Science (English Medium) कक्षा १२ - CBSE Question Bank Solutions for Mathematics

Show that lines: 

`vecr=hati+hatj+hatk+lambda(hati-hat+hatk)`

`vecr=4hatj+2hatk+mu(2hati-hatj+3hatk)` are coplanar 

Also, find the equation of the plane containing these lines.

Three persons A, B and C apply for a job of Manager in a Private Company. Chances of their selection (A, B and C) are in the ratio 1 : 2 :4. The probabilities that A, B and C can introduce changes to improve profits of the company are 0.8, 0.5 and 0.3, respectively. If the change does not take place, find the probability that it is due to the appointment of C

Find the particular solution of the differential equation x²dy = (2xy + y²) dx, given that y = 1 when x = 1.

If a line makes angles 90°, 60° and θ with x, y and z-axis respectively, where θ is acute, then find θ.

Find the local maxima and local minima, of the function f(x) = sin x − cos x, 0 < x < 2π.

Evaluate :

`∫(x+2)/sqrt(x^2+5x+6)dx`

Evaluate :   `∫1/(cos^4x+sin^4x)dx`

There are three coins. One is a two-headed coin (having head on both faces), another is a biased coin that comes up heads 75% of the times and the third is also a biased coin that comes up tails 40% of the time. One of the three coins is chosen at random and tossed and it shows heads. What is the probability that it was the two-headed coin?

Evaluate :

`int1/(sin^4x+sin^2xcos^2x+cos^4x)dx`

Find the distance between the planes 2x - y +  2z = 5 and 5x - 2.5y + 5z = 20

Show that the function f : R_* → R_* defined by f(x) = `1/x` is one-one and onto, where R_* is the set of all non-zero real numbers. Is the result true, if the domain R_* is replaced by N with co-domain being same as R?

Check the injectivity and surjectivity of the following function:

f : N → N given by f(x) = x²

Check the injectivity and surjectivity of the following function:

f : Z → Z given by f(x) = x²

Check the injectivity and surjectivity of the following function:

f : R → R given by f(x) = x²

Check the injectivity and surjectivity of the following function:

f : N → N given by f(x) = x³

Check the injectivity and surjectivity of the following function:

f : Z → Z given by f(x) = x³

Prove that the greatest integer function f : R → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.

Show that the modulus function f : R → R, given by f(x) = |x|, is neither one-one nor onto, where |x| is x, if x is positive or 0 and |x| is –x, if x is negative.

Show that the signum function f : R → R, given by

`f(x) = {(1", if"  x > 0), (0", if"  x  = 0), (-1", if"  x < 0):}`

is neither one-one nor onto.

Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function from A to B. Show that f is one-one.