28278 + Interview Questions in Maths in Class 12 Page 119 InterviewSolution

5901.	{:(" "Lt),(n rarr oo):} ((1)/(sqrt(n^(2)))+(1)/(sqrt(n^(2)+n))+(1)/(sqrt(n^(2)+2n))+(1)/(sqrt(n^(2)+3n))+....(1)/(sqrt(n^(2)+n(n-1)))))=
Answer» `2 SQRT(2)` 2 `2-2sqrt(2)` `2 sqrt(2)-2` ANSWER :D

Discussion

5902.	If p,q, in rand p^2+q^2-pq-q+1 le 0 + alpha + beta + gamma = 0 , then \|(1,cosy,cosbeta),(cosgamma,p,cosalpha),(cosbeta,cosalpha,q)\|=
Answer» 1 `-1` 0 None of these Answer :C

Discussion

5903.	If [[1,-x],[4,-3]] =[[1,8],[4,-3]] then value of x is :
Answer» 8 -4 3 -8 Answer :D

Discussion

5904.	A fair dice is rolled 6 times. If "getting an even number" is a success the probability of 5 successes is …………..
Answer» `(5)/(64)` `(3)/(32)` `(63)/(64)` `(5)/(6)` Answer :B

Discussion

5905.	There are 3 rows containing 2 seats in each row. In how many ways 3 persons can be seated such that no row remains empty.
Answer» ANSWER :48

Discussion

5906.	If a set A has 12 elements. Find the number of subsets of A having atmost 3 elements
Answer» ANSWER :299

Discussion

5907.	Solve the following Linear Programming Problems graphically : Minimise Z = x + 2y subject to 2x+y ge 3, x+2y ge 6, x, y ge 0
Answer» Answer :The minimum value of Z is 6 at the point A(6, 0) and B(0, 3)

Discussion

5908.	Find the number of ordered triples (a,b,c) of positive integers such that abc= 108
Answer» ANSWER :60

Discussion

5909.	2x-y+3z=1 3x-2y+5z=-4 5x-4y+9z=14
Answer»

Discussion

5910.	Find maximum and minimum value of the following functions in the given interval.f(x)=x^(2)log x, x in [0, 2]
Answer» ANSWER :MAXIMUM value `E^(2)` Minimum value 0

Discussion

5911.	Consider two regions R_(1):"points P are nearer to (1,0) than to "x=-1. R_(2): "Points P are nearer to (0,0) than to (8,0)" Find the area of the region common to R_(1) and R_(2).
Answer» <P> Solution :`R_(1) :` points P(X,y) are nearer to (1,0) than to x=-1 `therefore""sqrt((x-1)^(2)+y^(2))lt\|x+1\|` `rArr""y^(2)lt4x` `rArr"point P lie inside parabola "y^(2)=4x` `R_(2) :` Points P(x,y) are nearer to (0,0) than to (8,0) `therefore""\|x\|lt\|x-8\|` `rArr""x^(2)LTX^(2)-16x+64` `rArr""xlt4` `rArr""`point P lie to the left side of line x=4 The AREA of common region of `R_(1) and R_(2)` is area bounded by x=4 and `y^(2)=4x.` `therefore"Required area "=2overset(4)underset(0)int(4-(y^(2))/(4))dy` (Integrating along x-axis) `=2[4y-(y^(3))/(12)]_(0)^(4)` `=2[16-(64)/(12)]` `(64)/(3)` sq. units

Discussion

5912.	"if "f(x) ={ underset(5x, 1,2 le x le 3)(3x+4, 0 le x le 2}}, then evaluate int_(0)^(3) f(x) dx.
Answer» ANSWER :`(53)/(2)`

Discussion

5913.	There are 5 cards numbered 1 to 5, one number on one card. Two cards are drawn at random without replacement. Let X denote the sum of the numbers on two cards drawn. Find the mean and variance of X.
Answer» ANSWER :33

Discussion

5914.	The normal at 'P'cuts the axis of the parabola y^(2) =4ax in G and S is the focus of the parabola. IfDelta SPGis equilateral then each side is of length.
Answer» a 2a 3a 4a Answer :D

Discussion

5915.	Let O(0,0) , P(2,0) and Q (6,0) be the vertices of triangle OPQ . If R be a point inside the triangle OPQ such that areas of DeltaOPQ,DeltaPQRandDeltaOQRare equal , then the abscissa of R is -
Answer» ANSWER :3

Discussion

5916.	The ratio of the number of chickens to the number of pigs to the number of horses on Richard's farm is 33:17 : 21. What fraction of the animals are either pigs or horses?
Answer» `16//53` `17//54` `38//71` `25//31` ANSWER :C

Discussion

5917.	If f: R to R , f(x) = (3x -5)/(4), then f^(-1) (x) is equal to
Answer» `(5x-3)/(4)` `(3x+5)/(4)` `(4x-5)/(3)` `(4x+5)/(3)` ANSWER :D

Discussion

5918.	The vector, 2hat(i)+hat(j)+2hat(k),hat(i)+lambdahat(j)-hat(k)t" and "2hat(i)-lambdahat(k) areco-planar if
Answer» `lambda=-2` `lambda=1+sqrt(3)` `lambda=1-sqrt(3)` `lambda=-2,1pmsqrt(3)` ANSWER :A::B::C::D

Discussion

5919.	If z=((sqrt(3)+i)^(3)(3i+ 4)^(2))/((8+6i)^(2)), then \|z\| is equal to
Answer» `8` `2` `5` `4` ANSWER :B

Discussion

5920.	If a and b are positive integers such that b gt a then underset(n rarr oo)(lim) [(1)/(na) + (1)/(na + 1) + (1)/(na + 2) + ….+ (1)/(nb)]=
Answer» `LOG ((B)/(a))` `log ((a)/(b))` log (AB) `log (a + b)` ANSWER :A

Discussion

5921.	Find the integrals of the functions (cos2x+2sin^(2)x)/(cos^(2)x)
Answer» `2 secx+C` `2 TAN x+c` `tan x+c` NONE of these Answer :C

Discussion

5922.	Two arithmetic progressions have the same numbers. The reatio of the last term of the first progression to the first term of the second progression is equal to the ratio of the last term of the second progression to the first term of first progression is equal to 4. The ratio of the sum of the n terms of the first progression to the sum of the n terms of teh first progression to the sum of the n terms of the second progerssion is equal to 2.
Answer» `6//5` `7//2` `9//5` NONE of these Solution :Let the first TERM be a and COMMON difference be d for the first A.P. and the first term be b and common difference be e for the second A.P. and let the numbers of terms be n. Then `(a+(n-1)d)/b=(b+(n-1)e)/a=4` (1) `(n/2[2a+(n-1)d])/(n/2[2B+(n-1)e])=2` (2) From (1) and (2), we get a-4b+(n-1)d=0 (3) b-4a+(n-1)e=0 (4) 2a-4b+(n-1)d-2(n-1)e=0 (5) `4xx(3)+(4)` gives `-15b+4(n-1)d+(n-1)e=0` (6) `(4)+2xx(5)` gives `-7b+2(n-1)d-3(n-1)e=0` (7) Further, `15xx(7)-7xx(6)` gives `2(n-1)d-52(n-1)e=0` or d=26e`(becausengt1)` `therefored//e=26` Putting d=26e in (3) and solving it with (4), we get a=2(n-1)e,b=7(n-1)e Then, the ratio of their `n^(th)` terms is `(2(n-1)+(n-1)26e)/(7(n-1)e+(n-1)e)=7/2`

Discussion

5923.	Two arithmetic progressions have the same numbers. The reatio of the last term of the first progression to the first term of the second progression is equal to the ratio of the last term of the second progression to the first term of first progression is equal to 4. The ratio of the sum of the n terms of the first progression to the sum of the n terms of teh first progression to the sum of the n terms of the second progerssion is equal to 2. The ratio of their common difference is
Answer» 12 24 26 9 Solution :Let the first TERM be a and common difference be d for the first A.P. and the first term be B and common difference be e for the second A.P. and let the NUMBERS of terms be n. Then `(a+(n-1)d)/b=(b+(n-1)e)/a=4` (1) `(n/2[2a+(n-1)d])/(n/2[2B+(n-1)e])=2` (2) From (1) and (2), we get a-4b+(n-1)d=0 (3) b-4a+(n-1)e=0 (4) 2a-4b+(n-1)d-2(n-1)e=0 (5) `4xx(3)+(4)` gives `-15b+4(n-1)d+(n-1)e=0` (6) `(4)+2xx(5)` gives `-7b+2(n-1)d-3(n-1)e=0` (7) Further, `15xx(7)-7xx(6)` gives `2(n-1)d-52(n-1)e=0` or d=26e`(becausengt1)` `therefored//e=26` Putting d=26e in (3) and solving it with (4), we get a=2(n-1)e,b=7(n-1)e Then, the ratio of their `n^(th)` terms is `(2(n-1)+(n-1)26e)/(7(n-1)e+(n-1)e)=7/2`

Discussion

5924.	Two arithmetic progressions have the same numbers. The reatio of the last term of the first progression to the first term of the second progression is equal to the ratio of the last term of the second progression to the first term of first progression is equal to 4. The ratio of the sum of the n terms of the first progression to the sum of the n terms of teh first progression to the sum of the n terms of the second progerssion is equal to 2. The ratio of their first term is
Answer» `2//7` `3//5` `4//7` `2//5` Solution :Let the first term be a and common difference be d for the first A.P. and the first term be b and common difference be e for the second A.P. and let the numbers of terms be N. Then `(a+(n-1)d)/b=(b+(n-1)e)/a=4` (1) `(n/2[2a+(n-1)d])/(n/2[2B+(n-1)e])=2` (2) From (1) and (2), we GET a-4b+(n-1)d=0 (3) b-4a+(n-1)e=0 (4) 2a-4b+(n-1)d-2(n-1)e=0 (5) `4xx(3)+(4)` GIVES `-15b+4(n-1)d+(n-1)e=0` (6) `(4)+2xx(5)` gives `-7b+2(n-1)d-3(n-1)e=0` (7) Further, `15xx(7)-7xx(6)` gives `2(n-1)d-52(n-1)e=0` or d=26e`(becausengt1)` `therefored//e=26` Putting d=26e in (3) and solving it with (4), we get a=2(n-1)e,b=7(n-1)e Then, the ratio of their `n^(th)` terms is `(2(n-1)+(n-1)26e)/(7(n-1)e+(n-1)e)=7/2`

Discussion

5925.	Some standard forms of integration : intsqrt(x^(2)+7)dx=..........+C
Answer» `(X)/(2)SQRT(x^(2)+7)+(7)/(2)log\|x+sqrt(x^(2)+7)\|` `(x)/(2)sqrt(x^(2)+7)+(SQRT7)/(2)log\|x-sqrt(x^(2)+7)\|` `(x)/(2)sqrt(x^(2)+7)+(7)/(2)sin^(-1)""(x)/(sqrt7)` `log\|x+sqrt(x^(2)+7)\|` Answer :A

Discussion

5926.	If log_(e )k=2[(3)/(5)+(1)/(3)((3)/(5))^(3)+(1)/(5)((3)/(5))^(5)+….oo] then k =
Answer» 2 3 4 8 Answer :C

Discussion

5927.	If a, b and c are roots of the equation x^3+qx^2+rx +s=0 then the value of r is
Answer» 184 196 224 none of these Solution :We have, a+b+c=25 2a=b+2 `c^(2)=18b` Elminating a from (1) and (2), we have `b=16-(2c)/3` Then from (3), `c^(2)=18(16-(2c)/3)` or `c^(2)+12c-18xx16=0` or `(c-12)(c+24)=0` Now, c=-24 is not possible since it does not lie between 2 and 18. Hence, c=12. Then from (3), b=8 and finally from (2),a = 5. Thus, a=5,b=8 and c=12. Hence, abc`=5xx8xx12=480`. ALSO,equation `ax^(2)+bx+c=0 is 5x^(2)+8x+12=0`, which has imaginary ROOTS. If a,b,c are roots of the equation `x^(3)+QX^(2)+rx+s=0`, then the sum of PRODUCT of roots taken two at a time is `r=5xx8+5xx12+8xx12=196`.

Discussion

5928.	Number of integral solutions of (x+2)/(x^2 + 1) gt 1/2 is
Answer» 0 1 2 3 Answer :D

Discussion

5929.	Compute theintegrals I = int_(0)^(a) x^(2) sqrt((a - x)/( a + x)) dx, a gt 0
Answer» ANSWER :`a^(3) ((PI)/(4)- (2)/(3))` (SUBSTITUTION x = a COS t )

Discussion

5930.	Let f(x) = \|x-2\| AA x in R and g(x) =f(f(f(x))), then the number of solutions of g(x) = 3/2 is
Answer» 4 6 8 10 Answer :B

Discussion

5931.	If the circlex^(2)+ y ^(2)+ ax + by - 12=0has the centre at (2,3)then find a, b and the radius of the circle .
Answer» ANSWER :` a=-4 , b=-6 , r=5`

Discussion

5932.	If the vectors b, c, d are not coplanar, then the vector (axx b)xx(c xx d)+(a xx c)xx(d xx b)+(a xx d)xx(b xx c) is
Answer» PARALLEL to a parallel to b parallel to c perpendicular to a ANSWER :A

Discussion

5933.	If 1+x^(2)=sqrt(3)x, then sum_(n=1)^(24)(x^(n)-(1)/(x^n))^(2) is equal to
Answer» `0` `48` `-24` `-48` ANSWER :D

Discussion

5934.	Evaluate the following integrals (i) int_(1)^(2)((x+1)^(2))/(sqrt(x))dx
Answer» ANSWER :`(94sqrt(2)-56)/(15)`

Discussion

5935.	Sketch the graphs of the following functions. f(x) = 1 + 1/x^2
Answer» SOLUTION :

Discussion

5936.	Using binomial distribution find the mean and variance of X forthe following experiments A fair die is tossed 240 times, and X denote the number of times that four appeared.
Answer» ANSWER :MEAN = 40, VARIANCE = `100/3`

Discussion

5937.	The equation of plane containing the lines barr=(2hatj-3hatk)+lamda(hati+2hatj+3hatk) barr=(2hati+6hatj+3hatk)+lamda(2hati+3hatj+4hatk)
Answer» `(2,6,3)` `(0,2,-3)` `(2,3,4)` `(2,-6,4)` ANSWER :A

Discussion

5938.	If veca = vec b+ vec c, then is it true that\|veca\| = \|vec b\| + \|vec c\| ? Justify your answer.
Answer» ANSWER :No ; TAKE `veca,vecbandvecc` to REPRESENT the sides of a TRIANGLE.

Discussion

5939.	Find an anti derivative (or integral) of the following function. (ax+b)^(2)
Answer» ANSWER :`1/(3A)(ax+b)^(3)`

Discussion

5940.	Evaluate the following integrals intx^(2)cos^(-1)x dx
Answer» Answer :`(x^(3))/(3)cos^(-1)x-(1)/(3)sqrt(1-x^(2))+(1)/(9)(1-x^(2))^((3)/(2))+C`

Discussion

5941.	If sum_(i=1)^(18)(x_(i)-8)=9 and sum_(i=1)^(18)(x_(i)-8)^(2)=45 then find the standard deviation ofx_1 , x_2, ....,x_(18)
Answer» ANSWER :`(3)/(2)`

Discussion

5942.	If the ecentricity of a hyperbola is srqt3 then the eccentricity of its conjugate hyperbola is
Answer» `SQRT2` `SQRT3` `sqrt(3/2)` `2sqrt3` Answer :C

Discussion

5943.	Let p and be two propositions. Then the contrapositive of the implication p rarr q is
Answer» <P>`~Q RARR ~p` `~p rarr ~q` `q rarr p` `p HARR q` ANSWER :A

Discussion

5944.	The projection of a line segment on the axes are 9, 12 and 8. Then the length of the line segment is
Answer» 15 16 17 18 Answer :C

Discussion

5945.	A=[{:(1,-2,3),(-4,2,5):}]andB=[{:(2,3),(4,5),(2,1):}] then find AB and BA . Show that ABneBA.
Answer» ANSWER :`AB=[{:(0,-4),(10,3):}]BA=[{:(-10,2,21),(-16,2,37),(-2,-2,11):}]`

Discussion

5946.	In the following which is true:
Answer» `(p VV q) vv (-r) -= p vv (q vv r)` `p vv (q vv r) -= (p vv q) wedge (p vv r)` `p LEFTRIGHTARROW q -= (p to -q)` `p vv -q` is a tautology Answer :B

Discussion

5947.	If x and y are integers, and (x)/(y) is even, which of the following could be true? Indicate all that apply. A. xy is odd B. xy is even C. x+y is odd
Answer» ANSWER :B and C

Discussion

5948.	Evaluate the following integrals: (i) int_(2)^(3)x^(2)dx (ii) int_(4)^(9)(sqrtx)/((30-x^(3/2))^(2))dx (iii) int_(1)^(2)(xdx)/((x+1)(x+2)) (iv) int_(0)^(pi/4)sin^(3)2tcos2tdt
Answer» Answer :`(i) (19)/(3) (ii) (19)/(99), (III) log((32)/(27)), (iv) (1)/(8)`s

Discussion

5949.	Thevector (1)/(3) (2hat(i) - 2hat(k)) is
Answer» a unit vectors makes an angle `(pi)/(3)` with the vectors `2hat(i) - 4hat(j) - 3hat(k)` parallel to the VECTOR`-HAT(i) + hat(j) - (1)/(2)hat(k)` Perpendicularto the vectors `3hat(i) + 2hat(j) - 2hat(k)` Answer :A::C::D

Discussion

5950.	int (x dx)/((x-1)(x-2))" equals : "
Answer» `log\|((x-2)^(2))/(x-1)\|+C` `log \|((x-1)/(x-2))^(2)\|+C` `log \|((x-1)/(x-2))^(2)\|+C` SOLUTION :`" Let " (x)/((x-1)(x-2))=(A)/((x-1)) +(B)/((x-2))` `RARR "" x=A (x-2) +B(x-1)` x=1 then 1=A (1-2)+0 `rArr` A=-1 x=2 then 2=0 +B(2-1) `rArr` B=2 `:. (x)/((x-1)(x-2))=(1)/((x-1))+(2)/((x-2))` `:. int (x)/((x-1)(x-2)) dx= int ((-1))/(x-1)dx+ int(2)/(x-2)dx` ` =-log \|x-1\|+2 log \|x-2\|+c` `=- log \|x-1\|+log \|x-2\|^(2)+c` `=log \|((x-2)^(2))/(x-1)\|+c` `"(":' Log b- log a = log.(b)/(a)")"`

Discussion

Explore topic-wise InterviewSolutions in .

{:(" "Lt),(n rarr oo):} ((1)/(sqrt(n^(2)))+(1)/(sqrt(n^(2)+n))+(1)/(sqrt(n^(2)+2n))+(1)/(sqrt(n^(2)+3n))+....(1)/(sqrt(n^(2)+n(n-1)))))=

If p,q, in rand p^2+q^2-pq-q+1 le 0 + alpha + beta + gamma = 0 , then |(1,cosy,cosbeta),(cosgamma,p,cosalpha),(cosbeta,cosalpha,q)|=

If [[1,-x],[4,-3]] =[[1,8],[4,-3]] then value of x is :

A fair dice is rolled 6 times. If "getting an even number" is a success the probability of 5 successes is …………..

There are 3 rows containing 2 seats in each row. In how many ways 3 persons can be seated such that no row remains empty.

If a set A has 12 elements. Find the number of subsets of A having atmost 3 elements

Solve the following Linear Programming Problems graphically : Minimise Z = x + 2y subject to 2x+y ge 3, x+2y ge 6, x, y ge 0

Find the number of ordered triples (a,b,c) of positive integers such that abc= 108

2x-y+3z=1 3x-2y+5z=-4 5x-4y+9z=14

Find maximum and minimum value of the following functions in the given interval.f(x)=x^(2)log x, x in [0, 2]

Consider two regions R_(1):"points P are nearer to (1,0) than to "x=-1. R_(2): "Points P are nearer to (0,0) than to (8,0)" Find the area of the region common to R_(1) and R_(2).

"if "f(x) ={ underset(5x, 1,2 le x le 3)(3x+4, 0 le x le 2}}, then evaluate int_(0)^(3) f(x) dx.

There are 5 cards numbered 1 to 5, one number on one card. Two cards are drawn at random without replacement. Let X denote the sum of the numbers on two cards drawn. Find the mean and variance of X.

The normal at 'P'cuts the axis of the parabola y^(2) =4ax in G and S is the focus of the parabola. IfDelta SPGis equilateral then each side is of length.

Let O(0,0) , P(2,0) and Q (6,0) be the vertices of triangle OPQ . If R be a point inside the triangle OPQ such that areas of DeltaOPQ,DeltaPQRandDeltaOQRare equal , then the abscissa of R is -

The ratio of the number of chickens to the number of pigs to the number of horses on Richard's farm is 33:17 : 21. What fraction of the animals are either pigs or horses?

If f: R to R , f(x) = (3x -5)/(4), then f^(-1) (x) is equal to

The vector, 2hat(i)+hat(j)+2hat(k),hat(i)+lambdahat(j)-hat(k)t" and "2hat(i)-lambdahat(k) areco-planar if

If z=((sqrt(3)+i)^(3)(3i+ 4)^(2))/((8+6i)^(2)), then |z| is equal to

If a and b are positive integers such that b gt a then underset(n rarr oo)(lim) [(1)/(na) + (1)/(na + 1) + (1)/(na + 2) + ….+ (1)/(nb)]=

Find the integrals of the functions (cos2x+2sin^(2)x)/(cos^(2)x)

Some standard forms of integration : intsqrt(x^(2)+7)dx=..........+C

If log_(e )k=2[(3)/(5)+(1)/(3)((3)/(5))^(3)+(1)/(5)((3)/(5))^(5)+….oo] then k =

If a, b and c are roots of the equation x^3+qx^2+rx +s=0 then the value of r is

Number of integral solutions of (x+2)/(x^2 + 1) gt 1/2 is

Compute theintegrals I = int_(0)^(a) x^(2) sqrt((a - x)/( a + x)) dx, a gt 0

Let f(x) = |x-2| AA x in R and g(x) =f(f(f(x))), then the number of solutions of g(x) = 3/2 is

If the circlex^(2)+ y ^(2)+ ax + by - 12=0has the centre at (2,3)then find a, b and the radius of the circle .

If the vectors b, c, d are not coplanar, then the vector (axx b)xx(c xx d)+(a xx c)xx(d xx b)+(a xx d)xx(b xx c) is

If 1+x^(2)=sqrt(3)x, then sum_(n=1)^(24)(x^(n)-(1)/(x^n))^(2) is equal to

Evaluate the following integrals (i) int_(1)^(2)((x+1)^(2))/(sqrt(x))dx

Sketch the graphs of the following functions. f(x) = 1 + 1/x^2

Using binomial distribution find the mean and variance of X forthe following experiments A fair die is tossed 240 times, and X denote the number of times that four appeared.

The equation of plane containing the lines barr=(2hatj-3hatk)+lamda(hati+2hatj+3hatk) barr=(2hati+6hatj+3hatk)+lamda(2hati+3hatj+4hatk)

If veca = vec b+ vec c, then is it true that|veca| = |vec b| + |vec c| ? Justify your answer.

Find an anti derivative (or integral) of the following function. (ax+b)^(2)

Evaluate the following integrals intx^(2)cos^(-1)x dx

If sum_(i=1)^(18)(x_(i)-8)=9 and sum_(i=1)^(18)(x_(i)-8)^(2)=45 then find the standard deviation ofx_1 , x_2, ....,x_(18)

If the ecentricity of a hyperbola is srqt3 then the eccentricity of its conjugate hyperbola is

Let p and be two propositions. Then the contrapositive of the implication p rarr q is

The projection of a line segment on the axes are 9, 12 and 8. Then the length of the line segment is

A=[{:(1,-2,3),(-4,2,5):}]andB=[{:(2,3),(4,5),(2,1):}] then find AB and BA . Show that ABneBA.

In the following which is true:

If x and y are integers, and (x)/(y) is even, which of the following could be true? Indicate all that apply. A. xy is odd B. xy is even C. x+y is odd

Evaluate the following integrals: (i) int_(2)^(3)x^(2)dx (ii) int_(4)^(9)(sqrtx)/((30-x^(3/2))^(2))dx (iii) int_(1)^(2)(xdx)/((x+1)(x+2)) (iv) int_(0)^(pi/4)sin^(3)2tcos2tdt

Thevector (1)/(3) (2hat(i) - 2hat(k)) is

int (x dx)/((x-1)(x-2))" equals : "