Board Paper of Class 12-Science 2012 Maths (SET 2) - Solutions

Find ‘λ’when the projection of

If a line has direction ratios 2, − 1, − 2, then what are its direction cosines?

Let * be a ‘binary’ operation on N given by a * b = LCM (a, b) for all a, b ∈ N. Find 5 * 7.

Write the principal value of

Simplify:

Find the sum of the following vectors:

If , write the cofactor of the element a₃₂.

If are three vectors such that Find the value of

Solve the following differential equation:

How many times must a man toss a fair coin, so that the probability of having at least one head is more than 80%?

OR

If sin y = x sin(a + y), prove that

Let A = R − {3} and B = R − {1}. Consider the function f : A → B defined by

Show that is one-one and onto and hence find f⁻¹.

OR

Find the point on the curve y = x³ − 11x + 5 at which the equation of tangent is y = x − 11.

OR

Using differentials, find the approximated value of

OR

Using properties of determinants, prove the following:

If y = 3 cos (log x) + 4 sin(log x), show that

Find the equation of the line passing through the point (− 1, 3, − 2) and perpendicular to the lines

Find the particular solution of the following differential equation:

Using matrices, solve the following system of linear equations:

OR

Using elementary operations, find the inverse of the following matrix:

A manufacturer produces nuts and bolts. It takes 1 hour of work on machine A and 3 hours on machine B to produce a package of nuts. It takes 3 hours on machine A and 1 hour on machine B to produce a package of bolts. He earns a profit of Rs 17.50 per package on nuts and Rs 7 per package of bolts. How many packages of each should be produced each day so as to maximize his profits if he operates his machines for at the most 12 hours a day? From the above as a linear programming problem and solve it graphically.

Find the equation of the plane determined by the points A (3, −1, 2), B (5, 2, 4) and C (− 1, −1, 6) and hence find the distance between the plane and the point P (6, 5, 9)

OR

Show that the height of a closed right circular cylinder of given surface and maximum volume, is equal to the diameter of its base.

A girl throws a die. If she gets a 5 or 6, she tosses a coin three times and notes the number of heads. If she gets 1, 2, 3 or 4, she tosses a coin two times and notes the number of heads obtained. If she obtained exactly two heads, what is the probability that she threw 1, 2, 3 or 4 with the die?

Using the method of integration, find the area of the region bounded by the following lines: 3x − y − 3 = 0, 2x + y − 12 = 0, x − 2y − 1 = 0