28278 + Interview Questions in Maths in Class 12 Page 259 InterviewSolution

12901.	Let n in N such that n gt 1. Statement-1: int_(oo)^(0) (1)/(1+x^(n))dx=int_(0)^(1) (1)/((1-x^(n))^(1//n))dx Statement-2: int_a^b f(x)dx=int_(a)^(b) f(a+b-x)dx
Answer» Statement-1 is true, Statement-2 is True,Statement-2 is a correct EXPLANATION for Statement-1. Statement-1 is True, Statement-2 is not a correct explanation for Statement-1. Statement-1 is True, Statement-2 is False. Statement-1 is False, Statement-2 is True. Solution :Clearly, statement-2 is true (See page 44.17 Property IX). Putting `X^(N)=tan^(2)THETA`, we get `I_(1)=underset(0)overset(oo)int (1)/(1+x^(n))DX=(2)/(n)underset(0)overset(pi//2)int tan^((2)/(n)-1)theta d theta` Putting `x^(n)=sin^(2) theta`, we get `I_(2)=(2)/(n)underset(0)overset(pi//2)int tan^((2)/(n)-1) theta d theta` `:. underset(0)overset(oo)int(1)/(1+x^(n))dx=underset(0)overset(1)int(1)/((1-x^(n))^(1//n)dx` So, statement-1 is also true.

Discussion

12902.	A=[{:(2,-3),(4,6):}] verify (adjA)^(-1)=(adjA^(-1))
Answer» ANSWER :`(adjA)^(-1)=(adjA^(-1))`

Discussion

12903.	random variable X has the range {1,2,3,……….}. If P(X=r)=c^(r//r) for r=1,2,3,……… then c=
Answer» `E^(2)` `2^(e)` `log_(e)2` `1//2` ANSWER :C

Discussion

12904.	A diet for a sick person must contain at least 4000 units of vitamins, 50 units of minerals and 1400 calories. Two foods X and Y are available at the cost of रु 4 and रु 3 per unit respectively. One unit of the food X contains 200 units of vitamins, 1 unit of minerals and 40 calories, whereas one unit of the food Y contains 100 units of vitamins, 2 units of minerals and 40 calories. Find what combination of X and Y should be used to have least cost, satisfying the requirements.
Answer» Answer :5 UNIT of X and 30 UNITS of Y Minimum cost = रु 110

Discussion

12905.	Consider a binary opertion on N defined as a b=a ^(3) +b ^(3). Choose the correct answer.
Answer» Is ``both ASSOCIATIVE and COMMUTATIVE ? Is `` commutative but not associative ? Is `` associative but not commutative ? Is `` NEITHER comutative nor associative ? ANSWER :B

Discussion

12906.	If (0,6) and (0,3) are respectively the vertex and focus of a parabola then its equation is
Answer» `x^2 + 12Y = 72` `x^2 - 12y = 72` `y^2 - 12X = 72` `y^2 + 12x = 72` ANSWER :A

Discussion

12907.	Let a and b be unit vectors inclined at an angle 2alpha(0lealphalepi) each other, then (a+b) lt 1, if
Answer» `ALPHA=(pi)/(2)` `alphalt(pi)/(3)` `alphagt(2pi)/(3)` `(pi)/(3)ltalphalt(2pi)/(3)` Solution :Given , `(a+b)lt 1` ` IMPLIES SQRT(1+12cos 2 alpha )lt 1` ` implies sqrt(2(1+cos 2 alpha ))lt 1` `implies sqrt(4 cos^(2) alpha )lt 1` `implies \| cosalpha \|lt (1)/(2)` `implies (pi)/(3)lt alpha lt (2pi)/(3) [ :'0 le alpha le pi ]`

Discussion

12908.	Let A=B B^(T)+C C^(T), where B=[(cos theta), (sintheta)], C=[(sin theta), (-costheta)], theta in R. Then A is :
Answer» `[(0, 0),(0, 0)]` `[(0, 1), (1, 0)]` `[(1, 0), (0,1)]` `[(0,0),(0, 1)]` ANSWER :C

Discussion

12909.	Two dice are thrown. If it is known that the sum of numbers on the dice was less than 6, the probability of getting a sum 3, is .........
Answer» `(1)/(18)` `(5)/(18)` `(1)/(5)` `(2)/(5)` ANSWER :C

Discussion

12910.	Show that the poles of the tangents to the circle x^(2)+y^(2) = a^(2) with respect to the circle (x+a)^(2)+y^(2)=2a^(2) lie on y^(2) +4ax =0 .
Answer» ANSWER :` y ^(2) + 4AX =0 `

Discussion

12911.	If A is symmetric matrix , then B'AB is …….
Answer» ANSWER :`thereforeB'AB` is SYMMETRIC MATRIX.

Discussion

12912.	If (tanalpha-i[sin(alpha//2)+cos(alpha//2)])/(1+2isin(alpha//2)) is purely imaginary , is
Answer» `npi+pi/4` `npipmpi/2` `2npipmpi/4` `2npi+pi/2` ANSWER :A

Discussion

12913.	Find a if the coefficients of x^(2) and x^(3) in the expansion of (3 + ax)^(9) are equal.
Answer» ANSWER :`:. a = 9/7`

Discussion

12914.	The mass of a molecule of the compound C_(60) H_(122) is :-
Answer» `1.4 XX 10^(-21) g` `1.09 xx 10^(-21) g` `5.025 xx 10^(23) g` `16.023 xx 10^(23) g` ANSWER :A

Discussion

12915.	Let f(x) = [1- x^(2)], x in R, where [] is the greatest integer function. Then
Answer» F is increasing x= 0 is the POINT of MAXIMA of f f is CONTINUOUS at x=0 f is decreasing Answer :D

Discussion

12916.	If cot^(-1).(n)/(pi)gt(n)/(6),n in N, then the maximum value of n is :
Answer» 6 7 5 None of these Solution :`COT^(-1).(n)/(pi) GT(pi)/(6)` `rArr""(n)/(pi) LT cot.(pi)/(6)` [as `cot^(-1)x` is a DECREASING function] `rArr""(n)/(pi) lt sqrt3` `rArr""n lt sqrt3pi` `rArr""n lt 5.46` `rArr"MAXIMUM value of n is 5"`

Discussion

12917.	Let a , b, c be such that b (a + c) ne0. If\|{:(a, a+1, a- 1),(-b, b + 1, b- 1),(c, c -1, c+ 1):}\| + \|{:(a+ 1 , b+ 1,c -1),(a - 1,b - 1, c+ 1),((- 1)^(n+2)a, (-1)^(n+1)b, (-1)^(n) c):}\|= 0, then the value of n is
Answer» any INTEGER zero any EVEN integer any ODD integer ANSWER :D

Discussion

12918.	The projection of point P(vecp) on the plane vecr.vecn=q is (vecs), then
Answer» `vecs=((q-vecp.vecn)vecn)/(\|vecn\|^(2))` `vecs=vecp+((q-vecp.vecn)vecn)/(\|vecn\|^(2))` `vecs=vecp-((vecp.vecn)vecn)/(\|vecn\|^(2))` `vecs=vecp-((q-vecp.vecn)vecn)/(\|vecn\|^(2))` Solution :We have `vecs-vecp=lamdavecnandvecs.n=q.` THUS, `(lamdavecn+vecp).vecn=q` or `LAMDA=(q-vecp.vecn)/(\|vecn\|^(2))` `impliesvecs=vecp+((vecq-vecp.vecn)vecn)/(\|vecn\|^(2))`

Discussion

12919.	int_(0)^(pi) sin (n+(1)/(2))x. " cosec ".(x)/(2) dx=?
Answer» `(PI)/(2)` `pi` `2PI` ANSWER :C

Discussion

12920.	OABC is a regular tetrahedron of side unity, then:
Answer» The length of perpendicular from one vertex to opposite face is `SQRT((2)/(3))` The perpendicular distance from mid-point of `vec(OA)` to the plane ABC is `(1)/(sqrt6)` The angle between two skew EDGES is `pi/2` The distance of centroid of the TETRAHEDRON from any vertex is `sqrt((3)/(8))` ANSWER :A::B::C::D

Discussion

12921.	Using integration find the area of the region bounded by the linesy=1+\|x+1\| ,x=2 ,x=3 and y=0
Answer» ANSWER :`(27)/( 2)` SQUARE UNITS

Discussion

12922.	Investigate the following integrals for convergence : (a) int_(0)^(1)(dx)/(sqrt(sinx)), (b) int_(0)^(1)(dx)/(e^(x)-cosx), (c ) int_(0)^(1)(cos^(2)xdx)/((1-x)^(2)), (d) int_(0)^(1)(tanxdx)/(sqrt(1-x^(2))), (e ) int_(1//2)^(6//5)(sinxdx)/(sqrt(\|1-x^(2)\|)).
Answer» ANSWER :(a) It CONVERGES; (b) diverges; (c ) diverges; (d) converges; (E ) converges.

Discussion

12923.	If the normals drawn to the hyperbola xy = 4 at (alpha_(i), beta_(i)) (i = 1,2,3,4) are concurrentat the point (a, b), then ((alpha_(1) + alpha_(2) + alpha_(3) + alpha_(4)))/((beta_(1) + beta_(2) + beta_(3) + beta_(4)))(alpha_(1)alpha_(2)alpha_(3)alpha_(4)) =
Answer» `(-16B)/(a)` `(-16a)/(b)` `(4B)/(a)` `(4A)/(b)` Answer :B

Discussion

12924.	i+ 2i^(2) + 3i^(3) + 4i^(4) + …. + 100i^(100) =
Answer» `25( 1 +i)` `50 (1 - i)` `100 (1-i)` 0 Answer :B

Discussion

12925.	Prove that : Find the 4^("th") term from the end in the expansion (2a+5b)^(8).
Answer» SOLUTION :N/A

Discussion

12926.	Two dice are rolled. Let A be the event of getting sum 10 and B be the event of getting even on both the dice. Find whether A, B are independent or not.
Answer» ANSWER :A, B are not INDEPENDENT.

Discussion

12927.	The orbit of the Earth is an ellipse with eccentricity 1/60with the Sun at one of its foci, the major axis being approximately 186 xx 10^(6) miles in length. Find the shortest and longest distance of the Earth from the Sun.
Answer» ANSWER :`3145xx10^(4)` MILES

Discussion

12928.	The root of the equation tan^(-1)((x+1)/(x-1))+tan^(-1)((x-1)/(x))=pi-tan^(-1)7 is
Answer» `1/2` 1 2 `3/2` ANSWER :C

Discussion

12929.	The three sides of a trapezium are each of length 8 cm . Find the maximum area of the trapezium.
Answer» ANSWER :`48sqrt(3)`SQUARE CM

Discussion

12930.	A producer has 20 and 10 unit of labour and capital respectively which he can use to produce two kinds of goods X and Y. To produce one unit of x, 2 units of capital and 1 unit of labour is required. To produce one unit of Y,3 of labour and 1 unit of capital is required. If X and Y are priced at Rs. 80 and Rs. 100 per unit respectively, howshould the producer use his resources to maximise the total revenue?
Answer» ANSWER :At 25 km/h he should TRAVEL 50/3, at 40 km/h 40/3 km. Maximum distance 30 km in 1 hr.

Discussion

12931.	The vectors of two sides of the triangle are vec(a)=3hati+6hatj-2hatk and vec(b)=4hati-hatj+3hatk then find all the angles of the triangle.
Answer» Answer :`COS^(-1)((7)/(sqrt(75))),cos^(-1)((sqrt(26))/(sqrt(75))),90^(@)`

Discussion

12932.	Two braches of a hyperbola
Answer» have a COMMON tangent have a common normal do not have a common tangent do not have a common normal Answer :B::C

Discussion

12933.	Evaluate : int_((2x)/(1+x^(2)))dx
Answer»

Discussion

12934.	If alpha, beta are the roots of the equation x^(2)+x+1=0, then the equation whose roots are alpha^(22)" and "beta^(19), is
Answer» A. `X^(2)-x+1=0` B. `x^(2)+x+1=0` C. `x^(2)+x-1=0` D. `x^(2)-x-1=0` ANSWER :B

Discussion

12935.	tan{ilog((a-ib)/(a+ib))}=
Answer» ANSWER :`=(2b//a)/(1-(b/a)^2)=(2AB)/(a^2-b^2)`

Discussion

12936.	A and B are two students. Their chances of solving a problem correctly are (1)/(3) and (1)/(4),respectively. If the probability of their making a common error is (1)/(20)and they obtain the same answer, then the probability of their answer to be correct is ..........
Answer» `(1)/(12)` `(1)/(40)` `(13)/(120)` `(10)/(13)` Answer :D

Discussion

12937.	If sin ^(-1) .(2a)/(1+a^(2))+ cosec^(2) (cot ^(-1) 3)=?
Answer» `(a+b)/(1-ab)` `(a+b)/(1+ab)` `(a-b)/(1+ab)` NONEOF these Answer :A

Discussion

12938.	A,B are conjugate points w.r.t circle having centre O radius r then OA^(2)+OB^(2)-2r^(2)=AB^(2)
Answer» ANSWER :`AB^(2)`

Discussion

12939.	Let vec(a), vec(b), vec(c) be three vectors in the xyz space such that vec(a)xxvec(b)=vec(b)xxvec(c)=vec(c)xx vec(a) ne 0 If A, B, C are points with position vector vec(a), vec(b), vec(c) respectively, then the number of possible position of the centroid of triangle ABC is -
Answer» 1 2 3 6 Solution :`vec(a)xxvec(B)+vec(c)xxvec(b)=0` similarly `vec(b)+vec(c)=lambda_(2) vec(a)` `vec(a)+vec(c)=lambda_(1)vec(b)""vec(b)+vec(a)=lambda_(3)vec(c)`

Discussion

12940.	If X={8^(n)-7n-1:ninN}andY={49(n-1):ninN}, then
Answer» `XsubY` `YsubX` `X=Y` `XnnY=phi` ANSWER :A

Discussion

12941.	Statement -1 : Ifx in R, x ne 0 , then x^2 + 1/(x^(2)) cannot be equal to cos theta for any theta Statement -2 : Sum of a positive number and its reciprocal cannot be less than 2.
Answer» Statement-1 is TRUE, Statement-2 is True and Statement-2 is a CORRECT EXPLANATION for Statement-1. Statement-1 is True, Statement-2 is True and Statement-2 is NOT a correct explanation forStatement-1. Statement-1 is True, Statement-2 is FALSE Statement-1 is False, Statement-2 is True Answer :A

Discussion

12942.	Let alpha, beta be two distinct values of x lying in (0,pi) for which sqrt5 sin x, 10 sin x, 10 (4 sin ^(2) x+1) are 3 consecutive terms of a G.P. Then minimum value of \|alpha - beta\|=
Answer» `pi/10` `pi/5` `(2PI)/(5)` `(3PI)/(5 )` ANSWER :B

Discussion

12943.	Compute the integral I_(m,n) = int_(0)^(1) x^(m) (! - x)^(n)dx , where m and n are non-negative integers.
Answer» ANSWER :`I_(m,N) = (m!n!)/((m + n + 1)!)`

Discussion

12944.	The variance of first 10 multiples of 3 is
Answer» 64.25 54.25 70.25 74.25 Answer :D

Discussion

12945.	Let the sum of all divisiors of the form2^(p)*3^(q) (with p, q positive integers) of the number 19^(88)-1 be lambda. Find the unit digit of lambda.
Answer» ANSWER :`(4)`

Discussion

12946.	STATEMENT-1 :Ifx is positive , then least value ofx^(12) + (4)/(x^(2)) + (4)/(x)is 9. STATEMENT-2 :Ifa^(2) + b^(2) + c^(2) = 4 , x^(2) + y^(2) + z^(2) = 9then maximum value ofax + by + czis 6 STATEMENT-3 :2 + (3)/(2^(2)) + (4)/(2^(3)) + ......"upto" oo = 4.
Answer» F T F T T F T T T F F T ANSWER :3

Discussion

12947.	Q is a point on the auxiliary circle corresponding to the point P of the ellipse x^2/a^2 + y^2/b^2 =1 If T is the foot of the perpendicular dropped from the focus S onto the tangent to the auxiliary circle at Q then the Delta SPT is
Answer» isosceles equilateral right angled right isosceles Answer :A

Discussion

12948.	Differentiate w.r.t x the function sin^(3) x + cos^(6)x
Answer» ANSWER :`3 sin X COS x (sin x-2 cos^(4) x)`

Discussion

12949.	Point of concurrence of the normals drawn at (2,8) , (128,64) , (162, -72)to the parabola y^(2)=32 xis
Answer» A) (`2,8) ` B) ` (128, 64) ` C) ` ( 162, -72) ` D) ` ( 162, 72) ` ANSWER :C

Discussion

12950.	If z is a complex number such that z = ln(a – 1) + iln (b – 1) then arg(z) is :
Answer» `PI/4` `(3PI)/4` `(-pi)/4` `(-3pi)/4` ANSWER :C

Discussion

Explore topic-wise InterviewSolutions in .

Let n in N such that n gt 1. Statement-1: int_(oo)^(0) (1)/(1+x^(n))dx=int_(0)^(1) (1)/((1-x^(n))^(1//n))dx Statement-2: int_a^b f(x)dx=int_(a)^(b) f(a+b-x)dx

A=[{:(2,-3),(4,6):}] verify (adjA)^(-1)=(adjA^(-1))

random variable X has the range {1,2,3,……….}. If P(X=r)=c^(r//r) for r=1,2,3,……… then c=

Consider a binary opertion ** on N defined as a **b=a ^(3) +b ^(3). Choose the correct answer.

If (0,6) and (0,3) are respectively the vertex and focus of a parabola then its equation is

Let a and b be unit vectors inclined at an angle 2alpha(0lealphalepi) each other, then (a+b) lt 1, if

Let A=B B^(T)+C C^(T), where B=[(cos theta), (sintheta)], C=[(sin theta), (-costheta)], theta in R. Then A is :

Two dice are thrown. If it is known that the sum of numbers on the dice was less than 6, the probability of getting a sum 3, is .........

Show that the poles of the tangents to the circle x^(2)+y^(2) = a^(2) with respect to the circle (x+a)^(2)+y^(2)=2a^(2) lie on y^(2) +4ax =0 .

If A is symmetric matrix , then B'AB is …….

If (tanalpha-i[sin(alpha//2)+cos(alpha//2)])/(1+2isin(alpha//2)) is purely imaginary , is

Find a if the coefficients of x^(2) and x^(3) in the expansion of (3 + ax)^(9) are equal.

The mass of a molecule of the compound C_(60) H_(122) is :-

Let f(x) = [1- x^(2)], x in R, where [] is the greatest integer function. Then

If cot^(-1).(n)/(pi)gt(n)/(6),n in N, then the maximum value of n is :

Let a , b, c be such that b (a + c) ne0. If|{:(a, a+1, a- 1),(-b, b + 1, b- 1),(c, c -1, c+ 1):}| + |{:(a+ 1 , b+ 1,c -1),(a - 1,b - 1, c+ 1),((- 1)^(n+2)a, (-1)^(n+1)b, (-1)^(n) c):}|= 0, then the value of n is

The projection of point P(vecp) on the plane vecr.vecn=q is (vecs), then

int_(0)^(pi) sin (n+(1)/(2))x. " cosec ".(x)/(2) dx=?

OABC is a regular tetrahedron of side unity, then:

Using integration find the area of the region bounded by the linesy=1+|x+1| ,x=2 ,x=3 and y=0

Investigate the following integrals for convergence : (a) int_(0)^(1)(dx)/(sqrt(sinx)), (b) int_(0)^(1)(dx)/(e^(x)-cosx), (c ) int_(0)^(1)(cos^(2)xdx)/((1-x)^(2)), (d) int_(0)^(1)(tanxdx)/(sqrt(1-x^(2))), (e ) int_(1//2)^(6//5)(sinxdx)/(sqrt(|1-x^(2)|)).

If the normals drawn to the hyperbola xy = 4 at (alpha_(i), beta_(i)) (i = 1,2,3,4) are concurrentat the point (a, b), then ((alpha_(1) + alpha_(2) + alpha_(3) + alpha_(4)))/((beta_(1) + beta_(2) + beta_(3) + beta_(4)))(alpha_(1)alpha_(2)alpha_(3)alpha_(4)) =

i+ 2i^(2) + 3i^(3) + 4i^(4) + …. + 100i^(100) =

Prove that : Find the 4^("th") term from the end in the expansion (2a+5b)^(8).

Two dice are rolled. Let A be the event of getting sum 10 and B be the event of getting even on both the dice. Find whether A, B are independent or not.

The orbit of the Earth is an ellipse with eccentricity 1/60with the Sun at one of its foci, the major axis being approximately 186 xx 10^(6) miles in length. Find the shortest and longest distance of the Earth from the Sun.

The root of the equation tan^(-1)((x+1)/(x-1))+tan^(-1)((x-1)/(x))=pi-tan^(-1)7 is

The three sides of a trapezium are each of length 8 cm . Find the maximum area of the trapezium.

The vectors of two sides of the triangle are vec(a)=3hati+6hatj-2hatk and vec(b)=4hati-hatj+3hatk then find all the angles of the triangle.

Two braches of a hyperbola

Evaluate : int_((2x)/(1+x^(2)))dx

If alpha, beta are the roots of the equation x^(2)+x+1=0, then the equation whose roots are alpha^(22)" and "beta^(19), is

tan{ilog((a-ib)/(a+ib))}=

A and B are two students. Their chances of solving a problem correctly are (1)/(3) and (1)/(4),respectively. If the probability of their making a common error is (1)/(20)and they obtain the same answer, then the probability of their answer to be correct is ..........

If sin ^(-1) .(2a)/(1+a^(2))+ cosec^(2) (cot ^(-1) 3)=?

A,B are conjugate points w.r.t circle having centre O radius r then OA^(2)+OB^(2)-2r^(2)=AB^(2)

Let vec(a), vec(b), vec(c) be three vectors in the xyz space such that vec(a)xxvec(b)=vec(b)xxvec(c)=vec(c)xx vec(a) ne 0 If A, B, C are points with position vector vec(a), vec(b), vec(c) respectively, then the number of possible position of the centroid of triangle ABC is -

If X={8^(n)-7n-1:ninN}andY={49(n-1):ninN}, then

Statement -1 : Ifx in R, x ne 0 , then x^2 + 1/(x^(2)) cannot be equal to cos theta for any theta Statement -2 : Sum of a positive number and its reciprocal cannot be less than 2.

Let alpha, beta be two distinct values of x lying in (0,pi) for which sqrt5 sin x, 10 sin x, 10 (4 sin ^(2) x+1) are 3 consecutive terms of a G.P. Then minimum value of |alpha - beta|=

Compute the integral I_(m,n) = int_(0)^(1) x^(m) (! - x)^(n)dx , where m and n are non-negative integers.

The variance of first 10 multiples of 3 is

Let the sum of all divisiors of the form2^(p)*3^(q) (with p, q positive integers) of the number 19^(88)-1 be lambda. Find the unit digit of lambda.

STATEMENT-1 :Ifx is positive , then least value ofx^(12) + (4)/(x^(2)) + (4)/(x)is 9. STATEMENT-2 :Ifa^(2) + b^(2) + c^(2) = 4 , x^(2) + y^(2) + z^(2) = 9then maximum value ofax + by + czis 6 STATEMENT-3 :2 + (3)/(2^(2)) + (4)/(2^(3)) + ......"upto" oo = 4.

Q is a point on the auxiliary circle corresponding to the point P of the ellipse x^2/a^2 + y^2/b^2 =1 If T is the foot of the perpendicular dropped from the focus S onto the tangent to the auxiliary circle at Q then the Delta SPT is

Differentiate w.r.t x the function sin^(3) x + cos^(6)x

Point of concurrence of the normals drawn at (2,8) , (128,64) , (162, -72)to the parabola y^(2)=32 xis

If z is a complex number such that z = ln(a – 1) + iln (b – 1) then arg(z) is :

Consider a binary opertion on N defined as a b=a ^(3) +b ^(3). Choose the correct answer.