Arts (English Medium) Class 12 - CBSE Question Bank Solutions

If x, y, z ∈ R, then the value of determinant `|((2x^2 + 2^(-x))^2, (2^x - 2^(-x))^2, 1),((3^x + 3^(-x))^2, (3^x -3^(-x))^2, 1),((4^x + 4^(-x))^2, (4^x - 4^(-x))^2, 1)|` is equal to ______.

If cos2θ = 0, then `|(0, costheta, sin theta),(cos theta, sin theta,0),(sin theta, 0, cos theta)|^2` = ______.

If x = – 9 is a root of `|(x, 3, 7),(2, x, 2),(7, 6, x)|` = 0, then other two roots are ______.

If the value of a third order determinant is 12, then the value of the determinant formed by replacing each element by its co-factor will be 144.

`|(x + 1, x + 2, x + "a"),(x + 2, x + 3, x + "b"),(x + 3, x + 4, x + "c")|` = 0, where a, b, c are in A.P.

The determinant `|(sin"A", cos"A", sin"A" + cos"B"),(sin"B", cos"A", sin"B" + cos"B"),(sin"C", cos"A", sin"C" + cos"B")|` is equal to zero.

If the determinant `|(x + "a", "p" + "u", "l" + "f"),("y" + "b", "q" + "v", "m" + "g"),("z" + "c", "r" + "w", "n" + "h")|` splits into exactly K determinants of order 3, each element of which contains only one term, then the value of K is 8.

Let Δ = `|("a", "p", x),("b", "q", y),("c", "r", z)|` = 16, then Δ₁ = `|("p" + x, "a" + x, "a" + "p"),("q" + y, "b" + y, "b" + "q"),("r" + z, "c" + z, "c" + "r")|` = 32.

If x^y = e^x–y, prove that `("d"y)/("d"x) = logx/(1 + logx)^2`

The derivative of log₁₀x w.r.t. x is ______.

If x = `e^(x/y)`, then prove that `dy/dx = (x - y)/(xlogx)`.

If y^x = e^{y – x}, prove that `"dy"/"dx" = (1 + log y)^2/logy`

If y = `(cos x)^((cos x)^((cosx)....oo)`, show that `"dy"/"dx" = (y^2 tanx)/(y log cos x - 1)`

Find `"dy"/"dx"`, if y = `x^tanx + sqrt((x^2 + 1)/2)`

Verify the following using the concept of integration as an antiderivative

`int (x^3"d"x)/(x + 1) = x - x^2/2 + x^3/3 - log|x + 1| + "C"`

Evaluate the following:

`int x^2/(1 - x^4) "d"x` put x² = t

Evaluate the following:

`int (x^2"d"x)/(x^4 - x^2 - 12)`

Evaluate the following:

`int (x^2 "d"x)/((x^2 + "a"^2)(x^2 + "b"^2))`

Evaluate the following:

`int_"0"^pi  (x"d"x)/(1 + sin x)`

Evaluate the following:

`int (2x - 1)/((x - 1)(x + 2)(x - 3)) "d"x`