43030 + Interview Questions in Mathematics Page 82 InterviewSolution

4051.	Find the total number of real solutions to the equation 5^{z^2+12^{z^2=13^{z^2
Answer» Find the total number of real solutions to the equation 5^{z^2+12^{z^2=13^{z^2

Discussion

4052.	The locus of a point which is equidistant from the points (1,2,3) and (3,2,−1) is
Answer» The locus of a point which is equidistant from the points $(1, 2, 3)$ and $(3, 2, - 1)$ is

Discussion

4053.	11. The distance of the point (2,3) from the line x-2y + 5=0 measured in a direction parallel to the line x-3y=0 is
Answer» 11. The distance of the point (2,3) from the line x-2y + 5=0 measured in a direction parallel to the line x-3y=0 is

Discussion

4054.	If ∫11−cosx−sinxdx=ln∣∣∣tan[f(x)]−1tan[g(x)]∣∣∣+C, then the value of f(1)+g(1)−1=(where C is integration constant)
Answer» If $\int \frac{1}{1 - cos x - sin x} d x = ln ∣ ∣ ∣ \frac{tan [f (x)] - 1}{tan [g (x)]} ∣ ∣ ∣ + C,$ then the value of $f (1) + g (1) - 1 =$ (where $C$ is integration constant)

Discussion

4055.	If f(x)=cos[π2]x+cos[−π2]x, where [x] denotes the greatest integer less than or equal to x, then write the value of f(π).
Answer» If $f (x) = c o s [π^{2}] x + c o s [- π^{2}] x$ , where [x] denotes the greatest integer less than or equal to x, then write the value of $f (π)$ .

Discussion

4056.	12. (ax +b)"
Answer» 12. (ax +b)"

Discussion

4057.	Checkwhether the relation R in R defined as R = {(a, b):a ≤ b3}is reflexive, symmetric or transitive.
Answer» Check whether the relation R in R defined as R = {(a, b): a ≤ b³} is reflexive, symmetric or transitive.

Discussion

4058.	Solve for }x,9^{x+2}-6\cdot3^{x+1}+1=0
Answer» Solve for }x,9^{x+2}-6\cdot3^{x+1}+1=0

Discussion

4059.	If A and B are two events associated with a random experiment such that P(A)=0.5,P(B)=0.3 and P(A∩B)=0.3, find P(A∪B).
Answer» If A and B are two events associated with a random experiment such that $P (A) = 0.5, P (B) = 0.3$ and $P (A \cap B) = 0.3$ , find $P (A \cup B)$ .

Discussion

4060.	If in the expansion of (1+x)n, a, b, c are three consecutive coefficients, then n =
Answer» If in the expansion of $(1 + x)^{n}$ , a, b, c are three consecutive coefficients, then n =

Discussion

4061.	If the circles x2+y2=a and x2+y2−6x−8y+9=0, touch externally, then a=
Answer» If the circles $x^{2} + y^{2} = a$ and $x^{2} + y^{2} - 6 x - 8 y + 9 = 0$ , touch externally, then a=

4061.

If the circles x2+y2=a and x2+y2−6x−8y+9=0, touch externally, then a=

Answer»

If the circles $x^{2} + y^{2} = a$ and $x^{2} + y^{2} - 6 x - 8 y + 9 = 0$ , touch externally, then a=

Discussion

4062.	The domain of f(x)=√1−√x2−14x+49 is
Answer» The domain of $f (x) = \sqrt{1 - \sqrt{x^{2} - 14 x + 49}}$ is

Discussion

4063.	∫(xln2+1)x(1+x⋅2x)2 dx is equal to(where C is a constant of integration)
Answer» $\int \frac{(x ln 2 + 1)}{x (1 + x \cdot 2^{x})^{2}} d x$ is equal to (where $C$ is a constant of integration)

Discussion

4064.	If a→ and b→ are unit vectors, then find the angle between a→ and b→, given that 3a→-b→ is a unit vector. [CBSE 2014]
Answer» If $\vec{a}$ and $\vec{b}$ are unit vectors, then find the angle between $\vec{a}$ and $\vec{b}$ , given that $(\sqrt{3} \vec{a} - \vec{b})$ is a unit vector. [CBSE 2014]

Discussion

4065.	If a convex polygon has 35 diagonals, then the number of triangles in which exactly one side is common with that of polygon is
Answer» If a convex polygon has $35$ diagonals, then the number of triangles in which exactly one side is common with that of polygon is

Discussion

4066.	Let EC denote the complement of an event E. let E, F, G be pair wise independent events with P(G) > 0 and P(E∩F∩G)=0. Then P(EC∩FC)G) equals
Answer» Let $E^{C}$ denote the complement of an event E. let E, F, G be pair wise independent events with P(G) > 0 and $P (E \cap F \cap G) = 0$ . Then $P (\frac{E^{C} \cap F^{C})}{G})$ equals

Discussion

4067.	AB is a chord of length 16cm of circle of radius 10cm the tangents at A and B intersect at point P. Find the length of PA.
Answer» AB is a chord of length 16cm of circle of radius 10cm the tangents at A and B intersect at point P. Find the length of PA.

Discussion

4068.	Suppose a,b,c are such that the curve y=ax2+bx+c has tangent y=3x−3 at (1,0) and has tangent y=x+1 at (3,4), then the value of (2a−b−4c) equals to
Answer» Suppose $a, b, c$ are such that the curve $y = a x^{2} + b x + c$ has tangent $y = 3 x - 3$ at $(1, 0)$ and has tangent $y = x + 1$ at $(3, 4)$ , then the value of $(2 a - b - 4 c)$ equals to

Discussion

4069.	Find the principal values of each of the following:(i) tan-113(ii) tan-1-13(iii) tan-1cosπ2(iv) tan-12cos2π3
Answer» Find the principal values of each of the following: (i) $\tan^{- 1} (\frac{1}{\sqrt{3}})$ (ii) $\tan^{- 1} (- \frac{1}{\sqrt{3}})$ (iii) $\tan^{- 1} (\cos \frac{π}{2})$ (iv) $\tan^{- 1} (2 \cos \frac{2 π}{3})$

Discussion

4070.	The minimum value of f(x)=sin4x+cos4x,0≤x≤π2 is
Answer» The minimum value of $f (x) = s i n^{4} x + c o s^{4} x, 0 \leq x \leq \frac{π}{2}$ is

Discussion

4071.	The negative of a matrix is obtained by multiplying it by ______________.
Answer» The negative of a matrix is obtained by multiplying it by ______________.

Discussion

4072.	How to convert area into vector form
Answer» How to convert area into vector form

Discussion

4073.	If x = 2 + 3, find x-1x.
Answer» If x = 2 + $\sqrt{3}$ , find $x - \frac{1}{x}$ .

Discussion

4074.	For the given graph of the quadratic expression y=f(x),
Answer» For the given graph of the quadratic expression $y = f (x),$

Discussion

4075.	Find the values of x for which y=[x(x−2)]2 is an increasing function.
Answer» Find the values of $x$ for which $y = [x (x - 2)]^{2}$ is an increasing function.

Discussion

4076.	Let y=f(x), f:R→R be an odd differentiable function such that f′′′(x)>0 and g(α,β)=sin8α+cos8β+2−4sin2αcos2β. If f′′(g(α,β))=0, then sin2α+sin2β is equal to
Answer» Let $y = f (x)$ , $f : R \to R$ be an odd differentiable function such that $f^{'''} (x) > 0$ and $g (α, β) = s i n^{8} α + c o s^{8} β + 2 - 4 s i n^{2} α c o s^{2} β$ . If $f^{''} (g (α, β)) = 0$ , then $s i n^{2} α + s i n^{2} β$ is equal to

Discussion

4077.	limx→0xasinb xsin(xc),a,b,b ∈ R ~ {0} exists and has non-zero value, then
Answer» $lim x \to 0 \frac{x^{a} s i n^{b} x}{s i n (x^{c})}$ ,a,b,b $\in$ R ~ {0} exists and has non-zero value, then

Discussion

4078.	∫((lnx)−11+(lnx)2)2dx is equal to(where c is constant of integration)
Answer» $\int {(\frac{(ln x) - 1}{1 + (ln x)^{2}})}^{2} d x$ is equal to (where $c$ is constant of integration)

Discussion

4079.	For two data sets X and Y, each of size 5, the means are given to be 2 and 4 and the corresponding variances are 4 and 5, respectively. The variance of the combined data set is
Answer» For two data sets $X$ and $Y$ , each of size $5$ , the means are given to be $2$ and $4$ and the corresponding variances are $4$ and $5$ , respectively. The variance of the combined data set is

Discussion

4080.	If the variance of the data 2, 4, 5, 6, 8, 17 is 23.33, then the variance of 4, 8, 10, 12, 13, 34 is _____________________.
Answer» If the variance of the data 2, 4, 5, 6, 8, 17 is 23.33, then the variance of 4, 8, 10, 12, 13, 34 is _____________________.

Discussion

4081.	The value of positive integer n for which the quadratic equation, n∑k=1(x+k−1)(x+k)=10n has solutions α and α+1 for some α∈R, is
Answer» The value of positive integer $n$ for which the quadratic equation, $n \sum k = 1 (x + k - 1) (x + k) = 10 n$ has solutions $α$ and $α + 1$ for some $α \in R$ , is

Discussion

4082.	If f(x)=x3−1x3, show that f(x)+f(1x=0).
Answer» If $f (x) = x^{3} - \frac{1}{x^{3}}$ , show that $f (x) + f (\frac{1}{x} = 0)$ .

Discussion

4083.	There are exactly two distinct linear functions which map [-1,1] onto [0,3].Then the point of intersection of the two functions
Answer» There are exactly two distinct linear functions which map [-1,1] onto [0,3].Then the point of intersection of the two functions

Discussion

4084.	In a ΔABC, ∑(b+c)tanA2(tanB−C2)=
Answer» $In a Δ A B C, \sum (b + c) tan \frac{A}{2} (tan \frac{B - C}{2}) =$

Discussion

4085.	∫√tanx dx=
Answer» $\int \sqrt{t a n x} d x =$

Discussion

4086.	Evaluate the given limit :limx→πsin(π−x)π(π−x)
Answer» Evaluate the given limit : $lim x \to π \frac{sin (π - x)}{π (π - x)}$

Discussion

4087.	Question 1(iv)Check whether the following are quadratic equations:(iv)(x−3)(2x+1)=x(x+5)
Answer» Question 1(iv) Check whether the following are quadratic equations: $(i v) (x - 3) (2 x + 1) = x (x + 5)$

Discussion

4088.	Function f(x)=\|x\|−\|x−1\| is monotonically increasing when
Answer» Function $f (x) = \| x \| - \| x - 1 \|$ is monotonically increasing when

Discussion

4089.	Sin(17π/3) is equal to
Answer» Sin(17π/3) is equal to

Discussion

4090.	28. Find the equation of the circle which has its centre at the point (3,4) and touches the straight line 5x + 12y - 1= 0 Show the equation .
Answer» 28. Find the equation of the circle which has its centre at the point (3,4) and touches the straight line 5x + 12y - 1= 0 Show the equation .

Discussion

4091.	(i) Evaluate limx→π6√3 sin x−cos xx−π6. (ii) If f(x)=1+x+x22+...+x100100, then find the value of f'(1).
Answer» (i) Evaluate ${lim}_{x \to \frac{π}{6}} \frac{\sqrt{3} s i n x - c o s x}{x - \frac{π}{6}}$ . (ii) If $f (x) = 1 + x + \frac{x^{2}}{2} + . . . + \frac{x^{100}}{100},$ then find the value of f'(1).

Discussion

4092.	Evaluate ∫2x+4x2+2xdx(where C is constant of integration)
Answer» Evaluate $\int \frac{2 x + 4}{x^{2} + 2 x} d x$ (where $C$ is constant of integration)

Discussion

4093.	Suppose α,β,γ and δ are the interior angles of regular pentagon, hexagon, decagon and dodecagon respectively, then the absolute value of the product cosα.secβ.cosγ.cosecδ is:
Answer» Suppose $α, β, γ$ and $δ$ are the interior angles of regular pentagon, hexagon, decagon and dodecagon respectively, then the absolute value of the product $cos α . sec β . cos γ . cosec δ$ is:

Discussion

4094.	Suppose a population A has 100 observations 101, 102, . . . . 200 and another population B has 100 observations 151, 152, . . . . 250. If VA and VB represent the variances of the two populations, respectively then VAVB is
Answer» Suppose a population A has 100 observations 101, 102, . . . . 200 and another population B has 100 observations 151, 152, . . . . 250. If $V_{A}$ and $V_{B}$ represent the variances of the two populations, respectively then $\frac{V_{A}}{V_{B}}$ is

Discussion

4095.	436,382,337,301 ?
Answer» 436,382,337,301 ?

Discussion

4096.	if x is real, then the expression (x^2+34x-71)/(x^2+2x-7) (1) lies between 4 and 7 (2) lies between 5 and 9 (3) Has no value between 4 and 7 (4) Has no value between 5 and 9
Answer» if x is real, then the expression (x^2+34x-71)/(x^2+2x-7) (1) lies between 4 and 7 (2) lies between 5 and 9 (3) Has no value between 4 and 7 (4) Has no value between 5 and 9

Discussion

4097.	If x and yare connected parametrically by the equation, without eliminating theparameter, find.
Answer» If x and y are connected parametrically by the equation, without eliminating the parameter, find.

4097.

If x and yare connected parametrically by the equation, without eliminating theparameter, find.

Answer»

If x and y
are connected parametrically by the equation, without eliminating the
parameter, find.

Discussion

4098.	If the sum to n terms of the series 312×22+522×32+732×42+…… is 0.99, then the value of n is
Answer» If the sum to $n$ terms of the series $\frac{3}{1^{2} \times 2^{2}} + \frac{5}{2^{2} \times 3^{2}} + \frac{7}{3^{2} \times 4^{2}} + \dots \dots$ is $0.99$ , then the value of $n$ is

Discussion

4099.	Show that of all the rectangles inscribed in a given fixed circle, the squre has the maximum area.
Answer» Show that of all the rectangles inscribed in a given fixed circle, the squre has the maximum area.

Discussion

4100.	Number of solutions of the equation\|x2−2\|x\|\|=2x is
Answer» Number of solutions of the equation $\| x^{2} - 2 \| x \| \| = 2^{x}$ is

Discussion

Explore topic-wise InterviewSolutions in .

Find the total number of real solutions to the equation 5^{z^2+12^{z^2=13^{z^2

The locus of a point which is equidistant from the points (1,2,3) and (3,2,−1) is

11. The distance of the point (2,3) from the line x-2y + 5=0 measured in a direction parallel to the line x-3y=0 is

If ∫11−cosx−sinxdx=ln∣∣∣tan[f(x)]−1tan[g(x)]∣∣∣+C, then the value of f(1)+g(1)−1=(where C is integration constant)

If f(x)=cos[π2]x+cos[−π2]x, where [x] denotes the greatest integer less than or equal to x, then write the value of f(π).

12. (ax +b)"

Checkwhether the relation R in R defined as R = {(a, b):a ≤ b3}is reflexive, symmetric or transitive.

Solve for }x,9^{x+2}-6\cdot3^{x+1}+1=0

If A and B are two events associated with a random experiment such that P(A)=0.5,P(B)=0.3 and P(A∩B)=0.3, find P(A∪B).

If in the expansion of (1+x)n, a, b, c are three consecutive coefficients, then n =

If the circles x2+y2=a and x2+y2−6x−8y+9=0, touch externally, then a=

The domain of f(x)=√1−√x2−14x+49 is

∫(xln2+1)x(1+x⋅2x)2 dx is equal to(where C is a constant of integration)

If a→ and b→ are unit vectors, then find the angle between a→ and b→, given that 3a→-b→ is a unit vector. [CBSE 2014]

If a convex polygon has 35 diagonals, then the number of triangles in which exactly one side is common with that of polygon is

Let EC denote the complement of an event E. let E, F, G be pair wise independent events with P(G) &gt; 0 and P(E∩F∩G)=0. Then P(EC∩FC)G) equals

AB is a chord of length 16cm of circle of radius 10cm the tangents at A and B intersect at point P. Find the length of PA.

Suppose a,b,c are such that the curve y=ax2+bx+c has tangent y=3x−3 at (1,0) and has tangent y=x+1 at (3,4), then the value of (2a−b−4c) equals to

Find the principal values of each of the following:(i) tan-113(ii) tan-1-13(iii) tan-1cosπ2(iv) tan-12cos2π3

The minimum value of f(x)=sin4x+cos4x,0≤x≤π2 is

The negative of a matrix is obtained by multiplying it by ______________.

How to convert area into vector form

If x = 2 + 3, find x-1x.

For the given graph of the quadratic expression y=f(x),

Find the values of x for which y=[x(x−2)]2 is an increasing function.

Let y=f(x), f:R→R be an odd differentiable function such that f′′′(x)&gt;0 and g(α,β)=sin8α+cos8β+2−4sin2αcos2β. If f′′(g(α,β))=0, then sin2α+sin2β is equal to

limx→0xasinb xsin(xc),a,b,b ∈ R ~ {0} exists and has non-zero value, then

∫((lnx)−11+(lnx)2)2dx is equal to(where c is constant of integration)

For two data sets X and Y, each of size 5, the means are given to be 2 and 4 and the corresponding variances are 4 and 5, respectively. The variance of the combined data set is

If the variance of the data 2, 4, 5, 6, 8, 17 is 23.33, then the variance of 4, 8, 10, 12, 13, 34 is _____________________.

The value of positive integer n for which the quadratic equation, n∑k=1(x+k−1)(x+k)=10n has solutions α and α+1 for some α∈R, is

If f(x)=x3−1x3, show that f(x)+f(1x=0).

There are exactly two distinct linear functions which map [-1,1] onto [0,3].Then the point of intersection of the two functions

In a ΔABC, ∑(b+c)tanA2(tanB−C2)=

∫√tanx dx=

Evaluate the given limit :limx→πsin(π−x)π(π−x)

Question 1(iv)Check whether the following are quadratic equations:(iv)(x−3)(2x+1)=x(x+5)

Function f(x)=|x|−|x−1| is monotonically increasing when

Sin(17π/3) is equal to

28. Find the equation of the circle which has its centre at the point (3,4) and touches the straight line 5x + 12y - 1= 0 Show the equation .

(i) Evaluate limx→π6√3 sin x−cos xx−π6. (ii) If f(x)=1+x+x22+...+x100100, then find the value of f'(1).

Evaluate ∫2x+4x2+2xdx(where C is constant of integration)

Suppose α,β,γ and δ are the interior angles of regular pentagon, hexagon, decagon and dodecagon respectively, then the absolute value of the product cosα.secβ.cosγ.cosecδ is:

Suppose a population A has 100 observations 101, 102, . . . . 200 and another population B has 100 observations 151, 152, . . . . 250. If VA and VB represent the variances of the two populations, respectively then VAVB is

436,382,337,301 ?

if x is real, then the expression (x^2+34x-71)/(x^2+2x-7) (1) lies between 4 and 7 (2) lies between 5 and 9 (3) Has no value between 4 and 7 (4) Has no value between 5 and 9

If x and yare connected parametrically by the equation, without eliminating theparameter, find.

If the sum to n terms of the series 312×22+522×32+732×42+…… is 0.99, then the value of n is

Show that of all the rectangles inscribed in a given fixed circle, the squre has the maximum area.

Number of solutions of the equation|x2−2|x||=2x is

Let EC denote the complement of an event E. let E, F, G be pair wise independent events with P(G) > 0 and P(E∩F∩G)=0. Then P(EC∩FC)G) equals

Let y=f(x), f:R→R be an odd differentiable function such that f′′′(x)>0 and g(α,β)=sin8α+cos8β+2−4sin2αcos2β. If f′′(g(α,β))=0, then sin2α+sin2β is equal to