SALIENTFEATURES

● It is effectivebecause it produces

minimal amounts of friction.

● Hydraulic lifts are cheap-er to install than elevator-

type machines.

● They occupy less space in abuilding, requiring almost 10%

less area for the lift shaft.

● They are highly effective with heavyloads, as the hydraulic power providesa far greater lifting strength.

06 “Press forward. Do not stop, do not linger in yourjourney, but strive for the mark set before you.”

– George Whitefield

SECTION-AQ.1 Write the direction ratios of the vector

3 a + 2 b where a = i + j 2 k and b = 2i 4j+5k

Q.2 Find the projection of the vector

a = 2i + 3j + 2k on the vector b = 2i + 2j + k.

Q.3 In the interval /2< x < , find the value of

x for which the matrix ( 2sinx 3 ) is singular.1 2sinx

Q.4 State the reason why the RelationR={(a,b): a b2} on the set R of real numbers isnot reflexive.

SECTION-B 5x , 2 < x < 1 ,

Q.5 If y = tan-1 1 6x2 6 6dy = 2 + 3

then prove that dx 1+4x2 1+9x2

Q.6 If tan-1 x + tan-1 y = /4, xy < 1, then write thevalue of x+y+xy.

[ x y z ] = [ 1 4 ]Q.7 If 2x y w 0 5 , find the value of x+y.

^ ^ ^Q.8 Find a. (b x c ), if a = 2 i + j + 3k, ^ ^ ^ ^ ^ ^b = i + 2 j + k and c = 3 i + i + 2 k.

4 x dxQ.9 Evaluate: x2 + 1

2

Q.10 Write the integrating factor of the differential equation x dy + y = e 2 x.

dx

2 , 1 , 1 ,Q.11 If P (A) = 5 P(B) = 3 P(A B) = 5

_ _then find P(A| B).

Q.12 Find dy when x = a ( -sin y = a (1+cosdx

SECTION-CQ.13 Find the value of dy at

dx 4if x = ae (sin cos ) and y = ae (sin + cos ).

Q.14 Find the value(s) of x for which y = [x (x-2)]2 is an increasing function.

OR

Find the equations of the tangent and normal to the curve x2 y2

= 1 at the point ( 2 a,b).a2 b2

Q.15 Find the particular solution of the differential equation dy = 1+x+y+xy,

dxgiven that y=0 when x = 1.

Q.16 Prove that

OR

If tan-1 ( x 2 ) + tan-1 ( x + 2 ) = , find the value x.x 4 x + 4 4

Q.17 Find x dx.(x2 +1) (x-1)

OR½

Find sin-1 x dx.0 ( 1 x2 )3/2

Q.18 Show that the four points A, B, C and D with po-sition vectors

4 i + 5 j + k, j k, 3 i + 9 j + 4 k and (-i + j + k)respectively are coplanar.

Q.19 Four cards are drawn successively with re-placement from a well shuffled deck of 52 cards.What is the probability that

(i) all the four cards are spades?(ii) only 2 cards are spades?

OR

A pair of dice is thrown 4 times. If getting a dou-blet is considered a success, find the probabilitydistribution of number of successes. Hence findthe mean of the distribution.

Q.20 Find x+ 3 dx.5 4x 2x2

Q.21 Using Properties of determinants, prove that

| x+y x x |5x+4y 4x 2x = x3

10x+8y 8x 3x

Q.22 A line passes through (2, 1, 3) and is perpen-dicular to the lines

r = (i + j k) + (2 i 2 j + k) and

r = (2 i j 3 k) + (i + 2 j + 2 k). obtain its equation in vector and cartesian form.

Q.23 Find the values of a and b, if the function f de-fined by

{ x2 + 3x + a, x 1 }f (x) = bx + 2, x > 1is differentiable at x = 1.

SECTION-DQ.24 Two schools A and B want to award their se-lected students on the values of sincerity, truthful-ness and helpfulness. The school A wants to award` x each, ` y each and ` z each for the three respec-tive values to 3, 2 and 1 students respectively with atotal award money of ` 1,600. School B wants to spend` 2,300 to award its 4, 1 and 3 students on the re-spective values (by giving the same award money tothe three values as before). If the total amount ofaward for one prize on each value is ̀ 900, using ma-trices, find the award money for each value. Apartfrom these three values, suggest one more valuewhich should be considered for award.

If f, g: R R be two functions defined as f(x) =|x| x and g(x) = |x| x, x R. Then find fog andgof. Hence find fog ( 3), fog (5) and gof ( 2).

Q.25 Show that the altitude of the right circular coneof maximum volume that can be inscribed in asphere of radius r is 4r/3. Also show that the maxi-mum volume of the cone is 8/27 of the volume ofthe sphere.

Q.26 Using integration find the area of the regionbounded by the triangle whose vertices are (-1, 2),(1, 5) and (3, 4).

Q.27 A manufacturer produces nuts and bolts. Ittakes 2 hours work on machine A and 3 hours on ma-chine B to produce a package of nuts. It takes 3 hourson machine A and 2 hours on machine B to producea package of bolts. He earns a profit of ` 24 per pack-age on nuts and ̀ 18 per package on bolts. How manypackages of each should be produced each day so asto maximize his profit. if he operates his machinesfor at the most 10 hours a day. Make an L.P.P fromabove and solve it graphically?

Q.28 Find the vector and cartesian equations of theplane passing through the line of intersection of theplanes

r. (2 i + 2 j - 3 k)= 7, r. (2 i + 5 j + 3 k) = 9such that the intercepts made by the plane on x-axisand z-axis are equal.

Q.29 Solve the differential equation.

(x sin2 (y) y) dx+x dy = 0 given y = when x= 1.x 4OR

Solve the differential equation dy 3y cot x = sin 2x given y = 2 when x = dx 2

GENERAL INSTRUCTIONS(i) All questions are compulsory.

(ii) This question paper contains 29 questions.

(iii) Question 1-4 in Section A are very short-answer type questions carrying 1 mark each.

(iv) Question 5-12 in Section B are short-answertype questions carrying 2 marks each.

(v) Question 13-23 in Section C are long-answer-Itype questions carrying 4 marks each.

(vi) Question 24-29 in Section D are long-answer-II type questions carrying 6 marks each.

These questions and the marks alongside aremeant for practice purpose only. Students are

advised to check format, syllabus and marks forBoard test papers with their teachers. Questions

have been given by teachers and NIE is notresponsible for them.

EVALUATING PROBABILITIESEVALUATING PROBABILITIESEVALUATING PROBABILITIESThe faculty of Silver Oaks International School, Hyderabad, presents a Mathematics question paper for you to practice

MOCK PAPERMATHEMATICS

CLASS XII, CBSEMARKS: 100, TIME: 3Hrs

HYDRAULIC LIFTWHAT IS IT?● A hydraulic lift is a typeof machine that uses ahydraulic apparatus to liftor move objects.

● It uses the forcecreated when pres-sure is exerted onliquid or air in apiston.

● It works on a basic principle: to go up,a pump pushes liquid or air into thecylinder, pushing the piston upwards.To go down, the valve opens, andliquid or air is allowed to flowback into the reservoir. It ispushed back using the grav-itational force of the lift.

● One of the physics equations that applies to hydraulic lifttechnology is “Pressure x Area = Force”

● It determinesthe pressure

required on theliquid or air in a

piston to pro-duce enough

force to lift andmove an object.

WHAT I NEEDED● Cardboard:

30 sq cm● Two wooden beadings of

15 cm length● Four syringes

● Two small tubes of

20 cm in length

HOW I WENT ABOUT IT● I used the cardboard as the base

● Two walls were created from the basewith enough space between them using thewooden beadings

● I had to stick the syringes to the insideof the walls (one on each wall) with thehub of the syringe facing the top andplunger of the syringe completely pulledout and almost touching the base

● The other two syringes were stuck tothe base on either side of the walls

● The plunger of the syringes at the basemust be closed completely, not pulled out

● The tubes were used to connect thehubs of the base syringe and wall syringe(same for both sides). Now a small piece ofcardboard had to be placed for connectingboth the plunger flanges of the verticalsyringes, thus acting as the base to carrythe loads.

● A small load is placed on the load carry-ing base; when you pull the plungers ofboth the base syringes, the load goes upand when you push the plungers of boththe base syringes, the load comes down.

HOW DOES IT WORK?

AADHITYA A, MG School of Excellence, Bengaluru

QDescribe briefly thestudy plan you fol-

lowed while preparing forthe boards?My only mantra for study-ing was consistency. I wouldstudy for three to fourhours every day with a fo-cus on clarifying concepts.

QWhat were the keychallenges you faced

and how did you tacklethem?When I have to mug-up any-thing I struggle. I prefer toprepare notes to memorizeand most importantly get asolid grasp on concepts.

QHow did you deal withthe pressure to per-

form?While I was fortunate to havezero pressure from my par-ents to study, sometimes youjust can’t avoid stress. I reg-ularly invest time on sportsand music. I play basketballand pool table. I am also amusic enthusiast, and playthe drums whenever I get

time. I recently recorded twosongs with my friends.

QHow did you keepyourself motivated?

One’s objective is enough tokeep you motivated. In thelast few years I have been in-terested in robotics and ar-tificial intelligence. I wouldalways have this goal inmind while studying.

QHow did you balanceacademics with other

activities?Time management is the key.I used to plan my day in ad-vance in such a way thatthere would always be timefor each activity.

QWhat tips would yougive future aspirants?

Identify your objective, planyour schedule and stick to it.Optimal mix of study andhobbies is the key to success.

Total 492-98.4% Chemistry 100Mathematics 100Physics 99

Prakhar in his leisuretime, loves to play thedrums and basketball.

MARKS IN CBSE CLASS XII

Prakhar Goel, PSBB LearningLeadershipAcademy,Bengaluru,shares hissuccess mantra

A solid grasp ofconcepts helps

Don’tbreak intoa sweat!

The best approachto the board ex-ams, in my opin-

ion, is to treat them likeany other exam and notbreak into a cold sweatevery time they come tomind. Time managementis key. Preparing atimetable and sticking toit will ensure that onehas enough time for bothwork and play. Hard workis, of course, essential.But more than that, Ithink it is smart workthat counts. Also, payingattention in class helps –this way we can graspkey concepts in subjectslike maths in school it-self and reduce the studyload.

BOARDLUES

NidhiKolachana,class X B, Delhi PublicSchool,Hyderabad

ApoorvaAnand, class XI, G DGoenkaPublicSchool,Lucknow

MOCK PAPERS

Istudied for at leastone to two hoursevery day. I tried to

clear all concepts and dis-cussed doubts withfriends. Keeping awayfrom distraction was achallenge, but I discov-ered there is no harm ifeverything is done inmoderation. There wasno pressure from my

family but they expectedme to score according tomy own expectations. Ibelieve in investing qual-ity time rather thanquantity time. Preparingfor boards does not meanI barred myself fromother activities. I partic-ipated in competitionswhich helped me gain ex-perience, but yes, I gavestudies the most impor-tance. Following a regu-lar pattern of study willhelp you score goodmarks in boards.

Qualityover

quantity

● One packet salted biscuits (nocream /sugar) ● 5-6 cheese slices● Tomato ketchup ● Thin sev ● Finely chopped tomato, corian-der and onion ● Oregano andchilli flakes

INGREDIENTS● Spread the biscuits on a dish and apply a layer oftomato ketchup. (Different sauces such as mayon-naise, hot and sour or Schezwan sauce can be used)● Cut the cheese slices into small squares and place

them on each biscuit ● Top it with some finely chopped tomato, onionand coriander● Sprinkle oregano and chilli flakes as per taste● Finally add some sev

Voila! The dish is ready to be served as a study time snack

Note: It should be consumed immediately

PREPARATION

Study time can often be taxing and tiring. This is the time when I feel the most hungry and keep looking fortidbits. I am a foodie and I crave something different every few hours to satisfy my hunger pangs. So Ihave come up with quite a few recipes, my favourite one being the open faced biscuit sandwich. It

requires no cooking and takes just five minutes to prepare.

SAANCHI DESAI, class X, Gopal Sharma International

School, Powai

Humble biscuit gets a makeover