28278 + Interview Questions in Maths in Class 12 Page 126 InterviewSolution

6251.	A cruve is respresented by C=21x^(2)-6xy+29y^(2)+6x-58y-151=0 The center of the conic C is
Answer» (1,0) (0,0) (0,1) none of these Solution :`24X^(2)-6xy+29y+6x-58y-151=0` `2(x-3y+3)^(2)+2(3x+y-1)^(2)=180` or `((x-3y+3)^(2))/(60)+((3x+y-1)^(2))/(90)=1` or `((x-3y+3)/(sqrt(1+3^(2))sqrt(6)))^(2)+((3x+y-1)/(3sqrt(1+3^(2))))=1` Thus, C is an ellipse whose lengths of axes are `6,2sqrt(6)`. The minor and the major axes are `x-3y+3=0 and 3x+y-1=0`, respectively. Their POINT of INTERSECTION gives the center of the center of the conic. THEREFORE, Center `-=(o,1)`

Discussion

6252.	If the tangent at P(at^(2),2at) to theparabola y^(2)= ax intersects X-axis at A and the normal at P meets it at B then area of triangle PAB is
Answer» `4A^(2)\|t\|sqrt(1+r^(2))` `2A^(2)\|t\|(1+r^(2))` `4a^(2)\|t\|(1+t^2)` `(2a^(2)(1+t^(2)))/(\|t\|)` ANSWER :B

Discussion

6253.	In a bag there are infinitely many red, white and black balls which are identical. If Ten balls are selected at random then the probability that selection at random then the probability that selection includes atleast one ball from each colour is
Answer» `(5)/(11)` `(6)/(11)` `(7)/(11)` `(4)/(11)` Answer :B

Discussion

6254.	If f(x)..................
Answer» Solution :`C_(3) rarr C_(3)-(C_(1)+C_(2))=\|(1,x,0),(2X,x(x-1),0),(3x(x-1),2(x-1)(x-2),0)\|` `=0` `IMPLIES f(100)=0`

Discussion

6255.	Integrate the following functions with respect to x. =sqrt(1+sin2x,) x in((3pi)/(4),(7pi)/(4))
Answer» ANSWER :`cosx-sinx+c`

Discussion

6256.	Two cards are drawn at random and without replacement from a pack of 52 playing cards. Find the probability that both the cards are black.
Answer» ANSWER :`(25)/(102)`

Discussion

6257.	If the value of definite integral(int_(0)^(pi//2)(cos x)^(sqrt(2)+1)dx)/(int_(0)^(pi//2)(cos x)^(sqrt(2)-1)dx) is equal to (n - sqrt(n)), where n in N then find the value of n.
Answer» ANSWER :2

Discussion

6258.	Let a, b be integers such that all the roots of the equation (x^2 + ax + 20)(x^2 + 17x + b) = 0 are negative integers. What is the smallest possible value of a + b ?
Answer» ANSWER :25

Discussion

6259.	A particle acted on by constant forces vecf = 4 hati + 3 hatj - 3 hatk and vecg = 3 hati + hatj - hatk experiences a displacement from the point veca = hati + 2 hatj + 3 hatk to the point vecb = 5 hati + 4 hatj + hatk. The total work done by the forces is
Answer» ANSWER :44

Discussion

6260.	Find the probability of getting an even number in a single roll of the die.
Answer» <P> ANSWER :PROBABILITY of GETTING an even NUMBER `=P(2)+P(4)+P(6)` `2/21+4/21+6/21=12/21`

Discussion

6261.	Let a=5i+4j-k, b=-4i+j+5k, c=i+3j-k. Let alpha be a vector perpendicular to both a and b such that alpha*c=63," then "abs(alpha)^(2)//21^(2)=_______
Answer» ANSWER :3

Discussion

6262.	Find the number of 7 - digit numbers that can be formed using 2,2,2,3,3,4,4.
Answer» ANSWER :`(lfloor7)/(lfloor3lfloor2lfloor2)`

Discussion

6263.	Evaluate the following integral int e^(2x) cos x cos 3x dx
Answer» Answer :`e^(2X) [ (COS 4X + 2 SIN 4x)/(20) + ("cos 2x + sin 2x")/(8) ] + C`

Discussion

6264.	If x gt 0 then the sum of the series e^(-x) - e^(-2x) + e^(-3x) ......infty is
Answer» `1/(1 - E^(-X))` `1/(e^(x) - 1)` `1/(1 + e^(-x))` `1/(1 + e^(x))` ANSWER :D

Discussion

6265.	Sketch the curve y=f(x)x^(2)-x-6
Answer» Answer :`f((1)/(2))=-(25)/(4)`

Discussion

6266.	A fair coin is tossed n times. Let a_(n) denotes the number of cases in which no two heads occur consecutively. Then which of the following is not true ?
Answer» `a_(1)=2` `a_(2)=3` `a_(5)=13` `a_(8)=55` SOLUTION :`(c )` The cases for `a_(1){H,T}` i.e, `a_(1)=2` The cases for `a_(2){HT,TH,TT}`, `a_(2)=3` For `n ge 3` , if the first OUTCOME is `H`, then NEXT just `T` and then `a_(n-2)`. If the first out come is `T`, then `a_(n-1)` should follow. So, `a_(n)=1xx1xxa_(n-2)+1xxa_(n-1)impliesa_(n)=a_(n-2)+a_(n-1)` So, `a_(3)=a_(1)+a_(2)=5`, `a_(4)=3+5=8` and so on.

Discussion

6267.	Evaluate the definite integral in exercise overset((pi)/(4)) underset((pi)/(0)) int "cosec" x dx
Answer» ANSWER :`log_(E)((sqrt(2)-1)/(2-sqrt(3)))`

Discussion

6268.	Find the equation and length of the common chord of the following circles. x^2 + y^2 + 2x + 2y + 1 = 0 , x^2 + y^2 + 4x + 3y + 2 = 0 .
Answer» ANSWER :`2x+y+1=0, 2/sqrt5`

Discussion

6269.	int(x^(8)-9x^(2)+18)/(x^(4)-3x^(2)+ 13)dx=
Answer» `(X^(4))/(4)+X^(3)+6X^(2)+C` `(X^(5))/(5)+(X^(4))/(4)+6X+c` `(X^(5))/(5)+X^(3)+6X+c` `(X^(5))/(5)+X^(3)+6X+c` Answer :C

Discussion

6270.	Show that the line 2x-y+2=0 is a tangent to the parabola y^(2)=16x. Find the point of cotact also.
Answer» ANSWER :`=(1,4)`

Discussion

6271.	What is the approximate percent increase in total sales at Produce Stand P from January to June?
Answer» `19%` `24%` `28%` `32%` ANSWER :D

Discussion

6272.	The three sides of a right-angled triangle are in G.P. (gcometric progression). If the two acute angles be alpha and beta then tan alpha and tan beta are
Answer» `(SQRT(5)+1)/(2)"and"(sqrt(5)-1)/(2)` `sqrt((sqrt(5)+1)/(2))"and"sqrt((sqrt(5)-1)/(2))` `sqrt(5)` and `(1)/(sqrt(5)` `sqrt(5)/(2)"and(2)/(sqrt(5))` ANSWER :B

Discussion

6273.	Differentiate s^(sin x), x gt 0 w.r.t x
Answer» ANSWER :`X^(SIN x-1).sin x + x^(sin x). COS x.log x`

Discussion

6274.	Let T(k) be the statement 1+3+5+…+(2k-1)=k^2+10 Which of the following is correct ?
Answer» T(1) is true T(k) is true `rArr` T (k+1) is true T(N) is true for all `n in N` All above are CORRECT Answer :B

Discussion

6275.	Let f(x) ={{:(x^(3)+x^(2)-10x",",1lexlt0),(1+cosx,(pi)/(2)lexlepi),(1+cosx","(pi)/(2)lexlepi,):} then f(x) has,
Answer» LOCAL maxima at `X=(pi)/(2)` local minima at `x=(pi)/(2)` ABSOLUTE maxima at x =0 absolute maxima at `x=(pi)/(2)` Answer :A::C

Discussion

6276.	The solution of x(dy)/(dx) +y = x Cos x is
Answer» `XY = x Cos x + Sin x + C` `xy = x Sin x + Cos x + c` `xy = x Sin x - Cos x + c` `x + y = x Sin x - Cos x + c` ANSWER :B

Discussion

6277.	Find the co-ordinates of the point where the line through (5,1,6) and (3,4,1) crosses the ZX plane
Answer» Answer :From the example 10, the co-ordinates of any point on the LINE joining (5,1,6) and (3,4,1) may be ASSUMED as `(5-2lambda, 1+3lambda, 6-5lambda)` for some `LAMBDA`. If this point lies on ZX-plane, its y-coordinate = 0, `1+3lambda=0 rArr = (-1)/3` Hence, the required point = `(5+2/3, 0, 6+5/3) = ((1)/3, 0 (23)/3)

Discussion

6278.	f:R rarrR , f(x) =x^(2) , g : R rarr R , g(x) = 2^(x) , then {x\|(fog)(x) = (gof)(x)} = ............
Answer» `{0}` `{0,1}` R `{0,2}` SOLUTION :N/A

Discussion

6279.	Find the smaller of the area bounded by the parabola 4y^(2) – 3x – 8y + 7 = 0 andthe ellipse x^(2) + 4y^(2) – 2x – 8y + 1 = 0
Answer» Answer :`=2[sqrt(3)/2 int_(0)^(1) sqrt(X)dX + int_(0)^(2) (sqrt(4-X^(2)))/2dX]=[sqrt(3)/6+(2PI)/3]`

Discussion

6280.	If X is a random variable with the following distribution where p+q=1 then show that the variance of X is pq(a-b)^(2).
Answer» ANSWER :`=PQ(a-b)^(2)`

Discussion

6281.	A particular solution of (dy)/(dx) = (x(2 log x +1))/(sin y + y cos y) is
Answer» `y SIN y = X LOG x` `y^(2) sin y = x log x` `y sin y = x^(2) log x` none Answer :C

Discussion

6282.	The third term of G.P is 9. The product if its first five terms is
Answer» `3^(10)` `3^(5)` `3^(12)` `3^(9)` ANSWER :A

Discussion

6283.	Determine the largest 2-digit prime factor of the integer ((200),(100)) i.e., ""^(200)C_(100)
Answer» ANSWER :61

Discussion

6284.	Fromthe polynomial equation whose roots are 1+I,1-I,1+I,1-I
Answer» ANSWER :`x^4-4x^3 +8x^2 +4=0`

Discussion

6285.	Find all the points of discontinuity of f defined by f(x)=\|x\|-\|x+1\|
Answer» SOLUTION :LET (X)=\|x\| and `H(x)=x+1`

Discussion

6286.	Evaluate int_(0)^(1)e^(2-3x)dx .
Answer»

Discussion

6287.	if the right hand derivative ofh(x) = {x} ( {.} is fractional of x) exists at x =1 and it is equal to _____\
Answer» ANSWER :A

Discussion

6288.	Let f: R -{-4/3}rarrR be a function defined as f(x) =(4x)/(3x+4). The inverse of f is the map g : Range f rarrR - {-4/3} given by
Answer» `G(y) =(3Y)/(3-4Y)` `g(y) =(4y)/(4-3y)` `g(y) =(4y)/(3-4y)` `g(y) =(3y)/(4-3y)` SOLUTION :N/A

Discussion

6289.	State the converse, inverse and contrapositive of The ground being wet, there has been rainfall at nightpropositions. Stating it as a conditional, wherever necessary.
Answer» Solution :If the GROUND being wet then there has been rainfall at night. Con:If there has been rainfall at night then the ground being wet, INV : If the ground is not wet then there has been no RAIN FALL at night. Con :If there has been no rain fall at night then the ground is not wet.

Discussion

6290.	A book is published in three volumes, the pages being numbered from 1 onwards. The page numbers are continued from the first volume to the second volume to the third. The number of pages in the second volume is 50 more than that in the first volume, and the number pages in the third volume is one and a half times that in the second. The sum of the page numbers on the first pages of the three volumes is 1709. If n is the last page number, what is the largest prime factor of n ?
Answer» ANSWER :17

Discussion

6291.	Find the equation of the circle whose centre lies on the X- axis and passingthrough (-2, 3) and(4,5)
Answer» ANSWER :`X^(2) + y ^(2) - 8X -29 =theta`

Discussion

6292.	If the middle term of (1+x)^ (2n) is (1.3.5….(2n-1)k)/(n!) then k=
Answer» `(3X)^(N+1) ` `(2X)^(n+1)` `(2x)^n` `(3x)^n` ANSWER :C

Discussion

6293.	If theHCFof twonumbersis 18and their LCM is 360, allof thefollowingcannotbe the differencebetweenthe two numbers EXCEPT ?
Answer» 8 12 18 54 Answer :C

Discussion

6294.	A travelling wave is given by y - (0.8)/((3x^(2)+24xt + 48t^(2)+4)) where x and y are in minute andt in second. Find the velocity in m//s.
Answer» SOLUTION :`y - (0.8)/(3X^(2)+24xt + 48t^(2)+4)` ` = (0.8)/(3(x+4T)^(2)+4)` Acc. to wave EQUATION `x + 4t = x + vt` `implies v = 4 m//s`.

Discussion

6295.	Prove that the radical axis of the circles x^2 + y^2 + 2gx + 2fy + c = 0 and x^2 + y^2 + 2g'x + 2f'y + c' = 0is the diameter of the latter circle (or the former bisects the circumference fo the latter ) if 2g'(g-g') + 2f'(f-f') = c-c'.
Answer» ANSWER :(or) 2G (g-g') + 2F'(f-f') = c-c'

Discussion

6296.	If lim_(xrarroo)(ae^(x)+b cos x+c +dx)/(xsin^(2)x)=3, then the value of 272(abd)/(c^(3)) is equal to
Answer» ANSWER :34

Discussion

6297.	A circle of radius 4, is concentric with the ellipse 3x^2+13y^2=78. Prove that a common tangent is inclined to the major axis at an angle pi/4.
Answer» ANSWER :`THETA =pi/4`

Discussion

6298.	Show that the relation R defined in the set A of all triangles as R = {(T_1, T_2) : T_1 is similar to T_2}, is equivalence relation. Consider three right angle triangles T_1with sides 3, 4, 5, T_2 with sides 5, 12, 13 and T_3 with sides 6, 8, 10. Which triangles among T_1, T_2 and T_3 are related ?
Answer» Solution :Here A = set of all triangles and R =`{(T_1, T_2) : T_1` is similar to `T_2`} `because ` Every triangle is similar to itself. `therefore R` is reflexive. Let `T_1, T_2 in A and (T_1, T_2 ) in R` `therefore R` is symmetric. Let `""T_1, T_2, T_3 in A` and `""(T_1, T_2) in R and (T_2, T_3) in R` `rArr T_1` is similar to `T_2 and T_2` is similar to `T_3`. `rArr T_1` is similar to `T_3`. `rArr (T_1, T_3) in R` `therefore R` is tansitive. `because R` is reflexive, symmetric and transitive. `therefore ` R is an equivalence RELATION. Now `"" (3)/(6) = (4)/(8) = (5)/(10) = (1)/(2)` `rArr` The corresponding sides of `T_1and T_3` are proportional. `therefore T_1 and T_3` are similar. `rArr ` Triangle `T_1` is related to triangle `T_3`.

Discussion

6299.	A(x_(1),y_(1)), B(x_(2),y_(2)), C(x_(3),y_(3)) are three vertices of a triangle ABC. lx +my +n = 0 is an equation of the line L. If P divides BC in the ratio 2:1 and Q divides CA in the ratio 1:3 then R divides AB in the ratio (P,Q,R are the points as in problem 1)
Answer» 2:3 internally 2:3 externally 3:2 internally 3:2 externally Solution :From the above RESULT `rArr (2)/(1).(1)/(3).(AR)/(RB) =- 1` `:. (AR)/(RB) =- (3)/(2)` `:. R` DIVIDES `AB 3:2` externally.

Discussion

6300.	What is the number of moles of O-atom in 126 amu of HNO_(3) ?
Answer» 2 `(2)/(N_(A))` 6 `(6)/(N_(A))` ANSWER :A

Discussion

Explore topic-wise InterviewSolutions in .

A cruve is respresented by C=21x^(2)-6xy+29y^(2)+6x-58y-151=0 The center of the conic C is

If the tangent at P(at^(2),2at) to theparabola y^(2)= ax intersects X-axis at A and the normal at P meets it at B then area of triangle PAB is

In a bag there are infinitely many red, white and black balls which are identical. If Ten balls are selected at random then the probability that selection at random then the probability that selection includes atleast one ball from each colour is

If f(x)..................

Integrate the following functions with respect to x. =sqrt(1+sin2x,) x in((3pi)/(4),(7pi)/(4))

Two cards are drawn at random and without replacement from a pack of 52 playing cards. Find the probability that both the cards are black.

If the value of definite integral(int_(0)^(pi//2)(cos x)^(sqrt(2)+1)dx)/(int_(0)^(pi//2)(cos x)^(sqrt(2)-1)dx) is equal to (n - sqrt(n)), where n in N then find the value of n.

Let a, b be integers such that all the roots of the equation (x^2 + ax + 20)(x^2 + 17x + b) = 0 are negative integers. What is the smallest possible value of a + b ?

A particle acted on by constant forces vecf = 4 hati + 3 hatj - 3 hatk and vecg = 3 hati + hatj - hatk experiences a displacement from the point veca = hati + 2 hatj + 3 hatk to the point vecb = 5 hati + 4 hatj + hatk. The total work done by the forces is

Find the probability of getting an even number in a single roll of the die.

Let a=5i+4j-k, b=-4i+j+5k, c=i+3j-k. Let alpha be a vector perpendicular to both a and b such that alpha*c=63," then "abs(alpha)^(2)//21^(2)=_______

Find the number of 7 - digit numbers that can be formed using 2,2,2,3,3,4,4.

Evaluate the following integral int e^(2x) cos x cos 3x dx

If x gt 0 then the sum of the series e^(-x) - e^(-2x) + e^(-3x) ......infty is

Sketch the curve y=f(x)x^(2)-x-6

A fair coin is tossed n times. Let a_(n) denotes the number of cases in which no two heads occur consecutively. Then which of the following is not true ?

Evaluate the definite integral in exercise overset((pi)/(4)) underset((pi)/(0)) int "cosec" x dx

Find the equation and length of the common chord of the following circles. x^2 + y^2 + 2x + 2y + 1 = 0 , x^2 + y^2 + 4x + 3y + 2 = 0 .

int(x^(8)-9x^(2)+18)/(x^(4)-3x^(2)+ 13)dx=

Show that the line 2x-y+2=0 is a tangent to the parabola y^(2)=16x. Find the point of cotact also.

What is the approximate percent increase in total sales at Produce Stand P from January to June?

The three sides of a right-angled triangle are in G.P. (gcometric progression). If the two acute angles be alpha and beta then tan alpha and tan beta are

Differentiate s^(sin x), x gt 0 w.r.t x

Let T(k) be the statement 1+3+5+…+(2k-1)=k^2+10 Which of the following is correct ?

Let f(x) ={{:(x^(3)+x^(2)-10x",",1lexlt0),(1+cosx,(pi)/(2)lexlepi),(1+cosx","(pi)/(2)lexlepi,):} then f(x) has,

The solution of x(dy)/(dx) +y = x Cos x is

Find the co-ordinates of the point where the line through (5,1,6) and (3,4,1) crosses the ZX plane

f:R rarrR , f(x) =x^(2) , g : R rarr R , g(x) = 2^(x) , then {x|(fog)(x) = (gof)(x)} = ............

Find the smaller of the area bounded by the parabola 4y^(2) – 3x – 8y + 7 = 0 andthe ellipse x^(2) + 4y^(2) – 2x – 8y + 1 = 0

If X is a random variable with the following distribution where p+q=1 then show that the variance of X is pq(a-b)^(2).

A particular solution of (dy)/(dx) = (x(2 log x +1))/(sin y + y cos y) is

The third term of G.P is 9. The product if its first five terms is

Determine the largest 2-digit prime factor of the integer ((200),(100)) i.e., ""^(200)C_(100)

Fromthe polynomial equation whose roots are 1+I,1-I,1+I,1-I

Find all the points of discontinuity of f defined by f(x)=|x|-|x+1|

Evaluate int_(0)^(1)e^(2-3x)dx .

if the right hand derivative ofh(x) = {x} ( {.} is fractional of x) exists at x =1 and it is equal to _____\

Let f: R -{-4/3}rarrR be a function defined as f(x) =(4x)/(3x+4). The inverse of f is the map g : Range f rarrR - {-4/3} given by

State the converse, inverse and contrapositive of The ground being wet, there has been rainfall at nightpropositions. Stating it as a conditional, wherever necessary.

Find the equation of the circle whose centre lies on the X- axis and passingthrough (-2, 3) and(4,5)

If the middle term of (1+x)^ (2n) is (1.3.5….(2n-1)k)/(n!) then k=

If theHCFof twonumbersis 18and their LCM is 360, allof thefollowingcannotbe the differencebetweenthe two numbers EXCEPT ?

A travelling wave is given by y - (0.8)/((3x^(2)+24xt + 48t^(2)+4)) where x and y are in minute andt in second. Find the velocity in m//s.

Prove that the radical axis of the circles x^2 + y^2 + 2gx + 2fy + c = 0 and x^2 + y^2 + 2g'x + 2f'y + c' = 0is the diameter of the latter circle (or the former bisects the circumference fo the latter ) if 2g'(g-g') + 2f'(f-f') = c-c'.

If lim_(xrarroo)(ae^(x)+b cos x+c +dx)/(xsin^(2)x)=3, then the value of 272(abd)/(c^(3)) is equal to

A circle of radius 4, is concentric with the ellipse 3x^2+13y^2=78. Prove that a common tangent is inclined to the major axis at an angle pi/4.

Show that the relation R defined in the set A of all triangles as R = {(T_1, T_2) : T_1 is similar to T_2}, is equivalence relation. Consider three right angle triangles T_1with sides 3, 4, 5, T_2 with sides 5, 12, 13 and T_3 with sides 6, 8, 10. Which triangles among T_1, T_2 and T_3 are related ?

A(x_(1),y_(1)), B(x_(2),y_(2)), C(x_(3),y_(3)) are three vertices of a triangle ABC. lx +my +n = 0 is an equation of the line L. If P divides BC in the ratio 2:1 and Q divides CA in the ratio 1:3 then R divides AB in the ratio (P,Q,R are the points as in problem 1)

What is the number of moles of O-atom in 126 amu of HNO_(3) ?