.
Model test paper – 6
1.
2.
3.
Part – A
Without actual expansion, but by using properties of determinants only, prove that :
1 a a 2  bc
1 b b 2  ca  0
1 c c 2  ab
Write the statement for the following switching circuit. Simplify the statement. Construct the
switching circuit for a simplified statement.
a
a
a’
b’
a’
b’
There are three men aged 60, 65, 70 years. The probabilities to live 5 years more is 0.8, 0.6 and 0.3
respectively. Find the probability that at least two of three persons will remain alive 5 years hence.
4.
tan 3 x  2 x
x 0 3 x  sin 2 x
Evaluate Lt.
5.
Differentiate
x2  3x  1
w .r .t . x
x2  x  1
OR
sin 3 x
x 0
x
Evaluate Lt.
6.
Find the maximum value of sin x + cos x and value of x for which it is maximum.
7.
Evaluate:
8.
Slope of the tangent of a point P(x, y) on a curve is 
9.
10.
11.
sin 2 xdx
2
x  b sin 2 x
 a cos
x
. If the curve passes through the point (2,
y
3), find the equation of the curve.
 1 2 5 
1
Find A if A =  2 3 1  . Hence solve the following equations : –x  2 y  5z  2,
 1 1 1
2 x  3 y  z  15,  x  y  x  3
One shot is fired from each of three guns. Let A, B, C be the events that the target is hit by the
first, second and third gun respectively. Assuming that A, B, C are mutually independent events
and that P(A) = 0.5, P(B) = 0.6, P(C) = 0.8, find the probability that at least one hit is registered.
Let a function f be defined by
1  cos x
f  x 
when x  0  A, when x  0 Find A so that f is continuous at x = 0
x2




12.
If y  log x  x 2  a 2 , prove that a 2  x 2 . y2  xy1  0
13.
Find a point on y   x  2  where the tangent is parallel to the chord joining (2, 0) and (4, 4).
2
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.
14.
Evaluate  cos 1 xdx
OR
2
Evaluate
 xdx as the limit of a sum and show
that its value is the area of the trapezoidal region
1
15.
16.
17.
bounded by y  x , the x  axis and the two ordinates x  1 and x  2
Solve the differential equation :
d2 y
 sec 2 x  xe x
dx 2
1 x  1 x
Evaluate: Lt
x 0
sin 1 x
 1 3 4
1
T
Consider the matrix A =  3 1 6  . Prove that  AT    A1   T presents transpose 
 1 5 1 
/2
18.
Evaluate:
dx
 1  tan
3
0
x
OR
Find the area of the first quadrant bounded by the circle x 2  y 2  25 and the ellipse 9 x 2  16 y 2
 288. Find also the whole area between the curves.
PART – B
19.
20.
21.
22.
23.

 


Find the area of the parallelogram formed by the vectors 3i  2 j  k and i  3 j  5k
Find the equation of the sphere on the join of (2, 3, 5) and (4, 9, –3) as a diameter. Write the
coordinates of the centre and radius of the sphere.
OR
Show that the points (–3, –1, 0), (1, 2, 2), (1, –1, 1) and (–7, –1, –1) are coplanar. Also find the
equation of the plane containing these points.
Forces of magnitude 3, 6, 8, 11, 5 2 N act respectively along the sides AB, BC, CD, DA and the
diagonal AC of the square ABCD. Show that their resultant is a couple and find its moment.
A body is projected so that on its upward path, it passes through a point x ft, horizontally and y ft.
vertically from the point of projection. If R ft. is the range on the horizontal plane through the
R 
y
point of projection, show that angle of elevation of the projection is tan 1  .
.
 x R x


For two vectors a and b , prove that :
  
  2 2 2
  2 a .a a .b
a  b  a b  a .b     
a .b b .b

Show that the line L, whose vector equation is r  2iˆ  2 ˆj  3kˆ   iˆ  ˆj  4kˆ , is parallel to the
 
24.



plane  whose vector equation is r . iˆ  5 j  kˆ  5 , and find the distance between them.

25.

A stone dropped into a well, reaches the water with a velocity of 28 m/sec and the sound of its
55
seconds after it is let fall. Find the velocity of sound in air.
striking the water is heard
6
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.
26.
Three forces P, Q, R in one plane act on a particle, the angles between Q and R, R and P and P
and Q being between Q and R, R and P and P and Q being , ,  respectively. show that their
resultant is equal to (P² + Q² + R² + 2QR cos  + 2PR cos  +2PQ cos  )1/2.
Part –C
19.
20.
21.
22.
23.
24.
25.
26.
Four persons enter into partnership, the second has twice as large capital as the first and the third
as much as half the sum of the capital of first two and the fourth has a sum equal to capital of
other. Distribute total profits of Rs. 18,000 amongst the partners in proportion to capital.
The difference between the Banker’s discount and true discount on Rs 20, 200, paid 3 months
before it is due is Rs. 4. Find the rate percent.
OR
A bill for Rs. 4, 500 is due in 6 months. Find the difference in the amount between the Bankers
discount and true discount, the rate of interest being 5%. Show that the difference is equal to the
interest on the true discount for 6 months at 5%.
Minimize Z = 3x – y subject to constraints x  y  6, 2 x  y  8, 3 x  2 y  6, x  0, y  0
A man purchase a house and takes a mortgage on it for rs 60,000 to be paid off in 12 years by
equal annual payments. If the interest rate is 5% compounded annually, what amount will be
required to pay each year ?
A radio manufacturer finds that he can sell x radios per week at Rs.p. each, where p =

x2 
x

2  100   . His cost of production of x radios per week is Rs. 120 x 
 . show that his profit
2 
4


is maximum when the production is 40 radios per week. Find also his maximum profit per week.
The incidence of occupational disease in an industry is such that the workmen have a 20% chance
of suffering from it. What is the probability that out of 6 workers, 4 or more will contact the
disease ?
A garment manufacturer is planning production of new variety of shirts. It involves initially a
fixed cost of Rs. 1.5 lacs and a variable cost of Rs. 150 for producing each shirt. If each shirt can
be sold at Rs. 350, then find (i) cost function (ii) revenue function (iii) Profit function (iv) Breakeven point.
Between 4 and 5 P.M., the average number of phone calls per minute coming into the switch board
of a company is 3. Find the probability that during one particular minute there will be (a) No
Given e-3  0.498
phone call (b) Exactly 2 phone calls.
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