Download 3. What is the area of the parallelogram whose adjacent sides are

HSE II
Max : 80 Scores
1.
If A= x
MODEL EXAMINATION Feb-2012
MATHEMATICS
3 B=
1
-2
0
-3
C=
Time: 21⁄2 hrs
Cool off time : 15 mts
-1
x
(a) Find AB
(1)
(b) Find (AB)C
(1)
(c) If (AB)C=0,find the value of x.
(1)
3 3 -1
2. Express the matrix A = -2 -2 1 as the sum of a symmetric and a skew-symmetric
-4 -5 2
matrix.
(3)
3. What is the area of the parallelogram whose adjacent sides are given by
ā = 2î+2ĵ-5k , b = 2î+ĵ+3k.
(2)
4. Find a vector of magnitude 9 units perpendicular to both p+q and p-q where
p= î-3ĵ+3k and q= 3î-3ĵ+2k.
(3)
5. Given three points A(1,2,7), B(2,6,3) and C(3,10,-1)
(a) Write the position vectors of A, B and C.
(1)
(b) Find AB and AC.
(1)
(c)Prove that A, B and C are collinear points.
(1)
6. i) tan-1 x + tan -1 y = ..................
(1)
-1
-1
ii) Using the above property prove that tan 2 11 + tan 7 24 = tan 1⁄2
(2)
7. Find the principal value of cos-1(-½)
(1)
8. Find the shortest distance between the lines x + 1= y +1= z +1 and
x -3= y -5 = z -7
7
-6
1
1
-2
1
(3)
OR
9. Find the vector equation of the plane passing through the intersection of the
two planes r.(î+ĵ+k)=6 and r.(2î+3ĵ+4k)=-5 and through the point (1,1,1).
(3)
10. (a) Find the Cartesian equation of the plane passing through the points
A(2,2,-1) ,B(3,4,2) and C(7,0,6).
(3)
(b) Find the distance of the point (2,3,4) from the plane 3x-6y+2z+11=0.
(2)
11. Consider the binary operation ^ on the set {1,2,3,4,5} defined by
a ^ b = min{ a,b}. Write the operation table of the operation ^.
(2)
12. Consider the function f:A→B defined by f(x)=
x−2
.Show that f is
x −3
invertible. Find the inverse of f.
(3)
13.a) Examine the consistency of the system of equations, 2x + 5y = 13
x + 2y = 7. (1)
b) If consistent, solve x and y using matrix method.
(2)
x+a x x
14. Using properties of determinants prove that,
x x+a x =(3x+a)a 2 . (3)
x x x+a
15. Find the relationship between a and b so that the function f defined by
ax+1 , x ≤ 3
f(x) =
is continuous at x = 3.
(2)
bx+3 , x > 3
16. If y= Sin(Sinx) Prove that y2 + y1tanx + ycos2x =0.
(2)
17. A box is constructed from a rectangular metal sheet of 21 cm×16 cm by
cutting equal squares of sides x cm from the corners of the sheet and then turning
up the projected portions. For what value of x the volume of the box will be
maximum ?
(2)
2
18. Verify Rolle's theorem for the function f(x)=x +2x-8, x є [-4,2].
19. (a) Find the equation of the tangent line to the curve y=x2-2x+7 which is
perpendicular to the line 5y-15x=13.
3
(2)
(2)
2
(b) Find the intervals in which the function f(x)= 4x -6x -72x+30,is strictly
increasing.
(2)
20. (a)Fill in the blanks.
∫cosecx dx =..................
(1)
(b) Evaluate the integrals.
(i)∫sin2x cos4x dx
(ii) x
dx
(4)
(x+1) (x+2)
21. (a) Two persons Ram and Sam appear in an interview for two vacancies in the
1
same post. The probability of Ram's selection is 7 and that of Sam's
1
selection is 5. What is the probability that only one is selected ?
(3)
(b) A die is thrown twice. Let E be the event that ' the number 5 appears at least
once' and F be the event that ' the sum of the numbers appearing is 8'.Find
P ( E∩F ) and P ( F ⁄ E ).
(2)
OR
(a) In a bulb factory machines A, B and C manufacture 60%, 30% and10% bulbs
respectively. 1%, 2% and 3% of the bulbs produced by A, B and C
respectively are defective. A bulb is drawn at random from the total
production and found to be defective. Find the probability that this bulb has
been produced by machine A.
(3)
(b) A random variable X has the following probability distribution.
X
0
1
2
3
4
5
P(X) 0
k
2k
2k
3k
k
6
2
7
2
2k
Find the value of k and the mean of the random variable X.
π/2
22.(a) Evaluate
sin5x dx
5
sin x + cos5x
0
2
7k +k
(2)
(2)
(b) Find the area of the ellipse x2 + y2 = 1.
(4)
2
2
5
a
b
(c)Evaluate (x+1) dx as the limit of a sum.
(3)
0
23. (a) Consider the differential equation dy + y = x2
dx x
(i) Find its order and degree.
(1)
(ii) Find the integrating factor.
(1)
(iii) Solve the given equation.
(2)
(b) Form the differential equation representing the family of curves y=a sin(x+b),
where a and b are arbitrary constants.
(2)
24. Consider the linear programming problem :
Maximize z= 4x+3y subject to 2x+y ≤ 10, x+y ≤ 8, x ≥ 0, y≥ 0.
a) Draw the graph of the lines 2x+y= 10 and x+y= 8.
(2)
b) Solve this linear programming problem graphically.
(2)
25. A diet is to contain at least 80 units of vitamin A and 100 units of minerals. Two
foods F1 and F2 are available. Food F1 costs Rs.4 per unit and F2 costs Rs.6 per
unit. One unit of food F1 contains 3 units of vitamin A and 4 units of minerals. One
unit of food F2 contains 6 units of vitamin A and 3 units of minerals. If the problem
is to find the minimum cost for diet that consists of mixture of these two foods and
also meets the minimal nutritional requirements, Write the objective function and
constraints.
(2)
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