28278 + Interview Questions in Maths in Class 12 Page 133 InterviewSolution

6601.	Find the number of positive integers x satisfying the equation 1/x + 1/(x+1) + 1/(x+2) = 13/12.
Answer» ANSWER :1

Discussion

6602.	int x^(x)(1+logx)dx=....+c
Answer» `X^(x)LOGX` `E^(x^(x))` `x^(x)` NONE of these Answer :C

Discussion

6603.	If cos ""x/2cos "" (x )/(2 ^(2)) ….cos "" (x)/(2 ^(n)) = (sin x)/( 2 ^(n) sin (x //2^(n))) then ((1)/(2 ^(2))) sec ^(2) "" x/2 + ((1)/(2 ^(4))) ec ^(2) "" (x)/(2 ^(2)) …. ((1)/(2 ^(2n))) sec ^(2) "" (x)/(2 ^(n)) =
Answer» `((1)/(2 ^(2N)))COEC ^(2) (x)/(2 ^(n)) + cosec ^(2) x 2` `-((1)/(2 ^(2n)))coec ^(2) (x)/(2 ^(n)) + cosec ^(2) x 2` `cosec ^(2) (x)/(2 ^(n)) + cosec ^(2) x` none Answer :B

Discussion

6604.	If a variable x takes values 0,1,2,..,n with frequencies proportional to the binomial coefficients .^(n)C_(0),.^(n)C_(1),.^(n)C_(2),..,.^(n)C_(n), then var (X) is
Answer» `(n^(2)-1)/(12)` `(n)/(2)` `(n)/(4)` None of these Solution :We have `overline(X)=(0^(n)C_(0)+1^(n)C_(1)+2^(n)C_(2)+..+n^(n)C_(n))/(.^(n)C_(0)+.^(n)C_(1)+.^(n)C_(2)+..+.^(n)C_(n))=(underset(r=0)OVERSET(n)(sum r^(n)C_(r )))/(underset(r=0)overset(n)(sum .^(n)C_(r )))` `=(1)/(2^(n))underset(r=1)overset(n)(sum r)(n)/(r ).^(n-1)C_(r-1) "" [because underset(r=0)overset(n)(sum).^(n)C_(r )=2^(n), .^(n)C_(r )=(n)/(r ).^(n-1)C_(r-1)]` `=(n)/(2^(n))underset(r=1)overset(n)(sum).^(n-1)C_(r-1)=(n)/(2^(n))2^(n-1)=(n)/(2)[because underset(r=1)overset(n)(sum).^(n-1)C_(r-1)=2^(n-1)]` and `(1)/(N)sum f_(i)x_(1)^(2)=(1)/(2^(n))sum r^(2) .^(n)C_(r )=(1)/(2^(n)) underset(r=0)overset(n)(sum)[r(r-1)+r].^(n)C_(r )` `=(1)/(2^(n)){underset(r=0)overset(n)(sum)r-(r-1).^(n)C_(r )+underset(r=0)overset(n)(sum r) .^(n)C_(r )}` `=(1)/(2^(n)){underset(r=2)overset(n)(sum)r(r-1)(n)/(r )(n-1)/(r-1) .^(n-2)C_(r-2)+underset(r=1)overset(n)(sum r) (n)/(r ).^(n-1)C_(r-1)}` `=(1)/(2^(n)){n(n-1)2^(n-2)+N2^(n-1)}=(n(n-1))/(4)+(n)/(2)` `therefore "VAR"(X)=(1)/(N)sum f_(i)x_(i)^(2)-overline(X^(2))=(n(n-1))/(4)+(n)/(2)-(n^(2))/(4)=(n)/(4)`

Discussion

6605.	If a normal chord drawn at 't' on y^(2) = 4ax subtends a right angle at the focus then t^(2) =
Answer» 2 4 8 16 Answer :B

Discussion

6606.	Find the derivative of the following functions 'ab initio', that is, using the definition.s^2-bs+5
Answer» Solution :LET `y=s^2-bs+5` Then `(dy)/(DS)=lim_(hto0)((s+H)^2-b(s+h)+5-(s^2-bs-5))/h` `=lim_(hto0)((s^2+2hs+h^2-bs-bh+5-s^2+bs-5))/h` `=lim_(hto0)(2hs-bh+h^2)/h` `=lim_(hto0){2s-b+h}=2s-b`

Discussion

6607.	If each of the points (x,, 4), (-2, y,) lie on the-line joining the points (2, -1) and (5,-3) then the point P(x_(1), y_(1)) lies on the line
Answer» 6(x + y) - 25 = 0 2x + 6Y + 1 = 0 2x + 3Y - 6 = 0 6 (x + y) + 25 = 0 Answer :B

Discussion

6608.	Dice A has 4 red and 2 white faces whereas dice B has 3 red and 3 white faces. A coin is tossed once, if it falls head then the game continues by throwing the dice A and if it falls tail then the dice B is to be used. If red turns up at first 3 throws, then the probability that dice A is being used is
Answer» `(7)/(37)` `(64)/(91)` `(9)/(41)` `(27)/(35)` ANSWER :B

Discussion

6609.	be binary operation defined on a set R by ab=a+b-(ab)^2. Show that is commutative, but it is not associative. Find the identity element for .
Answer» SOLUTION :N/A

Discussion

6610.	Write the value of int(e^x-1)/(1-e^(-x))dx
Answer» SOLUTION :`INT(e^X-1)/(1-e^(-x))dx=intcot^2xdx=int("COSEC"^2x-1)dx=-cot x-x+c`

Discussion

6611.	Write the value of int(e^x-1)/(1-e^-x)dx
Answer» SOLUTION :`INT(e^x-1)/(1-e^-x)dx=int(e^x-1)/(1-1/e^x)dx=int(e^x-1)/(e^x-1)xxe^xdx=inte^xdx=e^x+c`

Discussion

6612.	Ifsin 18^(@)=(sqrt(5)-1)/(4) , thensin 81^(@)=
Answer» `(SQRT(5)+1)/(4sqrt(2))+(sqrt(10-2sqrt(5)))/(4sqrt(2))` `(sqrt(5)+1)/(4 sqrt(2))/(sqrt(10+2sqrt5))/(4sqrt(2))` `(sqrt(5)-1)/(4sqrt(2))+(sqrt(10+2sqrt(5)))/(4sqrt(2))` `(sqrt(5)-1)/(4sqrt(2))+(sqrt(10-2sqrt(5)))/(4sqrt(2))` Answer :A

Discussion

6613.	Approximately how many dollar's worth of vegetables were sold in September. October, and November combined by Produce Stand P?
Answer» `$5,724` `$6,230` `$6,6,21` `$7,130` ANSWER :D

Discussion

6614.	Number of ways of allotting 30 marks to 10 questions so that each question to get atleast two marks is
Answer» `""^(19)C_(6)` `""^(19)C_(10)` `""^(30)C_(10)` `""^(30)C_(15)` Answer :C

Discussion

6615.	Solve the following linear programming problem graphically: Maximize : z=30x+25y Subject to: x+hle6 3x+2yle15 xge0 yge0
Answer» SOLUTION :NA

Discussion

6616.	Form the differential equation of the family of circles in the first quadrant which touch the coordinate axes.
Answer» ANSWER :`(X + YY')^(2) = ( x - y)^(2)(1 + (y')^(2))`

Discussion

6617.	Prove that if the graph of the function y=f(x), defined throughout the number scale, is symmetrical about two vertical axes x =a and x=b (a lt b), then this function is a periodic one.
Answer» ANSWER :`F[a+(a-X)]=f(x)`

Discussion

6618.	((1+i)x-i)/(2+i)+((1+2i)y+i)/(2-i)=1rArr(x,y)=
Answer» (0,0) (3,1) (3,-1) (-3,1) ANSWER :C

Discussion

6619.	int (sin x.cos x)/(cos^(2) x + 3 cos x + 2)dx =
Answer» `log"" ((COS X + 2)^(2))/(\| cos x + 1\|) + C ` `log \|(cos x + 2)/( cos x + 1)\| + c ` log`(\|1 + cos x \|)/((2 + cos x)^(2)) + c` `log \|(1 + cos x)/(2 + cos x)\| +c ` ANSWER :C

Discussion

6620.	Find two positive numbers whose sum is 15 and the sum of whose squares is minimum.
Answer» ANSWER :`(15)/(2)` and `(15)/(2)`

Discussion

6621.	IF x in Rthen(2a(x-1) sin^2alpha ) /( x^2- sinalpha )connotliebetween
Answer» `a SIN ^2ALPHA, acos^2alpha ` `a sin^2( ALPHA//2) a,cos^2( alpha //2)` `2asin^2 alpha, 2acos ^2 alpha ` `2asin ^2(alpha //2 , 2acos ^2 (alpha //2)` ANSWER :D

Discussion

6622.	If vec(a),vec(b),vec(c) are position vectors of the vertices of the triangle ABC, then \|(vec(a)-vec(c))xx(vec(b)-vec(a))\|/((vec(c)-vec(a)).(vec(b)-vec(a))) is equal to
Answer» COT A cot C `-TAN C` tan A Answer :D

Discussion

6623.	Locus of feet of perpendicular from (5,0) to the tangents of (x^(2))/(16)-y^(2)/(9)=1 is
Answer» `x^(2)+y^(2)=4` `x^(2)+y^(2)=16` `x^(2)+y^(2)=9` `x^(2)+y^(2)=25` ANSWER :B

Discussion

6624.	Solve the following linear programming problem graphically: Maximize : z=200x+500y Subject to: x+2yge10 3x+4yle24 xge0 yge0
Answer» SOLUTION :NA

Discussion

6625.	intx^(5)3sqrt((1+x^(3))^(2))dx is equal to
Answer» `(1)/(8)U^(8//3)-(1)/(5)u^(5//3)+C,u=1+x^(3)` `(3)/(8)u^(8//3)-(5)/(3)u^(5//3)+C,u=1+x^(3)` `-(3)/(8)u^(8//3)+(3)/(5)u^(5//3)+C,u=1+x^(3)` `-(1)/(8)u^(8//3)+(1)/(5)u^(5//3)+C,u=1+x^(3)` ANSWER :A

Discussion

6626.	If 1,alpha_1,alpha_2,…..alpha_(n-1) are the n^(th) roots of unity then (1-alpha_1)(1-alpha_2)…..(1-alpha_(n-1))=
Answer» n-1 n -1 1 Answer :B

Discussion

6627.	If x and y are the sides of two squares such that y=x-x^(2). Then the rate of change, if the area of second square with respect to the first square, when x = 2, is ……….
Answer» 1 4 3 6 Answer :C

Discussion

6628.	((2x+3y)/5)+(2f(x)+3f(y))/5 and f^(')(0)=p and f(0)=q. Then f^(")(0) is. (where f(x) is a polynomial function)
Answer» SOLUTION :Obviously, `f(x)` is a linear function ALSO from `f^(')(0)=P` and `f(0)=q, f(x)px+q` or `f^(")(0)=0`

Discussion

6629.	If the function f:[0,4] rightarrow R is differentiable then show that: (i) For a,b epsilon(0,4), (f(4))^(2) - (f(0))^(2) = 8f(a) f(b) (ii)int_0^4 f(t)dt =2 [alphaf(alpha^^(2) + betaf(beta^(2))]for some alpha, beta, such that 0
Answer» Answer :(i) `f(a)f(B) for a,b,in (0,4)` (II) `int_0^4 f(t)dt =2 [alphaf(ALPHA^^(2) + betaf(BETA^(2))]`for some alpha, beta such that `0 ltalpha, beta lt 2`.

Discussion

6630.	I. underset(x to 0)"Lt" {:(,(1+x)^(2)-(1-x)^(2)-2),(,(1+x)^(3)-(1-x)^(3)-3):} II. If f(x)=(4-7x)/(3x+4) and underset(x to 2)"Lt" f(x)=l, underset(x to 0)"Lt" f(x)=m then the equation whose roots are 1/l, 1//m" is "x^(2)-1=0
Answer» only I is true only II is true both I and II are true neither I nor II are true Answer :C

Discussion

6631.	inte^(x)(1+sinx)/(1+cosx)dx is equal to
Answer» `log\|tanx\|+C` `E^(X)tanx//2+C` `e^(x)cotx+C` `sinlogx+C` ANSWER :B

Discussion

6632.	(a xx b) xx c + (b xx c) xx a + (C xx a) xx b =
Answer» 0 a b c Answer :A

Discussion

6633.	John spends (1)/(3) of his waking hours working, (1)/(5) of his waking hours eating meals, (3)/(10) of his waking hours at the gym, and 2hours going ot and from work. He engages in no other activities while awake. How many hours is John awake?
Answer» ANSWER :12 HOURS

Discussion

6634.	Let m be a natural number such that 2000lt m lt 60000 and let k be the sum of all the digits in m. Then the number of numbers m for which k is even is
Answer» 19909 19989 18999 19999 Answer :D

Discussion

6635.	For any natural number n, expressed in base 10, let S(n) denote the sum of all digits of n. Find all natural numbers n such that n^3 = 8S(n)^3 + 6nS(n) + 1.
Answer» ANSWER :17

Discussion

6636.	State when the equality will hold, \|veca - vecb\|ge\|veca\|-\|vecb\|
Answer» Solution : LET ABC be a triangle such that `vec(AB) = veca, vec(CB) = vecb` Then `vec(AC) = veca-vecb` be triangle LAW In the `triangleABC, AC+CB ge AB` `IMPLIES \|veca-vecb\|+\|vecb\| ge veca` `implies \|veca-vecb\| ge \|veca\| - \|vecb\|` THEREFORE The equality holds when `veca` and `vecb` are collinear vectors.

Discussion

6637.	int(cosx+xsinx)/(x(x+cosx))dx
Answer» ANSWER :`log\|x\|-log\|x+cosx\|+c`

Discussion

6638.	bar(a),bar(b) and bar( c ) are unit vectors. bar(a).bar(b)=0=bar(a).bar( c ) and the angle between bar(b) and bar( c ) is (pi)/(3). Then \|bar(a)xx bar(b)-bar(a)xx bar( c )\|=…………..
Answer» `(1)/(2)` 1 2 0 Answer :B

Discussion

6639.	P,Q,R and four point with the position vectors 3i-4j+5k, -4i+5j+k and -3i+4j+3k, respectively. Then the line PQ meets the line RS at the point
Answer» 3i+4j+3k `-3i+4j+3k` `-i+4j+k` `i+j+k` ANSWER :B

Discussion

6640.	If alpha, beta, gamma are the angles which a line makes with positive direction of the axes, then match the quantities in Column-I to their values in column-II :
Answer» ANSWER :A::B::C::D

Discussion

6641.	For each of the differential equations in find the particular solution satisfying the given condition : (x+y)dy+(x-y)dx=0,y=1 when x=1
Answer» Answer :`LOG (x^(2) + y^(2)) + 2 tan^(-1) (y)/(x) = (PI)/(2) + log 2`

Discussion

6642.	Find (dy)/(dx), " if "x= a cos theta, y= a sin theta.
Answer» ANSWER :`= -COT THETA`.

Discussion

6643.	y = tan^(-1)((3x-x^3)/(1-3x^2)), 1/(sqrt3) , x , 1/(sqrt3).
Answer» ANSWER :`(3)/(1+x^(2))`

Discussion

6644.	Let S be the set of all real values of lamda for which the system of linear equations lamdax+y+z=5lamda 2lamdax+2y-z=1 3y+z=9 has infinitely solutios. Then S
Answer» equal to R is a singleton contains exactly TWO elements is an EMPTY set Answer :D

Discussion

6645.	If the sum of n terms of the series 2^(3) + 4^(3) + 6^(3) + ......... is 3528 then n =
Answer» 10 6 8 9 Answer :B

Discussion

6646.	For the quadratic polynomial f (x) =4x ^(2)-8ax+a. the statements (s) which hold good is/are:
Answer» There is only one integral 'a' gor which f (x) is NON- NEGATIVE `AA x in R` For `a lt 0,` the number zero lies between the zeroes of the polynomial `f (x) =0` has two DISTINCT solutions in `(0,1) ` for `a in ((1)/(7), (4)/(7))` The MINIMUM value of `f (x)` for minimum value of a for which f (x) is non-negative `AA x in R` is 0 ANSWER :A::B::C::D

Discussion

6647.	If a matrix has 5 elements, what are the possible orders it can have ?
Answer» ANSWER :`1xx5,5xx1`

Discussion

6648.	Let h(x) = f_(1)(x) + f_(2) (x) + f_(3)(x)+...+f(n)(x), where f_(1)(x), f_(2)(x),f_(3)(x),....,f_(n)(x) are real valued functions of x. Statement I f(x) = \|cos\|x\|\| + cos^(-1)("sgn x")+\|"In x"\| is not differentiable at 3 points in (0, 2pi) Statement II Exactly one function, is f_(i) (x), i = 1, 2,...,n is not differentiable and the rest of the function is differentiable at x = a makes h(x) not differentiable at x = a.
Answer» Statement I is CORRECT, Statement II is ALSO correct, Statement II is the correct explanation of Statement I Statement I is correct, Statement II is also correct, Statement II is not the correct explanation of Statement I Statement I is correct, Statement II is INCORRECT Statement I is incorrect, Statement II is correct. ANSWER :A

Discussion

6649.	Find the angle between the lines whose direction ratios are a, b, c and b-c, c-a, a-b
Answer» Answer :SINCE a(b-c)+b(c-a)+c(a-b) = ab-ac+bc-ba_ca-cd = 0, the lines are PERPENDICULAR `therefore` The angle between the lines = `90^@`

Discussion

6650.	The sum of 15^(2) + 16^(2) + 17^(2) + ....... + 30^(2) =
Answer» 8840 8440 8540 8450 Answer :B

Discussion

Explore topic-wise InterviewSolutions in .

Find the number of positive integers x satisfying the equation 1/x + 1/(x+1) + 1/(x+2) = 13/12.

int x^(x)(1+logx)dx=....+c

If cos ""x/2cos "" (x )/(2 ^(2)) ….cos "" (x)/(2 ^(n)) = (sin x)/( 2 ^(n) sin (x //2^(n))) then ((1)/(2 ^(2))) sec ^(2) "" x/2 + ((1)/(2 ^(4))) ec ^(2) "" (x)/(2 ^(2)) …. ((1)/(2 ^(2n))) sec ^(2) "" (x)/(2 ^(n)) =

If a variable x takes values 0,1,2,..,n with frequencies proportional to the binomial coefficients .^(n)C_(0),.^(n)C_(1),.^(n)C_(2),..,.^(n)C_(n), then var (X) is

If a normal chord drawn at 't' on y^(2) = 4ax subtends a right angle at the focus then t^(2) =

Find the derivative of the following functions 'ab initio', that is, using the definition.s^2-bs+5

If each of the points (x,, 4), (-2, y,) lie on﻿ the-line joining the points (2, -1) and﻿ (5,-3) then the point P(x_(1), y_(1)) lies on the line﻿

** be binary operation defined on a set R by a**b=a+b-(ab)^2. Show that ** is commutative, but it is not associative. Find the identity element for **.

Write the value of int(e^x-1)/(1-e^(-x))dx

Write the value of int(e^x-1)/(1-e^-x)dx

Ifsin 18^(@)=(sqrt(5)-1)/(4) , thensin 81^(@)=

Approximately how many dollar's worth of vegetables were sold in September. October, and November combined by Produce Stand P?

Number of ways of allotting 30 marks to 10 questions so that each question to get atleast two marks is

Solve the following linear programming problem graphically: Maximize : z=30x+25y Subject to: x+hle6 3x+2yle15 xge0 yge0

Form the differential equation of the family of circles in the first quadrant which touch the coordinate axes.

Prove that if the graph of the function y=f(x), defined throughout the number scale, is symmetrical about two vertical axes x =a and x=b (a lt b), then this function is a periodic one.

((1+i)x-i)/(2+i)+((1+2i)y+i)/(2-i)=1rArr(x,y)=

int (sin x.cos x)/(cos^(2) x + 3 cos x + 2)dx =

Find two positive numbers whose sum is 15 and the sum of whose squares is minimum.

IF x in Rthen(2a(x-1) sin^2alpha ) /( x^2- sinalpha )connotliebetween

If vec(a),vec(b),vec(c) are position vectors of the vertices of the triangle ABC, then |(vec(a)-vec(c))xx(vec(b)-vec(a))|/((vec(c)-vec(a)).(vec(b)-vec(a))) is equal to

Locus of feet of perpendicular from (5,0) to the tangents of (x^(2))/(16)-y^(2)/(9)=1 is

Solve the following linear programming problem graphically: Maximize : z=200x+500y Subject to: x+2yge10 3x+4yle24 xge0 yge0

intx^(5)3sqrt((1+x^(3))^(2))dx is equal to

If 1,alpha_1,alpha_2,…..alpha_(n-1) are the n^(th) roots of unity then (1-alpha_1)(1-alpha_2)…..(1-alpha_(n-1))=

If x and y are the sides of two squares such that y=x-x^(2). Then the rate of change, if the area of second square with respect to the first square, when x = 2, is ……….

((2x+3y)/5)+(2f(x)+3f(y))/5 and f^(')(0)=p and f(0)=q. Then f^(")(0) is. (where f(x) is a polynomial function)

If the function f:[0,4] rightarrow R is differentiable then show that: (i) For a,b epsilon(0,4), (f(4))^(2) - (f(0))^(2) = 8f(a) f(b) (ii)int_0^4 f(t)dt =2 [alphaf(alpha^^(2) + betaf(beta^(2))]for some alpha, beta, such that 0

I. underset(x to 0)"Lt" {:(,(1+x)^(2)-(1-x)^(2)-2),(,(1+x)^(3)-(1-x)^(3)-3):} II. If f(x)=(4-7x)/(3x+4) and underset(x to 2)"Lt" f(x)=l, underset(x to 0)"Lt" f(x)=m then the equation whose roots are 1/l, 1//m" is "x^(2)-1=0

inte^(x)(1+sinx)/(1+cosx)dx is equal to

(a xx b) xx c + (b xx c) xx a + (C xx a) xx b =

John spends (1)/(3) of his waking hours working, (1)/(5) of his waking hours eating meals, (3)/(10) of his waking hours at the gym, and 2hours going ot and from work. He engages in no other activities while awake. How many hours is John awake?

Let m be a natural number such that 2000lt m lt 60000 and let k be the sum of all the digits in m. Then the number of numbers m for which k is even is

For any natural number n, expressed in base 10, let S(n) denote the sum of all digits of n. Find all natural numbers n such that n^3 = 8S(n)^3 + 6nS(n) + 1.

State when the equality will hold, |veca - vecb|ge|veca|-|vecb|

int(cosx+xsinx)/(x(x+cosx))dx

bar(a),bar(b) and bar( c ) are unit vectors. bar(a).bar(b)=0=bar(a).bar( c ) and the angle between bar(b) and bar( c ) is (pi)/(3). Then |bar(a)xx bar(b)-bar(a)xx bar( c )|=…………..

P,Q,R and four point with the position vectors 3i-4j+5k, -4i+5j+k and -3i+4j+3k, respectively. Then the line PQ meets the line RS at the point

If alpha, beta, gamma are the angles which a line makes with positive direction of the axes, then match the quantities in Column-I to their values in column-II :

For each of the differential equations in find the particular solution satisfying the given condition : (x+y)dy+(x-y)dx=0,y=1 when x=1

Find (dy)/(dx), " if "x= a cos theta, y= a sin theta.

y = tan^(-1)((3x-x^3)/(1-3x^2)), 1/(sqrt3) , x , 1/(sqrt3).

Let S be the set of all real values of lamda for which the system of linear equations lamdax+y+z=5lamda 2lamdax+2y-z=1 3y+z=9 has infinitely solutios. Then S

If the sum of n terms of the series 2^(3) + 4^(3) + 6^(3) + ......... is 3528 then n =

For the quadratic polynomial f (x) =4x ^(2)-8ax+a. the statements (s) which hold good is/are:

If a matrix has 5 elements, what are the possible orders it can have ?

Find the angle between the lines whose direction ratios are a, b, c and b-c, c-a, a-b

The sum of 15^(2) + 16^(2) + 17^(2) + ....... + 30^(2) =

If each of the points (x,, 4), (-2, y,) lie on the-line joining the points (2, -1) and (5,-3) then the point P(x_(1), y_(1)) lies on the line

be binary operation defined on a set R by ab=a+b-(ab)^2. Show that is commutative, but it is not associative. Find the identity element for .