This document provides information about a crash course for IIT-JEE, BITS, UPTU and AIPMT exams, including details about faculty, study material, practice problems, mock tests, and registration. It mentions that the course includes over 240 hours of training by experienced faculty, concise chapter-wise theory, 3000 practice problems, 10 full-length tests, and a 30% discount on registration until March 10th. The batch is scheduled to commence on March 15th, 22nd, and 29th.

1. Stable & Best Faculty
Best System
Proven Result
Ultimate Care
Near Jaipuriya Mall
CRASH COURSE for IIT-JEE,BITS,UPTU& AIPMT
Tips + Tricks + Time management
Training of more than 240 hrs. by highly experienced and best in class faculty.
Study material having concise chapterwise theory for clarity on key concepts & formulae.
Specialized Practice Booklets containing 3000 problems, forms the Course.
Expert's tips on time management and strategic planning will help students to focus and plan well.
10 full length tests on IIT‐Main pattern along with detail discussion.
Course Features
discount on
registration
till 10th March
30%
MOCK TEST PAPER
MATHEMATICS
(class12th)
Registration Fee : 2000/-
Registration Open :
Batch Commencement Data : 15th, 22nd & 29th March
Boost your IIT-JEE / AIPMT Preparation

2. Class – XII
Subject – Mathematics
(Three hours)
Section A – Answer Question 1 (compulsory) and Five other questions.
Section B & Section C – Answer two questions from either Section B or Section C.
All working, including rough work, should be done on the same sheet as, and adjacent to, the rest of the answer.
The intended marks for questions or parts of questions are given in brackets [ ].
SECTION A
Question 1. [103 = 30]
i. Find the value of x such that
1 x 1





15
2
1
3
5
3





2
1
3










x
2
1
=0
ii. Evaluate the following limit :
2
3
24
3
lim
xe
xe
x
x
x



iii. Find the equation of hyperbola if it passes through the points (6, 4) and (-3, 1).
iv. Differentiate 2
1
1
2
tan
x
x
y

 
w.r.t. x.
v. Evaluate the integral  
dx
xba
x
222
sin
2sin
vi. Find the equation of the ellipse whose centre is the origin, major axis is 9/2 and eccentricity
3
1
, where the
major axis is the horizontal axis.
vii. There are 10 persons who are to be seated around a circular table. Find the probability that two particular
persons will always sit together.
viii. The coefficient of rank correlation of marks obtained by 10 students in English and Economics was found
to be 0.5. it was later discovered that the difference in ranks in the two subjects obtained by one of the students
was wrongly taken as 3 instead of 7. find the correct coefficient of rank correlation.
ix. Solve the equation:
;
3
)32(
3
2)1(
i
i
iyi
i
ixi






Ryx ,,
x. Solve differential equations:
x
ey
dx
dy
x
1
tan2
)1(



3. Question 2.
(a)Using properties of determinants, show that
x
x
x
2
2
2
sin
sin
sin1
x
x
x
2
2
2
cos
cos1
cos

x
x
x
2sin41
2sin4
2sin4

= x2sin42  [5]
(b) Solve the following system of equations using matrices:
02
52752
9



zyx
zyx
zyx
[5]
Question 3.
(a) Find the equation to the pair of lines through the origin which are perpendicular to the lines represented
by .02 22
 byhxyax
[5]
(b) In Boolean algebra, prove that if caba  and ,''' caba  then b = c.
[5]
Question 4.
(a) If ,sinsinsin 111
 
zyx prove that
012 2222
 xyzzyx [5]
(b) Given that : ,
cossin
cossin
xx
xx
y


 show that
2
1 y
dx
dy
 [5]
Question 5:
(a) Verify Rolle’s theorem for the function
F(x) =
2/
)3( x
exx 
 in [-3, 0]. [5]
(b) Prove that the perimeter of a right angled triangle of given hypotenuse is maximum when the triangle is
isosceles. [5]
Question 6:
(a) Prove that 2log
81
)1log(
1
0
2




 dx
x
x
[5]
(b) Find the area cut off from the parabola
2
34 xy  by the line .1232  xy [5]
Question 7:
(a) Calculate Karl Pearson’s correlation coefficient between the marks in English and Hindi obtained by 10
students.
Marks in English 10 25 13 25 22 11 12 25 21 20
Marks in Hindi 12 22 16 15 18 18 17 23 24 17
(b) Find the regression coefficient yxb between x and y for the following data:
,24x ,44y ,306xy ,1642
x ,5742
y 4n [5]

4. Question 8:
(a) Two cards are drawn from a pack of 52 cards. What is the probability that either both are red or both
are kings? [5]
(b) The probability that a contractor will get a plumbing contract is 2/3, and the probability that he will not
get an electric contract in 5/9. If the probability of getting at least one contract is 4/5. What is the
probability that he will get both? [5]
Question 9:
(a) Given that:
idc
iba
in
in
i
i
i
i











2
2
2
2
2
)2(1
)12(1
................
41
31
21
1
Show that: 22
22
4
4
1)2(
1)12(
.............
257
82
17
2
dc
ba
n
n





 [5]
(b) Solve the differential equation
yx
yx
dx
dy
22
22
cos)sin1(
sin)cos1(


 [5]
SECTION B
Question 10.
(a) Prove by vector method that in any ABC , .coscos BaAbc  [5]
(b) The chances of X, Y, Z becoming managers of certain company are 4:2:3. The probabilities that bonus
scheme will be introduce if X, Y, Z becomes managers, are 0.3, 0.5 and 0.8 respectively. If the bonus
scheme has been introduced, what is the probability that X is appointed as the manager.
[5]
Question 11.
(a) Prove that  ba cb ac  = a b c 2
[5]
(b) Show that four points whose position vectors are ,ˆ7ˆ6 ji  ,ˆ4ˆ29ˆ16 kji 
,ˆ6ˆ3 kj  kji ˆ10ˆ5ˆ2  are coplanar. [5]
Question 12:
(a) In an automobile factory, certain parts are to be fixed to the chasis in a section before it moves into another
section. On a given day, one of the three persons A, B and C carries out this tasks A has 45%, B has 35% and C
has 20% chance of doing it. The probability that A, B and C will take more than the allotted time are ,
16
1
10
1
and
20
1
respectively. If it is found that none of them has taken more than what is the probability that A has
taken more time? [5]
(b) A and B play a game in which A’s chance of winning the game is .
5
3
In a series of 6 games, find the
probability that A will win atleast 4 games. [5]

5. SECTION C
Question 13:
(a) The banker’s gain on a certain bill due 6 months hence, is Rs. 10, the rate of interest being 10% per
annum. Find the face value of the bill. [5]
(b) A sum of Rs. 2522 is borrowed from a money lender at 5% p.a. compounded annually. If this amount is
to be paid back is 3 equal annually installments find the annual installments.
[5]
Question 14:
(a) Solve the LPP graphically
Min. 21 43 xxZ 
Subject to 321  xx
42 21  xx
0, 21 xx [5]
(b) The relation between total cost y and the output x is given by 5)
5
7
(3 



x
x
xy . Prove that the
marginal cost falls continuously as the output increase. [5]
Question 15:
(a) Calculate an index number for the second year, taking the first year as base, taking into account the
prices of the four commodities (in Rupees per kg) and the weight given here under:
A B C D
I Year 30 28 36 28
II Year 42 35 45 42
Weight 24 14 6 25
[5]
(b) The profits of a soft drink firm in thousands of rupees during each month of a year were.
Plot these on a graph. Calculate four-monthly moving averages and plot these on the same graph.
[5]
Jan. Feb. Mar. Apr May Jun. Jul Aug. Sep. Oct. Nov Dec.
1.2 0.8 1.4 1.6 2.0 2.4 3.6 4.8 3.4 1.8 0.8 1.2

6. Prakhar Goel
Student of APEX
One year classroom Program.
He is studying MBBS in Maulana
Azad Medical college Delhi.
Siddhant Rathore
CBSE : 91.6%
Student of APEX
ree year classroom Program.
He is studying Mechanical
Engineering in BITS Goa.
Dhwani Jain
Student of APEX
ree year classroom Program.
She is studying Chemical
Engineering in NUS Singapore.
Selected in NUS
National University of Singapore: 1st Rank in Asia
AIPMT AIR 427(GE)
Prerna Kashyap
CBSE : 96%
Student of APEX
Two year classroom Program.
She is studying in NIT Kurukshetra.
IIT-Main AIR 521(GE)
Apeksha Garg
CBSE : 96%
Student of APEX
One year classroom Program.
Vibhav Singh
CBSE : 96%
Student of APEX
One year classroom Program.
M-95, C-98, P-95 B-98, C-97, P-95
School : Cambridge Indirapuram
School : Amity Noida
Student of APEX
Four year classroom Program.
She is studying in Delhi Technical
Univercity(DTU) .
IIT-Main Score: 186
School : DPS Noida
School : Ryan Noida
STAR PERFORMERS OF THE YEAR
Arindham Roy
CBSE : 95%
Student of APEX
Two year classroom Program.
He is studying in NIT Patna.
IIT-Main Score : 158
School : Cambridge Indirapuram
BITS SCORE: 329
Sharvi Tomar
CBSE : 95.4%
Shubham Mukharjee
Student of APEX
One year classroom Program.
He is studying in IIT Guahati.
IIT(Adv.)AIR 1823(GE)
Shaurya Tomar
CBSE : 93.2%
Student of APEX
Four year classroom Program.
He is studying in Thapar
Institute Patiyala.
IIT-Main Score : 149
School : DPS Noida
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CBSE : 92.8%
Student of APEX
Four year classroom Program.
He is studying in Government
Engineering College Bilaspur.
IIT-Main Score :158
School : KV Noida
CBSE TOPPERS
68% Selections in IIT-Main
Selections in IIT-Advanced56%
60%Selections in AIPMT
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CBSE : 94%
Student of APEX
Twor year classroom Program.
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CBSE : 95.4%
Student of APEX
Four year classroom Program.
M-95, C-96, P-95 M-95, C-95, P-91
School : DPS Noida School : Modern school Noida
Shaurya Tomar
CBSE : 93.2%
Student of APEX
Four year classroom Program.
M-95, C-97, P-92
School : DPS Noida
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CBSE : 90.4%
Student of APEX
Two year classroom Program.
Areesha
CBSE : 91%
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Deepti
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Aarushi
CBSE : 88%
School : St. Teresa Indirapuram School : Ralli Int. schoolSchool : St. Francis Indirapuram
Student of APEX
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Student of APEX
Four year classroom Program.
Student of APEX
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