \)

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

Find the general equation of the circle whose diameter is the line segment joining the points (−4, −2)and (1, 1) is x²+y²+5x+3y+6=0

Find the vertices, foci for the hyperbola 9x²−16y² = 144.

Find the locus of a point which divides so that the sum of its distances from (-4, 0) and (4, 0) is 10 units.

Find the equation of the hyperbola whose vertices are (0, ±7) and e = \(\frac { 4 }{ 3 } \)

If \(\vec{ a } =\hat { -3i } -\hat { j } +\hat { 5k } \), \(\vec{b}=\hat{i}-\hat{2j}+\hat{k} \),  \(\vec{c}=\hat{4j}-\hat{5k} \ \) find\( \ {\vec a } .(\vec { b } \times \vec { c } )\)

Find the volume of the parallelepiped whose coterminus edges are given by the vectors \(\hat { 2i } -\hat { 3j } +\hat { 4k } \), \(\hat { i } +\hat { 2j } -\hat { k } \) and \(\hat {3 i } -\hat { j } +\hat { 2k } \)

Find the area of the triangle whose vertices  are A(3, -1, 2) B(1, -1, -3) and C(4, -3, 1)

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

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The temperature T in celsius in a long rod of length 10 m, insulated at both ends, is a function of length x given by T = x(10 − x). Prove that the rate of change of temperature at the midpoint of the rod is zero.

Prove that the function f (x) = x² + 2 is strictly increasing in the interval (2,7) and strictly decreasing in the interval (−2, 0)

Find the point at which the curve y-e^xy+x=0 has a vertical tangent.

Determine the domain of concavity of the curve y=2-x²

Let f, g : (a, b)→R be differentiable functions. Show that d(fg) = fdg + gdf

If U(x, y, z) = \(\frac { { x }^{ 2 }+{ y }^{ 2 } }{ xy } +3{ z }^{ 2 }y\), find \(\frac { \partial U }{ \partial x } ;\frac { \partial U }{ \partial y } \) and \(\frac { \partial U }{ \partial z } \)

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

Find, by integration, the volume of the solid generated by revolving about the x-axis, the region enclosed by y = e−^2x y = 0, x = 0 and x = 1

Evaluate \(\int _{ 1 }^{ 2 }{ \frac { { e }^{ x } }{ 1+{ e }^{ 2x } } dx } \)

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=x²+2, x-axis, x=1 and x=2.

Find value of m so that the function y = e^mx is a solution of the given differential equation, y''− 5y' + 6y = 0

Determine the order and degree (if exists) of the following differential equations: 
\(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +3{ \left( \frac { dy }{ dx } \right) }^{ 2 }={ x }^{ 2 }log\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) \)

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

Form the D.E corresponding to y=e^mx by eliminating 'm'.

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.

Find the probability mass function and cumulative distribution function of number of girl child in families with 4 children, assuming equal probabilities for boys and girls.

Let p: Jupiter is a planet and q: India is an island be any two simple statements. Give verbal sentence describing each of the following statements.
(i) ¬p
(ii) p ∧ ¬q
(iii) ¬p ∨ q
(iv) p➝ ¬q
(v) p↔q

Fill in the following table so that the binary operation ∗ on A = {a, b, c} is commutative.

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