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ISC Class 12 Mathematics Solved Question Paper 2013

Long Answer Type

31.
If $\vec{a} and \vec{b}$ are unit vectors and $θ$ is the angle between them, then show that $|\vec{a} - \vec{b}| = 2 \sin (\frac{θ}{2})$

$|\vec{a} - \vec{b}| = {|\vec{a}|}^{2} - 2 |\vec{a}| |\vec{b}| \cos (θ) + {|\vec{b}|}^{2}$

$\vec{a}, \vec{b}$ are unit vectors

$|\vec{a}| = |\vec{b}| = 1$

${|\vec{a} - \vec{b}|}^{2} = 1 - 2 \cos (θ) + 1$

= $2 - 2cos(θ)$

= $2 (1 - \cos (θ))$

= 2 $\times 2 \sin^{2} (\frac{θ}{2})$

$∴ |\vec{a} - \vec{b}| = 2 \sin (\frac{θ}{2})$

Hence proved.

32.

Find the equation of line passing through the point (- 1, 3, - 2) and perpendicular to the lines: $\frac{x}{1} = \frac{y}{2} = \frac{z}{3}$ and $\frac{x + 2}{- 3} = \frac{y - 1}{2} = \frac{z + 1}{5}$

33.

Find the equations of planes parallel to the plane 2x - 4y + 4z = 7 and which are at a distance of five units from the point (3, - 1, 2)

34.

For A, B and C, the chances of being selected as manager of a firm are 4 : 1 : 2, repectively. The probabilities for them to introduce a radical change in the marketing strategy are 0.3, 0.8 and 0.5 respectively. If a change takes place; find the probability that it is due to the appontment of B.

35.

Mr. Nirav deposits Rs.250 at the beginning of each month in an account that pays an interest of 6 % per annum compounded monthly, how many months will be required for the deposit to amount to at least Rs. 6390?

36.

A mill owner buys two types of machines A and B for his mill. Machine A occupies 1000sqm of area and requires 12 men to operate it; while machine B occupies 1200sqm of area and requires 8 men to operate it. The qwner has 7600 sqm of area available and 72 men to operate the machines. If machine A produces 50 units B produces 40 units daily, how many machines of each type should he buy to maximise the daily output? Use Linear Programming to find the solutions.

37.

A bill of Rs. 60,000 was drawn on 1^st April 2011 at 4 months and discounted for Rs. 58560 at a bank. If the rate of interest was 12% per annum, on what date was the bill discounted?

38.

A company produces a commodity with Rs.24,000 fixed cost. The variable cost is estimated to be 25 % of the total revenue recovered on selling the product at a rate of Rs. 8 per unit. Find the following:

(i) Cost function

(ii) Revenue function

(iii) Breakeven point