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

If sin y = xsin(a + y) prove that `(dy)/(dx) = sin^2(a + y)/sin a`

If A = `[[0 , 2],[3, -4]]` and kA = `[[0 , 3"a"],[2"b", 24]]` then find the value of k,a and b.

If tan^-1 x - cot^-1 x = tan^-1 `(1/sqrt(3)),`x> 0 then find the value of x and hence find the value of sec^-1 `(2/x)`.

`"If y" = (sec^-1 "x")^2 , "x" > 0  "show that"  "x"^2 ("x"^2 - 1) (d^2"y")/(d"x"^2) + (2"x"^3 - "x") (d"y")/(d"x") - 2 = 0`

Find: ∫ sin x · log cos x dx

Solve: tan^-1 4 x + tan^-1 6x `= π/(4)`.

Prove that the function f : N → N, defined by f(x) = x² + x + 1 is one-one but not onto. Find the inverse of f: N → S, where S is range of f.

Solve the differential equation: `(d"y")/(d"x") - (2"x")/(1+"x"^2) "y" = "x"^2 + 2`

Solve the differential equation:  ` ("x" + 1) (d"y")/(d"x") = 2e^-"y" - 1; y(0) = 0.`

If `"A" = [(1,1,1),(1,0,2),(3,1,1)]`, find A^-1. Hence, solve the system of equations x + y + z = 6, x + 2z = 7, 3x + y + z = 12.

Solve for x : `tan^-1 ((2-"x")/(2+"x")) = (1)/(2)tan^-1  ("x")/(2), "x">0.`

Solve the differential equation : `("x"^2 + 3"xy" + "y"^2)d"x" - "x"^2 d"y" = 0  "given that"  "y" = 0  "when"  "x" = 1`.

Find the intervals in which the function `f("x") = (4sin"x")/(2+cos"x") -"x";0≤"x"≤2pi` is strictly increasing or strictly decreasing. 

Using integration find the area of the triangle formed by negative x-axis and tangent and normal to the circle `"x"^2 + "y"^2 = 9  "at" (-1,2sqrt2)`.

If A and B are square matrices of the same order 3, such that ∣A∣ = 2 and AB = 2I, write the value of ∣B∣.

If f(x) = x + 1, find `d/dx (fof) (x)`

if  `vec"a"= 2hat"i" + 3hat"j"+ hat"k", vec"b" = hat"i" -2hat"j" + hat"k" and vec"c" = -3hat"i" + hat"j" + 2hat"k", "find" [vec"a" vec"b" vec"c"]`

Choose the correct option from the given alternatives :

Let f(x) = x³ – 6x² + 9x + 18, then f(x) is strictly decreasing in ______.

Let the function f: R → R be defined by f(x) = 4x – 1, ∀ x ∈ R. Then, show that f is one-one.

Let f: R → R be the function defined by f(x) = 4x – 3 ∀ x ∈ R. Then write f^–1