# 12th Maths Paper Solutions Set 1 : CBSE Abroad Previous Year 2014

General Instructions: i. All questions are compulsory. ii. The question paper consists of 29 questions divided into three sections A, B and C. Section A comprises of 10 questions of one mark each, Section B comprises of 12 questions of four marks each, and Section C comprises of 7 questions of six marks each. iii. All questions in section A are to be answered in one word, one sentence or as per the exact requirements of the question. iv. There is no overall choice. However, internal choice has been provided in 4 questions of four marks each and 2 questions of six marks each. You have to attempt only one of the alternatives in all such questions. v. Use of calculators is not permitted. You may ask for logarithmic tables, if required.
Q1 :

Let R = {(a, a3) : a is a prime number less than 5} be a relation. Find the range of R.

R = {(a, a3) : a is a prime number less than 5}

Prime numbers less than 5 are 2 and 3. So, a can take values 2 and 3.

âˆ´ R  = {(2, 23), (3, 33)} = {(2, 8), (3, 27)}

Hence, the range of R is {8, 27}.

Q2 :

Write the value of cos-1(-12) + 2 sin-1 (12).

For any x ∈ [-1, 1], cos-1x represents an angle in [0, π] whose cosine is x.

cos-1(-12) = any angle in [0, π] whose cosine is -12.

cos-1(-12)=2π3

Similarly,

sin-1(12) = an angle in [-π2,π2] whose sine is 12.

sin-1(12)=π6

cos-1(-12) + 2 sin-1 (12) = 2π3+2(π6)=4π+2π6=π

Hence, cos-1(-12) + 2 sin-1 (12)=π .

Q3 :

Use elementary column operations  C2  â†’C2 - 2C1 in the matrix equation (4323) = (1023)(21

Disclaimer: The question is incorrect. After applying the given operation, L.H.S is not equal to R.H.S

Q4 :

If (a + 483b-6) = (2a + 28b + 2a - 8b), write the value of a - 2b.

Q5 :

If A is a 3 x 3 matrix, |A|0 and |3A|=k|A|, then write the value of k.

Q6 :

Evaluate :

dxsin2 x cos2 x

Q7 :

Evaluate :

0π4tan x dx

Q8 :

Write the projection of vector iË†+jË†+kË† along the vector jË†.

Q9 :

Find a vector in the direction of vector 2 iË†-3 jË†+6 kË† which has magnitude 21 units.

Q10 :

Find the angle between the lines râƒ— =2iË†-5jË†+kË†+Î»(3iË†+2jË†+6kË†) and râƒ— =7iË†-6kË†+Î¼(iË†+2

Q11 :

Let f : W → W be defined as f(x) = x − 1 if x is odd and f(x) = x + 1 if x is even. Show that f is invertible. Find the inverse of f, where W is the set of all whole numbers.

Q12 :

Solve for x :
cos (tan-1x)=sin (cot-134)

OR

Prove that:
cot-1 7 + cot​-1 8 + cot​-1 18 = cot​-1 3

Q13 :

Using properties of determinants, prove that

âˆ£âˆ£âˆ£âˆ£a+xxxya+yyzza+zâˆ£âˆ£âˆ£âˆ£=a2(a+x+y+z)

Q14 :

If x = a cos Î¸ + b sin Î¸ and y = a sin Î¸ - b cos Î¸, show that

y2d2ydx2-xdydx+y=0.

Q15 :

If xmyn = (x + y)m+n, prove that dydx=yx.

Q16 :

Find the approximate value of f(3.02), up to 2 places of decimal, where f(x) = 3x2 + 5x + 3.

OR

Find the intervals in which the function f(x)=32x4-4x3-45x2+51 is
(a) strictly increasing
(b) strictly decreasing

Q17 :

Evaluate :
x cos-1 x1-x2âˆšdx

OR

Evaluate :
(3x-2) x2+x+1 ---------âˆšdx

Q18 :

Solve the differential equation (x2 − yx2) dy + (y2 + x2y2) dx = 0, given that y = 1, when x = 1.

Q19 :

Solve the differential equation dydx + y cot x = 2 cos x, given that y = 0 when x = π2.

Q20 :

Show that the vectors a,â†’ b,â†’ c â†’ are coplanar if and only if a â†’+ b â†’, b â†’+ c â†’ and c â†’+ a â†’

Q21 :

Find the shortest distance between the lines whose vector equations are
and

Q22 :

Three cards are drawn at random (without replacement) from a well-shuffled pack of 52 playing cards. Find the probability distribution of the number of red cards. Hence, find the mean of the distribution.

Q23 :

Two schools P and Q want to award their selected students on the values of tolerance, kindness and leadership. School P wants to award Rs x each, Rs y each and Rs z each for the three respective values to 3, 2 and 1 students, respectively, with a total award money of Rs 2,200. School Q wants to spend Rs 3,100 to award 4, 1 and 3 students on the respective values (by giving the same award money to the three values as school P). If the total amount of award for one prize on each value is Rs 1,200, using matrices, find the award money for each value.

Q24 :

Show that a cylinder of a given volume, which is open at the top, has minimum total surface area when its height is equal to the radius of its base.

Q25 :

Evaluate :

Q26 :

Find the area of the smaller region bounded by the ellipse and the line

Q27 :

Find the equation of the plane that contains the point (1, –1, 2) and is perpendicular to both the planes 2x + 3y – 2z = 5 and x + 2y – 3z = 8. Hence, find the distance of point P (–2, 5, 5) from the plane obtained

OR

Find the distance of the point P (–1, –5, –10) from the point of intersection of the line joining the points A (2, –1, 2) and B (5, 3, 4) with the plane x – y + z = 5.

Q28 :

A cottage industry manufactures pedestal lamps and wooden shades, each requiring the use of a grinding/cutting machine and a sprayer. It takes 2 hours on the grinding/cutting machine and 3 hours on the sprayer to manufacture a pedestal lamp. It takes 1 hour on the grinding/cutting machine and 2 hours on the sprayer to manufacture a shade. On any day, the sprayer is available for at the most 20 hours and the grinding/cutting machine for at most 12 hours. The profit from the sale of a lamp is Rs 25 and that from a shade is Rs 15. Assuming that the manufacturer can sell all the lamps and shades that he produces, how should he schedule his daily production in order to maximise his profit? Formulate an LPP and solve it graphically.

Q29 :

An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probabilities of an accident for them are 0.01, 0.03 and 0.15, respectively. One of the insured persons meets with an accident. What is the probability that he is a scooter driver or a car driver?

OR

Five cards are drawn one by one, with replacement, from a well-shuffled deck of 52 cards. Find the probability that

(i) all the five cards diamonds
(ii) only 3 cards are diamonds
(iii) none is a diamond