28278 + Interview Questions in Maths in Class 12

1.	Find the locus of the point of intersection of two tangents to the hyperbola x^(2)/a^(2)-y^(2)/b^(2)=1. which makes an angle alpha with one another.
Answer» Answer :`(X^(2)+y^2-a^(2)+B^(2))^(2)=(4a^(2)y^(2)-4B^(2)x^(2)+4a^(2)b^(2)) COT^(2) alpha`

Discussion

2.	Determine the truth of falsity of theA sub A for any set A propositions with reasons.
Answer» Solution :`A sub A ` is FALSE, as A is an improper SUBSET of itself, not a proper subset.

Discussion

3.	There are three coins. One is a two headed coin (having head on both faces), another is a biased coin that comes up heads 75% of the time and third is an unbiased coin. One of the three coins is chosen at random and tossed, it shows heads, what is the probability that it was the two headed coin ?
Answer» ANSWER :`(4)/(9)`

Discussion

4.	Let P and Q be point on the line joining A(5,6) and B (3, -4) such that AP=PQ=QB. Then the midpoint of PQ is
Answer» (4,2) (4,1) (3, 1) (2, 2) ANSWER :B

Discussion

5.	If \|z-2/z\|=2then the greatest value of \|z\| is
Answer» `SQRT3 - 1` `sqrt3` `sqrt3 + 1` `sqrt3 + 2` ANSWER :C

Discussion

6.	Show that [[a_1,b_1,-c_1],[-a_2,b_2,c_2],[a_3,b_3,-c_3]]= [[a_1,b_1,c_1],[a_2,b_2,c_2],[a_3,b_3,c_3]]
Answer» SOLUTION :`[[a_1,b_1,-c_1],[-a_2,-b_2,c_2],[a_3,b_3,-c_3]]=[[a_1,b_1,-c_1],[a_2,b_2,-c_2],[a_3,b_3,-c_3]](-1)` (TALING COMMON (-1) from `R_2`) `(-)(-)[[a_1,b_1,c_1],[a_2,b_2,c_2],[a_3,b_3,c_3]]=[[a_1,b_1,c_1],[a_2,b_2,c_2],[a_3,b_3,c_3]]` (TAKING common (-1) from `C_3`)

Discussion

7.	If f(x)=(x-2)/(x+2),x!= -2 then f^(-1)(x), is equal to
Answer» `(4(x+2))/(x-2)` `((x+2))/(4(x-2))` `(x+2)/(x-2)` `(2(1+x))/(1-x)` ANSWER :D

Discussion

8.	For the reaction CO(g) + CI_(2)(g) hArr COCI_(2)(g) the value of (K_(c))/(K_(P))is equal to :-
Answer» `SQRT(RT)` RT `(1)/(RT)` 1.0 Answer :A

Discussion

9.	Find theareaof the regionboundedby thecirclex^2+y^2=32,X - axisand the liney=x in the firstquadrant .
Answer» ANSWER :` 4 PI` SP . UNITS

Discussion

10.	Find the coefficient of x^3 in ( sqrt(x^5) + (3)/(sqrt(x^3)) )^6
Answer» ANSWER :540

Discussion

11.	Find the area bounded between the curves y=x^(2), y=sqrt(x)
Answer» ANSWER :`(1)/(3)`

Discussion

12.	If f'(a)=2 and f(a)=4 then lim_(xtoa)(xf(a)-af(x))/(x-a) equals to
Answer» `2a-4` `4-2a` `2a+4` `4+2a` ANSWER :B

Discussion

13.	For what values o m the equation (1+m)x^(2)-2(1+3m)x+(1+8m)=0 has (m epsilonR) (i) both roots are imaginary? (ii) both roots are equal? (iii) both roots are real and distinct? (iv) both roots are positive? (v) both roots are negative? (vi) roots are opposite in sign? (vii)roots are equal in magnitude but opposite in sign? (viii) atleast one root is positive? (iv) atleast one root is negative? (x) roots are in the ratio 2:3?
Answer» Solution :`:'(1+m)x^(2)-2(1+3m)x+(1+8m)=0` `impliesD=4(1+3m)^(2)-4(1+m)(1+8m)=4m(m-3)` (i) Both roots are IMAGINARY. `:.Dlt0` `implies4m(m-3)lt0` `implies0ltmlt3` or ` m epsilon (0,3)` (ii) Both roots are equal `:.D=0` `implies4m(m-3)=0` `impliesm=0,3` (iii) Both roots are real and distinct `:.Dgt0` `implies4x(m-3)gt0` `impliesmlt0` or `mgt3` `:.m epsilon(-oo,0)uu(3,oo)` (iv) Both roots are positive. CASE I Lsum of the roots `gt0` `implies(2(1+3m))/((1+m))gt0` `impliesm epsilon (-oo,-1)uu(-1/3,oo)` Case II Produc of the roots `gt0` `implies((1+8m))/((1+m))gt0` `m epsilon (-oo,-1)uu(-1/8,oo)` Case III` D ge0` `implies 4m(m-3)ge0` `m epsilon (-oo,0]uu[3,oo)` Combining all Cases, we get `m epsilon (-oo,-1)uu[3,oo)` (v) Both roots are negative. Consider the following cases: Case I Sum of the roots `lt 0implies(2(1+3m))/((1+m))lt0` `impliesm epsilon (-1,-1/3)` Case II PRODUCT of the roots `gt0implies((1+8m))/((1+m))gt0` `impliesm epsilon (-oo,1)uu(-1/8,oo)` Case III `Dge0` `4m(m-3)ge0impliesm epsilon (-oo,0]uu[3,oo)` Combining all cases we get `m epsilon phi` (vi) Roots are opposite in sign, then Case I Consider the following cases: Product of the roots `lt0` `implies((1+8m))/((1+m))lt0` ` m epsilon (-1,-1/8)` Case II `D gt0implies4m(m-3)gt0` `implies m epsilon (-oo,0)uu(3,oo)` Combining all cases we get `m epsilon (-1,-1/8)` (vii) Roots are equal in magnitude but opposite in sign, then Consider the following cases: Case I Sum of the roots `=0` `implies(2(1+3m))/((1+m))=0` `impliesm=-1/3, m !=1` Case `Dgt0implies4m(m-3)gt0` `impliesm epsilon (-oo,0)uu(3,oo)` Combining all cases we get `m=-1/3` (viii) Atleast one root is positive, then either one root si positive or bothh roots are positive i.e. (d) `uu`(f)` or `m epsilon (-oo,-1)uu(-1.-1/8)uu[3,oo)` (ix) Atleast one root is negative, then either one root is negative or both roots are negative. i.e. (e) `DUU` (f) or `m epsilon (-1,-1/8)` (x) Let roots are `2 alpha` are `3 alpha`. Then Consider the following cases: Case I Sum of the roots `=2alpha+3 alpha=(2(1+3m))/((1+m))` `implies alpha=(2(1+3m))/(5(1+m))` Case II Product of the roots `=2 alpha.3 alpha=((1+8m))/((1+m))` `implies6 alpha^(2)=((1+8m))/((1+m))` From Eqs i and ii we get `6{(2(1+3m))/(5(1+m))}^(2)=((1+8))/((1+m))` `implies24(1+3m)^(2)=25(1+8m)(1+m)` `implies24(9m^(2)+6m+1)=25(8m^(2)+9m+1)` `=16m^(2)-81m-1=0` or `m=(81+-sqrt((-81)^(2)+64))/32` `impliesm=(81+-sqrt(6625))/32`

Discussion

14.	The values of (1-sqrt3i)^(1//3) are
Answer» `2^(1//3)CIS[(2kpi+pi//3)/(3)],k=0,1,2` `2^(1//5)cis[(2kpi+pi//6)/(3)],k=0,1,2` `2^(1//3)cis[(2kpi-pi//3)/(3)],k=0,1,2` `2^(1//5)cis[(2kpi-pi//6)/(3)],k=0,1,2` ANSWER :C

Discussion

15.	A,B,C are 3 newspaper from a city. 20% of the population red A, 16% read B, 14% read C, 8% both A and B, 5% both A and C, 4% both B and C, 2% all the three. Find the percentage of the populations who read atleast one newspaper.
Answer» ANSWER :0.35

Discussion

16.	y = sin^(-1) x + sin^(-1)sqrt(1 - x^2), 0 lt x lt 1 find (dy)/(dx).
Answer» ANSWER :`0`

Discussion

17.	If a line y = x touches the curve y=x^(2)+bx+c at (1, 1) then ………..
Answer» B=1, c = 2 `b=-1, c=1` b = 1, c = 1 b = 0, c = 1 ANSWER :B

Discussion

18.	{:(,"Column-I",,"Column-II"),((A),BCl_(3),(P),3C -4e^(-)"bond is present in dimer"),((B),BeCl_(2),(Q),"Planar"),((C),PCl_(3),(R),"Change in hybridisation of the central atom during dimerisation"),((D),AsCl_(3),(S),"Tautomerism is observed during final product formation in hydrolysis process"),(,,(T),"Basicty of one of the hydrolysed product is three"):}
Answer» Solution :(A) `BCl_(3)` does not dimerise, it is `sp^(2)` hybridised so planar on hydrolysis it gives `H_(3)BO_(3)` whose basicity is one. (B) sp hybridised planar `BeCl_(2)` dimerise give `Be_(2)Cl_(4)` in which Be is `sp^(2)` hybridised and hydrolysis product is `Be(OH)_(2).` (C ) `sp^(3)` hybridised `AsCl_(3)` is pyramidal in shape which gives `H_(3)AsO_(3)` which does not show tautomerism due to large size of As and basicity of`H_(3)AsO_(3)` is THREE. ]

Discussion

19.	Find the order of the differential equation corresponding to (i) y = c(x-c)^(2) where c is an arbitrary constant. (ii) y = Ae^(x) + Be^(3x) + Ce^(5x) where A, B, C are arbitrary constant. (iii) xy = c e^(x) + b e^(-x) + x^(2) where b, c are arbitrary constants. (iv) The family of all circles in the xy-plane with centre at the origin.
Answer» Answer :(i) Since there is only one arbitrary constant, the order is 1 (ii) since there are 3 independent arbitrary constants order 3. (iii) Since there are 2 independent arbitrary constants order 2. (IV) `X^(2) + y^(2) = a^(2)`, a is a PARAMETER has only one arbitrary constant, order 1.

Discussion

20.	Let a and b natural numbers such that 2a - b, a - 2b and a + b are all distinct squares. What is the smallest possible value of b ?
Answer» ANSWER :21

Discussion

21.	Let A and B be two matrices different from identify matrix such that AB=BA and A^(n)-B^(n) is invertible for some positive integer 'n'. IfA^(n)-B^(n)=A^(n+1)-B^(n+1)=A^(n+2)-B^(n+2), then
Answer» I-A is non singular I-B is non singular I-A is singular I-B is singular SOLUTION :`A^(n+2)-B^(n+2)=(A+B)(A^(n+1)-B^(n+1))-AB(A^(n)-B^(n))` `impliesA^(n)-B^(n)=(A+B)(A^(n)-B^(n))-AB(A^(n)-B^(n))` `impliesI=A+B-AB""[becauseA^(n)-B^(n)"is invertible"]` `implies(I-A)(I-B)=0` As A, `B ne l`, we GET I-A and I-B are singular matrices.

Discussion

22.	(1 + x^(2))dy + 2xy dx = cot x dx (x ne 0)
Answer» ANSWER :`y = (1 + x)^(-1) LOG \|sin x\|+ C(1 + x^(2))^(-1)`

Discussion

23.	State with reason whether following functions have inverse : g:{5,6,7,8} rarr {1,2,3,4}with g:{(5,4),(6,3),(7,4),(8,2)}
Answer» SOLUTION :N/A

Discussion

24.	"If"f(x) ={underset(x^(2)+1.2 le x le 3)(2x+1.1 le x le 2), then evaluate int_(1)^(3) f(x) dx.
Answer» ANSWER :`(34)/(3)`

Discussion

25.	If the mean of poisson distribution is 2.56 them find the standard deviation.
Answer» ANSWER :1.6

Discussion

26.	Solve as directed: 5x le 20 in positive integers, in integers.
Answer» Solution :`5X LE 20` `rArr (5x)/5 le (20)/5 rArr X le 4` Ifis apositive INTEGER, then the solution set is {1,2,3,4} If x is a integer,then the solution set is : S = { x:x in Z and x `le` 4]

Discussion

27.	Two planes barr_1.barn_1=p_1" and "barr_2.barn_2=p_2 are parallel if
Answer» `barn_1.barn_2=1` `barn_1.barn_2=0` `barn_1=lambda barn_2` `barn_1-barn_2=0` ANSWER :C

Discussion

28.	The vertices of a rectangle are (-1, -2), (4,2), (4,3) and (-1,3). When the rantangle is graphed in the standard (x, y) coordinate plane below, what precent of the total area of the rectangle lies in Quadrant III?
Answer» 0.08 0.12 0.125 0.32 Answer :A

Discussion

29.	Find the area of the region bounded by the line y = 3x +2, the x-axis and the ordinates x = -1 and x = 1.
Answer» ANSWER :`(13)/3`

Discussion

30.	By using the properties of definite integrals, evaluate the integrals int_(0)^(pi)(xdx)/(1+sinx)
Answer»

Discussion

31.	Solve cosx+cos3x-2cos2x=0
Answer»

Discussion

32.	If y = tan^(-1) x, provethat (1 + x^2)y_2 + 2xy_1 = 0
Answer» SOLUTION :`y = TAN^(-1) IMPLIES y_1 = 1/(1 + x^2) implies (1 + x^2)y_1 = 1 implies (1 + x^2)y_2 + 2xy_1 = 0`

Discussion

33.	The points of intersection of the perpendicular tangents drawn to the ellipse 4x^(2) + 9y^(2) = 36 lie on the curve.
Answer» `X^(2) + y^(2) = 13` `x^(2) - y^(2) = 5` `x + y = 5` `(x^(2))/(9) + (y^(2))/(4) = 1` ANSWER :A

Discussion

34.	The most phagocytic WBC, are :-
Answer» Neutrophill and MONOCYTES Neutrophill and eosinophill Lymphocytes and macrophages Eosinophils and lymphocytes Answer :A

Discussion

35.	In the group (Z,) if a * b= a + b -n AA a, b in Z, where n is a fixed integer, then the inverse of (-n) is
Answer» N `-n` `-3N` `3n` ANSWER :D

Discussion

36.	If : int(2x^(2)+3)/((x^(2)-1)(x^(2)-4))dx=log[((x-2)/(x+))^(a).((x+1)/(x-1))^(b)]+c then : (a, b)-=
Answer» `1/2, 3/4` `-1, 3/2` `1, 3/2` `-1/2, 3/4` ANSWER :A

Discussion

37.	intx^(5)cos^(2)logsqrtxdx=
Answer» `(x^(6))/(12)+(x^(6))/(74)(6coslogx+sinlogx)+c` `(x^(6))/(12)-(x^(6))/(37)(6coslogx+sinlogx)+c` `((x^(3)-1)(x^(3)+1))/(12)+(x^(6))/(37)(6coslogx+sinlogx)+c` `((x^(3)-1)(x^(3)+1))/(12)+(x^(6))/(74)(6coslogx+sinlogx)+c` ANSWER :A::D

Discussion

38.	In the function f(x)= 2x^(3) + bx^(2) + qx satisfies conditions of Rolle's theorem in [-1, 1] and c= (1)/(2) then the value of 2b + q is…….
Answer» 0 1 `-(1)/(2)` `-1` ANSWER :D

Discussion

39.	Optimization of the objective function is a process of
Answer» MAXIMIZING the OBJECTIVE function maximizing or MINIMIZING the objective function minimizing the objective function only minimizing the objective function Answer :B

Discussion

40.	Find the vectorequtionsof a linepassingthroughthe point(2,3,2)andparallelto theline vec(r )=(-2hat(i)+3hat(j) ) + lambda (2hat(i) - 3hat(j) + 6hat(k))Alsofind thedistancebetweentheselines .
Answer» Solution : The givenline is`L_(1) : vec(r ) =(-2hat(i) +3hat(j)) + lambda (2hat(i) -3hat(j) +6hat(k))` Therequired line is `L_(2) :vec( r)=(2hat(i) +3hat(j) +2hat(k))+ MU (2hat(i)-3hat(j) +6hat(k))` Now, FIND thedistancebetweenthe PARALLELLINES `L_(1)" and" L_(2)`.

Discussion

41.	Differentiate the functions given in Exercises 1 to 11 w.r.t. x. cos x. cos 2x. Cos 3x.
Answer» ANSWER :`-COS x cos 2x cos 3x [tan x +2 tan 2x+3 tan 3x]`

Discussion

42.	If f(x)=1+x^(2)+x^(4)+x^(6)+…..oo then int_(0)^(x)f(x)dx=
Answer» `LOG(1-x)` `log(1+x)` `log((1+x)/(1-x))` `(1)/(2)log((1+x)/(1-x))` ANSWER :A

Discussion

43.	Evaluate the following integrals inte^(x)((cosx-sinx)/(1-cos2x))dx
Answer» ANSWER :`-(E^(X))/(2)cosec+C`

Discussion

44.	Acoording to Newton's law of cooling, the body cools from 110^(@)C" to "60^(@)C at room temperature of 10^(@)C in 1 hour. The body coolsto 30^(@)C after another
Answer» `(LOG5)/(LOG2)-1"HOURS"` `(log5)/(log2)+1"hours"` `(log5)/(log2)"hours"` `(log2)/(log5)"hours"` ANSWER :A

Discussion

45.	Write the negation of each of the following statements and write equivalent statement after simplification. Justify each step : (1) p^^(qtor) (2) ~pvv(qto~r) (3) (~pvv~q)^^(p^^~q).
Answer» ANSWER :(1) `~(qvvp)` (2) `~pvvq` (3) `~p^^Q` (4) `(p^^~q)VV(~r)` (5) `~q^^(rvv~p)` (6) `(pvv~q)vv(~r)`

Discussion

46.	Evaluate : (i) (dx)/(4cosx+3sinx) (ii) int(dx)/((2sinx+3cosx)^(2))
Answer» SOLUTION :(i) Put `4=rsinthetaand3=rcostheta" so that"` `r^(2)=25andtheta=TAN^(-1)((4)/(3))`. `:.int(dx)/(4cosx+3sinx)=int(dx)/(rsinthetacosx+rcosthetasinx)` `=(1)/(r)int(dx)/(sin(theta+X))=(1)/(r)int"cosec"(theta+x)dx` `=(1)/(r)log{tan((theta+x)/(2))}+C` `=(1)/(5)log{:[tan{(1)/(2)tan^(-1)((4)/(3))+(x)/(2)}]:}+C`. (II) Put `2=rcosthetaand3=rsintheta" so that "r^(2)=13andtheta=tan^(-1)((3)/(2))`. `becauseint(dx)/((2sinx+3cosx)^(2))=int(dx)/((rcosthetasinx+rsinthetacosx)^(2))` `=(1)/(r^(2))int(dx)/(sin^(2)(theta+x))=(1)/(13)*int"cosec^(2)(theta+x)dx` `=-(1)/(13)cot(theta+x)+C` `=-(1)/(13)cot{tan^(-1)((3)/(2))+x}+C`.

Discussion

47.	if the system of equations: {:(2x+3y-z=0),(3x+2y+kz=0),(4x+y+z=0):} have a set of non-zero integral solutions then , find the smallest positive value of z.
Answer» ANSWER :5

Discussion

48.	a xx (b xx c) =
Answer» (a.B) C = (a.c) b (a.c) b - (a.b) c (c.a) b - (c.b) a (c.b)a - (c.a) t Answer :B

Discussion

49.	Epress matrix A = [(1,2),(2,-1)] as the sum of a symmetric and skew-symmetric matrix.
Answer» <P> ANSWER :`P + Q = A`

Discussion

50.	Leta_(1), a_(2), a_(3), ………be terms of an A.P. if (a_(1) + a_(2) + ...... a_(p))/(a_(1) + a_(2) + ...... a_(q))= p^(2)/q^(2),p != q, then a_(6)/a_(21) equals
Answer» `41/11` `7/2` `2/7` `11/41` ANSWER :D

Discussion

Explore topic-wise InterviewSolutions in Current Affairs.

Find the locus of the point of intersection of two tangents to the hyperbola x^(2)/a^(2)-y^(2)/b^(2)=1. which makes an angle alpha with one another.

Determine the truth of falsity of theA sub A for any set A propositions with reasons.

Let P and Q be point on the line joining﻿ A(5,6) and B (3, -4) such that AP=PQ=QB.﻿ Then the midpoint of PQ is﻿

If |z-2/z|=2then the greatest value of |z| is

Show that [[a_1,b_1,-c_1],[-a_2,b_2,c_2],[a_3,b_3,-c_3]]= [[a_1,b_1,c_1],[a_2,b_2,c_2],[a_3,b_3,c_3]]

If f(x)=(x-2)/(x+2),x!= -2 then f^(-1)(x), is equal to

For the reaction CO(g) + CI_(2)(g) hArr COCI_(2)(g) the value of (K_(c))/(K_(P))is equal to :-

Find theareaof the regionboundedby thecirclex^2+y^2=32,X - axisand the liney=x in the firstquadrant .

Find the coefficient of x^3 in ( sqrt(x^5) + (3)/(sqrt(x^3)) )^6

Find the area bounded between the curves y=x^(2), y=sqrt(x)

If f'(a)=2 and f(a)=4 then lim_(xtoa)(xf(a)-af(x))/(x-a) equals to

The values of (1-sqrt3i)^(1//3) are

A,B,C are 3 newspaper from a city. 20% of the population red A, 16% read B, 14% read C, 8% both A and B, 5% both A and C, 4% both B and C, 2% all the three. Find the percentage of the populations who read atleast one newspaper.

y = sin^(-1) x + sin^(-1)sqrt(1 - x^2), 0 lt x lt 1 find (dy)/(dx).

If a line y = x touches the curve y=x^(2)+bx+c at (1, 1) then ………..

Let a and b natural numbers such that 2a - b, a - 2b and a + b are all distinct squares. What is the smallest possible value of b ?

Let A and B be two matrices different from identify matrix such that AB=BA and A^(n)-B^(n) is invertible for some positive integer 'n'. IfA^(n)-B^(n)=A^(n+1)-B^(n+1)=A^(n+2)-B^(n+2), then

(1 + x^(2))dy + 2xy dx = cot x dx (x ne 0)

State with reason whether following functions have inverse : g:{5,6,7,8} rarr {1,2,3,4}with g:{(5,4),(6,3),(7,4),(8,2)}

"If"f(x) ={underset(x^(2)+1.2 le x le 3)(2x+1.1 le x le 2), then evaluate int_(1)^(3) f(x) dx.

If the mean of poisson distribution is 2.56 them find the standard deviation.

Solve as directed: 5x le 20 in positive integers, in integers.

Two planes barr_1.barn_1=p_1" and "barr_2.barn_2=p_2 are parallel if

The vertices of a rectangle are (-1, -2), (4,2), (4,3) and (-1,3). When the rantangle is graphed in the standard (x, y) coordinate plane below, what precent of the total area of the rectangle lies in Quadrant III?

Find the area of the region bounded by the line y = 3x +2, the x-axis and the ordinates x = -1 and x = 1.

By using the properties of definite integrals, evaluate the integrals int_(0)^(pi)(xdx)/(1+sinx)

Solve cosx+cos3x-2cos2x=0

If y = tan^(-1) x, provethat (1 + x^2)y_2 + 2xy_1 = 0

The points of intersection of the perpendicular tangents drawn to the ellipse 4x^(2) + 9y^(2) = 36 lie on the curve.

The most phagocytic WBC, are :-

In the group (Z,*) if a ** b= a + b -n AA a, b in Z, where n is a fixed integer, then the inverse of (-n) is

If : int(2x^(2)+3)/((x^(2)-1)(x^(2)-4))dx=log[((x-2)/(x+))^(a).((x+1)/(x-1))^(b)]+c then : (a, b)-=

intx^(5)cos^(2)logsqrtxdx=

In the function f(x)= 2x^(3) + bx^(2) + qx satisfies conditions of Rolle's theorem in [-1, 1] and c= (1)/(2) then the value of 2b + q is…….

Optimization of the objective function is a process of

Find the vectorequtionsof a linepassingthroughthe point(2,3,2)andparallelto theline vec(r )=(-2hat(i)+3hat(j) ) + lambda (2hat(i) - 3hat(j) + 6hat(k))Alsofind thedistancebetweentheselines .

Differentiate the functions given in Exercises 1 to 11 w.r.t. x. cos x. cos 2x. Cos 3x.

If f(x)=1+x^(2)+x^(4)+x^(6)+…..oo then int_(0)^(x)f(x)dx=

Evaluate the following integrals inte^(x)((cosx-sinx)/(1-cos2x))dx

Acoording to Newton's law of cooling, the body cools from 110^(@)C" to "60^(@)C at room temperature of 10^(@)C in 1 hour. The body coolsto 30^(@)C after another

Write the negation of each of the following statements and write equivalent statement after simplification. Justify each step : (1) p^^(qtor) (2) ~pvv(qto~r) (3) (~pvv~q)^^(p^^~q).

Evaluate : (i) (dx)/(4cosx+3sinx) (ii) int(dx)/((2sinx+3cosx)^(2))

if the system of equations: {:(2x+3y-z=0),(3x+2y+kz=0),(4x+y+z=0):} have a set of non-zero integral solutions then , find the smallest positive value of z.

a xx (b xx c) =

Epress matrix A = [(1,2),(2,-1)] as the sum of a symmetric and skew-symmetric matrix.

Leta_(1), a_(2), a_(3), ………be terms of an A.P. if (a_(1) + a_(2) + ...... a_(p))/(a_(1) + a_(2) + ...... a_(q))= p^(2)/q^(2),p != q, then a_(6)/a_(21) equals

Let P and Q be point on the line joining A(5,6) and B (3, -4) such that AP=PQ=QB. Then the midpoint of PQ is