If adj(A) = \(\left[ \begin{matrix} 2 & -4 & 2 \\ -3 & 12 & -7 \\ -2 & 0 & 2 \end{matrix} \right] \), find A.

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 rank of the matrix math \(\left[ \begin{matrix} 4 \\ -2 \\ 1 \end{matrix}\begin{matrix} 4 \\ 3 \\ 4 \end{matrix}\begin{matrix} 0 \\ -1 \\ 8 \end{matrix}\begin{matrix} 3 \\ 5 \\ 7 \end{matrix} \right] \).

The complex numbers u, v, and w are related by \(\frac { 1 }{ u } =\frac { 1 }{ v } +\frac { 1 }{ w } \) If v = 3−4i and w = 4+3i, find u in rectangular form.

If (x₁ + iy₁)(x₂ + iy₂)(x3 + iy₃)...(x_n+ iy_n) = a + ib, show that
(x₁² + y₁²)(x₂² + y₂²)(x₃² + y₃²)...(x_n² + y_n²) = a² + b²​

Find the principal value of -2i.

Solve the equation 2x³+11x²−9x−18 = 0.

Solve: \({ tan }^{ -1 }\left( \cfrac { x-1 }{ x-2 } \right) +{ tan }^{ -1 }\left( \cfrac { x+1 }{ x+2 } \right) =\cfrac { \pi }{ 4 } \)

Find the condition for the line lx + my + n = 0 is tangent to the circle x² + y² = a²

Prove that \(\left[ \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } ,\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } ,\overset { \rightarrow }{ c } \right] \)=\(\left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] \)

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Find the equation of normal to the cure y = sin²x at \(\left( \frac { \pi }{ 3 } ,\frac { 3 }{ 4 } \right) \).

Find the area of the region bounded by the y-axis and the parabola x = 5 − 4y − y².

Evaluate \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { \sqrt { cot \ x } }{ \sqrt { cot \ x } +\sqrt { tan \ x } } dx } \)

Show that the differential equation representing the family of curves \({ y }^{ 2 }=2a\left( x+a^\frac { 2 }{ 3 } \right) \) where a is a positive parameter, is \({ \left( { y }^{ 2 }-2xy\frac { 2 }{ 3 } \right) }^{ 3 }=8{ \left( y\frac { dy }{ dx } \right) }^{5 }\).

Solve the Linear differential equation:
\(x\frac { dy }{ dx } +y=xlogx\)

In a pack of 52 playing cards, two cards are drawn at random simultaneously. If the number of black cards drawn is a random variable, find the values of the random variable and number of points in its inverse images.

Suppose a pair of unbiased dice is rolled once. If X denotes the total score of two dice, write down
(i) the sample space
(ii) the values taken by the random variable X,
(iii) the inverse image of 10, and
(iv) the number of elements in inverse image of X.

Verify the
(i) closure property,
(ii) commutative property,
(iii) associative property
(iv) existence of identity and
(v) existence of inverse for the arithmetic operation + on Z_e = the set of all even integers

Verify whether the following compound propositions are tautologies or contradictions or contingency
( p ⟶ q) ↔️ (~ p ⟶ q)

Let G = {1, i, -1, -i} under the binary operation multiplication. Find the inverse of all the elements.