43030 + Interview Questions in Mathematics Page 25 InterviewSolution

1201.	The range of parameter a for which the variable line y=2x+a lies between the circles x2+y2−2x−2y+1=0 and x2+y2−16x−2y+61=0 without intersecting or touching either circle is
Answer» The range of parameter $a$ for which the variable line $y = 2 x + a$ lies between the circles $x^{2} + y^{2} - 2 x - 2 y + 1 = 0$ and $x^{2} + y^{2} - 16 x - 2 y + 61 = 0$ without intersecting or touching either circle is

Discussion

1202.	find the general solution of tan x+tan 2x+tan 3x=tan xtan2x tan3x
Answer» find the general solution of tan x+tan 2x+tan 3x=tan xtan2x tan3x

Discussion

1203.	Find the equation of a curve passing through the point (0,-2) given that at any point (x, y)on the curve the product of the slope of its tangent and y co-ordinate of the point is equal to the x co-ordinate of the point.
Answer» Find the equation of a curve passing through the point (0,-2) given that at any point (x, y)on the curve the product of the slope of its tangent and y co-ordinate of the point is equal to the x co-ordinate of the point.

Discussion

1204.	The number of pairs (a,b) of real numbers, such that whenever α is a root of the equation x2+ax+b=0, α2−2 is also a root of this equation, is
Answer» The number of pairs $(a, b)$ of real numbers, such that whenever $α$ is a root of the equation $x^{2} + a x + b = 0, α^{2} - 2$ is also a root of this equation, is

Discussion

1205.	Interval of r for which x2 + y2 = r2 and x2 +y2 -6x-8y +9 0 cut at two distinct points is
Answer» Interval of r for which x2 + y2 = r2 and x2 +y2 -6x-8y +9 0 cut at two distinct points is

Discussion

1206.	The solution of the differential equation dydx−2y tan 2x=e2xsec 2x is
Answer» The solution of the differential equation $\frac{d y}{d x} - 2 y t a n 2 x = e^{2 x} s e c 2 x$ is

Discussion

1207.	If f:(12,∞)→R defined as f(x)=log5(2x−1), then f−1(x)=
Answer» If $f : (\frac{1}{2}, \infty) \to R$ defined as $f (x) = {log}_{5} (2 x - 1),$ then $f^{- 1} (x) =$

Discussion

1208.	The line 2x+3y=6 cuts x-axis at A and y-axis at B, the line Kx+8y =11 cuts x -axis at A' and y -axis at B´, then points A, B, A´, B´, will be con cyclic for
Answer» The line 2x+3y=6 cuts x-axis at A and y-axis at B, the line Kx+8y =11 cuts x -axis at A' and y -axis at B´, then points A, B, A´, B´, will be con cyclic for

Discussion

1209.	Let d1,d2,d3 be three mutually exclusive diseases. Let S be the set of observable symptoms of these diseases. A doctor has the following information from a random sample of 5000 patients: 1800 had disease d1, 2100 has disease d2, and others had disease d3. 1500 patients with disease d1, 1200 patients with disease d2, and 900 patients with disease d3 showed the symptom. Which of the diseases is the patient most likely to have?
Answer» Let $d_{1}, d_{2}, d_{3}$ be three mutually exclusive diseases. Let S be the set of observable symptoms of these diseases. A doctor has the following information from a random sample of 5000 patients: 1800 had disease d_1,2100 has disease d₂, and others had disease d₃. 1500 patients with disease d₁_,1200 patients with disease d₂, and 900 patients with disease d₃ showed the symptom. Which of the diseases is the patient most likely to have?

Discussion

1210.	The value of cot π4+x cot π4-x is ___________.
Answer» The value of $\cot (\frac{π}{4} + x) \cot (\frac{π}{4} - x)$ is ___________.

Discussion

1211.	Find the vector andCartesian equation of the planes (a) that passes throughthe point (1, 0, −2) and the normal to the plane is .(b) that passes through the point (1, 4, 6) and the normal vector tothe plane is .
Answer» Find the vector and Cartesian equation of the planes (a) that passes through the point (1, 0, −2) and the normal to the plane is . (b) that passes through the point (1, 4, 6) and the normal vector to the plane is .

1211.

Find the vector andCartesian equation of the planes (a) that passes throughthe point (1, 0, −2) and the normal to the plane is .(b) that passes through the point (1, 4, 6) and the normal vector tothe plane is .

Answer»

Find the vector and
Cartesian equation of the planes

(a) that passes through
the point (1, 0, −2) and the normal to the plane is
.

(b) that passes through the point (1, 4, 6) and the normal vector to
the plane is
.

Discussion

1212.	Let S be the set of all triangles in the xy-plane, each having one vertex at the origin and the other two vertices lie on coordinate axes with integral coordinates. If each triangle in S has area 50 sq. units, then the number of elements in the set S is :
Answer» Let $S$ be the set of all triangles in the $x y$ -plane, each having one vertex at the origin and the other two vertices lie on coordinate axes with integral coordinates. If each triangle in $S$ has area $50$ sq. units, then the number of elements in the set $S$ is :

Discussion

1213.	The points on the parabola y2=36x whose ordinate is three times the absicca are
Answer» The points on the parabola $y^{2} = 36 x$ whose ordinate is three times the absicca are

Discussion

1214.	Find the second order derivative of the function y=x.cos x
Answer» Find the second order derivative of the function $y = x . cos x$

Discussion

1215.	A school awarded 30 medals in tennis, 14 in carrom and 25 in badminton. If these medals were bagged by a total of 50 students and only 5 students got medals in all the three sports, then the number of medals received by students in exactly two of the three games is
Answer» A school awarded $30$ medals in tennis, $14$ in carrom and $25$ in badminton. If these medals were bagged by a total of $50$ students and only $5$ students got medals in all the three sports, then the number of medals received by students in exactly two of the three games is

Discussion

1216.	If A and B are symmetric matrices of the same order, then(i) AB – BA is a _________.(ii) BA – 2BA is a _________.
Answer» If A and B are symmetric matrices of the same order, then (i) AB – BA is a _________. (ii) BA – 2BA is a _________.

Discussion

1217.	The tangent at a point whose eccentric angle is 60∘ on the ellipse x2a2+y2b2=1 (a>b), meets the auxiliary circle at L and M. If LM subtends a right angle at the centre, then eccentricity of the ellipse is
Answer» The tangent at a point whose eccentric angle is $60^{\circ}$ on the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b),$ meets the auxiliary circle at $L$ and $M$ . If $L M$ subtends a right angle at the centre, then eccentricity of the ellipse is

Discussion

1218.	The value(s) of m such that the roots of the quadratic equation (m+1)x2+(m+1)x−m+1=0 are equal is
Answer» The value(s) of $m$ such that the roots of the quadratic equation $(m + 1) x^{2} + (m + 1) x - m + 1 = 0$ are equal is

Discussion

1219.	If α,β are roots of x2−5x−8=0 and tn=αn−βn, then the value of t6−8t45t5 is
Answer» If $α, β$ are roots of $x^{2} - 5 x - 8 = 0$ and $t_{n} = α^{n} - β^{n}$ , then the value of $\frac{t_{6} - 8 t_{4}}{5 t_{5}}$ is

Discussion

1220.	Two dice are thrown. The events A, B and C are as follows: A: getting an even number on the first die. B: getting an odd number on the first die. C: getting the sum of the numbers on the dice ≤ 5 State true or false: (give reason for your answer) (i) A and B are mutually exclusive (ii) A and B are mutually exclusive and exhaustive (iii) (iv) A and C are mutually exclusive (v) A and are mutually exclusive (vi) are mutually exclusive and exhaustive.
Answer» Two dice are thrown. The events A, B and C are as follows: A: getting an even number on the first die. B: getting an odd number on the first die. C: getting the sum of the numbers on the dice ≤ 5 State true or false: (give reason for your answer) (i) A and B are mutually exclusive (ii) A and B are mutually exclusive and exhaustive (iii) (iv) A and C are mutually exclusive (v) A and are mutually exclusive (vi) are mutually exclusive and exhaustive.

Discussion

1221.	Findthe values of k sothat the function fis continuous at the indicated point.
Answer» Find the values of k so that the function f is continuous at the indicated point.

1221.

Findthe values of k sothat the function fis continuous at the indicated point.

Answer»

Find
the values of k so
that the function f
is continuous at the indicated point.

Discussion

1222.	The value of limx→0+cos−1(x−[x]2)⋅sin−1(x−[x]2)x−x3, where [x] denotes the greatest integer ≤x is:
Answer» The value of $lim x \to 0^{+} \frac{{cos}^{- 1} (x - [x]^{2}) \cdot {sin}^{- 1} (x - [x]^{2})}{x - x^{3}},$ where $[x]$ denotes the greatest integer $\leq x$ is:

Discussion

1223.	The function g(x) = \|x – 1\| + \|x + 1\| is not differentiable at x = ____________.
Answer» The function g(x) = \|x – 1\| + \|x + 1\| is not differentiable at x = ____________.

Discussion

1224.	Let L be a normal to the parabola y2=4x. If L passes through the point (9,6), then L is given by
Answer» Let $L$ be a normal to the parabola $y^{2} = 4 x$ . If $L$ passes through the point $(9, 6)$ , then $L$ is given by

Discussion

1225.	Examinethe following functions for continuity.(a) (b) (c) (d)
Answer» Examine the following functions for continuity. (a) (b) (c) (d)

1225.

Examinethe following functions for continuity.(a) (b) (c) (d)

Answer»

Examine
the following functions for continuity.

(a)

(b)

(c) (d)

Discussion

1226.	sinAcosC+cosasinC=? , if tanA=3
Answer» sinAcosC+cosasinC=? , if tanA=3

Discussion

1227.	Find the coefficient of:(i) x10 in the expansion of 2x2-1x20(ii) x7 in the expansion of x-1x240(iii) x-15 in the expansion of 3x2-a3x310(iv) x9 in the expansion of x2-13x9(v) xm in the expansion of x+1xn(vi) x in the expansion of 1-2x3+3x5 1+1x8.(vii) a5b7 in the expansion of a-2b12.(viii) x in the expansion of 1-3x+7x2 1-x16.
Answer» Find the coefficient of: (i) x¹⁰ in the expansion of ${(2 x^{2} - \frac{1}{x})}^{20}$ (ii) x⁷ in the expansion of ${(x - \frac{1}{x^{2}})}^{40}$ (iii) $x^{- 15}$ in the expansion of ${(3 x^{2} - \frac{a}{3 x^{3}})}^{10}$ (iv) $x^{9}$ in the expansion of ${(x^{2} - \frac{1}{3 x})}^{9}$ (v) $x^{m}$ in the expansion of ${(x + \frac{1}{x})}^{n}$ (vi) x in the expansion of $(1 - 2 x^{3} + 3 x^{5}) {(1 + \frac{1}{x})}^{8}$ . (vii) $a^{5} b^{7}$ in the expansion of ${(a - 2 b)}^{12}$ . (viii) x in the expansion of $(1 - 3 x + 7 x^{2}) {(1 - x)}^{16}$ .

Discussion

1228.	If either vector , then . But the converse need not be true. Justify your answer with an example.
Answer» If either vector , then . But the converse need not be true. Justify your answer with an example.

Discussion

1229.	A fair die is tossed repeatedly until a six is obtained. Let X denote the number of tosses required.
Answer» A fair die is tossed repeatedly until a six is obtained. Let X denote the number of tosses required.

Discussion

1230.	Explanation of ECG.
Answer» Explanation of ECG.

Discussion

1231.	whats the maximum value of 1/sec theta , 0
Answer» whats the maximum value of 1/sec theta , 0

Discussion

1232.	How to draw graph of f(x) = [2 sin X] ?
Answer» How to draw graph of f(x) = [2 sin X] ?

Discussion

1233.	prove that sin(x+y)sin(x−y)=tanx+tanytanx−tany
Answer» prove that $\frac{sin (x + y)}{sin (x - y)} = \frac{tan x + tan y}{tan x - tan y}$

Discussion

1234.	Find A2−5A+6I IfA=⎡⎢⎣2012131−10⎤⎥⎦
Answer» Find $A^{2} - 5 A + 6 I I f A = ⎡ ⎢ ⎣ \begin{matrix} 2 & 0 & 1 2 & 1 & 3 1 & - 1 & 0 \end{matrix} ⎤ ⎥ ⎦$

Discussion

1235.	If ∫dx(x2+5x+6)√x+1=Atan−1(√x+1)+Btan−1(√x+1−B)+C, then which of the following is/are true?(where A,B are fixed constants and C is integration constant).
Answer» If $\int \frac{d x}{(x^{2} + 5 x + 6) \sqrt{x + 1}} = A {tan}^{- 1} (\sqrt{x + 1}) + B {tan}^{- 1} (\frac{\sqrt{x + 1}}{- B}) + C,$ then which of the following is/are true? $($ where $A, B$ are fixed constants and $C$ is integration constant $) .$

Discussion

1236.	34. let R be the relation on Z defined by R = {(a,b):a,b belongs to Z , a-b is an integer } . find the domain and range of R
Answer» 34. let R be the relation on Z defined by R = {(a,b):a,b belongs to Z , a-b is an integer } . find the domain and range of R

Discussion

1237.	If the length of the perpendicular from the point (β,0,β)(β≠0) to the line, x1=y−10=z+1−1 is √32, then β is equal to :
Answer» If the length of the perpendicular from the point $(β, 0, β) (β \neq 0)$ to the line, $\frac{x}{1} = \frac{y - 1}{0} = \frac{z + 1}{- 1}$ is $\sqrt{\frac{3}{2}}$ , then $β$ is equal to :

Discussion

1238.	Solve for x:(x−2)3(x−3)(x−5)2<0
Answer» $Solve for x : (x - 2)^{3} (x - 3) (x - 5)^{2} < 0$

Discussion

1239.	If a, ß are roots of the equation x² + x + 2 = 0 and gamma , delta are the roots of the equation x²-x+7=0, then the equation whose roots are alpha gaama + beta delta and having unity as leading coefficient will have constant term k, then value of (50 + k) is
Answer» If a, ß are roots of the equation x² + x + 2 = 0 and gamma , delta are the roots of the equation x²-x+7=0, then the equation whose roots are alpha gaama + beta delta and having unity as leading coefficient will have constant term k, then value of (50 + k) is

Discussion

1240.	The value of \|c\| for which the set {(x,y):x2+y2+2x≤1}∪{(x,y):x−y+c≥0} contains only one point is equal to ___
Answer» The value of \|c\| for which the set ${(x, y) : x^{2} + y^{2} + 2 x \leq 1} \cup {(x, y) : x - y + c \geq 0}$ contains only one point is equal to ___

Discussion

1241.	The value of tan xtan 3x whenever defined never lie between
Answer» The value of $\frac{t a n x}{t a n 3 x}$ whenever defined never lie between

Discussion

1242.	If the lengths of transverse and conjugate axis of the hypberbola are 4,2 then the distance Between the foci is
Answer» If the lengths of transverse and conjugate axis of the hypberbola are 4,2 then the distance Between the foci is

Discussion

1243.	A normal to the hyperbola, 4x2−9y2=36 meets the co-ordinate axes x and y at A and B, respectively. If the parallelogram OABP (O being the origin) is formed, then the locus of P is :
Answer» A normal to the hyperbola, $4 x^{2} - 9 y^{2} = 36$ meets the co-ordinate axes $x$ and $y$ at $A$ and $B$ , respectively. If the parallelogram $O A B P$ ( $O$ being the origin) is formed, then the locus of $P$ is :

Discussion

1244.	Calculate the mean, median and standard deviation of the following distribution : textClass−interval:31−3536−4041−4546−5051−5556−6061−6566−70Frequency:2381216523
Answer» Calculate the mean, median and standard deviation of the following distribution : $\begin{matrix} t e x t C l a s s - i n t e r v a l : & 31 - 35 & 36 - 40 & 41 - 45 & 46 - 50 & 51 - 55 & 56 - 60 & 61 - 65 & 66 - 70 Frequency : & 2 & 3 & 8 & 12 & 16 & 5 & 2 & 3 \end{matrix}$

1244.

Calculate the mean, median and standard deviation of the following distribution : textClass−interval:31−3536−4041−4546−5051−5556−6061−6566−70Frequency:2381216523

Answer»

Calculate the mean, median and standard deviation of the following distribution :

$\begin{matrix} t e x t C l a s s - i n t e r v a l : & 31 - 35 & 36 - 40 & 41 - 45 & 46 - 50 & 51 - 55 & 56 - 60 & 61 - 65 & 66 - 70 Frequency : & 2 & 3 & 8 & 12 & 16 & 5 & 2 & 3 \end{matrix}$

Discussion

1245.	Polynomials in Real LifePolynomials are everywhere. It is found in a roller coaster of an amusement park, the slope of a hill, the curve of a bridge or the continuity of a mountain range. They play a key role in the study of algebra, in analysis and on the whole many mathematical problems involving them.Based on the given information, answer the following question:Which of the following polynomials has exactly one zero?
Answer» Polynomials in Real Life Polynomials are everywhere. It is found in a roller coaster of an amusement park, the slope of a hill, the curve of a bridge or the continuity of a mountain range. They play a key role in the study of algebra, in analysis and on the whole many mathematical problems involving them. Based on the given information, answer the following question: Which of the following polynomials has exactly one zero?

1245.

Polynomials in Real LifePolynomials are everywhere. It is found in a roller coaster of an amusement park, the slope of a hill, the curve of a bridge or the continuity of a mountain range. They play a key role in the study of algebra, in analysis and on the whole many mathematical problems involving them.Based on the given information, answer the following question:Which of the following polynomials has exactly one zero?

Answer»

Polynomials in Real Life

Polynomials are everywhere. It is found in a roller coaster of an amusement park, the slope of a hill, the curve of a bridge or the continuity of a mountain range. They play a key role in the study of algebra, in analysis and on the whole many mathematical problems involving them.

Based on the given information, answer the following question:

Which of the following polynomials has exactly one zero?

Discussion

1246.	If the two curves y2=4ax and xy=c2 cut orthogonally such that c4=λa4, then the value of λ is
Answer» If the two curves $y^{2} = 4 a x$ and $x y = c^{2}$ cut orthogonally such that $c^{4} = λ a^{4},$ then the value of $λ$ is

Discussion

1247.	Prove that tan−1x+tan−12x1−x2=tan−1(3x−x31−3x2),\|x\|<1√3
Answer» Prove that ${tan}^{- 1} x + {tan}^{- 1} \frac{2 x}{1 - x^{2}} = {tan}^{- 1} (\frac{3 x - x^{3}}{1 - 3 x^{2}}), \| x \| < \frac{1}{\sqrt{3}}$

Discussion

1248.	Examine the continuity of the function .
Answer» Examine the continuity of the function .

Discussion

1249.	The latusrectum of a hyperbola subtends a right angle at its centre,then its e =
Answer» The latusrectum of a hyperbola subtends a right angle at its centre,then its e =

Discussion

1250.	Find the area bounded by the curve x 2 = 4 y and the line x = 4 y – 2
Answer» Find the area bounded by the curve x 2 = 4 y and the line x = 4 y – 2

Discussion

Explore topic-wise InterviewSolutions in .

The range of parameter a for which the variable line y=2x+a lies between the circles x2+y2−2x−2y+1=0 and x2+y2−16x−2y+61=0 without intersecting or touching either circle is

find the general solution of tan x+tan 2x+tan 3x=tan xtan2x tan3x

Find the equation of a curve passing through the point (0,-2) given that at any point (x, y)on the curve the product of the slope of its tangent and y co-ordinate of the point is equal to the x co-ordinate of the point.

The number of pairs (a,b) of real numbers, such that whenever α is a root of the equation x2+ax+b=0, α2−2 is also a root of this equation, is

Interval of r for which x2 + y2 = r2 and x2 +y2 -6x-8y +9 0 cut at two distinct points is

The solution of the differential equation dydx−2y tan 2x=e2xsec 2x is

If f:(12,∞)→R defined as f(x)=log5(2x−1), then f−1(x)=

The line 2x+3y=6 cuts x-axis at A and y-axis at B, the line Kx+8y =11 cuts x -axis at A' and y -axis at B´, then points A, B, A´, B´, will be con cyclic for

The value of cot π4+x cot π4-x is ___________.

Find the vector andCartesian equation of the planes (a) that passes throughthe point (1, 0, −2) and the normal to the plane is .(b) that passes through the point (1, 4, 6) and the normal vector tothe plane is .

Let S be the set of all triangles in the xy-plane, each having one vertex at the origin and the other two vertices lie on coordinate axes with integral coordinates. If each triangle in S has area 50 sq. units, then the number of elements in the set S is :

The points on the parabola y2=36x whose ordinate is three times the absicca are

Find the second order derivative of the function y=x.cos x

A school awarded 30 medals in tennis, 14 in carrom and 25 in badminton. If these medals were bagged by a total of 50 students and only 5 students got medals in all the three sports, then the number of medals received by students in exactly two of the three games is

If A and B are symmetric matrices of the same order, then(i) AB – BA is a _________.(ii) BA – 2BA is a _________.

The tangent at a point whose eccentric angle is 60∘ on the ellipse x2a2+y2b2=1 (a&gt;b), meets the auxiliary circle at L and M. If LM subtends a right angle at the centre, then eccentricity of the ellipse is

The value(s) of m such that the roots of the quadratic equation (m+1)x2+(m+1)x−m+1=0 are equal is

If α,β are roots of x2−5x−8=0 and tn=αn−βn, then the value of t6−8t45t5 is

Findthe values of k sothat the function fis continuous at the indicated point.

The value of limx→0+cos−1(x−[x]2)⋅sin−1(x−[x]2)x−x3, where [x] denotes the greatest integer ≤x is:

The function g(x) = |x – 1| + |x + 1| is not differentiable at x = ____________.

Let L be a normal to the parabola y2=4x. If L passes through the point (9,6), then L is given by

Examinethe following functions for continuity.(a) (b) (c) (d)

sinA*cosC+cosa*sinC=? , if tanA=3

If either vector , then . But the converse need not be true. Justify your answer with an example.

A fair die is tossed repeatedly until a six is obtained. Let X denote the number of tosses required.

Explanation of ECG.

whats the maximum value of 1/sec theta , 0

How to draw graph of f(x) = [2 sin X] ?

prove that sin(x+y)sin(x−y)=tanx+tanytanx−tany

Find A2−5A+6I IfA=⎡⎢⎣2012131−10⎤⎥⎦

If ∫dx(x2+5x+6)√x+1=Atan−1(√x+1)+Btan−1(√x+1−B)+C, then which of the following is/are true?(where A,B are fixed constants and C is integration constant).

34. let R be the relation on Z defined by R = {(a,b):a,b belongs to Z , a-b is an integer } . find the domain and range of R

If the length of the perpendicular from the point (β,0,β)(β≠0) to the line, x1=y−10=z+1−1 is √32, then β is equal to :

Solve for x:(x−2)3(x−3)(x−5)2&lt;0

If a, ß are roots of the equation x² + x + 2 = 0 and gamma , delta are the roots of the equation x²-x+7=0, then the equation whose roots are alpha gaama + beta delta and having unity as leading coefficient will have constant term k, then value of (50 + k) is

The value of |c| for which the set {(x,y):x2+y2+2x≤1}∪{(x,y):x−y+c≥0} contains only one point is equal to ___

The value of tan xtan 3x whenever defined never lie between

If the lengths of transverse and conjugate axis of the hypberbola are 4,2 then the distance Between the foci is

A normal to the hyperbola, 4x2−9y2=36 meets the co-ordinate axes x and y at A and B, respectively. If the parallelogram OABP (O being the origin) is formed, then the locus of P is :

Calculate the mean, median and standard deviation of the following distribution : textClass−interval:31−3536−4041−4546−5051−5556−6061−6566−70Frequency:2381216523

If the two curves y2=4ax and xy=c2 cut orthogonally such that c4=λa4, then the value of λ is

Prove that tan−1x+tan−12x1−x2=tan−1(3x−x31−3x2),|x|&lt;1√3

Examine the continuity of the function .

The latusrectum of a hyperbola subtends a right angle at its centre,then its e =

Find the area bounded by the curve x 2 = 4 y and the line x = 4 y – 2

If A and B are symmetric matrices of the same order, then(i) AB – BA is a _.(ii) BA – 2BA is a _.

The tangent at a point whose eccentric angle is 60∘ on the ellipse x2a2+y2b2=1 (a>b), meets the auxiliary circle at L and M. If LM subtends a right angle at the centre, then eccentricity of the ellipse is

sinAcosC+cosasinC=? , if tanA=3

Solve for x:(x−2)3(x−3)(x−5)2<0

Prove that tan−1x+tan−12x1−x2=tan−1(3x−x31−3x2),|x|<1√3