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M P Keshari » ISC Mathematics Guess Paper for 2012

ISC Mathematics Guess Paper for 2012

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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 y2 = – 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 – x2)/(1 + x2)]                                                                        [3]

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

(viii) Solve the following differential equation : x dy/dx + y = 3x2 – 2, x > 0.        [3]

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

i/(1 + i).                                                                                    [3]

(x) Evaluate the following limit : limx→0[(ax – 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) = x2 – 5x – 14, find f(A). Hence obtain A3.                                           [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 (x2 + x + 1).                                         [5]

(b) If y = √{(1 – x)/(1 + x)}, show that (1 – x2)dy/dx + y = 0.                           [5]

Q.5. (a) Find the point on the curve y = x3 – 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 : 0π/2x sin2 x dx.                                                             [5]

(b) Find the area bounded by the curve y2 = 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 p2 + q2 = (a2 + b2)/(c2 + d2).                  [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 4th January, 2009 at 5 months, was discounted on 26th 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 – x2/5. 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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