GURU HARKRISHAN PUBLIC SCHOOL
TILAK NAGAR
:
NEW DELHI
MOCK EXAMINATION (2014-15)
CLASS-XII
M.M.-100
SUB-.MATHEMATICS
TIME-3Hr.
General Instructions:The paper consists of three sections.
Section A has six questions of one mark each.
Section B has thirteen questions of four marks each.
Section C has seven questions of six marks each.
Q.1
Q.2
Q.3
Q.4
SECTION –A
Find the value of Sin -1 (Sin (3π) )
5
Find a vector in the direction of the vector 2 L-J+ 2 K which has magnitude 7 units.
Construct a 2 X 3 matrix, whose elements are
Aij= 1 | – 3 i + j|
2
The total revenue in Rupee received from the sale of x units of a product is given by
R (x) = 13 x2 + 26x + 15
Find the marginal revenue when x=7
Q.5
Find the value of cot ( tan-1 a + cot -1 a )
Q.6
Evaluate ∫Sec 2 (7-4x) dn.
SECTION-B
Q.7
-1
Prove that tan
Where
Q.8
Using elementary transformations, find the inverse of
2
[
−1
Q.9
−3
]
2
Differentiate y= x Sin x + (Sin x) Cos x w.r.t ‘x’
Q.10 Using properties of determinants, show that
1
x
x2
2
x
1
x
= (1 – x 3) 2
2
x x
1
Q.11 Find the value of K so that the function
f (x) = k x + 1
x≤π
Cos x
is continous at x= π
x>π
Q.12 If y = Sin -1 x, show that
(1-x2) d2y – x dy = o
dx2
dx
Q.13 If Cos y = x Cos (a+ y)
; Cos a ≠ + 1
Prove that dy = Cos 2 (a + y)
dx
Sin a
Q.14 Find the interval in which the function
f (x) = -4x3 + 6x2+ 72 x -30
is (i) strictly increasing
(ii) strictly decreasing
Show that the function y= 4 Sin θ – θ is
2+Cos θ
Increasing function of θ in [0, π ]
2
2
Q.15 Prove that the curves x=y and x y = K cut at right angles if 8K2 = 1
OR
Find the equation of the tangent to the curve
Y= x-7
(x-2) (x-3)
at the point where it cuts the x axis.
Q.16 Evaluate
Q.17 Evaluate
Q.18 Find the shortest distance between the lines
r = (1- t L + ( t-2) J + ( 3 -2 t) K
r = (S+1) L + ( 2 s-1) J – ( 2s+ 1) K
Q.19 Find the distance between the point P ( 6, 5, 9 ) and the plane determined by the
points A (3, -1, 2); B (5, 2, 4) and C (-1, -1, 6)
SECTION C
Q.20 Evaluate
Q.21 Evaluate as a limit of Sum:
Q.22 Find the equation of the plane which contains the line of intersection of the planes
r. (l + 2 J + 3K)-4=0
r. (2L+J-K)+ 5=0
and which is perpendicular to the plane
r.(5L+3J-6K)+ 8=0
Q.23 Two adjacent sides of a parallelogram are
2L-4J+5K and L-2J+3K.
Find the unit vector parallel to its diagonal. Also, find the area of the Parallelogram.
Q.24 A company produces three products every day. These production on a certain day 45
tone. It is found that the production of the third product exceeds the production of
the first product by 8 tones while the total production of the first and third product is
twice the production of second product. Determine the level of production of each
product using matrix method.
Q.25 A dietician has to develop a special diet using two foods P and Q. Each packet
(containing 30 g) of food P contains 12 units of calcium, 4 units of iron, 6 units of
cholesterol and 6 units of vitamin A. Each packet of the same quantity of food Q
contains 3 units of calcium, 20 units of iron, 4 units of cholesterol and 3 units of
Vitamin A. The diet requires at least 240 units of calcium, at least 460 units of iron
and at most 300 units of cholesterol. How many packets of each food should be used
to minimize the amount of vitamin A in the diet?
Q.26 Show that the Semi-vertical angle of the cone of maximum volume and of given slant
height is
OR
A point on the hypotenuse of a triangle is at a distance ‘a’ and ‘b’ from the sides of
the triangle. Show that the minimum length of the hypotenuse is
(a 2/3 + b 2/3 ) 3/2