Quiz about Maths in Class 12

Question 1

For any two complex numbers z_(1) & z_(2) and any two real numbers a b |az_(1)- bz_(2)|^(2) + |bz_(1) + az_(2)|^(2) =

Accepted Answer

NA

Question 2

How many isomer of molecular formula C_(3)H_(6)BrCl (Including stereo isomer)?

Accepted Answer

NA

Question 3

if cosx=-45 wherex in[ 0pi] then the valueofcos(x2) isequal to

Accepted Answer

NA

Question 4

A vertical tower standing at O has marks P Q R S at heights of 1m 2m 3m and 4m from the foot O and A is a point on the horizontal plane through O. If PQ and RS subtend angles alpha and beta respectively at A where OA = 2m then cos (alpha + beta) =

Accepted Answer

NA

Question 5

If the line lx+my=1 is a normal to the hyperbola x^(2)a^(2)-y^(2)b^(2)=1 then a^(2)l^(2) -b^(2)m^(2) is equal to

Accepted Answer

NA

Question 6

Compute the integrals int_(0)^(1) x I n (1 + x^(2)) dx

Accepted Answer

NA

Question 7

The value of int_I^a [x] f (x) dx a > 1 where [x] denotes the greatest integer not exceeding x is

Accepted Answer

NA

Question 8

The value of int_(0)^(pi2)(1+2cosx)((2+cosx)^(2))dx is

Accepted Answer

NA

Question 9

Solve y - x (dy)( dx) = 2 (1 + x^(2) (dy)( dx))

Accepted Answer

NA

Question 10

If xyz are real and x+iy=(x+iz)(x-iy) then

Accepted Answer

NA

Question 11

If p=sin(989^(@))cos(991^(@))then which one of the following is correct ?

Accepted Answer

NA

Question 12

Prove that the coefficient of x^(n) in the expansion of (1 + x)^(2n) is twice the coefficient of x^(n) in the expansion of (1 + x)^(2n - 1).

Accepted Answer

NA

Question 13

If higher powers of x^(2) are neglected then the value of log(1+x^(2))-log(1+x)-log(1-x)=

Accepted Answer

NA

Question 14

In the given sets the binary operation * is defined. Check the commutativity and associativityis each case for *  (i) In Z a*b=a-b  (ii) In Q a** b=1+ab  (iii) In Q a**b=(ab)(2)  (iv) In Z^(+) a**b =2^(ab) (v) In Z^(+) a**b=a^(b) (vi) In R--1 a **b=(a)(b+1)

Accepted Answer

NA

Question 15

If a_(1) a_(2)…a_(n) are in H.P. then a_(1).a_(2) + a_(2).a_(3) + a_(3).a_(4) + …+ a_(n-1) .a_(n)=

Accepted Answer

NA

Question 16

The solution of 3e^(x)cos^(2) ydx+(1-e^(x)) cot y dy = 0 is

Accepted Answer

NA

Question 17

Let f(x) = sgn (sgn (sgn x)). Then lim_(x rarr 0) f(x) is

Accepted Answer

NA

Question 18

The probability of a man hitting a target is (2)(5) . He fires at the target k times (k a given number) . Then the minimum k so that the probabilityof hitting the targer at least once is more than (7)(10) is

Accepted Answer

NA

Question 19

The system of equations  x+y+z=5 x+2y+lamdaz=mu x+2y+3z=9 has  (i) unique solution of lamda!=3  (ii) infinitely many solutions if lamda=3 mu=9  (iii) no solution if lamda=3 mu!=9 From the above statements number of correct statements is

Accepted Answer

NA

Question 20

Integrate the following functions  intsqrt(tanx)(1+tan^(2)x)dx

Accepted Answer

NA

Question 21

Find the area of the region BOBRESB is enclosed by the ellipse and the lines x = 0 and x = ae where b^2 = a^2 (1- e^2) and e

Accepted Answer

NA

Question 22

While finding the Sine of a certain angle x an absent minded professor failed to notice hat his calculator was not in the correct angular mode. However he was lucky to get the right answer. The two least positive values of x for which the Sine of x degrees is the same as the Sine of x radians were found by him as ( m pi )( n - pi )and ( p pi )( q+ pi ) where mn p and q are positive integers . Suppose ( mn )( pq) be denoted by the quantity L. Now answer the following questions. Assume that f(x) is differentiable for all x.The sign of f(x) is as follows  f(x) gt 0on ( - oo -4)  f(x) lt 0on ( - 4 6)  f (x ) gt 0 on ( 6 oo)  Let g(x) = f ( 10-2x ) . The value of g ( L) is

Accepted Answer

NA

Question 23

While finding the Sine of a certain angle x an absent minded professor failed to notice hat his calculator was not in the correct angular mode. However he was lucky to get the right answer. The two least positive values of x for which the Sine of x degrees is the same as the Sine of x radians were found by him as ( m pi )( n - pi )and ( p pi )( q+ pi ) where mn p and q are positive integers . Suppose ( mn )( pq) be denoted by the quantity L. Now answer the following questions.  If x is measured in radians and underset( x rarroo)(Lim) ( sqrt( Ax^(2) + Bx ) - Cx) = L thenthe value of ( BC)(A) equals( ABC in R )

Accepted Answer

NA

Question 24

While finding the Sine of a certain angle x an absent minded professor failed to notice hat his calculator was not in the correct angular mode. However he was lucky to get the right answer. The two least positive values of x for which the Sine of x degrees is the same as the Sine of x radians were found by him as ( m pi )( n - pi )and ( p pi )( q+ pi ) where mn p and q are positive integers . Suppose ( mn )( pq) be denoted by the quantity L. Now answer the following questions.  The value of (m+n+p+q) is equal to

Accepted Answer

NA

Question 25

A box of oranges is inspected by examining three randomly selected oranges drawn without replacement. Ifall the three oranges are good the box is approved for sale otherwise it is rejected. Find the probability that a box containing 15 oranges out of which 12 are good and 3 are bad one will be approved for sale .

Accepted Answer

NA

Question 26

If A = 1pi [(sin^(-1)(xpi)tan^(-1)(xpi))(sin^(-1)(xpi)cot^(-1)(pix))] B = 1pi [(-cos^(-1)(xpi)tan^(-1)(xpi))(sin^(-1)(xpi)-tan^(-1)(pix))] then A-Bis equal to

Accepted Answer

NA

Question 27

Let ABC are three events such that P(A)=0.2P(B)=0.5P(C )=0.6P(A nn B)=0.15P(A nn C)=0.25P(AnnBnnC)=0.04.If P(AuuBuuCge0.7)then

Accepted Answer

NA

Question 28

The volume of a parallelepiped with edges vec(OA)=(31.4)vec(OB)=(123)vec(OC)=(215) is ……………

Accepted Answer

NA

Question 29

The value oflim_(n to oo)(1)(n^(4))(sum_(k=1)^(n)k^(2)int_(k)^(k+1)x ln[(x-k)(k+1-x)]dx)is negative inverse of

Accepted Answer

NA

Question 30

Two forces are equal to 2vec(OA) and 3vec(BO) their resultant being lambda vec(OG) where G is the point on AB such that (BG)(AG)=-(2)(3). What is the value of lambda?

Accepted Answer

NA

Question 31

Let X denote the sum of the numbers obtained when two fair dice are rolled.﻿ Find the variance and standard deviation of X.

Accepted Answer

NA

Question 32

The set of all values of a for which the functioln f(x) =(a^2-3a+2) (cos ^2x4- sin ^2x4)+ (a-1) x + sin 1 does not possess critical points is

Accepted Answer

NA

Question 33

Passage 1  The second section gives power to the President by and with the advice and consent of the Senate to make treaties provided two thirds of the Senators present concur…  Some are displleased with it not on account of any errors or defects in it but because they say the treaties will have the force of laws and thus should be made only by the legislature. These gentlemen seem not to consider that the judgement of our courts and the commissions constitutionally given by our governor are as valid and as binding on all persons whom they concern as the laws passed by our legislature. All constitutional acts of power whether in the executive or in the judicial department have as much legal validity and obligation as if they proceeded from the legislature. it surely does not follow that because the people have given the power of makin laws to the legislature they should therefore likewise give the legislature the power to do every other act of government by which the citizens are to the bound and affected. The President is to have power by and with the advice and consent of the Senate to make treaties provided two thirds of the senators present concur....  Though several writers o teh subject of government place the power of making treatis in the class of executive aunthorities this is evidently an arbitrary classicication for if we attend carefully to its operation it will be found to partake more of the legislative than of the executive character though it does not seem strictly to fall within the definition of either of them. The essence of teh legislative branch is to enact laws or in other words to prescribe rules for the regulation of teh society while the execution of the laws and the employment of the common strength either for this purpose of for the common defense seem to comprise all the functions of the executive branch. The power of making treaties is plainly neither the one nor the other. It relates neither to the execution of the existing laws nor to the creation of new ones and still less to an exertion of the common strength. Its objects are CONTRACTS with foreign nations which have the force of law but derive it from the obligations of good faith. They are not rules prescribed by the government of the citizen but agreements between two governments. The power in questin seems therefore to form a distinct department and to belong properly neither to the legislative nor to the exective branch. The qualties elsewhere detailed as indispensable in the management of foreign negotiations point to the Exective as the best agent in those transactions while the vast importance of the trust and the operation of treaties as laws plead strongly for the participation of the whole or a portion of the legislative body in the office of making them.  In Passage 1 Jays statemetn at lines 7-9 (they say ...legislature) and in Passage 2 hamiltons statement at lines 36-39 (The essence...cociety) both serve to help the authors

Accepted Answer

NA

Question 34

Out of any seats located symmetrically about the middle of the row at least one is empty then total of ways of seating

Accepted Answer

NA

Question 35

Passage 1  The second section gives power to the President by and with the advice and consent of the Senate to make treaties provided two thirds of the Senators present concur…  Some are displleased with it not on account of any errors or defects in it but because they say the treaties will have the force of laws and thus should be made only by the legislature. These gentlemen seem not to consider that the judgement of our courts and the commissions constitutionally given by our governor are as valid and as binding on all persons whom they concern as the laws passed by our legislature. All constitutional acts of power whether in the executive or in the judicial department have as much legal validity and obligation as if they proceeded from the legislature. it surely does not follow that because the people have given the power of makin laws to the legislature they should therefore likewise give the legislature the power to do every other act of government by which the citizens are to the bound and affected. The President is to have power by and with the advice and consent of the Senate to make treaties provided two thirds of the senators present concur....  Though several writers o teh subject of government place the power of making treatis in the class of executive aunthorities this is evidently an arbitrary classicication for if we attend carefully to its operation it will be found to partake more of the legislative than of the executive character though it does not seem strictly to fall within the definition of either of them. The essence of teh legislative branch is to enact laws or in other words to prescribe rules for the regulation of teh society while the execution of the laws and the employment of the common strength either for this purpose of for the common defense seem to comprise all the functions of the executive branch. The power of making treaties is plainly neither the one nor the other. It relates neither to the execution of the existing laws nor to the creation of new ones and still less to an exertion of the common strength. Its objects are CONTRACTS with foreign nations which have the force of law but derive it from the obligations of good faith. They are not rules prescribed by the government of the citizen but agreements between two governments. The power in questin seems therefore to form a distinct department and to belong properly neither to the legislative nor to the exective branch. The qualties elsewhere detailed as indispensable in the management of foreign negotiations point to the Exective as the best agent in those transactions while the vast importance of the trust and the operation of treaties as laws plead strongly for the participation of the whole or a portion of the legislative body in the office of making them.  Which choice best states the relationship between the two passages ?

Accepted Answer

NA

Question 36

Passage 1  The second section gives power to the President by and with the advice and consent of the Senate to make treaties provided two thirds of the Senators present concur…  Some are displleased with it not on account of any errors or defects in it but because they say the treaties will have the force of laws and thus should be made only by the legislature. These gentlemen seem not to consider that the judgement of our courts and the commissions constitutionally given by our governor are as valid and as binding on all persons whom they concern as the laws passed by our legislature. All constitutional acts of power whether in the executive or in the judicial department have as much legal validity and obligation as if they proceeded from the legislature. it surely does not follow that because the people have given the power of makin laws to the legislature they should therefore likewise give the legislature the power to do every other act of government by which the citizens are to the bound and affected. The President is to have power by and with the advice and consent of the Senate to make treaties provided two thirds of the senators present concur....  Though several writers o teh subject of government place the power of making treatis in the class of executive aunthorities this is evidently an arbitrary classicication for if we attend carefully to its operation it will be found to partake more of the legislative than of the executive character though it does not seem strictly to fall within the definition of either of them. The essence of teh legislative branch is to enact laws or in other words to prescribe rules for the regulation of teh society while the execution of the laws and the employment of the common strength either for this purpose of for the common defense seem to comprise all the functions of the executive branch. The power of making treaties is plainly neither the one nor the other. It relates neither to the execution of the existing laws nor to the creation of new ones and still less to an exertion of the common strength. Its objects are CONTRACTS with foreign nations which have the force of law but derive it from the obligations of good faith. They are not rules prescribed by the government of the citizen but agreements between two governments. The power in questin seems therefore to form a distinct department and to belong properly neither to the legislative nor to the exective branch. The qualties elsewhere detailed as indispensable in the management of foreign negotiations point to the Exective as the best agent in those transactions while the vast importance of the trust and the operation of treaties as laws plead strongly for the participation of the whole or a portion of the legislative body in the office of making them.  Based on Passage 2 how would Homilton most likely respond to jays statement that the decisions of the judicial and executive branches are as valid and as binding on all persons whom they concern as the laws passed by our legislature (lines 12-14)?

Accepted Answer

NA

Question 37

Passage 1  The second section gives power to the President by and with the advice and consent of the Senate to make treaties provided two thirds of the Senators present concur…  Some are displleased with it not on account of any errors or defects in it but because they say the treaties will have the force of laws and thus should be made only by the legislature. These gentlemen seem not to consider that the judgement of our courts and the commissions constitutionally given by our governor are as valid and as binding on all persons whom they concern as the laws passed by our legislature. All constitutional acts of power whether in the executive or in the judicial department have as much legal validity and obligation as if they proceeded from the legislature. it surely does not follow that because the people have given the power of makin laws to the legislature they should therefore likewise give the legislature the power to do every other act of government by which the citizens are to the bound and affected. The President is to have power by and with the advice and consent of the Senate to make treaties provided two thirds of the senators present concur....  Though several writers o teh subject of government place the power of making treatis in the class of executive aunthorities this is evidently an arbitrary classicication for if we attend carefully to its operation it will be found to partake more of the legislative than of the executive character though it does not seem strictly to fall within the definition of either of them. The essence of teh legislative branch is to enact laws or in other words to prescribe rules for the regulation of teh society while the execution of the laws and the employment of the common strength either for this purpose of for the common defense seem to comprise all the functions of the executive branch. The power of making treaties is plainly neither the one nor the other. It relates neither to the execution of the existing laws nor to the creation of new ones and still less to an exertion of the common strength. Its objects are CONTRACTS with foreign nations which have the force of law but derive it from the obligations of good faith. They are not rules prescribed by the government of the citizen but agreements between two governments. The power in questin seems therefore to form a distinct department and to belong properly neither to the legislative nor to the exective branch. The qualties elsewhere detailed as indispensable in the management of foreign negotiations point to the Exective as the best agent in those transactions while the vast importance of the trust and the operation of treaties as laws plead strongly for the participation of the whole or a portion of the legislative body in the office of making them.  As used in line 33 partkes most nearly means

Accepted Answer

NA

Question 38

10 boys are randomly divided into two grops containing 5 boys each. The probability that the two tallest boys are in the different groups is

Accepted Answer

NA

Question 39

Passage 1  The second section gives power to the President by and with the advice and consent of the Senate to make treaties provided two thirds of the Senators present concur…  Some are displleased with it not on account of any errors or defects in it but because they say the treaties will have the force of laws and thus should be made only by the legislature. These gentlemen seem not to consider that the judgement of our courts and the commissions constitutionally given by our governor are as valid and as binding on all persons whom they concern as the laws passed by our legislature. All constitutional acts of power whether in the executive or in the judicial department have as much legal validity and obligation as if they proceeded from the legislature. it surely does not follow that because the people have given the power of makin laws to the legislature they should therefore likewise give the legislature the power to do every other act of government by which the citizens are to the bound and affected. The President is to have power by and with the advice and consent of the Senate to make treaties provided two thirds of the senators present concur....  Though several writers o teh subject of government place the power of making treatis in the class of executive aunthorities this is evidently an arbitrary classicication for if we attend carefully to its operation it will be found to partake more of the legislative than of the executive character though it does not seem strictly to fall within the definition of either of them. The essence of teh legislative branch is to enact laws or in other words to prescribe rules for the regulation of teh society while the execution of the laws and the employment of the common strength either for this purpose of for the common defense seem to comprise all the functions of the executive branch. The power of making treaties is plainly neither the one nor the other. It relates neither to the execution of the existing laws nor to the creation of new ones and still less to an exertion of the common strength. Its objects are CONTRACTS with foreign nations which have the force of law but derive it from the obligations of good faith. They are not rules prescribed by the government of the citizen but agreements between two governments. The power in questin seems therefore to form a distinct department and to belong properly neither to the legislative nor to the exective branch. The qualties elsewhere detailed as indispensable in the management of foreign negotiations point to the Exective as the best agent in those transactions while the vast importance of the trust and the operation of treaties as laws plead strongly for the participation of the whole or a portion of the legislative body in the office of making them.  As used in line 64 office most nearly means

Accepted Answer

NA

Question 40

Find the number of permutations of the letters of the word CONTRADICTORY such that neither the pattern CON nor RAD nor ORY appears

Accepted Answer

NA

Question 41

Find the equation of a curve passing through the point (02) given that thesum ofthe coordinates of any point on the curve exceeds the magnitude of the slope ofthe tangent to the curveat that point by 5.

Accepted Answer

NA

Question 42

int dxsqrt(2-x^(2))

Accepted Answer

NA

Question 43

int dxsqrt(3+4x^(2))

Accepted Answer

NA

Question 44

Let f_1 (xy)-=ax^2+2hxy+by^2=0and let f_(i+1) (xy)=0 denote the equation of the bisectors of f_i (xy)=0for all i=123….. f_3 (xy) =0 is

Accepted Answer

NA

Question 45

The equation of the hyperbola with is transverse axis parallel to x-axis and its centre is(3-2)the length of axes are 86 is

Accepted Answer

NA

Question 46

Let f_1 (xy)-=ax^2+2hxy+by^2=0and let f_(i+1) (xy)=0 denote the equation of the bisectors of f_i (xy)=0for all i=123…..  If f_(i+1) (xy)=0 repersents the equation of a pair of perpendicular lines then f_3(xy) =0 is same as

Accepted Answer

NA

Question 47

Let f_1 (xy)-=ax^2+2hxy+by^2=0and let f_(i+1) (xy)=0 denote the equation of the bisectors of f_i (xy)=0for all i=123…..  The value of sum_(n=2)^(5) (f_(n+2)(xy))(f_n(xy)) is

Accepted Answer

NA

Question 48

int dxsqrt(5+x^(2))

Accepted Answer

NA

Question 49

arg z= (pi)(2) if and only if z is a

Accepted Answer

NA

Question 50

arg z= (-pi)(2) if and only if z is a

Accepted Answer

NA

1.	For any two complex numbers z_(1) & z_(2) and any two real numbers a , b \|az_(1)- bz_(2)\|^(2) + \|bz_(1) + az_(2)\|^(2) =
Answer» `(a^(2) - B^(2)) (\|z_(1)\|^(2) + \|z_(2)\|^(2))` `(a^(2) + b^(2)) (\|z_(1)\|^(2) + \|z_(2)\|^(2))` `(a^(2) + b^(2)) (\|z_(1)\|^(2) - \|z_(2)\|^(2))` `(a^(2) -b^(2)) (\| z_(1)\|^(2) - \|z_(2)\|^(2))` ANSWER :B

2.	How many isomer of molecular formula C_(3)H_(6)BrCl (Including stereo isomer)?
Answer» ANSWER :`8.00`

3.	if cosx=-4/5 ,wherex in[ 0,pi] then the valueofcos(x/2) isequal to
Answer» `(1)/(10)` `(2)/(5)` `(1)/(sqrt(10))` `-(2)/(5)` Answer :C

4.	A vertical tower standing at O has marks P, Q, R, S at heights of 1m, 2m, 3m and 4m from the foot O and A is a point on the horizontal plane through O. If PQ and RS subtend angles alpha and beta respectively at A where OA = 2m then cos (alpha + beta) =
Answer» `5//sqrt(26)` `24//sqrt(650)` `23//sqrt(650)` `1//sqrt(26)` ANSWER :C

5.	If the line lx+my=1 is a normal to the hyperbola x^(2)/a^(2)-y^(2)/b^(2)=1" then "a^(2)/l^(2) -b^(2)/m^(2) is equal to
Answer» `a^(2)-B^(2)` `a^(2)+b^(2)` `(a^(2)+b^(2))^(2)` `(a^(2)-b^(2))^(2)` ANSWER :C

6.	Compute the integrals :int_(0)^(1) x I n (1 + x^(2)) dx
Answer» ANSWER :In `2 - (1)/(2)`

7.	The value of int_I^a [x] f' (x) dx, a > 1, where [x] denotes the greatest integer not exceeding x is :
Answer» `AF(a) - {F(1) + f(2) + …….+ f([a])}` `[a]f(a) - {f(1) + f(2) + …….+ f([a])}` `[a]f([a]) - {f(1) + f(2) + …….+ f([a])}` `af([a]) - {f(1) + f(2) + …….+ f([a])}` ANSWER :B

8.	The value of int_(0)^(pi//2)(1+2cosx)/((2+cosx)^(2))dx is
Answer» `-(1)/(2)` 2 `(1)/(2)` NONE of these Answer :C

9.	Solve : y - x (dy)/( dx) = 2 (1 + x^(2)"" (dy)/( dx))
Answer» ANSWER :`y - 2 = A* (2X )/(2x + 1)`

10.	If x,y,z are real and x+iy=(x+iz)(x-iy) then
Answer» `X^(2)+y^(2)=1` `y^(2)+Z^(2)=1` `x^(2)+z^(2)=1` `x^(2)+y^(2)=z^(2)` Answer :C

11.	If p=sin(989^(@))cos(991^(@)),then which one of the following is correct ?
Answer» <P>p is finite and positive p is finite and negative p =0 p is undefined Solution :GIVEN , p`=sin(989^(@))cos(991^(@))` Which can be WRITTEN as `=sin(1080^(@)-91)cos(1080^(@)-89^(@))` `=-sin91^(@)cos89^(@)` `=-sin(90^(@)+1^(@))cos89^(@)` `-=cos1^(@)cos89^(@)` As`cos1^(@)andcos89^(@)`are positive. therefore their product is also + ve Hence, p isfinite and negative .

15.	If a_(1), a_(2),…a_(n) are in H.P. then a_(1).a_(2) + a_(2).a_(3) + a_(3).a_(4) + …+ a_(n-1) .a_(n)=
Answer» `na_(1)a_(N)` `(n -1) a_(1)a_(n)` `n(a_(1) - a_(n))` `(n - 1) (a_(1) - a_(n))` ANSWER :B

16.	The solution of 3e^(x)cos^(2) ydx+(1-e^(x)) cot y dy = 0 is
Answer» `TAN y = c(e^(X) - 1)^(3)` `Tan y = c(e^(x) +1)^(3)` `Tan y = c(e^(x) - 1)^(2)` `COS y = c(e^(x) - 1)^(3)` Answer :A

19.	The system of equations x+y+z=5, x+2y+lamdaz=mu, x+2y+3z=9 has (i) unique solution of lamda!=3 (ii) infinitely many solutions if lamda=3, mu=9 (iii) no solution if lamda=3, mu!=9 From the above statements, number of correct statements is
Answer» 1 2 3 0 Solution :The system of equations `x+y+z=5` `x+2y+lamdaz=mu,x+2y+3z=9` is EQUIVALENT of `AX=B`, where `A=[(1,1,1),(1,2,lamda),(1,2,3)],X=[(x),(y),(z)],B=[(5),(mu),(9)]` Augmented matrix `[A\|B]=[(1,1,1),(1,2,lamda),(1,2,3)][(5),(mu),(9)]` `=[(1,1,1),(0,1,lamda-1),(0,1,2)][(5),(mu-5),(4)]` `{:((R_(2)toR_(2)-R_(1)),(R_(3)toR_(3)-R_(1))):}` `=[(1,1,1),(0,1,lamda-1),(0,0,3-lamda)][(5),(mu-5),(9-mu)]` `:.` Rank A=` rank `[A\|B]` if lamda!=3`, `:.` The system has UNIQUE solution if `lamda!=3` `:.` (i) is correct if `lamda=3,mu=9,` rand `A=` rank `[A\|B]=2` `lt` number of unknowns. `:.` The sytem has infinitely many solutions if `lamda=3,mu=9` If `lamda=3, mu!=9` rank `A=2`, rank `[A\|B]=3` `:.` The system has no solution.

12.	Prove that the coefficient of x^(n) in the expansion of (1 + x)^(2n) is twice the coefficient of x^(n) in the expansion of (1 + x)^(2n - 1).
Answer» ANSWER :The coefficient of `x^(N)` in the expansion of `(1 + x)^(2N)` is twice the coefficient of `x^(n)` in the expansion of `(1 + x)^(2n -1)`

13.	If higher powers of x^(2) are neglected, then the value of log(1+x^(2))-log(1+x)-log(1-x)=
Answer» `X^(2)` `(1)/(2)x^(2)` `2x^(2)` `3x^(2)` ANSWER :C

14.	In the given sets the binary operation * is defined. Check the commutativity and associativityis each case for : (i) In Z, ab=a-b (ii) In Q, a b=1+ab (iii) In Q, ab=(ab)/(2) (iv) In Z^(+), ab =2^(ab) (v) In Z^(+), ab=a^(b) (vi) In R-{-1}, a **b=(a)/(b+1)
Answer» Answer :(II), (III),(IV) are COMMUTATIVE `""` (iii) is associative.

17.	Let f(x) = sgn (sgn (sgn x)). Then lim_(x rarr 0) f(x) is :
Answer» 0 1 2 None of these ANSWER :D

18.	The probability of a man hitting a target is (2)/(5) . He fires at the target k times (k, a given number) . Then the minimum k, so that the probabilityof hitting the targer at least once is more than (7)/(10), is
Answer» 3 5 2 4 Answer :A

20.	Integrate the following functions : intsqrt(tanx)(1+tan^(2)x)dx
Answer» ANSWER :`(2)/(3)TAN^(3/2)x+c`

21.	Find the area of the region BOB'RESB is enclosed by the ellipse and the lines x = 0 and x = ae , where b^2 = a^2 (1- e^2) and e
Answer» ANSWER :`AB[esqrt(1-E^2)+SIN^(-1)e]`

22.	While finding the Sine of a certain angle x, an absent minded professor failed to notice hat his calculator was not in the correct angular mode. However he was lucky to get the right answer. The two least positive values of x for which the Sine of x degrees is the same as the Sine of x radians were found by him as ( m pi )/( n - pi )and ( p pi )/( q+ pi ) where m,n , p and q are positive integers . Suppose ( mn )/( pq) be denoted by the quantity 'L'. Now answer the following questions. Assume that f(x) is differentiable for all x.The sign of f'(x) is as follows : f'(x) gt 0on ( - oo , -4) f'(x) lt 0on ( - 4, 6) f' (x ) gt 0 on ( 6, oo) Let g(x) = f ( 10-2x ) . The value of g' ( L) is
Answer» POSITIVE negative zero the function g is not differentiable at X= 5 Answer :C

23.	While finding the Sine of a certain angle x, an absent minded professor failed to notice hat his calculator was not in the correct angular mode. However he was lucky to get the right answer. The two least positive values of x for which the Sine of x degrees is the same as the Sine of x radians were found by him as ( m pi )/( n - pi )and ( p pi )/( q+ pi ) where m,n , p and q are positive integers . Suppose ( mn )/( pq) be denoted by the quantity 'L'. Now answer the following questions. If x is measured in radians and underset( x rarroo)("Lim") ( sqrt( Ax^(2) + Bx ) - Cx) = L, thenthe value of ( BC)/(A) equals( A,B,C in R )
Answer» 4 2 `(1)/(2)` NONE of these Answer :B

24.	While finding the Sine of a certain angle x, an absent minded professor failed to notice hat his calculator was not in the correct angular mode. However he was lucky to get the right answer. The two least positive values of x for which the Sine of x degrees is the same as the Sine of x radians were found by him as ( m pi )/( n - pi )and ( p pi )/( q+ pi ) where m,n , p and q are positive integers . Suppose ( mn )/( pq) be denoted by the quantity 'L'. Now answer the following questions. The value of (m+n+p+q) is equal to
Answer» 720 900 1080 1260 Answer :B

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Let X denote the sum of the numbers obtained when two fair dice are rolled. Find the variance and standard deviation of X.

25.	A box of oranges is inspected by examining three randomly selected oranges drawn without replacement. Ifall the three oranges are good, the box is approved for sale otherwise it is rejected. Find the probability that a box containing 15 oranges out of which 12 are good and 3 are bad one will be approved for sale .
Answer» ANSWER :`(44)/(91)`

26.	If A = 1/pi [(sin^(-1)(xpi),tan^(-1)(x/pi)),(sin^(-1)(x/pi),cot^(-1)(pix))] B = 1/pi [(-cos^(-1)(xpi),tan^(-1)(x/pi)),(sin^(-1)(x/pi),-tan^(-1)(pix))] then A-Bis equal to :
Answer» I 0 2I `(1)/(2)I` ANSWER :D

27.	Let A,B,C are three events such that P(A)=0.2,P(B)=0.5,P(C )=0.6,P(A nn B)=0.15,P(A nn C)=0.25,P(AnnBnnC)=0.04.If P(AuuBuuCge0.7),then,
Answer» `0.05leP(BNNC)le0.5` `0.03leP(BnnC)le0.38` `0.06leP(BnnC)le0.24` `0.04leP(BnnC)le0.22` ANSWER :C

28.	The volume of a parallelepiped with edges vec(OA)=(3,1.4),vec(OB)=(1,2,3),vec(OC)=(2,1,5) is ……………
Answer» 10 `(5)/(3)` `-10` 1 Answer :A

29.	The value oflim_(n to oo){(1)/(n^(4))(sum_(k=1)^(n)k^(2)int_(k)^(k+1)x ln[(x-k)(k+1-x)]dx)}is negative inverse of
Answer» ANSWER :`0.5`

30.	Two forces are equal to 2vec(OA) and 3vec(BO), their resultant being lambda vec(OG), where G is the point on AB such that (BG)/(AG)=-(2)/(3). What is the value of lambda?
Answer» 1 -1 2 None of the above Solution :`vec(OG)=(2vec(OA)-3vec(OB))/(2-3)` `vec(OG)=(2vec(OA)-3vec(OB))/(-1)` `-vec(OG)=2vec(OA)-3vec(OB) " " ...(1)` `lambda vec(OG)=2vec(OA)+3vec(OB) "" ....(2)` ADDING (1) and (2) `(lambda-1)vec(OG)=4vec(OA)` `rArr vec(OA)=((lambda-1)/(4))vec(OG) " " ...(3)` SUBTRACTING(2) from (1) `(-1-lambda)vec(OG)=-6vec(OB)=((1+lambda))/(6)vec(OG) " " ...(4)` From EQU(2), (3) and (4) `lambdavec(OG)=2((lambda-1)/(4))vec(OG)+3((lambda+1)/(6))vec(OG)` `rArr lambda(lambda-1)/(2)=(lambda+1)/(3)` `:. lambda=(lambda-1)/(2) or lambda=(lambda+1)/(3)` `rArr 2lambda-lambda=-1 or 3lambda-lambda=1` `rArr lambda=-1 or lambda=(1)/(2)`

31.	Let X denote the sum of the numbers obtained when two fair dice are rolled. Find the variance and standard deviation of X.
Answer» ANSWER :VAR(X)=5.833, S.D=2.415

32.	The set of all values of a for which the functioln f(x) =(a^2-3a+2) (cos ^2x//4- sin ^2x//4)+ (a-1) x + sin 1 does not possess critical points is
Answer» `[1,OO]` `(0,1)CUP(1,4)` `(-2,4)` `(1,3) cup (3,5)` ANSWER :B

33.	Passage 1 The second section gives power to the President, "by and with the advice and consent of the Senate, to make treaties, provided two thirds of the Senators present concur…" Some are displleased with it, not on account of any errors or defects in it, but because they say the treaties will have the force of laws, and thus should be made only by the legislature. These gentlemen seem not to consider that the judgement of our courts, and the commissions constitutionally given by our governor, are as valid and as binding on all persons whom they concern as the laws passed by our legislature. All constitutional acts of power, whether in the executive or in the judicial department, have as much legal validity and obligation as if they proceeded from the legislature. it surely does not follow that because the people have given the power of makin laws to the legislature, they should therefore likewise give the legislature the power to do every other act of government by which the citizens are to the bound and affected. The President is to have power, "by and with the advice and consent of the Senate, to make treaties, provided two thirds of the senators present concur."... Though several writers o teh subject of government place the power of making treatis in the class of executive aunthorities, this is evidently an arbitrary classicication, for if we attend carefully to its operation, it will be found to partake more of the legislative than of the executive character, though it does not seem strictly to fall within the definition of either of them. The essence of teh legislative branch is to enact laws, or, in other words, to prescribe rules for the regulation of teh society, while the execution of the laws, and the employment of the common strength, either for this purpose of for the common defense, seem to comprise all the functions of the executive branch. The power of making treaties is, plainly, neither the one nor the other. It relates neither to the execution of the existing laws, nor to the creation of new ones, and still less to an exertion of the common strength. Its objects are CONTRACTS with foreign nations, which have the force of law, but derive it from the obligations of good faith. They are not rules prescribed by the government of the citizen, but agreements between two governments. The power in questin seems therefore to form a distinct department, and to belong, properly, neither to the legislative nor to the exective branch. The qualties elsewhere detailed as indispensable in the management of foreign negotiations point to the Exective as the best agent in those transactions, while the vast importance of the trust, and the operation of treaties as laws, plead strongly for the participation of the whole or a portion of the legislative body in the office of making them. In Passage 1, Jay's statemetn at lines 7-9 ("they say ...legislature") and in Passage 2, hamilton's statement at lines 36-39 ("The essence...cociety") both serve to help the authors
Answer» argue that the power to make treaties should reside exclusively with the LEGISLATURE. critique those who place in the power to make treaties in the class of executive authorities. DEFINE the WIDELY accepted role of the LEGISLATIVE BRANCH to make laws. support the contention that treaties carry the same legal authority as laws. Answer :C

34.	Out of any seats located symmetrically about the middle of the row at least one is empty then total of ways of seating
Answer» <P>`.^(m//2)C_(N).2^(n).n!` `.^(m)P_(n)` `.^(m//2)P_(n)` `.^(m)P_(n)-.^(m//2)P_(n)` ANSWER :a