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Differentiate...

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Differentiate the following w.r.t. x : 1. √x-1/√x 2. 1/(x+2) 3. Sin nx + nx cos nx

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∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 1 c o s (n x) & c o s (n + 1) x & c o s (n + 2) x s i n (n x) & s i n (n + 1) x & s i n (n + 2) x \end{matrix} ∣ ∣ ∣ ∣ ∣

does not depend

Q. The determinant

Δ = ∣ ∣ ∣ ∣ ∣ \begin{matrix} a^{2} & a & 1 cos (n x) & cos (n + 1) x & cos (n + 2) x sin (n x) & sin (n + 1) x & sin (n + 2) x \end{matrix} ∣ ∣ ∣ ∣ ∣

is independent of

Q. The value of the determinant

∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & a & a^{2} c o s (n - 1) x & c o s n x & c o s (n + 1) x s i n (n - 1) x & s i n n x & s i n (n + 1) x \end{matrix} ∣ ∣ ∣ ∣ ∣ (a \neq 1)

Q. Solve the equation for

x

.

∣ ∣ ∣ ∣ ∣ \begin{matrix} a^{2} & a & 1 sin (n + 1) x & sin n x & sin (n - 1) x cos (n + 1) x & cos n x & cos (n - 1) x \end{matrix} ∣ ∣ ∣ ∣ ∣ = 0

a > 0

cos x + cos 3 x + . . . . . . + cos (2 n - 1) x = \frac{sin 2^{n} x}{2 sin x}, x \neq k π

.

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Standard XII Mathematics