14949 + Interview Questions in Maths in Class 11 Page 84 InterviewSolution

4151.	Using binomial theorem prove that 8^(n) - 7n always leaves remainder f when divided by49 for all positive integer n.
Answer» ANSWER :1

Discussion

4152.	If A and B are acute angles such that sinA = sin^(2) B,cos^(2) A = 3 cos^(2) B then
Answer» `A = pi//6` `A = pi//2` `B = pi//4` `B = pi//3` ANSWER :A::C

Discussion

4153.	The minute hand of a watch is 1.5cm long. How far does it tip move in 40 minute? (Use pi=3.14)
Answer» ANSWER :`6.28cm`

Discussion

4154.	Evaluate the following limits in Exercises lim_(xto(pi)/(2))(tan2x)/(x-(pi)/(2))
Answer» ANSWER :2

Discussion

4155.	Find the component statements of the following compound statements: Number 7 is prime and odd.
Answer» <P> ANSWER :p: PRIME number q: odd number.

Discussion

4156.	In trianle ABC, CD is the bisector of the angle C. if cos . C/2 = 1/3and CD = 6, then the value of (1/a + 1/b) = 1/lambda , " then " lambda is
Answer» 3 6 9 12 Answer :C

Discussion

4157.	Show that x^(2)-y^(2)+x-3y-2=0repersent a pair of straight lines. Find also angle between the lines
Answer» ANSWER :`90^(@)`

Discussion

4158.	The mean and standard deviation of 20 observations are found to boe 10 and 2 respectively.On rechecking it was found that on observation 8 was incorrect.Calculate the correct mean and the standard deviation in each of the following cases.(ii)if it replaced by 12.
Answer» ANSWER :(i) 10,1, 1.99 (II) 10.2, 1.98

Discussion

4159.	Find the number of solutions of tanx + secx = 2cosx in [0,2pi]
Answer» Solution :Here, `tanx + secx = 2cosx rArr SINX +1=2cos^(2)x` `rArr 2SIN^(2)x+sinx-1=0 rArr sinx=1/2, -1` But `sinx=-1 rArr x=(3pi)/(2)` for which `tanx+secx =2 cosx` is not DEFINED. THUS, `sinx=1/2 rArr x=pi/6, (5pi)/(6)` `rArr` number of solutions of `tanx+secx = 2cos x` is 2.Ans.

Discussion

4160.	Differentiate the following w.r.t. x or t or u as the case may be: 11. y= (x^2 + 2x +5)/( x^3 + 2x + 4)
Answer» ANSWER :`(dy)/( dx) = (-X^(4) - 4X^(3) - 13X^(2) + 8x -2)/( (x^(3) + 2x +4)^2)`

Discussion

4161.	Determine if f defined by f(x)={{:(x sin""(1)/(x)," if "x ne 0),(0," if "x= 0):} is a continuous function?
Answer» ANSWER :h(X) is continuous in the entire INTERVAL.

Discussion

4162.	If 1/2 le x le 1, thencos^(-1)x+cos^(-1)(x/2+1/2sqrt(3-3x^2)) is equal to
Answer» `(PI)/2` `(pi)/4` `(pi)/3` `2COS^(-1)x-(pi)/3` Answer :C

Discussion

4163.	Find the conditions that the line (i) y = mx + c may touch the circle x^(2) +y^(2) = a^(2) , (ii) y = mx + cmay touch the circle x^(2) + y^(2) + 2gx + 2fy + c = 0.
Answer» ANSWER :`a^(2)`

Discussion

4164.	0.7625 xx 0.000357
Answer» ANSWER :0.0002723

Discussion

4165.	Find the values of sin (-(11pi)/(3)).
Answer» ANSWER :`(SQRT3)/(2)`

Discussion

4166.	Solve: tan 2x = - cot (x + (pi)/(3)).
Answer» Answer :`X = N PI + (5pi)/(5)`

Discussion

4167.	Find the equation of line intersecting the x-axis at a distance of 3 units to the left of origin with slope -2.
Answer» ANSWER :`2X + y + 6 = 0 `

Discussion

4168.	Write the first five terms of each of the sequenceswhose n^(th)terms are: a_(n) =n(n^2+5)/4
Answer» Answer :`(3)/(2),(9)/2,(21)/(2) 21 and (75)/2`

Discussion

4169.	A person standing at a junction (crossing) of twostraight paths represented by the equations 2x-3y+4=0 and 3x+4y-5=0 seek to reach the path whose equation is 6x-7y+8=0 in the least times. Find the equation of the path that he should follow.
Answer» ANSWER :`119 X + 102y = 125`

Discussion

4170.	Evaluate the following limits in Exercises lim_(xto-2)((1)/(x)+(1)/(2))/(x+2)
Answer» ANSWER :`-(1)/(4)`

Discussion

4171.	If (r)/(r_(2))=(r_(1))/(r_(3)) then
Answer» `A=90^(@)` `B=90^(@)` `C=90^(@)` `C=60^(@)` ANSWER :C

Discussion

4172.	x + y ge 24,2x + y ge 32, x ge 0, y ge 0
Answer» SOLUTION :

Discussion

4173.	The orthocentre of the triangle formed by the lines x+3y-10 and 6x^(2)+xy-y^(2)=0 is
Answer» (1,3) (3,1) (-1,3) (1,-3) ANSWER :A

Discussion

4174.	Consider the lines L _(1) : (x +1)/( 3) = (y+2)/(1) = (z+1)/(2), L_(2): (x-2)/(1) = (y+2)/(2) = (z-3)/(3) The unit vector perpendicular to both L _(1) and L _(2) is
Answer» `(-hati + 7 hatj + 7 HATK)/(sqrt99)` `(- hati - 7 hatj + 5 hatk)/(5sqrt3)` `(- hati + 7 hatj + 5 hatk)/(5sqrt3)` `(7 hati - 7 hatj - hatk)/(sqrt99)` ANSWER :B

Discussion

4175.	150 workers were engaged to finish a job in a certain number of days. 4 workers dropped out on second day, 4 more workers dropped out on third day and so on. It took 8 more days to finish the work. Find the number of days in which the work was completed.
Answer» ANSWER :25 DAYS

Discussion

4176.	A plane pi passes through the point (1,1,1) If b ,c,a are the dr' s of a normal to the pane, where a,b,c(a lt b lt c) are the prime factors of 2001, then the equation of the plane pi is
Answer» `29 x + 31 y + 3Z = 63` `23 x + 29Y - 3z = 23 ` `23 x + 29 y + 3z = 55` `31 x + 27 y + 3z = 71` Answer :C

Discussion

4177.	Evaluate the following limits : Lim_(x to 0) (a^(x) - b^(x))/(sin x)
Answer» ANSWER :`LOG(a/b)`

Discussion

4178.	Prove the following identieties using the theory of permutation where C_(0),C_(1),C_(2),……C_(n) are the combinatorial coefficents in the expansion of (1+x)^n,n in N: ""^(100)C_(10)+5.""^(100)C_(11)+10 .""^(100)C_(12)+ 10.""^(100)C_(13)+ 10.""^(100)C_(14)+ 10.""^(100)C_(15)=""^(105)C_(90)
Answer»

Discussion

4179.	Let T= {x\|(x+5)/(x-7)-5 = (4x-40)/(13-x)}. Is T an empty set ? Justify your answer.
Answer» ANSWER :T is not an EMPTY SET.

Discussion

4180.	Find the sum of: 24 terms and n terms of 2 (1)/(2) , 3 (1)/(3) , 4 (1)/(6) ,5 ,......,
Answer» ANSWER :`290 (5N(n+5))/(12)`

Discussion

4181.	If u_(n)=2Cos^(n) theta then show that u_(1)u_(n)-u_(n-1)= u_(n+1)
Answer»

Discussion

4182.	If (1+tantheta)(1+tanphi)=2 then theta+phi =
Answer» `30^(@)` `45^(@)` `60^(@)` `75^(@)` ANSWER :B

Discussion

4183.	Let a, b, g be the distinct real numbers. The points with position vectors abar(i)+b""bar(j)+gbar(k), b""bar(i)+gbar(j)+abar(k), gbar(i)+abar(j)+b""bar(k) are
Answer» Collinear form an ISOSCELES triangle fom a RIGHT ANGLED triangle form an EQUILATERAL triangle Answer :D

Discussion

4184.	If f: R to R, g : R to R are defined by f(x) = 4x-1 and g(x) = x^(2)+2, then find (i) (gof)(x) (ii) (gof)(a+1)/4, (iii) (fof)(x), (iv) (go(fof))(0)
Answer» ANSWER :(i) `16x^(2) -8x +3`, (ii) `a^(2)+2`, (III) `16x-5`, (IV) 27

Discussion

4185.	Evaluate the following limits : Lim_(x to 2) (x-2)/(sqrt(x)-sqrt(2))
Answer» ANSWER :`2sqrt(2)`

Discussion

4186.	Let P be (5,3) and a point R on y=x and Q on the x-axis be such that PQ+QR+RP is minimum. The the coordinates of Q are
Answer» `(17//4,0)` (17,0)` `(17//2,0)` NONE of these Answer :A

Discussion

4187.	y = "tan"^(-1)((sqrt(1 + a^2x^2 ) - 1)/(ax)) implies (1 + a^2x^2)y^('') + 2a^2 xy^'=
Answer» `a^2` `2a^2` `0` `-2a^2` ANSWER :C

Discussion

4188.	Show that the vectors 2hati-hatj+hatk,hati-3hatj-5hatkand3hati-4hatk form the vertices of a right angled triangle.
Answer» ANSWER :`COS^(-1)(SQRT(6/41)),cos^(-1)(sqrt(35/41))`

Discussion

4189.	If y=8x^(3)-60x^(2)+144x+57 is a decreasing function in the interval (a,b) , then (a,b) is
Answer» `(-3,2)` `(2,3)` `(5,6)` `(3,2)` ANSWER :B

Discussion

4190.	The set of values of 'b' for which the origin and the point (1,1) lie on the same side of the striaght line a^2x+aby+1=0,AAa in R, bgt0 are
Answer» `B in (2,4)` `b in (0,2)` `BIN[0,2]` NONE of these Answer :B

Discussion

4191.	d/(dx)[log_(e) {(e^x + 2) + sqrt(e^(2x )+ 4e^x+ 5))}] =
Answer» `1/(sqrt(E^(2x) + 4e^(x) + 5))` `(e^x)/(sqrt(e^(2x) + 4e^(x) + 5))` `(e^x)/(sqrt(e^(2x) + 4e^(x) + 3))` `(-e^x)/(sqrt(e^(2x) + 4e^(x) + 3))` Answer :B

Discussion

4192.	The semivertical angle of the cone of maximum volume and of given slant height is
Answer» ANSWER :`thereforealpha=Tan^(-1)SQRT2`

Discussion

4193.	If barb=4bari+3barj and barc be two perpendicular vectors in xy-plane. The vector in the same plane having components 1, 2 along barb and barc respectively, is
Answer» `2bari-barj` `bari-2barj` `bari+2barj` `2bari+barj` ANSWER :A

Discussion

4194.	If bar(a), bar(b), bar(c) are non coplanar vectors and lambda is a real number, then the vectors bar(a)+2bar(b)+3bar(c), lambdabar(b)+4bar(c) and (2lambda-1)bar(c) are non-coplanar for
Answer» All VALUES of `lamda` No VALUE of `lamda` All except TWO values of `lamda` All except ONE values of `lamda` ANSWER :C

Discussion

4195.	A tower, of x metres high, has a flagstaff at its top. The tower and the flagstaff subtend equal angles at a point distant y metres from the foot of the tower. Then the length of the flagstaff in metresis
Answer» `(y(X^(2)-y^(2)))/((x^(2)-y^(2)))` `(y(x^(2)+y^(2)))/((y^(2)-x^(2)))` `(x(x^(2)+y^(2)))/((x^(2)-y^(2)))` `(x(x^(2)-y^(2)))/((x^(2)+y^(2)))` ANSWER :B

Discussion

4196.	If Tan70^(@) - Tan20^(@) = k Cot40^(@) then k =
Answer» -2 -1 1 2 Answer :D

Discussion

4197.	Which of the above statements is correct. IIn a DeltABC if 4s(x-a)(s-b)(s-c)=a^(2)b^(2) then it is angled triangle. In a DeltABC the expression sinA+sinB+sinC is maximum then the triangle is equilaterl
Answer» onlyI only II both I and II neither I nor II Answer :C

Discussion

4198.	For all positive integers n gt 1, { x(x^(n-1) - n,a^(n-1) ) } + a^(n) (n-1) } is divisible by
Answer» `(x-a)^(2)` `x-a` `2(x-a)` `(x+a)` ANSWER :A

Discussion

4199.	What will Rs. 500 amounts to in 10 years after its deposit in a bank which pays annual interest rate of 10% compounded annually?
Answer» ANSWER :RS `500 (1.1)^(10)`

Discussion

4200.	AA t in R, …….is the point lies on parabola x^(2) = 4ay.
Answer» `X = at^(2) , y = 2AT` `x = 2at, y = at^(2)` `x = 2at^(2), y = at` `x = 2at, y = at^(2)` ANSWER :D

Discussion

Explore topic-wise InterviewSolutions in .

Using binomial theorem prove that 8^(n) - 7n always leaves remainder f when divided by49 for all positive integer n.

If A and B are acute angles such that sinA = sin^(2) B,cos^(2) A = 3 cos^(2) B then

The minute hand of a watch is 1.5cm long. How far does it tip move in 40 minute? (Use pi=3.14)

Evaluate the following limits in Exercises lim_(xto(pi)/(2))(tan2x)/(x-(pi)/(2))

Find the component statements of the following compound statements: Number 7 is prime and odd.

In trianle ABC, CD is the bisector of the angle C. if cos . C/2 = 1/3and CD = 6, then the value of (1/a + 1/b) = 1/lambda , " then " lambda is

Show that x^(2)-y^(2)+x-3y-2=0repersent a pair of straight lines. Find also angle between the lines

The mean and standard deviation of 20 observations are found to boe 10 and 2 respectively.On rechecking it was found that on observation 8 was incorrect.Calculate the correct mean and the standard deviation in each of the following cases.(ii)if it replaced by 12.

Find the number of solutions of tanx + secx = 2cosx in [0,2pi]

Differentiate the following w.r.t. x or t or u as the case may be: 11. y= (x^2 + 2x +5)/( x^3 + 2x + 4)

Determine if f defined by f(x)={{:(x sin""(1)/(x)," if "x ne 0),(0," if "x= 0):} is a continuous function?

If 1/2 le x le 1, thencos^(-1)x+cos^(-1)(x/2+1/2sqrt(3-3x^2)) is equal to

Find the conditions that the line (i) y = mx + c may touch the circle x^(2) +y^(2) = a^(2) , (ii) y = mx + cmay touch the circle x^(2) + y^(2) + 2gx + 2fy + c = 0.

0.7625 xx 0.000357

Find the values of sin (-(11pi)/(3)).

Solve: tan 2x = - cot (x + (pi)/(3)).

Find the equation of line intersecting the x-axis at a distance of 3 units to the left of origin with slope -2.

Write the first five terms of each of the sequenceswhose n^(th)terms are: a_(n) =n(n^2+5)/4

A person standing at a junction (crossing) of twostraight paths represented by the equations 2x-3y+4=0 and 3x+4y-5=0 seek to reach the path whose equation is 6x-7y+8=0 in the least times. Find the equation of the path that he should follow.

Evaluate the following limits in Exercises lim_(xto-2)((1)/(x)+(1)/(2))/(x+2)

If (r)/(r_(2))=(r_(1))/(r_(3)) then

x + y ge 24,2x + y ge 32, x ge 0, y ge 0

The orthocentre of the triangle formed by the lines x+3y-10 and 6x^(2)+xy-y^(2)=0 is

Consider the lines L _(1) : (x +1)/( 3) = (y+2)/(1) = (z+1)/(2), L_(2): (x-2)/(1) = (y+2)/(2) = (z-3)/(3) The unit vector perpendicular to both L _(1) and L _(2) is

150 workers were engaged to finish a job in a certain number of days. 4 workers dropped out on second day, 4 more workers dropped out on third day and so on. It took 8 more days to finish the work. Find the number of days in which the work was completed.

A plane pi passes through the point (1,1,1) If b ,c,a are the dr' s of a normal to the pane, where a,b,c(a lt b lt c) are the prime factors of 2001, then the equation of the plane pi is

Evaluate the following limits : Lim_(x to 0) (a^(x) - b^(x))/(sin x)

Prove the following identieties using the theory of permutation where C_(0),C_(1),C_(2),……C_(n) are the combinatorial coefficents in the expansion of (1+x)^n,n in N: ""^(100)C_(10)+5.""^(100)C_(11)+10 .""^(100)C_(12)+ 10.""^(100)C_(13)+ 10.""^(100)C_(14)+ 10.""^(100)C_(15)=""^(105)C_(90)

Let T= {x|(x+5)/(x-7)-5 = (4x-40)/(13-x)}. Is T an empty set ? Justify your answer.

Find the sum of: 24 terms and n terms of 2 (1)/(2) , 3 (1)/(3) , 4 (1)/(6) ,5 ,......,

If u_(n)=2Cos^(n) theta then show that u_(1)u_(n)-u_(n-1)= u_(n+1)

If (1+tantheta)(1+tanphi)=2 then theta+phi =

Let a, b, g be the distinct real numbers. The points with position vectors abar(i)+b""bar(j)+gbar(k), b""bar(i)+gbar(j)+abar(k), gbar(i)+abar(j)+b""bar(k) are

If f: R to R, g : R to R are defined by f(x) = 4x-1 and g(x) = x^(2)+2, then find (i) (gof)(x) (ii) (gof)(a+1)/4, (iii) (fof)(x), (iv) (go(fof))(0)

Evaluate the following limits : Lim_(x to 2) (x-2)/(sqrt(x)-sqrt(2))

Let P be (5,3) and a point R on y=x and Q on the x-axis be such that PQ+QR+RP is minimum. The the coordinates of Q are

y = "tan"^(-1)((sqrt(1 + a^2x^2 ) - 1)/(ax)) implies (1 + a^2x^2)y^('') + 2a^2 xy^'=

Show that the vectors 2hati-hatj+hatk,hati-3hatj-5hatkand3hati-4hatk form the vertices of a right angled triangle.

If y=8x^(3)-60x^(2)+144x+57 is a decreasing function in the interval (a,b) , then (a,b) is

The set of values of 'b' for which the origin and the point (1,1) lie on the same side of the striaght line a^2x+aby+1=0,AAa in R, bgt0 are

d/(dx)[log_(e) {(e^x + 2) + sqrt(e^(2x )+ 4e^x+ 5))}] =

The semivertical angle of the cone of maximum volume and of given slant height is

If barb=4bari+3barj and barc be two perpendicular vectors in xy-plane. The vector in the same plane having components 1, 2 along barb and barc respectively, is

If bar(a), bar(b), bar(c) are non coplanar vectors and lambda is a real number, then the vectors bar(a)+2bar(b)+3bar(c), lambdabar(b)+4bar(c) and (2lambda-1)bar(c) are non-coplanar for

A tower, of x metres high, has a flagstaff at its top. The tower and the flagstaff subtend equal angles at a point distant y metres from the foot of the tower. Then the length of the flagstaff in metresis

If Tan70^(@) - Tan20^(@) = k Cot40^(@) then k =

Which of the above statements is correct. IIn a DeltABC if 4s(x-a)(s-b)(s-c)=a^(2)b^(2) then it is angled triangle. In a DeltABC the expression sinA+sinB+sinC is maximum then the triangle is equilaterl

For all positive integers n gt 1, { x(x^(n-1) - n,a^(n-1) ) } + a^(n) (n-1) } is divisible by

What will Rs. 500 amounts to in 10 years after its deposit in a bank which pays annual interest rate of 10% compounded annually?

AA t in R, …….is the point lies on parabola x^(2) = 4ay.