28278 + Interview Questions in Maths in Class 12 Page 175 InterviewSolution

8701.	If f'(x)=sec^(2)x+cosec^(2)x-1 and f((pi)/(4))=1
Answer» ANSWER :`tanx-cotx-x+(PI)/(4)+1`

Discussion

8702.	A reversible reaction is which :-
Answer» PROCEEDS in ONE direction Proceeds in both directions Proceeds spontaneously All STATEMENTS are wrong Answer :A

Discussion

8703.	Consider tetrahedron with faces F_(1),F_(2),F_(3),F_(4). Let vec(V_(1)),vec(V_(2)),vec(V_(3)),vec(V_(4)), be vectors, whose magnitudes are respectively equal to areas ofF_(1),F_(2),F_(3),F_(4) & whose directions are perpendicular to their faces in outward directon, then \|vec(V_(1))+vec(V_(2))+vec(V_(3))+vec(V_(4))\|
Answer» 1 4 0 None of these Solution :Point on line `C(3+2lambda,1+4lambda,2+5lambda)` DR's of DC `(2lambda-1,4lambda-2,5lambda-5)` `DC_\|_"line"` `THEREFORE LAMBDA=(7)/(9)impliesDR's"of"DC ((5)/(9),(10)/(9),(-10)/(9))` EQUATION of plane : `(5)/(9)(x-3)+(10)/(9)(y-1)-(10)/(9)(z-2)=0` `therefore` Equation of plane `x+2y-2z=1`

Discussion

8704.	If C_(r)= ""^(n)C_(r), then C_(0)+C_(4)+C_(8)+C_(12)+cdots=
Answer» `(2^((n)/(2))[SIN""(NPI)/(4)+2^(n/(2)-1)]]/(2)` `2^(n/(2))sin""(npi)/(2)` `2^(n-1)COS""(npi)/(4)` `(2^(n/(2))[cos ""(npi)/(4)+2^(n/2-1)])/(2)` Answer :A

Discussion

8705.	A:In Delta ABC, if r_1,r_2,r_3 are in H.P. then r_2//r=1//3 R: In DeltaABC,1/r_1+1/r_2+1/r_3=2/r
Answer» A is true, R is true and R is CORRECT EXPLANATION of A A is true, R is true and R is not correct explanation of A A is true, R is false A is false, R is true Answer :D

Discussion

8706.	If m = ( 2013) ! then the value of (1)/(log_(2)m) + ( 1)/( log_(3)m ) + "........"+ ( 1)/( log_(2013)m) is equal to
Answer» 1 `(2013)!` `(1)/((2013)!)` NONE of these Answer :A

Discussion

8707.	Evaluate the integral underset(0)overset(1)int (log(1+x))/(1+x^(2))dx
Answer» `PI/4 LN 2` `pi/2 ln 2` `pi/8 ln 2` `pi ln 2` ANSWER :C

Discussion

8708.	If ina triangle ABC , a=15, b= 36, c = 39 thensin"" (C )/(2)=
Answer» `( SQRT(3))/(2)` `1/2` `(1)/(sqrt(2))` `-(1)/(2)` ANSWER :D

Discussion

8709.	On all chords (parallel to one and the same direction) a circle of radius R symmetrical parabolic segments of the same altitude h are constructed. The planes of the segments are perpendicular to the plane of the circle. Find the volume of the solid thus obtained
Answer» ANSWER :`(2)/(3) PI HR^(2)`

Discussion

8710.	Defineeach of thefollowing: "(i)injective function """"(ii) surjectivefunction " "(iii) bijective function """"(iv) many -onefunction" "(v) intofunction """ Give anexampleof each typeof functions.
Answer» SOLUTION : (i) `f : N to N : f (x) =2x` isinjectivefunctionas ` f(x_(1))=f(x_(2))rArr 2x_(1)=2x_(2)rArr x_(1) = x_(2)` (ii)`A ={1, -1, 2,3}" and"= { 1, 4,9}` Then`f : A toB : f(x) = x^(2)` is surjectivesince eachelementof Bhas atleastonepre -imagein A. (III) Let E be theset of all evennaturalnumbers Then`f N to E : f(x) =2x` is one-oneand ONTO . f is one-onesince `f (x_(1)) = f(x_(2))rArr2x_(1) = 2x_(2) rArr x_(1) =x_(2)` f is ontosince foreach`y in E` thereexists` (1)/(2) y in N ` such that ` f ((1)/(2)y) =y` (iv)Examplegiven in (ii)is many -one . (v)Let `A = {1,2,3} " and" B = {1,4,9,16}` Then `f : A toB : f (x) = x^(2)` is anintofunction sincerange(f) = `{1,4,9 } sub B`

Discussion

8711.	int (1)/(2x^(2)-3)dx=
Answer» `(1)/(SQRT(6)) "LOG " \|(sqrt(2)X - sqrt(3))/(sqrt(2) x + sqrt(3)) \| + C ` `(1)/(2 sqrt(6)) "log " \|(sqrt(2)x - sqrt(3))/(sqrt(2) x+ sqrt(3)) \| + C ` `(1)/(sqrt(3)) "log " \|(sqrt(2)x - sqrt(3))/(sqrt(2) x + sqrt(3)) \| + C ` `(1)/(sqrt(2)) "log " \|(sqrt(2)x - sqrt(3))/(sqrt(2) x + sqrt(3)) \| + C ` Answer :B

Discussion

8712.	int ((x+3)/((x+4)^(2)))e^(x)dx=.....+C
Answer» `E^(X)((1)/(x+4))` `e^(-x)((1)/(x+4))` `e^(-x)((1)/(x-4))` `e^(2X)((1)/(x-4))` ANSWER :A

Discussion

8713.	Given three non-coplanar vectors OA=a, OB=b,OC=c. Let S be the centre of the spheres passing through the points, O,A,B,C if OS=x, then:
Answer» X must be LINEAR combinationof a,B and c x must be linear calculation of `b XX c, c xx a` and `a xx b` `x=(a^(2)(b xx c)+b^(2)(c xx a)+c^(2)(a xx b))/(2[abc]), a=\|a\|,b=\|b\|,c=\|c\|` x=a+b+c Answer :A::B::C

Discussion

8714.	Draw the graph of y= [x]+ sqrt(x-[x]), where[*] denotes the greatest ineger function.
Answer» Solution :`y= f(X) = [x] + sqrt(x-[x])` For `0 le x LT 1, y = sqrtx`,the graph of which is PART of parabola `y^(2)=x` lying above the x-axis. For `1 le x lt 2, y=1 + sqrt(x-1)`, the graph of which is part of the parabola `(y-1)^(2)=x-1` lying above the axis of the parabola `y=1`. THUS, all the branches of `y=f (x) ` are part of different parabolas. Also we can see that in each interval `y= f(x)` is an increasing function. Hence the graph of`y= f(x)` is as SHOWN in the following figure.

Discussion

8715.	IFS=overset ( m) sum"" ^(n+r ) C_(k),then
Answer» `S+""^(N)C_(K+1) =-""^(n)C_(K+1)` `S+""^(n)C_(K+1)=""^(n+m+1) C_(K+1)` `S= ""^(n)C_K =""^(n+m)C_(K)` NONE of these ANSWER :B

Discussion

8716.	If the equation ................
Answer» Solution :Since the angle between the pair of straight LINES represented by the GIVEN EQUATION is `PI`. `:. h^(2)=ab` i.e. `(6/2)^(2)=3 m` i.e. `m=3`

Discussion

8717.	int_(1//3)^(1) ((x-x^(3))^(1//3))/(x^(4))dx=
Answer» 4 6 8 12 Answer :B

Discussion

8718.	If (x^(4))/((x-1)(x+2))=(1)/(3(x-1))-(16)/(3(x+2))+x^(2)-x+k then k=
Answer» 0 1 2 3 Answer :D

Discussion

8719.	If a parabola passess through the points (-2, 1), (1, 2) and (-1, 3) having horizontal axis, then the length of the latus rectum of that parabola is
Answer» 5 `5/2` `2/5` `1/5` ANSWER :C

Discussion

8720.	If in A.P., 3rd term is 18 and 7 term is 30, then find sum of its first 17 terms
Answer» ANSWER :`=612`

Discussion

8721.	If f(x)=log_(e^(2))(logx), then f'(e) is :
Answer» 0 `(1)/(2E)` `(E )/(2)` `(2)/(e )` ANSWER :B

Discussion

8722.	Expand the expression (2x - 3)^(6).
Answer» Answer :`= 64X^(6) - 576x^(5) + 2160x^(4) - 4320x^(3) + 4860x^(2) - 2916x + 729`

Discussion

8723.	Solution of the differential equation (dy)/(dx)=(1+y^(2))/(1+x^(2)) is
Answer» 1+xy+C (y-x)=0 x+y=c (1-xy) y-x=c (1+xy) 1+xy=c(x+y) ANSWER :C

Discussion

8724.	By using the properties of definite integrals, evaluate the integrals int_(0)^(pi/2)cos^(2)xdx
Answer» ANSWER :`pi/4`

Discussion

8725.	If a, b, c arein AP amd b, c, a are in GP , thenshow that c, a, b are in H.P. Finda : b : c .
Answer» ANSWER :`4:1:2(-2)`

Discussion

8726.	If Z is a complex number such that Z=-overlineZ,then
Answer» Z is any complex number Real PART of Z is the same as its IMAGINARY part Z is PURELY real Z is purely imaginary Answer :B

Discussion

8727.	There are 4 mangoes, 3 apples 2 oranges in bag, fruits of the same variety being identical. In how many different ways can a selection of fruits be made if atleast one mango is to be selected.
Answer» ANSWER :48

Discussion

8728.	The area of the region bounded by the curves y=\|x-1\|andy=3-\|x\| is
Answer» ANSWER :4

Discussion

8729.	Evaluate the following lim_(xto0)((3+x)^3-27)/x
Answer» SOLUTION :`lim_(xto0)((3+X)^3-27)/x` `=lim_(xto0)(27+x^3+27x+9x^2-27)/x` `=lim_(xto0)(x^3+27x+9x^2)/x` `=lim_(xto0)(x(x^2+27+9x))/x` `=lim_(xto0)(x^2+9x+27)=27`

Discussion

8730.	If the system of equation (b+c)x+ay+a^(2)z=0,(c+a)x+by+b^(2)z=0,(a+b)x+cy+c^(2)z=0 has a non trivial solution and a ne b ne c prove that a+b+c=0
Answer» ANSWER :`a+b+c=0`

Discussion

8731.	Consider the sequence in the form of group (1),(2,2)(3,3,3),(4,4,4,4),(5,5,5,5,5…..) The 2000^(th) term of the sequence is not divisible by
Answer» 3 9 7 none of these Solution :1,2,2,3,3,3,4,4,4,4,.. Let us write the terms in the groups as follows: 1,(2,2),(3,3,3),(4,4,4,4),… CONSISTING of 1,2,3,4,.. Terms. Let 2000th term fall in nth group. Then, `((n-1)n)/2lt2000le(n(n+1))/2` or n(n-1)`lt4000len(n+1)` Let us consider, `n(n-1)lt4000` or `n^(2)-n-4000lt0` or `nlt(1+sqrt(16001))/2` r `nlt64` We have `n(n+1)ge4000` or `n^(2)+n-4000ge0` or `nge63` That means 2000th term falls in 63rd group, which means that the 2000th term is 63. Now, the total NUMBER of terms up to 62ND group is `(62xx63)//2=1953`. Hence, the sum of first 2000 terms is `1^(2)+2^(2)+..+62^(2)+63(2000-1953)` `=(62(63)125)/6+63xx47=84336` Sum of the remaining terms=`63xx16=1008`.

Discussion

8732.	If 36, 84, 126 are three successive binomial coefficients in the expansion of (1+x)^(n), find n.
Answer» ANSWER :N= 9

Discussion

8733.	If bar(a) and bar(b) are adjacent sides of a rhombus, then bar(a).bar(b)=0.
Answer» ANSWER :FALSE STATEMENT

Discussion

8734.	Consider the determinant Delta=\|[0,2,6],[1,5,0],[3,7,1]\|Write the minors of the element in the first column ?
Answer» SOLUTION :The MINORS are GIVEN below: `M_11=\|[5,0],[7,1]\|=5-0=5,` `M_21=\|[2,6],[7,1]\|=2-42=-40` `M_31=\|[2,6],[5,0]\|=0-30=-30`

Discussion

8735.	Choose the correct answer. If theta is the angle between two vectors veca"and"vecb, then veca.vecb ge 0 only whena)0 lt theta lt pi/2b)0 le theta le pi/2c)0 lt theta lt pid) 0 le theta le pi
Answer» `0 LT THETA lt pi/2` `0 LE theta le pi/2` `0 lt theta lt pi` ` 0 le theta le pi` ANSWER :B

Discussion

8736.	What is the interval in which log_5xis increasing ?
Answer» Solution :LET `y=log_5x=(In5)/(In5)RARR(DY)/dx=1/(xIn5)` y is decreasing for `(dy)/dxlt0rArrxlt0

Discussion

8737.	A bag contains 4 white balls and 3 black balls. If two balls are drawn at random, then mean of number of white balls is
Answer» `(8)/(7)` `(7)/(8)` `(1)/(4)` `(1)/(5)` Answer :A

Discussion

8738.	If two cards are drawn from pack 52 cards at random, then find the probability of getting both red or both kings.
Answer» ANSWER :`(55)/(221)`

Discussion

8739.	int_(0)^(2a) f(x) dx = 2 int_(0)^(a) f (x) dx if
Answer» `F(2a-X) =f(x)` `f(a-x) = f(x)` `f(x) = - f(-x)` `f(-x) = f(x)` ANSWER :`f(2a-x) =f(x)`

Discussion

8740.	If f :Rto R is definedby f (x)f(x)= [ x -3]+ \| x -4\| forx inR thenlim_(x to 3)f(x)is equal to
Answer» `-2` `-1` 0 1 Answer :C

Discussion

8741.	Solvexsec((y)/(x))(y dx+x dy)=y" cosec"((y)/(x))(x dy-y dx)
Answer» ANSWER :`C SIN ((y)/(2))`

Discussion

8742.	If alpha, beta are the roots of the equation x^(2)+7x+12=0, then the equation whose roots are (alpha+beta)^(2) and (alpha-beta)^(2) is
Answer» `X^(2)+50x+49=0` `x^(2)-50x+49=0` `x^(2)-50x-49=0` `x^(2)+12x+7=0` ANSWER :B

Discussion

8743.	I : If a,b,c,d are four vectors then [b c d] a + [c a d] b + [a b c] c = [a b c]d II : The points with position vectors a,b,c,d are coplanar then [a b d] + [b c d] + [c a d] = [a b c]
Answer» only I is ture Only II is ture both I and II are true Neither I nor II are true Answer :C

Discussion

8744.	If alpha, beta ,gamma are roots of a cubic equationx^(3) + 2x^(2) - 3x - 1 = 0, then sum alpha^(-5) =
Answer» -365 518 365 -518 Answer :4

Discussion

8745.	Evaluate: int(x^2)/(x^4+x^2-2)\ dx
Answer» ANSWER :`sqrt2/3tan^(-1)(x/sqrt2)+1/6log\|(x-1)/(x+1)\|+C`

Discussion

8746.	Derivative of tan^(-1)2xsqrt(1-x^(2))/(1-2x^(2))w.r.t.sec^(-1)" "(1)/(2x^(2)-1)" at "x=(1)/(2)
Answer» `(1)/(2)` `-(1)/(2)` `-1` 1 Answer :C

Discussion

8747.	IF A={:[(1,2,5),(1,-1,-1),(2,3,-1):}] find A^-1 Hence solve the following system of equations: x+2y+5z=10,x-y-z=-2,2x+3y-z=-11
Answer» ANSWER :x=-1,y=-1,z=3

Discussion

8748.	Area bounded by the curve y=x^3, the x-axis and the ordinates x = -2 and x = 1 is
Answer» -9 `-15/4` `15/4` `17/4` ANSWER :D

Discussion

8749.	Write relations in tabular form and determine their type for R={(x,y):x "divides" y} on A ={1,2,3,4,5,6}
Answer» SOLUTION :R`={(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,2),(2,4),(2,6),(3,3),(3,6),(4,4),(5,5),(6,6)}` R is REFLEXIVE TRANSITIVE but not SYMMETRIC.

Discussion

8750.	Find for what values of a and b the system of linear equation.x+2y+2z=5 , 2x+5y+3z=10 and 3x+y+lambda z=mu has (i) no solution,(ii)a uniquesolution, (iii)an infinite no of solution.
Answer» ANSWER :`LAMBDA = 11`and`MU = 15`

Discussion

Explore topic-wise InterviewSolutions in .

If f'(x)=sec^(2)x+cosec^(2)x-1 and f((pi)/(4))=1

A reversible reaction is which :-

If C_(r)= ""^(n)C_(r), then C_(0)+C_(4)+C_(8)+C_(12)+cdots=

A:In Delta ABC, if r_1,r_2,r_3 are in H.P. then r_2//r=1//3 R: In DeltaABC,1/r_1+1/r_2+1/r_3=2/r

If m = ( 2013) ! then the value of (1)/(log_(2)m) + ( 1)/( log_(3)m ) + "........"+ ( 1)/( log_(2013)m) is equal to

Evaluate the integral underset(0)overset(1)int (log(1+x))/(1+x^(2))dx

If ina triangle ABC , a=15, b= 36, c = 39 thensin"" (C )/(2)=

On all chords (parallel to one and the same direction) a circle of radius R symmetrical parabolic segments of the same altitude h are constructed. The planes of the segments are perpendicular to the plane of the circle. Find the volume of the solid thus obtained

Defineeach of thefollowing: "(i)injective function """"(ii) surjectivefunction " "(iii) bijective function """"(iv) many -onefunction" "(v) intofunction """ Give anexampleof each typeof functions.

int (1)/(2x^(2)-3)dx=

int ((x+3)/((x+4)^(2)))e^(x)dx=.....+C

Given three non-coplanar vectors OA=a, OB=b,OC=c. Let S be the centre of the spheres passing through the points, O,A,B,C if OS=x, then:

Draw the graph of y= [x]+ sqrt(x-[x]), where[*] denotes the greatest ineger function.

IFS=overset ( m) sum"" ^(n+r ) C_(k),then

If the equation ................

int_(1//3)^(1) ((x-x^(3))^(1//3))/(x^(4))dx=

If (x^(4))/((x-1)(x+2))=(1)/(3(x-1))-(16)/(3(x+2))+x^(2)-x+k then k=

If a parabola passess through the points (-2, 1), (1, 2) and (-1, 3) having horizontal axis, then the length of the latus rectum of that parabola is

If in A.P., 3rd term is 18 and 7 term is 30, then find sum of its first 17 terms

If f(x)=log_(e^(2))(logx), then f'(e) is :

Expand the expression (2x - 3)^(6).

Solution of the differential equation (dy)/(dx)=(1+y^(2))/(1+x^(2)) is

By using the properties of definite integrals, evaluate the integrals int_(0)^(pi/2)cos^(2)xdx

If a, b, c arein AP amd b, c, a are in GP , thenshow that c, a, b are in H.P. Finda : b : c .

If Z is a complex number such that Z=-overlineZ,then

There are 4 mangoes, 3 apples 2 oranges in bag, fruits of the same variety being identical. In how many different ways can a selection of fruits be made if atleast one mango is to be selected.

The area of the region bounded by the curves y=|x-1|andy=3-|x| is

Evaluate the following lim_(xto0)((3+x)^3-27)/x

If the system of equation (b+c)x+ay+a^(2)z=0,(c+a)x+by+b^(2)z=0,(a+b)x+cy+c^(2)z=0 has a non trivial solution and a ne b ne c prove that a+b+c=0

Consider the sequence in the form of group (1),(2,2)(3,3,3),(4,4,4,4),(5,5,5,5,5…..) The 2000^(th) term of the sequence is not divisible by

If 36, 84, 126 are three successive binomial coefficients in the expansion of (1+x)^(n), find n.

If bar(a) and bar(b) are adjacent sides of a rhombus, then bar(a).bar(b)=0.

Consider the determinant Delta=|[0,2,6],[1,5,0],[3,7,1]|Write the minors of the element in the first column ?

Choose the correct answer. If theta is the angle between two vectors veca"and"vecb, then veca.vecb ge 0 only whena)0 lt theta lt pi/2b)0 le theta le pi/2c)0 lt theta lt pid) 0 le theta le pi

What is the interval in which log_5xis increasing ?

A bag contains 4 white balls and 3 black balls. If two balls are drawn at random, then mean of number of white balls is

If two cards are drawn from pack 52 cards at random, then find the probability of getting both red or both kings.

int_(0)^(2a) f(x) dx = 2 int_(0)^(a) f (x) dx if

If f :Rto R is definedby f (x)f(x)= [ x -3]+ | x -4| forx inR thenlim_(x to 3)f(x)is equal to

Solvexsec((y)/(x))*(y dx+x dy)=y" cosec"((y)/(x))*(x dy-y dx)

If alpha, beta are the roots of the equation x^(2)+7x+12=0, then the equation whose roots are (alpha+beta)^(2) and (alpha-beta)^(2) is

I : If a,b,c,d are four vectors then [b c d] a + [c a d] b + [a b c] c = [a b c]d II : The points with position vectors a,b,c,d are coplanar then [a b d] + [b c d] + [c a d] = [a b c]

If alpha, beta ,gamma are roots of a cubic equationx^(3) + 2x^(2) - 3x - 1 = 0, then sum alpha^(-5) =

Evaluate: int(x^2)/(x^4+x^2-2)\ dx

Derivative of tan^(-1)2xsqrt(1-x^(2))/(1-2x^(2))w.r.t.sec^(-1)" "(1)/(2x^(2)-1)" at "x=(1)/(2)

IF A={:[(1,2,5),(1,-1,-1),(2,3,-1):}] find A^-1 Hence solve the following system of equations: x+2y+5z=10,x-y-z=-2,2x+3y-z=-11

Area bounded by the curve y=x^3, the x-axis and the ordinates x = -2 and x = 1 is

Write relations in tabular form and determine their type for R={(x,y):x "divides" y} on A ={1,2,3,4,5,6}

Find for what values of a and b the system of linear equation.x+2y+2z=5 , 2x+5y+3z=10 and 3x+y+lambda z=mu has (i) no solution,(ii)a uniquesolution, (iii)an infinite no of solution.

Solvexsec((y)/(x))(y dx+x dy)=y" cosec"((y)/(x))(x dy-y dx)