Commerce (English Medium) Class 12 - CBSE Question Bank Solutions for Mathematics

\[\int\limits_1^3 \left( 2 x^2 + 5x \right) dx\]

\[\int\limits_1^3 \left( x^2 + 3x \right) dx\]

\[\int\limits_0^2 \left( x^2 + 2 \right) dx\]

\[\int\limits_0^3 \left( x^2 + 1 \right) dx\]

Find : `∫_a^b logx/x` dx

Prove that `int_a^b ƒ ("x") d"x" = int_a^bƒ(a + b - "x") d"x" and "hence evaluate" int_(π/6)^(π/3) (d"x")/(1+sqrt(tan "x")`

If A is 3 × 3 invertible matrix, then show that for any scalar k (non-zero), kA is invertible and `("kA")^-1 = 1/"k" "A"^-1`

Find inverse, by elementary row operations (if possible), of the following matrices

`[(1, 3),(-5, 7)]`

Find inverse, by elementary row operations (if possible), of the following matrices

`[(1, -3),(-2, 6)]`

x = `"t" + 1/"t"`, y = `"t" - 1/"t"`

x = `"e"^theta (theta + 1/theta)`, y= `"e"^-theta (theta - 1/theta)`

x = 3cosθ – 2cos³θ, y = 3sinθ – 2sin³θ

sin x = `(2"t")/(1 + "t"^2)`, tan y = `(2"t")/(1 - "t"^2)`

x = `(1 + log "t")/"t"^2`, y = `(3 + 2 log "t")/"t"`

If x = e^cos2t and y = e^sin2t, prove that `"dy"/"dx" = (-y log x)/(xlogy)`

If x = asin2t (1 + cos2t) and y = b cos2t (1–cos2t), show that `("dy"/"dx")_("at  t" = pi/4) = "b"/"a"`

If x = 3sint – sin 3t, y = 3cost – cos 3t, find `"dy"/"dx"` at t = `pi/3`

Differentiate `x/sinx` w.r.t. sin x

Differentiate `tan^-1 ((sqrt(1 + x^2) - 1)/x)` w.r.t. tan^–1x, when x ≠ 0

If x = sint and y = sin pt, prove that `(1 - x^2) ("d"^2"y")/("dx"^2) - x "dy"/"dx" + "p"^2y` = 0