28278 + Interview Questions in Maths in Class 12 Page 89 InterviewSolution

4401.	Relation R = {(4,5),(1,4),(4,6),(7,6),(3,7)} then R^(-1)OR = .......
Answer» `{(1,1),(4,4),(7,4),(4,7),(7,7)}` `{(1,1),(4,4),(4,7),(7,4),(7,7),(3,3)}` `{(1,5),(1,6),(3,6)}` NONE of these Solution :N/A

Discussion

4402.	Find the values of the following integrals (i) int_(0)^(pi/2) sin^(4) x cos^(4) x dx
Answer» ANSWER :`(3PI)/(256)`

Discussion

4403.	Let f : R to Rbe a continuous periodic function and T be the period of it. Then prove that for any positive integer n,int_(0)^(nT) f(x) dx=n int_(0)^(T) f(x) dx
Answer» Answer :Thus, eq. (1) is true for any POSITIVE INTEGER n, by the principle of MATHEMATICAL INDUCTION.

Discussion

4404.	A graph G has 'm' vertices of odd degree and 'n' vertices of even degree.Then which of the following statements is necessarily true?
Answer» ` m+ n` is an ODD number ` m+ n` is an EVEN number `n+1` is an even number ` m+ 1` is an odd number Answer :B

Discussion

4405.	Construct a2xx2 matrix ,A=[a_(ij)], whose elements are given by : (i)a_(ij)=((i+j)^(2))/(2), (ii)a_(ij)=(i)/(j) (iii)a_(ij)=((i+2j)^(2))/(2)
Answer» ANSWER :(i) `=[{:(2,(9)/(2)),((9)/(2),8):}]` (ii) `=[{:(1,(1)/(2)),(2,1):}]` (III) `=[{:((9)/(2),(25)/(2)),(8,18):}]`

Discussion

4406.	If lim_(n rarr infty ) [(1+ 1/x^(2))(1+ 2^(2)/n^(2))...(1+n^(2)/n^(2))]^(1//n)=k, then log k =
Answer» `LOG 4 + pi/2 -1` `log 2 + pi/2 +1` `log 2 + pi/2 -2` `log 2 + pi/2 -1` ANSWER :C

Discussion

4407.	Regression equation of y on x and y be x + 2y - 5 = 0 and 2x + 3y - 8 = 0 respectively and the variance of x is 12. find the variance of y.
Answer» ANSWER :4

Discussion

4408.	Find the coefficient of 1/y^10 in the expansion of (y^3+a^7/y^5)^10
Answer» SOLUTION :LET (R+1)th term contains `1/y^10`. ` THEREFORE (r+1)`term in the expansion of `(y^3+a^7/y^5)^10` is `"^10C_r (y^3)^(10-r)(a^7/y^5)^r` `"^10C_r y^(30-r) a^(7r) y^(-5R) = ^10C_r a^(7r) y^(30-8r)` `therefore y^(30-8r) = 1/y^10 = y^-10` or , 30-8r = -10 or, 8r = 40 or, r = 5 The coefficient of `1/y^10` is `"^10C_5 a^(7xx5) = (10!)/(5!5!) a^35 = 252a^35`

Discussion

4409.	Find the number of irrational terms in the expansion of (5^(1//6)+2^(1//8))^(100).
Answer» ANSWER :97

Discussion

4410.	In thermodynamic process pressure of a fixedmass of gas is changed in such a manner that the gas releases 30 joule of heat and 18 joule of work was done on the gas. If the initial internal energy of the gas was 60 joule, then, the final internal energy (in J) will be :
Answer» SOLUTION :`dQ = dU + DW` `-30 = -18 +U_(2) -60` `U_(2) = 60-30+18 = 48`.

Discussion

4411.	Approximately what percent of average annualGDP of Province P from 1996 - 2000 came from copper production?
Answer» `3%` `6%` `9%` `14%` ANSWER :A

Discussion

4412.	Find the most likely price in Mumbai (x) corresponding to the price of ₹ 70 at Kolkata (y) from the following data : {:(,"Mumbai", "Kolkata"),("Average price ",""67,""65),("Standard deviation ",""3 . 5 ,""2 . 5 ):} Correlation of coefficient = 0 . 5
Answer» ANSWER :`28 . 5`

Discussion

4413.	Express the matrix [{:(2,3,1),(1,-1,2),(4,1,2):}] as the sum of a symmetric and a skew symmetric matrix.
Answer» ANSWER :`=[{:(2,3,1),(1,1,2),(4,1,0):}]`

Discussion

4414.	If f (x ) =x ^(2) + (x ^(2))/((1 + x ^(2))) + (x ^(2))/( (1 + x ^(2))) + …+ (x ^(2) ((1 + x ^(2)) + … then x =0
Answer» `F (x)` has no LIMIT `f (x)` is discotinous `f (x)` is continous but not DIFFERENTIABLE `f (x)` is differentiable ANSWER :B

Discussion

4415.	The probability distribution of a random variable X is given below: (i) Determine the value of k. (ii) Determine P(X le 2) and P(X gt 2) (ii) Find P(X le 2) +P(X gt 2).
Answer» ANSWER :`(i) (8)/(15), (II) (1)/(15), (III) 1 `

Discussion

4416.	Find adjoint of each of the matrices [{:(4,2),(1,5):}]
Answer» ANSWER :`[{:(5,-2),(-1,4):}]`

Discussion

4417.	The value of 2^(1/4), 4^(1/8), 8^(1/16) ...... infty is
Answer» 1 2 `3/2` 4 Answer :B

Discussion

4418.	Three coins are tossed simultaneously. Consider the event E 'three heads or three tails'F 'at least two heads' and G 'at the most two heads'. Of the pairs (E, F), (E, G) and (F, G) (i) Which are independent?, "" (ii) Which are dependent?
Answer» ANSWER :The events E and F are INDEPENDENT and the events (E and G) and (F and G) are DEPENDENT.

Discussion

4419.	Twelve identical machines, running continuously at the same constant rate, take 8 days to complete a shipment. How many additional machines, each running at the same constant rate, would be needed to reduce the time required to complete a shipment by 2 days?
Answer» 2 3 4 6 Answer :C

Discussion

4420.	Equation of the plane which bisects the line segment joining (-1,2,3) and (3,-5,6) perpendicularly, is
Answer» 4x+2y-3z=28 4x-7y-3z=28 4x-7y+3z=28 4x-7y-3z=8 Answer :C

Discussion

4421.	If range of f(x) = (x^(2) - 3x + 2)/(x^(2) - ax+ 4) is R- (1) then sum of all possible real value(s) of 'a' is
Answer» 4 3 5 None of these SOLUTION :`f(X) = ((x -1)(x -2))/(x^(2) - ax + 4)` for range of `f(x)` to be R - {1} `x^(2) - ax + 4 = (x - 1)^(2) or x^(2) - ax + 4 = (x - 2)^(2)` not possible `""a = 4`.

Discussion

4422.	If a, b,c gt 0& x,y,z in R" then the determinant"[{:(,(a^(2)+a^(-2))^(2),(a^(x)-a^(x))^(2),1),(,(b^(y)+b^(-y))^(2),(b^(y)-b^(y))^(2),1),(,(c^(x)+c^(-2))^(2),(c^(x)-c^(-z))^(2),1):}]=
Answer» `a^(x)B^(y)C^(Z)` `a^(-x)b^(-y)c^(-z)` `a^(2x)b^(2Y)c^(2z)` zero Answer :D

Discussion

4423.	Verify that A^(2)=I when A=[{:(0,1,-1),(4,-3,4),(3,-3,4):}].
Answer» ANSWER :`=[{:(1,0,0),(0,1,0),(0,0,1):}]=I`

Discussion

4424.	If ninN,"then "2.4^(2n+1)+3^(3n+1) is divisible by
Answer» 2 9 11 27 Answer :C

Discussion

4425.	P is a point inside a Delta ABC, D, E, F are the feet or perpendicular from P to the line BC, AB respectively. Show that (BC)/(PD)+ (CA)/(PE) + (AB)/(PF) ge (2s ^(2))/(Delta). Prove that equality holds if P is its incentre, s = semipermeter, Delta = area of triangle.
Answer» ANSWER :EQUALITY holds when `p = q=r=>P` is incentre of TRIANGLE.

Discussion

4426.	A random variable x has the following probability distribution. Determine (i) k(ii) p(xlt3)
Answer» ANSWER :`3/10`

Discussion

4427.	If three unit vectors veca, vecb, vecc" satisfy "overset(-)a+overset(-)b+overset(-)c=overset(-)0, then the angle between overset(-)a and overset(-)b is
Answer» `(2PI)/(3)` `(5PI)/(6)` `pi/3` `pi/6` ANSWER :A

Discussion

4428.	Two forces F_(1)={2, 3} and F_(2)={4, 1} are specified relative to a general cartesian form. Their points of application are respectivel, A=(1, 1) and B=(2, 4). Find the coordinates of the resultantand the equation of the straight line l containing it.
Answer» ANSWER :`{6, 4} and 4x-6y+13=0`

Discussion

4429.	If the variance of the distribution is 45.8, then the variance of the distribution.
Answer» 93.6 `SQRT(93.9)` 183.2 `sqrt183.2` ANSWER :C

Discussion

4430.	Assertion (A) : If alpha, beta are the roots of ax^(2) + bx + c = 0 then the equation whose roots are(alpha-1)/(alpha), (beta-1)/(beta) is c(1-x)^(2)+ b(1-x)+a=0Reason (R): If alpha, beta are the roots of f(x) = 0then the equation whose roots are (alpha-1)/(alpha) and (beta-1)/(beta) is f((1)/(1-x))=0
Answer» Both A, R are true and R EXPLAIN Assertion Both A, R are true but R does't explain A A is true R is FALSE A is false R is true Answer :A

Discussion

4431.	If f : N to Z is defined by f(x)={{:("2 if "n = 3k", " k in Z),("10 if " n = 3k+1 " , "k in Z),("0 if " n = 3k +2", " k in Z):} then {n in N : f (n) gt 2) is equal to
Answer» {3, 6, 4} {1, 4, 7} {4, 7} {7} ANSWER :B

Discussion

4432.	Let f:R to R be a positive increasing function with underset(x to oo) (f(3x))/(f(x)=1
Answer» 1 `2//3` `3//2` 3 Answer :A

Discussion

4433.	If A+B+C=pi then prove that sin2A-sin2B+sin2C=4cosAsinBcosC
Answer» ANSWER :RHS

Discussion

4434.	If G is the centroid of the triangle PQR, where vec(GP)=2hat(i)+hat(j)+3hat(k),vec(GQ)=hat(i)-hat(j)+2hat(k), then the area of the triangle PQR is
Answer» `SQRT(35)sq.units` `(3sqrt(35))/(2)sq.units` `sqrt(35)/(2)sq.units` `(5sqrt(35))/(2)sq.units` ANSWER :B

Discussion

4435.	if f (x) =2 x +cot^(-1) x + en (sqrt(1+x^(2)) -x) the f(x):
Answer» INCREASING in `[0,oo]` decreasing in `[0,oo]` NEITHER increases nordecreases in`[0,oo]` increases in `[-oo,oo]` ANSWER :A::D

Discussion

4436.	If alpha" and "beta are the roots of x^(2)-ax+b^(2)=0, then alpha^(2)+beta^(2) is equal to
Answer» `a^(2)+2B^(2)` `a^(2)-2b^(2)` `a^(2)-2b` `a^(2)+2b` ANSWER :B

Discussion

4437.	The triangle formed by the tangent to the curve f(x)=x^2+bx-b at the point (1,1) and the coordinate axes, lies in the first quadrant , if its area is 2, then the value of b is :
Answer» `-1` `3` `-3` 1 Answer :C

Discussion

4438.	If D_1=\|{:(1,yz,x),(1,zx,y),(1,xy,z):}\|and D_2=\|{:(1,1,1),(x,y,z),(x^2,y^2,z^2):}\| then ,………
Answer» `D_1+2D_2=0` `2D_1+D_2=0` `D_1+D_2=0` `D_1=D_2` ANSWER :C

Discussion

4439.	If A = [(a^(2),ab,ac),(ab,b^(2),bc),(ac,bc,c^(2))] and B = [(0,c,-b),(-c , 0,a),(b,-a,0)]then the product AB equals:
Answer» I O A B ANSWER :B

Discussion

4440.	int(dx)/(xsqrt(2ax-x^2))=
Answer» `asqrt((2a-X)/(x))+C` `(1)/(a)SQRT((2a-x)/(x))+c` `-asqrt((2a-x)/(x))+c` `(-1)/(a)sqrt((2a-x)/(x))+c` ANSWER :D

Discussion

4441.	In triangle ABC, AD is prependicular to BC and DE is perpendicular to AB
Answer» `{:(a,b,c,d),(p,r,Q,q):}` `{:(a,b,c,d),(q,r,p,s):}` `{:(a,b,c,d),(s,p,q,r):}` `{:(a,b,c,d),(r,p,s,q):}` Solution : a. Area of `DeltaADB = (1)/(2) AD XX BD` `= (1)/(2) c sin B xx c COS B` `= (c^(2))/(4) sin 2B` b. Area of `DeltaADC = (1)/(2) AD xx CD` `= (1)/(2) b sin C xx b cos C = (b^(2))/(4) sin 2C` c. Area of `DELTAADE = (1)/(2) AE xx DE` `= (1)/(2) AD cos ((pi)/(2) - B) AD sin ((pi)/(2) - B)` `= (1)/(4) AD^(2) sin 2B` `= (1)/(4) c^(2) sin^(2) B sin 2B` d. Area of `DeltaBDE = (1)/(2) BE xx DE` `= (1)/(2) BD cos B xx AD sin ((pi)/(2) -B)` `= (1)/(2) c cos B cos B xx c sin B cos B` `= (1)/(4) c^(2) cos^(2) B sin 2B`

Discussion

4442.	In the group G={1,5,7,11} under multiplication modulo 12, the solution of 7^(-1)ox_(12)(x ox_(12)11)=5 is x =
Answer» 5 1 7 11 Answer :B

Discussion

4443.	If alpha, beta are two different values of theta which satisfy is bc cos theta cos phi + ac sin theta sin phi=ab, then prove that (b^(2)+c^(2)-a^(2)) cos alpha cos beta+ ac sin alpha sin beta= a^(2)+b^(2)-c^(2).
Answer» ANSWER :` (a^(2)+B^(2)-C^(2))=RHS`

Discussion

4444.	1+(1)/(4) + (1.3)/(4.8) + (1.3.5)/(4.8.12)+…...=
Answer» `SQRT2` `(1)/(sqrt2)` `SQRT3` `(1)/(sqrt3)` ANSWER :A

Discussion

4445.	A = {x : x in R, \|x\| lt 1}, B = {x : x in R,\| x - 1\| ge 1} " and " A cup B = R - D " then " D =
Answer» `{x :1 LT x le 2}` `{x : 1 le x lt 2}` `{x : 1 le x le 2}` none Answer :B

Discussion

4446.	If \|vec(a)\|=2\|vec(b)\|=5 and \|vec(a)xx vec(b)\|=8 then find vec(a).vec(b).
Answer» ANSWER :6

Discussion

4447.	Let f(x)=x^(3)-3x^(2)+6AA x in R " and "g(x)={{:(max.f(t), x+1 le t le x+2","-3 le x le 0),(1-x " for " x ge 0):} Then find y=g(x) " for " x in [-3, 1].
Answer» Solution :`f(X)=x^(3)-3x^(2)+6` If `f'(x)=3x^(2)-6x=0`, then `x=0, 2` are the critical points of `f(x)`. `x=0` is the POINT of LOCAL MAXIMA and x=2 is the point of local minima. Clearly, `f(x)` is increasing in `(-oo,0)` and `(2, oo)` and decreasingin (0,2). Case 1 : ` x+2 le 0 rArr x le -2` `rArr " " g(x)=f(x+2), -3 le x le -2` Case2: `x+1 LT 0 ` and `0 lt x+2 lt 2` `x lt -1 ` and `-2 lt x lt 0` i.e., `-2 lt x lt -1 " " :. g(x)=f(0)` Case 3: `0 le x+1, x+2 le 2` `rArr -1 le x le 0, g(x)=f(x+1)` `rArr g(x)={{:(f(x+2)", " -3 le x lt -2),(f(0)"," -2le x lt -1),(f(x+1)"," -1 le x lt 0),(1-x ","0 lexlt1):}`

Discussion

4448.	Match the following colum -I with column - II.
Answer» <P>`{:(P,Q,R,S),(2,3,4,1):}` `{:(P,Q,R,S),(4,1,3,2):}` `{:(P,Q,R,S),(1,4,2,3):}` `{:(P,Q,R,S),(3,4,1,2):}` ANSWER :D

Discussion

4449.	Integrate the following rational functions : int(2x-1)/((x-1)(x+2)(x-3))dx
Answer» Answer :`:. I=log\|(sqrt(x-3))/((x-1)^((1)/(6))*(x+2)^((1)/(2)))\|+C`

Discussion

4450.	(a xx b) xx (c xx d) + (a xx c) xx (d xx b) + (a xx d) xx (b xx c) =
Answer» [B C d]a 2 [b c d] a `-[b c d] a` `-2 [b c d] a` ANSWER :D

Discussion

Explore topic-wise InterviewSolutions in .

Relation R = {(4,5),(1,4),(4,6),(7,6),(3,7)} then R^(-1)OR = .......

Find the values of the following integrals (i) int_(0)^(pi/2) sin^(4) x cos^(4) x dx

Let f : R to Rbe a continuous periodic function and T be the period of it. Then prove that for any positive integer n,int_(0)^(nT) f(x) dx=n int_(0)^(T) f(x) dx

A graph G has 'm' vertices of odd degree and 'n' vertices of even degree.Then which of the following statements is necessarily true?

Construct a2xx2 matrix ,A=[a_(ij)], whose elements are given by : (i)a_(ij)=((i+j)^(2))/(2), (ii)a_(ij)=(i)/(j) (iii)a_(ij)=((i+2j)^(2))/(2)

If lim_(n rarr infty ) [(1+ 1/x^(2))(1+ 2^(2)/n^(2))...(1+n^(2)/n^(2))]^(1//n)=k, then log k =

Regression equation of y on x and y be x + 2y - 5 = 0 and 2x + 3y - 8 = 0 respectively and the variance of x is 12. find the variance of y.

Find the coefficient of 1/y^10 in the expansion of (y^3+a^7/y^5)^10

Find the number of irrational terms in the expansion of (5^(1//6)+2^(1//8))^(100).

In thermodynamic process pressure of a fixedmass of gas is changed in such a manner that the gas releases 30 joule of heat and 18 joule of work was done on the gas. If the initial internal energy of the gas was 60 joule, then, the final internal energy (in J) will be :

Approximately what percent of average annualGDP of Province P from 1996 - 2000 came from copper production?

Find the most likely price in Mumbai (x) corresponding to the price of ₹ 70 at Kolkata (y) from the following data : {:(,"Mumbai", "Kolkata"),("Average price ",""67,""65),("Standard deviation ",""3 . 5 ,""2 . 5 ):} Correlation of coefficient = 0 . 5

Express the matrix [{:(2,3,1),(1,-1,2),(4,1,2):}] as the sum of a symmetric and a skew symmetric matrix.

If f (x ) =x ^(2) + (x ^(2))/((1 + x ^(2))) + (x ^(2))/( (1 + x ^(2))) + …+ (x ^(2) ((1 + x ^(2)) + … then x =0

The probability distribution of a random variable X is given below: (i) Determine the value of k. (ii) Determine P(X le 2) and P(X gt 2) (ii) Find P(X le 2) +P(X gt 2).

Find adjoint of each of the matrices [{:(4,2),(1,5):}]

The value of 2^(1/4), 4^(1/8), 8^(1/16) ...... infty is

Three coins are tossed simultaneously. Consider the event E 'three heads or three tails'F 'at least two heads' and G 'at the most two heads'. Of the pairs (E, F), (E, G) and (F, G) (i) Which are independent?, "" (ii) Which are dependent?

Twelve identical machines, running continuously at the same constant rate, take 8 days to complete a shipment. How many additional machines, each running at the same constant rate, would be needed to reduce the time required to complete a shipment by 2 days?

Equation of the plane which bisects the line segment joining (-1,2,3) and (3,-5,6) perpendicularly, is

If range of f(x) = (x^(2) - 3x + 2)/(x^(2) - ax+ 4) is R- (1) then sum of all possible real value(s) of 'a' is

If a, b,c gt 0& x,y,z in R" then the determinant"[{:(,(a^(2)+a^(-2))^(2),(a^(x)-a^(x))^(2),1),(,(b^(y)+b^(-y))^(2),(b^(y)-b^(y))^(2),1),(,(c^(x)+c^(-2))^(2),(c^(x)-c^(-z))^(2),1):}]=

Verify that A^(2)=I when A=[{:(0,1,-1),(4,-3,4),(3,-3,4):}].

If ninN,"then "2.4^(2n+1)+3^(3n+1) is divisible by

P is a point inside a Delta ABC, D, E, F are the feet or perpendicular from P to the line BC, AB respectively. Show that (BC)/(PD)+ (CA)/(PE) + (AB)/(PF) ge (2s ^(2))/(Delta). Prove that equality holds if P is its incentre, s = semipermeter, Delta = area of triangle.

A random variable x has the following probability distribution. Determine (i) k(ii) p(xlt3)

If three unit vectors veca, vecb, vecc" satisfy "overset(-)a+overset(-)b+overset(-)c=overset(-)0, then the angle between overset(-)a and overset(-)b is

Two forces F_(1)={2, 3} and F_(2)={4, 1} are specified relative to a general cartesian form. Their points of application are respectivel, A=(1, 1) and B=(2, 4). Find the coordinates of the resultantand the equation of the straight line l containing it.

If the variance of the distribution is 45.8, then the variance of the distribution.

If f : N to Z is defined by f(x)={{:("2 if "n = 3k", " k in Z),("10 if " n = 3k+1 " , "k in Z),("0 if " n = 3k +2", " k in Z):} then {n in N : f (n) gt 2) is equal to

Let f:R to R be a positive increasing function with underset(x to oo) (f(3x))/(f(x)=1

If A+B+C=pi then prove that sin2A-sin2B+sin2C=4cosAsinBcosC

If G is the centroid of the triangle PQR, where vec(GP)=2hat(i)+hat(j)+3hat(k),vec(GQ)=hat(i)-hat(j)+2hat(k), then the area of the triangle PQR is

if f (x) =2 x +cot^(-1) x + en (sqrt(1+x^(2)) -x) the f(x):

If alpha" and "beta are the roots of x^(2)-ax+b^(2)=0, then alpha^(2)+beta^(2) is equal to

The triangle formed by the tangent to the curve f(x)=x^2+bx-b at the point (1,1) and the coordinate axes, lies in the first quadrant , if its area is 2, then the value of b is :

If D_1=|{:(1,yz,x),(1,zx,y),(1,xy,z):}|and D_2=|{:(1,1,1),(x,y,z),(x^2,y^2,z^2):}| then ,………

If A = [(a^(2),ab,ac),(ab,b^(2),bc),(ac,bc,c^(2))] and B = [(0,c,-b),(-c , 0,a),(b,-a,0)]then the product AB equals:

int(dx)/(xsqrt(2ax-x^2))=

In triangle ABC, AD is prependicular to BC and DE is perpendicular to AB

In the group G={1,5,7,11} under multiplication modulo 12, the solution of 7^(-1)ox_(12)(x ox_(12)11)=5 is x =

If alpha, beta are two different values of theta which satisfy is bc cos theta cos phi + ac sin theta sin phi=ab, then prove that (b^(2)+c^(2)-a^(2)) cos alpha cos beta+ ac sin alpha sin beta= a^(2)+b^(2)-c^(2).

1+(1)/(4) + (1.3)/(4.8) + (1.3.5)/(4.8.12)+…...=

A = {x : x in R, |x| lt 1}, B = {x : x in R,| x - 1| ge 1} " and " A cup B = R - D " then " D =

If |vec(a)|=2|vec(b)|=5 and |vec(a)xx vec(b)|=8 then find vec(a).vec(b).

Let f(x)=x^(3)-3x^(2)+6AA x in R " and "g(x)={{:(max.f(t), x+1 le t le x+2","-3 le x le 0),(1-x " for " x ge 0):} Then find y=g(x) " for " x in [-3, 1].

Match the following colum -I with column - II.

Integrate the following rational functions : int(2x-1)/((x-1)(x+2)(x-3))dx

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