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1151.

Show that cos12^(@)+cos84^(@)+cos132^(@)+cos156^(@)=-1/2

Answer»


ANSWER :RHS
Discussion
1152.

A metal 'M' having atomic mass 31.25 crystallizes in cubic close packing and it shows "Schottky defects". If the edge length of the cubic lattice is 500 pm and density of the metal is 1.6075 gm//ml then calculate number of moles of metal atom 'M' missing per litre of the crystal. Given : 1 amu = 1.67xx10^(-24)gm

Answer»


SOLUTION : `rho_("ideal")=(ZXX(M_(0))/(N_(A)))/(a^(3))`
`(4xx(31.25)/(N_(A)))/((500xx10^(-10))^(3))=(4xx31.25xx1.67xx10^(-24))/(124xx10^(-24))=1.67g//ml`
`rho_("actual")=1.6075g//ml`
Difference DUE to SCHOTTKEY effect `= 1.67-1.6075rArr0.0625g//ml`
`=0.0625xx10^(3)g//L`
`=62.5 g//L`
`=(62.5)/(31.25)` Mole/L
`=2 "mole"//L`
Discussion
1153.

If a3 xx 3square matrix satisfiesA^(3)-23A - 40 I = 0, " then "A^(-1)equals

Answer»

` (1)/(40)[23 -A^(2)]`
` -(1)/(40)[23 I- A^(2)]`
` (1)/(40)[23I -A^(2)]`
` (1)/(40)[A^(2) - 23]`

ANSWER :C
Discussion
1154.

Two third of the students of a class are boys and rest are girls . It is known that the probability of a girl getting a first division in Board examination is 0.4 and a boy getting first division is 0.35.Find the probability that a students chosen at random will get first division in Board examination.

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ANSWER :` (11)/(30)`
Discussion
1155.

The value of (sec x + tan x -1)/(tan x - sec x +1), if cosec 2x =2 is

Answer»

0
1
1
3

Answer :D
Discussion
1156.

Find the equation of the circle passing through the points (1,1), (2,-1), and (3,2).

Answer»


ANSWER :`x^2 + y^2 - 2X + 6Y - 40 = 0`
Discussion
1157.

Which of the following is correct combination?

Answer»

<P>(I) (ii) (Q)
(IV) (i) (S)
(II) (III) (R )
(III) (ii) (P)

ANSWER :B
Discussion
1158.

Which of the following is correct combination?

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(II) (III) (S)
(III) (ii) (P)
(I) (iv) (Q)
(IV) (i) (R )

Answer :C
Discussion
1159.

Let z_(1),z_(2),z_(3) be three complex numbers satisfying 1/z_(1)+1/z_(2)+1/z_(3)=0. Let z_(k)=r_(k)(cosalpha_(k)+isinalpha_(k)) and omega_(k)=(cos2alpha_(k)+isin2alpha_(k))/(z_(k)) for k=1,2,3. If omega_(1),omega_(2) and omega_(3) are the affixes of points A_(1),A_(2) and A_(3) respectively in the Argand plane, then DeltaA_(1) A_(2) A_(3) has its

Answer»

INCENTER at the ORIGIN
centroid at the origin
circumcenter at the origin
orthocenter at the origin

ANSWER :B
Discussion
1160.

Evaluate the following integrals. int(1)/(sin2x+sin^(2)x)dx

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Answer :`(1)/(2)LOG|(TANX)/(tanx+2)|+c`
Discussion
1161.

Solve the equation |[x+a,x,x],[x,x+a,x],[x,x,x+a]|=0, a ne0

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SOLUTION :`X=-%`
TAKE COMMON factor (3x+a) factor `R_1`
Then apply `C_2rarrC_2` and `C_3rarrC_3-C_1`
Discussion
1162.

If pair of tangents are drawn from any point (p) on the circle x^(2) + y^(2) = 1to the hyperbola x^(2)/2 - y^(2)/1 = 1 such that locus of circumcentre of triangle formed by pair of tangents and chord of contact is lambda_(1) x^(2) - 2lambda_(2)y^(2) = 2 (x^(2)/2 - y^(2))^(2), then

Answer»

`lambda_(1) = 2, lambda_(2)= 1`
`lambda_(1)^(2) + lambda_(2)^(2) = 5`
`lambda_(1) = 1, lambda_(2) = - 1`
`lambda_(1)^(2) + lambda_(2)^(2) = 2`

SOLUTION :N/A
Discussion
1163.

x^(n)+ y^(n) is divisible by

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`x- " y for all "n in N`
`x+ " y for all "n in N`
`x + " y for all "n=2M -1, m in N`
`x + " y for all "n = 2m, m in N`

ANSWER :C
Discussion
1164.

If( sinx ) /( cosx)xx( secx )/( cosec x )xx ( tanx )/( cotx) = 9, wherex in( 0,(pi)/(2)) thenthevalueof x isequal to

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`PI/4`
`pi/3`
`pi/2`
`pi`

ANSWER :B
Discussion
1165.

Evaluate the following define integrals as limit of sums : int_(0)^(4) x^(2)dx

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ANSWER :`64/3`
Discussion
1166.

A pointon theplanethatpassesthroughthepoints(1,-1,6), (0,0,7)andperpendiculartothe planex- 2y + z =6is

Answer»

`(1,-1,2)`
`(1,1,2)`
`(-1,1,2)`
` (1,1,-2)`

ANSWER :B
Discussion
1167.

If vectors : vec(p)=ahat(i)+hat(j)+hat(k),vec(q)=hat(i)+bhat(j)+hat(k),vec(r)=hat(i)+hat(j)+chat(k)(ane1,b ne1,c ne1) are coplanar, then value of (1)/(1-a)+(1)/(1-b)+(1)/(1-c)=

Answer»

0
1
-1
2

Solution :`I=underset(1)overset(2)(int)(e^(X))/(x)dx`
`f(x)=(e^(x))/(x),f'(x)=((x-1)e^(x))/(x^(2))gt 0 [AA x in (1,2)]`
`rarrIgtAr.[ACDE]`
` I gt e xx 1 IMPLIES I gt e ....(1)`
`rarr I lt "AREA of trapezium ABCDE"`
`ILT (1)/(2)(1)[e+(e^(2))/(2)]`
`I lt (e)/(2)+(e^(2))/(4)"".......(2)`
`(I gt 3+(e)/(2))` is false from (2)
Discussion
1168.

Differentiate tan^(-1)((2^(x+1))/(1-4^(x))) with respect to x.

Answer»


ANSWER :`(2^(X+1)LOG2)/(1+4^(x))`
Discussion
1169.

The mean of 5 observations is 5 and their variance is 124. If three of the observations are 1,2 and 6, then the mean deviation from the mean of the data is

Answer»

2.5
2.6
2.8
2.4

Answer :C
Discussion
1170.

Find the equation of tangentto the hyperbola (x^(2))/(3)-(y^(2))/(2)=1drawn from the point (3, 2)

Answer»


ANSWER :X - y = 1
Discussion
1171.

Approximate theintegral I = int_(0)^(1) (dx)/(1 + x) using thetrapezoidal formulaat n = 10

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ANSWER : In 2 = 0.69315
Discussion
1172.

If O is the origin OP, OQ are the tangent to the circle x^(2)+y^(2)+2gx+2fy+c=0 then the circumcentre of the triangleOPQ is

Answer»

`(-G,-F)`
`(-f,-g)`
`(-g//2,-f//2)`
`(-f//2,-g//2)`

ANSWER :C
Discussion
1173.

Find the number of integral solutions of x_1+x_2+x_3+x_4=4 where each x_ige-10

Answer»


ANSWER :`""^(47)C_3`
Discussion
1174.

Findthe areaof theregionboundedbyparabola y=2x^2 +10and the liney=4x+16.

Answer»


ANSWER :`(64)/(3) ` SQ. UNITS
Discussion
1175.

Determine whether a ** b =a+b ("mod" 7) "on" {0,1,2,3,4,5,6}operations as defined by * are binary operations on the sets specified in each case. Give reasons if it is not a binary operation.

Answer»

SOLUTION :`a**B=a +b ("MOD" 7) in {0,1,2,3,4,5,6}`
for "a,b in 7`
` :. **` is a , binary operation on the given SET.
Discussion
1176.

Resolve (x^(4)+x^(2)+x+2)/((x^(2)-x+1)^(2)) into partial fractions

Answer»


ANSWER :`1+(2X)/(X^(2)-x+1)+(x+1)/((x^(2)-x+1)^(2))`
Discussion
1177.

If I=int_(0)^(1)cos(2Cot^(-1)""(sqrt (1-x)/(1+x)))dx then

Answer»

`pi/2`
`- pi/2`
`1/2`
`- 1/2`

ANSWER :D
Discussion
1178.

Define f(x) = {{:(x,(0le x le 1)),(2-x,(1 lex le 2)):} Then Rolle's theorem is not applicable to f(x) because

Answer»

F(x) is not DEFINED everywhere on [0, 2]
f(x) is not continuous on [0, 2]
f(x) is not DIFFERENTIABLE on (1, 2)
f(x) is not differentiable on (0, 2)

Answer :D
Discussion
1179.

Two cards are randomly selected from a pack of 52 playing cards. The probability that both cards are greater than 3 and lessthan 8 is

Answer»

`(20)/(221)`
`(45)/(221)`
`(16)/(169)`
`(81)(169)`

Answer :A
Discussion
1180.

x = log (1/y +sqrt(1+1/(y^(2))))rArr yis equal to

Answer»

`TANHX`
`cothx`
`sechx`
`"COSECH"X`

ANSWER :d
Discussion
1181.

If the three lines x – 3y = p, ax + 2y = q and ax + y = r from a right – angled triangle then:

Answer»

`a^2-6a-12=0`
`a^2-9a+12=0`
`a^2-9a+18=0`
`a^2-6a-18=0`

Solution :EQUATION x-3y=p and AX +=r are only POSSIBLE to FORM a RIGHT angle triangle
`therefore` value of a=3
satisfy the equation `a^2 - 9a +18=0`
Discussion
1182.

Resolve (3x-1)/(x^(3)+1) into partial fractions.

Answer»


ANSWER :`:.` PARTIAL fractions are, `(-4)/(3(X+1))+(4x+1)/(3(x^(2)-x+1))`
Discussion
1183.

There are five cities A, B, C, D, E on a certain island. Each city is connected to every other city by road. In how many ways can a person starting from city A come back to A after visiting some cities without visiting a city more than once and without taking the same road more than once ? (The order in which he visits the cities also matters. e.g., the routes A rarr B rarr C A and A rarr C rarr B rarr A are different).

Answer»


ANSWER :60
Discussion
1184.

Angle between the tangents drawn to parabola y^(2)+4a^(2)-4ax=0, from origin is :

Answer»

`30^(@)`
`tan^(-1)(2)`
`90^(@)`
`tan^(-1)((1)/(2))`

Solution :
`C_(1):(x+1)^(2)+y^(2)=9,C_(2):(x-2)^(2)+y^(2)=49`
`because QR = R+3`
`PR = 7-r`
`impliesPR+QR=10`
`therefore` Locus of point R is an ELLIPSE for which 2a = 10 and P and Q are its FOCUS.
`implies2a=10" and "2ae = PQ =3impliese=(3)/(10)`
Discussion
1185.

Evaluate the following integrals inte^(x)cos^(2)xdx

Answer»


ANSWER :`(1)/(2)e^(x)[1+(1)/(5)(cos2x+2sin2x)]+C`
Discussion
1186.

If Delta_1=|{:(x,b,b),(a,x,b),(a,a,x):}|and Delta_2=|{:(x,b),(a,x):}| are given determinants then,

Answer»

`Delta_1=3(Delta_2)^2`
`d/(DX)(Delta_1)=3Delta_2`
`d/(dx)(Delta_1)=3Delta_2^2`
`Delta_1=3(Delta_2)^(3/2)`

ANSWER :B
Discussion
1187.

IfDelta= a^2- ( b-c)^2whereDeltais theareaof triangleABCthen tanA isequalto

Answer»

`(15 )/( 16)`
`(8)/( 17)`
` (8)/(15)`
`1/2`

ANSWER :C
Discussion
1188.

Area of the region bounded by two parabolas y = x^(2) and x = y^(2) is

Answer»

`(1)/(4)`
`(1)/(3)`
4
3

Answer :B
Discussion
1189.

lim_(ntooo)(((n+1)(n+2)....3n)/(n^(2n)))^(1//n) is equal to

Answer»

`(27)/(e^(2))`
`(9)/(e^(2))`
`3log 3-2`
`(18)/(e^(4))`

ANSWER :1
Discussion
1190.

If A(n)=sin^(n)alpha+cos^(n)alpha, then A(1)A(4)+A(2)+A(5)=

Answer»

`A(1)A(2)+A(4)A(5)`
`A(1)A(6)+A(2)A(3)`
`A(1)A(3)+A(2)A(6)`
`A(1)A(2)+A(3)A(6)`

ANSWER :B
Discussion
1191.

Evaluate the following definite integrals . int_(1)^(4)(x^(2)-x)dx

Answer»


ANSWER :`27/2`
Discussion
1192.

Evaluate the following determinants: [[0,a^2,b],[b^2,0,a^2],[a,b^2,0]]

Answer»

SOLUTION :`[[0,a^2,B],[b^2,0,a^2],[a,b^2,0]]`
=`-a^2[[b^2,a^2],[a,0]]+b[[b^2,0],[a,b^2]]`
=`-a^2(0-a^2)+b(b^4-0)=a^5+b^5`
Discussion
1193.

A student appears for tests I, II and III. The student is successful if he passes either in tests I and II or tests I and III. The probabilities of the student passing in tests I, II, and III are, respectively, p, q, and 1/2. then p(1+q)=

Answer»

`1//2`
1
`3//2`
`3//4`

Solution :If A,B, C represent events that the student is SUCCESSFUL in tests `I,II,III,` respectively, the the proability that the student is successful is
`P[ANNB nnC')uu(AnnB'C')uu(AnnBnnC)]`
`=P(AnnBnnC')+P(AnnB'nnC)+P(AnnBnnC)`
`=P(A)P(B)P(C')+P(A)P(B')P(C')+P(A)P(B)P(C)`
`""[thereforeA,B,C "are independent events"]`
`=pq(1-(1)/(2))+p(1-q)1/2+pq1/2`
`=pq+1/2p-1/2pq`
`=1/2(pq+p)`
`therefore1/2(pq+q)=1/2`
`orp(1+q)=1`
Discussion
1194.

Integrate the following functions tan^-1x

Answer»

Solution :`int tan^-1x DX `
=`int 1 XX tan^-1x dx`
=`tan^-1x xx x-int1/(1+x^2) C dx`
=`xtan^-1x -1/2 int (2x)/(1+x^2) dx`
=`x tan^-1 x-1/2 LOG (1+x^2)+c`
Discussion
1195.

In three dimensional space , the equation xy = 0 represents

Answer»

a PAIR of LINES
a PLANE
a pair of perpendicular planes
a parie of parallel planes

Answer :C
Discussion
1196.

I: The equation obtained by removing second term from x^(4) + 8x^(3) + x - 5 = 0 is x^(4) - 24x^(2) + 65x - 55 = 0 . II: The equation whose roots are exceed by 2 than those of x^(3) - 2x^(2) + 3x - 1= 0 is x^(3) - 8x^(2) + 23x - 23 = 0.

Answer»

only I is TRUE
only II is true
both I and II are true
NEITHER I nor II true

Answer :3
Discussion
1197.

The solution of (dy)/(dx) = e^(x+y)+ x^(2)e^(x^(3)+y) is

Answer»

`E^(x) - e^(-y) + (1)/(3)e^(x^(3)) = c`
`e^(x) + e^(-y) + (1)/(3)e^(x^(3)) = c`
`e^(-x) - e^(-y) + (1)/(3)e^(x^(3)) = c`
`e^(x) + e^(y) + (1)/(3)e^(x^(3)) = c`

Answer :B
Discussion
1198.

Analyse the following pairs of compounds. Weite 1, if they are Chain isomers Write 2, if they are Metamers Write 3, if they are Position isomers Write 4, if they are Homologues Write5, if they are Ring chain isomers Write6, if they are Resonating structure Write7, if they are Functional isomers Write8, if they are Tautomers

Answer»




SOLUTION :`(B) as I^(st)is3^(@)"AMINE"&II^(nd)is 2^(@)"amine."`
`(c) "as parent CABON chain is different"""]`
Discussion
1199.

Integrate the function in Exercise. x log x

Answer»


Answer :`(X^(2))/(2)logx-(x^(2))/(4)+C`
Discussion
1200.

Evalute the following integrals int (1)/(cos (x -a ) cos (x - b))dx

Answer»


Answer :`(1)/(SIN(b-a))log|(COS(x-b))/(cos(x-a))|+C`
Discussion