# ISC Mathematics Guess Paper for 2012

**ICSE – Guess Paper – XII – for the year – 2012**

**Mathematics **

**Time – 3 Hours. ** **Full Marks – 100. **

** Section – A **

**[Question 1 is compulsory and answer ANY FIVE questions]**

Q.1. (i) X and Y are matrices and if

X – Y = 1 1 1 and X + Y = 3 5 7

1 1 0 – 1 1 4

1 0 0 11 8 0

find X and Y. [3]

(ii) In the parabola y^{2} = – 4 x, find the length of latus rectum, co-ordinates of focus and the equation of directrix. [3]

(iii) A point P(x, y) moves so that the sum of its distances from the points F(4, 2) and F’(– 2, –2) is 8 units. Find the equation of its locus and show that it is an ellipse. [3]

(iv) Find the equation of the hyperbola with centre at the origin, the length of transverse axis 6 and one focus at (0, 4). [3]

(v) Prove that : tan ^{–1} (1/4) + tan ^{–1} (1/9) = cos ^{–1} (2/√5). [3]

(vi) Differentiate following w.r.t. x :

cos ^{–1} [(1 – x^{2})/(1 + x^{2})] [3]

(vii) Evaluate : ∫[sin x/√(3 + 2 cos x)] dx. [3]

(viii) Solve the following differential equation : x dy/dx + y = 3x^{2} – 2, x > 0. [3]

(ix) Express the following in the standard form of a + ib :

i/(1 + i). [3]

(x) Evaluate the following limit : lim_{x→0}[(a^{x} – 1)/sin x]. [3]

Q.2. (a) Prove that the determinant :

1 + a 1 1

1 1 + b 1

1 1 1 + c = abc(1 + 1/a + 1/b + 1/c) [5]

(b) If matrix A = 3 –5

–4 2

and f(x) = x^{2} – 5x – 14, find f(A). Hence obtain A^{3}. [5]

Q.3. (a) Write the Boolean expression for the following switching circuit :

– a – b – c’ –

––a – b’ – c ––

– a – b’ – c’ –

Simplify the expression. Construct the switching circuit for the simplified expression. [5]

(b) Simplify the Boolean expression abc + ab’c + a’b'c and construct an equivalent switching circuit. [5]

Q.4. (a) Prove that ^{–1} x + cot ^{–1} (x + 1) = tan ^{–1} (x^{2} + x + 1). [5]

(b) If y = √{(1 – x)/(1 + x)}, show that (1 – x^{2})dy/dx + y = 0. [5]

Q.5. (a) Find the point on the curve y = x^{3} – 3x where the tangent to the curve is parallel to the chord joining (1, – 2) and (2, 2). [5]

(b) Show that the right circular cone of least curved surface area and given volume has an altitude equal to √2 times the radius of the base. [5]

Q.6. (a) Evaluate the following : _{0}∫^{π/2}x sin^{2} x dx. [5]

(b) Find the area bounded by the curve y^{2} = 4ax and the lines y = 2a and y-axis. [5]

Q.7. (a) Find the Spearsman’s rank correlation between marks in Mathematics and Statistics obtained by 10 students : [5]

Mathematics | 80 | 38 | 95 | 30 | 74 | 84 | 91 | 60 | 66 | 40 |

Statistics | 85 | 50 | 92 | 58 | 70 | 65 | 88 | 56 | 52 | 46 |

(b) Given the observations (10, – 5), (10, – 3), (11, – 2), (11, 0), (12, 1), (15, 6), (16, 4), (11, – 2), predict the value of Y corresponding to the value 14 of X and predict the value of X when the value of Y is 3.

8. (a) An urn contains 9 red, 7 white and 4 black balls. If two balls are drawn at random, find the probability that:

(i) Both balls are red.

(ii) One ball is white.

(iii) One is white and the other red.

(iv) Both balls are of the same colour. [5]

(b) Two cards are drawn at random one by one without replacement from a pack of 52 playing cards. Find the probability that both the cards are black. [5]

Q.9. (a) If (a + ib)/(c + id) = p + iq, prove that p^{2} + q^{2} = (a^{2} + b^{2})/(c^{2} + d^{2}). [5]

(b) Solve the following differential equation : dy/dx = y/x + tan (y/x). [5]

**Section – B. **

**(Answer ANY TWO questions) **

Q.10. (a) If a^{→} and b^{→} are two unit vectors such that |a^{→} + b^{→}| = √3, find the value of (2a^{→} – 5b^{→}).(3a^{→} + b^{→}). [5]

(b) Find the area of parallelogram whose adjacent sides are i – 3j + k and i + j + k. Verify your result by calculating the area using diagonals. [5]

Q.11. (a) Determine the equations of the line passing through the point (1, 2, – 4) and perpendicular to the two lines :

(x – 8)/3 = (y + 9)/–16 = (z – 10)/7 and (x – 15)/3 = (y – 29)/8 = (z – 5)/–5. [5]

(b) If a plane meets the co-ordinate axes in points A, B, C and the centroid of the triangle ABC is (α, β, γ), find the equation of the plane. [5]

Q.12. (a) A man is known to speak truth 3 out of 4 times. He throws a die and reports that it is a six. Find the probability that it is actually a six. [5]

(b) A die is thrown 4 times. Getting a ‘1 or 6’ is considered a success. Find the probability of getting :

(i) Exactly 3 success.

(ii) Exactly 4 success.

(iii) At most two success.

**Section – C. **

**(Answer ANY TWO questions) **

Q.13. (a) A bill of exchange drawn on 4^{th} January, 2009 at 5 months, was discounted on 26^{th} March, 2009. If the banker’s discount at 3% is Rs603.60, find the face value of the bill. [5]

(b) A machine costing Rs2 lacs has effective life of 7 years and its scrap value is Rs30000. What amount should the company put into a sinking fund earning 5% p.a. so that it can replace the machine after its useful life? Assume that a new machine will cost Rs3 lacs after 7 years. [5]

Q.14. (a) Two tailors, A and B, charge Rs150 and Rs200 per day respectively. A can stitch 6 shirts and 4 pants while B can stitch 10 shirts and 4 pants per day. How many days shall each work if it is desired to produce at least 60 shirts and 32 pants at a minimum labour cost? [5]

(b) The total revenue in rupees received from the sale of x units of a product is given by

R(x) = 300x – x^{2}/5. Find :

(i) the average revenue,

(ii) the marginal revenue, and

(iii) the total revenue when MR = 0. [5]

Q.15. (a) Consider the following data:

Items | Units | Price in 2006 (in Rs) | Price in 2010 (in Rs) |

Wheat | 1 kg | 5.60 | 7.20 |

Rice | 1 kg | 17.20 | 24.80 |

Pulses | 1 kg | 36.00 | 44.00 |

Milk | 1l |
24.00 | 30.00 |

Clothing | 1 m | 199.00 | 130.00 |

Using 2006 as the base year, calculate the index for 2010 correct up to one decimal using

(i) simple aggregate method (ii) simple average of relatives method. [5]

(b) Assuming a four yearly cycle, calculate the trend by the method of moving average from the following data :

Year | 1994 | 1995 | 1996 | 1997 | 1998 | 1999 | 2000 | 2001 | 2002 | 2003 |

Value | 12 | 25 | 39 | 54 | 70 | 87 | 105 | 100 | 82 | 65 |

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