EINSTEIN CLASSES P R E S E N T S C B S E XIIth Board PRACTICE ASSIGNMENT MATHEMATICS NOTE THE FOLLOWING POINTS : Einstein Classes is primarily concerned with the preparation of JEE-ADVANCE /JEE-MAIN/BITS/PMT/AIIMS We also support the School Syllabus & XIIth Board Preparation in the same program This assignment is meant to help for those students who are appearing for XIIth CBSE Board Eamination Einstein Classes, Unit No -, Vardhman Ring Road Plaza, Vikas Puri Etn, Outer Ring Road, New Delhi 8 Ph : 9695 ()
Each question carry mark SECTION A Which of the following graphs represent a function? Write the range of one branch of cos, other than the principal branch 5 If A and B, find A B 4 7 4 If a b, find the values of a and b 5 cos75 5 Find the value of : sin 75 sin 75 cos75 6 If the radius of a circle is increasing at the rate of 7 mm/sec, at what rate is the circumference changing? 7 Evaluate : e log 8 What are the direction cosines of the vector î 6ĵ kˆ 9 Find a unit vector parallel to the sum of the vectors a î 4ĵ 5kˆ and b î ĵ kˆ The cartesian equation of a line AB are the line AB 4 y z Find the direction ratios of 7 5 If A, determine whether A + A is symmetric or skew symmetric 7 If A is a square matri of order such that adj A =, find A Give an eample of two non-zero matrices A and B such that AB = 4, Is the function f defined by f () continuous at =?, Einstein Classes, Unit No -, Vardhman Ring Road Plaza, Vikas Puri Etn, Outer Ring Road, New Delhi 8 Ph : 9695 ()
5 Evaluate : 9 6 6 Write the order and degree of the differential equation : / dy d y a 7 Find P(A B) if P(B) = 5 and P(A B) = 5 8 Two balls are drawn from a bag containing 4 red, 4 blue and 5 green balls What is the probability that both are green? 9 Write the value of î (ĵ kˆ ) ĵ (kˆ î) kˆ (ĵ î) If A = {,, } and B = {, 5}, then find the number of functions from A to B Find the value of cot (sin + cos ), If A, B are square matrices of equal order and B is skew symmetric, then show that ABA is also skew symmetric If A is a square matri or order and A =, find the value of 5A sin cos 4 If A, < < and A + A = I, where I is a unit matri, find the value cos sin of 5 Evaluate : 6 If 6 sin (5 7 ) = k cos(5 7 ) C 5, then find the value of k 7 The probabilities that a husband and wife will be alive years from now are 7 and 8 Find the probability that in years, husband will be widower 8 A number if chosen from the first natural numbers What is the probability that it is divisible both by and? 9 Find the differential equation of the family of concentric circles + y = r, where r (> ) is an arbitrary constant If f () 4 and f( ) =, find f() 4 tan, If f () k, find the value of k So that the function is continuous at =, Einstein Classes, Unit No -, Vardhman Ring Road Plaza, Vikas Puri Etn, Outer Ring Road, New Delhi 8 Ph : 9695 ()
k For what value of k, the matri 5 has no inverse? If is a cube root of unity, find the value of Each equation carry 4 marks SECTION B Show that the function f : R R defined by 5 for all R is bijective If f : R R and g : R R are given by f() = and g() = 5, find gof and fog Also show that gof fog Prove that : tan cos 4 a a bc 4 Without epanding show that : b b ca c c ab a b c a a 5 Prove that : b b c a b (a b c) c c c a b 6 If cos () f sin, k, is continuous at = Find the value of k 7 Differentiable sin sin tan wrt sin sin n 8 If y ( a ), prove that dy ny a 9 Find the equation of the tangent to the curve y a at the point a a, 4 4 Evaluate : sin 4 Evaluate : ( ) Einstein Classes, Unit No -, Vardhman Ring Road Plaza, Vikas Puri Etn, Outer Ring Road, New Delhi 8 Ph : 9695 (4)
Evaluate : f(), where f() = + + + + 5 5 In OAB, kˆ OA î ĵ and OB î ĵ kˆ Find the area of OAB 4 Show that the line r î ĵ (7î 5kˆ ) lies in the plane r (5î ĵ 7kˆ ) 5 A problem in mathematics is given to three candidates whose chances of solving it are, and respectively If all the three try to solve the problem simultaneously, find the 7 8 probability that eactly one of them can solve it 6 A can hit a target 4 times in 5 shots, B times in 4 shots and C twice in shots They fire a volley What is the probability that atleast shots hit the target? 7 Evaluate : sincos 5 8 If A and I, then find k so that A = ka I 4 6 5 9 If A, show that A A + I = O Hence find A 7 6 For what value of k, is the function cos4, f() 8 continuous at = k, If y = d y dy y, show that y Differentiate (log ) cos wrt A particle moves along the curve 6y = + Find the points on the curve at which y coordinate is changing 8 times as fast as the -coordinate sin(tan ) 4 Evaluate : 8 5 Evaluate : (e ) 6 Prove that : 7 Prove that : / 4 4 log( tan)d log 8 log log Einstein Classes, Unit No -, Vardhman Ring Road Plaza, Vikas Puri Etn, Outer Ring Road, New Delhi 8 Ph : 9695 (5)
8 Solve the differential equation cos dy + y sin = sin cos 9 Find the projection of b c on a where a î ĵ kˆ, b î ĵ kˆ and c î ĵ 4 kˆ Find the relation between and such that a b is perpendicular to c where a î ĵ kˆ, b î ĵ kˆ and c î ĵ kˆ Find the probability distribution of the number of heads in a single throw of three coins? The Central Board of Secondardy Education has a list of eaminers of 5 persons Out of these 5 are women 5 of them know Hindi and the remaining do not know Hindi 9 of them are working teachers and the remaining are retired teachers What is the probability of selecting a Hindi knowing working woman teacher as eaminer? Let N be the set of natural numbers and R be the relation on N N defined by (a, b) R (c, d) if ad = bc for all a, b, c, d N Show that R is an equivalence relation 4 Prove that tan tan cos 4 9 5 5 Using properties of determinants, solve for a a a a a a a a a If, y, z are different and y y y, show that yz + = z z z 6 Determine the values of a and b so that the function a b, f(), a b, may be continuous 7 If y sin tan, prove that dy 8 Discuss the applicability of Rolle s theorem for the function f() = in [, ] sin cos 9 Evaluate : a cos b sin, a ± b 4 Evaluate : tan Einstein Classes, Unit No -, Vardhman Ring Road Plaza, Vikas Puri Etn, Outer Ring Road, New Delhi 8 Ph : 9695 (6)
4 Solve the differential equation : ( + e )dy + ( + y )e = given that y = when = dy 4 Solve the differential equation sin( y) cos( y) 4 Form the differential equation of the family of curves y = A cos + B sin, where A and B are arbitrary constants 44 If a and b are two vectors, then prove that a b a b 45 If a,b, c are vectors such that a b a c,ab a c and a, then prove that b c 46 Find the point where the line joining the points (,, 4) and (, 5, ) intersects the plane rˆ (î ĵ kˆ ) Is this point equidistant from the given points? 47 A and B toss a coin alternately till one of them tosses a head and wins the game If A starts the game, find their respective probabilities of winning 4 48 Prove that cot cot 7 49 If f() =, find f(a) when A Hence find A 5 For what value of k is the function, k cos, if f () continuous at, if 5 Differentiate tan wrt sin cos 5 Differentiate log e wrt 5 Find the maimum and minimum values of f() = + in 5 Find the points of maima and minima also (log( ) log ) 54 Evaluate :, 4 / 55 Evaluate : / 4 sin 56 Evaluate : 8 6 Einstein Classes, Unit No -, Vardhman Ring Road Plaza, Vikas Puri Etn, Outer Ring Road, New Delhi 8 Ph : 9695 (7)
dy 57 Solve the differential equation log( ) 58 Form the differential equation not containing any arbitrary constant and satisfied by the equation y = a(b ), where a and b are arbitrary constants 59 Find a vector whose magnitude is units and which is perpendicular to the vector a and b where a î ĵ 4kˆ and b 6î 5ĵ kˆ 6 If a and b are any two vectors, prove that a a a b (a b) a b b b 6 Two cards are drawn from a deck of 5 cards one after the other without replacement Find the probability that one of these cards is an ace and the other is king of other colour 6 A machine operates if all its three components function The probability that the first component fails during the year is 4, the second component fails is and the third component fails is 5 What is the probability that the machine will fail during the year? 6 A random variable X has the following probability distribution Find the mean and variance : X P(X) /8 /8 /8 /8 Each question carry 6 marks SECTION C Find the inverse of the matrics p p and q q and hence find the inverse of the matri pq q p pq q p Find all the points of local maima and minima and the corresponding local maimum 4 9 and local minimum values of the function f() 4 A given rectangular area is to be fenced off in a field whose length lies along a straight river If no fencing is needed along the river, show that the least amount of fencing will be required when the length of the field is twice its breadth 4 Make a rough sketch of the region given below and find its area using the methods of integration : {(, y); y +, y +, } Einstein Classes, Unit No -, Vardhman Ring Road Plaza, Vikas Puri Etn, Outer Ring Road, New Delhi 8 Ph : 9695 (8)
y 5 Solve the differential equation : y log dy dy 6 A house wife wishes to mi together two kinds of food F and F in such a way that the miture will contain atleast units of vitamin A, units of vitamin B and 8 units of vitamin C The vitamin contents of each kilogram of food F and F are as under : Food Vitamin A Vitamin B Vitamin C Food F Food F On kg of food F costs Rs 6 and one kg of food F costs Rs Formulate the above problem as a linear programming problem, and use corner point method to find the least cost of the miture which will produce the diet 7 Find the equation of the plane passing through the line of intersection of the planes y + z = and + y + z = 8 and parallel to the line with direction ratios,, Also find the distance of P (,, ) from this plane measured along a line parallel to y z 5 8 Find the foot of the perpendicular drawn from P(,, ) on the line Also obtain the equations and length of perpendicular 6 y 7 z 7 9 A fair die is rolled If turns up, a ball is picked up at random from bag A If and turns up, a ball is picked up from bag B If 4, 5 or 6 turns up, a ball is picked up from bag C Bag A contains red and white balls; bag B contains red and 4 white balls bag C contains 4 red and 5 white balls The die is rolled and a bag is picked up and a ball is drawn (i) What are the chances of drawing a red ball? (ii) If the ball drawn is red, what are the chances that bag B was picked up? Let A = N N and * be a binary operation on A defined by (a, b) * (c, d) = (a + c, b + d) Show that * is commutative and associative Find the identity element for * on A, if any If If f() 4, 6 4 A 4 and, show that (fof) () = for all B 4 equations : y =, + y + 4z = 7 and y + z = 7 What is the inverse of f? 4 4, find AB and hence solve the system of 5 Find the interval in which the function f() = + 6 + 7 is strictly (a) increasing (b) decrasing Also find the points at which the tangents are parallel to -ais Einstein Classes, Unit No -, Vardhman Ring Road Plaza, Vikas Puri Etn, Outer Ring Road, New Delhi 8 Ph : 9695 (9)
4 Find the coordinates of the foot of the perpendicular drawn from the point A(, 8, 4) to the line joining the points B(,, ) and C(,, ) 5 Find the image of the point (,, ) in the plane + y + 4z = 8 6 Find the equation of the plane passing through the point (,, 5) and perpendicular to each of the planes r (î ĵ kˆ ) and r (î 4ĵ kˆ ) 5 7 A dealer wishes to purchase a number of fans and sewing machines He has only Rs 576 to invest and has space for at most items A fan and a sewing machine cost Rs 6 and Rs 4 respectively He can sell a fan at a profit of Rs and a sewing machine at a profit of Rs 8 Assuming that he can sell whatever he buys, how would he invest his money in order to maimise his profit? Translate the problem into LPP and solve it graphically 8 Find the equation of the line passing through the point (,, ) and parallel to the line r î ĵ (î ĵ 6kˆ ) Also find the distance between these lines 9 Find the area of the region bounded by = y, the y-ais and the lines y = and y = Find the smaller of the two areas in which the circle + y = 4 is divided by the parabola y = ( ) Evaluate ( ) as limit of sums An enemy helicopter is moving along the curve y = + 7 A soldier, placed at (, 7), wants to shoot down the helocopter when it is nearest to him Find the nearest distance Also find the coordinates of the helocopter when it is nearest to the soldier A company has two factories at P and Q and has three depots situated at A, B and C The weekly requirement at the depots A, B and C is respectively 5, 5 and 4 units, while the production capacity of the factories at P and Q are respectively 8 and 6 units The cost (in Rs) of transportation is given below : How many units should be transported from each factory to each depot in order that the transportation cost is minimum? 4 Find the matri A, satisfying the matri equation A 5 5 Three coins are tossed Consider the events E : Three heads or three trails, F : atleast two heads and G : at most two heads Of the pairs (E, F), (E, G) and (F, G), find which are dependent and which are independent Einstein Classes, Unit No -, Vardhman Ring Road Plaza, Vikas Puri Etn, Outer Ring Road, New Delhi 8 Ph : 9695 ()
6 Let X denote the number of hours you study on a Sunday Also it is known that where k is a constant (a) Find the value of k, if k, if or P(X ) k(5 ), if or 4, otherwise (b) What is the probability that you study atleast two hours? Eactly two hours? Atmost two hours? 7 Consider the function f : R + [ 5, ) defined by f() = 9 + 6 5, where R + is the set of all non-negative real number Show that f is invertible and find its inverse 8 Using matrices, solve the following system of equations : 4, y z 4 6 5, y z 6 9 y z 9 A given quantity of metal is to be cast into a half cylinder (ie, with rectangular base and semicircular ends) Show that in order that the total surface area may be minimum, the ratio of the length of the cylinder to the diameter of its circular ends is : ( + ) Show that the surface area of a closed cuboid with square base and given volume is minimum when it is a cubic Prove that curves y = 4 and = 4y divide the area of the square bounded by =, = 4, y =, y = 4 into three equal parts Show that the lines r î ĵ kˆ (î ĵ) and r 4î kˆ µ(î kˆ ) intersect Find their point of intersection Find the equation of the plane through the point (,, ) and passing through the intersection of r (î 7ĵ 4kˆ ) and r (î 5ĵ 4kˆ ) 4 y z y 7 z 7 Prove that the lines and are coplanar Also find the equation of the plane containing them 5 Kellogg is a new cereal formed of a miture of bran and rice that contains atleast 88 grams of protein and atleast 6 milligrams of iron per kilogram Knowing that bran contains 8 grams of protein and 4 milligrams of iron per kilogram and that rice contains grams of protein and milligrams of iron per kilogram Find the minimum cost of producing this new cereal if bran costs Rs 5 per kilogram and rice costs Rs 4 per kilogram Einstein Classes, Unit No -, Vardhman Ring Road Plaza, Vikas Puri Etn, Outer Ring Road, New Delhi 8 Ph : 9695 ()