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

Evaluate the following:

`int "e"^(-3x) cos^3x  "d"x`

Evaluate the following:

`int sqrt(tanx)  "d"x`  (Hint: Put tanx = t²)

If `int "dx"/((x + 2)(x^2 + 1)) = "a"log|1 + x^2| + "b" tan^-1x + 1/5 log|x + 2| + "C"`, then ______.

The determinant `abs (("a","bc","a"("b + c")),("b","ac","b"("c + a")),("c","ab","c"("a + b"))) =` ____________

If A `= [(1,2),(2,1)]` and f(x) = (1 + x) (1 - x), then f(a) is ____________.

If A `= [(2"x", 0),("x","x")] "and A"^-1 = [(1,0),(-1,2)],` then x equals ____________.

If a, b, c are the roots of the equation x³ - 3x² + 3x + 7 = 0, then the value of `abs((2 "bc - a"^2, "c"^2, "b"^2),("c"^2, 2 "ac - b"^2, "a"^2),("b"^2, "a"^2, 2 "ab - c"^2))` is ____________.

`abs(("x", -7),("x", 5"x" + 1))`

If `abs ((2"x",5),(8, "x")) = abs ((6,-2),(7,3)),`  then the value of x is ____________.

The value of the determinant `abs ((alpha, beta, gamma),(alpha^2, beta^2, gamma^2),(beta + gamma, gamma + alpha, alpha + beta)) =` ____________.

Using properties of determinants `abs ((1, "a", "a"^2 - "bc"),(1, "b", "b"^2 - "ca"),(1, "c", "c"^2 - "ab")) =` ____________.

If `"y" = ("x" + sqrt(1 + "x"^2))^"n",  "then" (1 + "x"^2)  ("d"^2 "y")/"dx"^2 + "x" ("dy")/("dx")` is ____________.

If `"y = a"^"x", "b"^(2"x" -1), "then" ("d"^2"y")/"dx"^2` is ____________.

If `"y" = (varphi "n x")/"x",` then the value of y'' (e) is ____________.

If `"x" = "a" ("cos"  theta + theta  "sin"  theta), "y = a" ("sin"  theta - theta  "cos"  theta), "then" ("d"^2 "y")/("dx"^2) =` ____________.

If `"y"^2 = "ax"^2 + "bx + c", "then"  "d"/"dx" ("y"^3 "y"_"z") =` ____________.

If `sqrt(("x + y")) + sqrt (("y - x")) = "a", "then"  "dy"/"dx" =` ____________.

If `"xy"^2 = "ax"^2 + "bxy" + "y"^2, "then find"  "dy"/"dx"`

If `"y = tan"^-1 [("sin x + cos x")/("cos x - sin x")], "then"  "dy"/"dx"` is equal to ____________.

If f(x) = `"log"_("x"^2) ("log x")`, then f(e) is ____________.