Ravish write letters to his five friends and address the corresponding envelopes. In how many ways can the letters be placed in the envelopes so that at least two of them are in the wrong envelopes -

In how many ways can we get a sum of at most 17 by throwing six distinct dice -

The number of non negative integral solutions of equation 3x + y + z = 24

Sum of divisors of 2⁵ ·3⁷ ·5³ · 7² is –

The length of the perpendicular from the pole to the straight line is

The condition for the lines and to be perpendicular is

If f : R →R; f(x) = sin x + x, then the value of (f^-1 (x)) dx, is equal to

Here we used the concept of integration and the inverse functions to solve the question. Finding an antiderivative of a function is the procedure known as integration. The process of adding the slices to complete it is comparable. The process of integration is the opposite of that of differentiation. So the final answer is .

The polar equation of the straight line with intercepts 'a' and 'b' on the rays and respectively is

The polar equation of the straight line parallel to the initial line and at a distance of 4 units above the initial line is

The polar equation of axy is

If x, y, z are integers and x  0, y  1, z  2, x + y + z = 15, then the number of values of the ordered triplet (x, y, z) is -

If then the equation whose roots are 

Here we used the concept of quadratic equations and solved the problem. We also understood the concept of discriminant and used it in the solution to find the intervals. Therefore, the equation is .

Let p, q  {1, 2, 3, 4}. Then number of equation of the form px² + qx + 1 = 0, having real roots, is

Here we used the concept of quadratic equations and solved the problem. We also understood the concept of discriminant and used it in the solution to find the intervals. Therefore, total 7 seven solutions are possible so 7 equations can be formed.

ax² + bx + c = 0 has real and distinct roots null. Further a > 0, b < 0 and c < 0, then –

Here we used the concept of quadratic equations and solved the problem. The concept of sum of roots and product of roots was also used here. Therefore, 0 < ∣α∣ < β.

The cartesian equation of is

Here we used the concept of quadratic equations and solved the problem. We also understood the concept of trigonometric ratios and used the formula to find the equation. So the equation is .

The castesian equation of is

Here we used the concept of quadratic equations and solved the problem. We also understood the concept of trigonometric ratios and used the formula to find the equation. So the equation is .