43030 + Interview Questions in Mathematics Page 146 InterviewSolution

7251.	∫π3π21-cos2xdx
Answer» $\int_{π}^{\frac{3 π}{2}} \sqrt{1 - \cos 2 x} d x$

Discussion

7252.	The minimum value of √ex2−1 is
Answer» The minimum value of $\sqrt{e^{x^{2}} - 1}$ is

Discussion

7253.	Area of the region bounded by the curves y=16−x24 and y=sec−1[−sin2x],(where [⋅] denotes greatest integer function ) is (in sq. units)
Answer» Area of the region bounded by the curves $y = \frac{16 - x^{2}}{4}$ and $y = {sec}^{- 1} [- {sin}^{2} x],$ $($ where $[\cdot]$ denotes greatest integer function $)$ is $($ in sq. units $)$

Discussion

7254.	if 1,α1,α2,α3................α7 are the roots of the equation x8 - 1 = 0.Find of the value of ∑7i=1(αi)2 __
Answer» if 1, $α_{1}$ , $α_{2}$ , $α_{3}$ ................ $α_{7}$ are the roots of the equation $x^{8}$ - 1 = 0.Find of the value of $\sum_{i = 1}^{7} (α_{i})^{2}$ __

Discussion

7255.	Find a point on the x-axis, which is equidistant from the points (5, 4) and (2, 3).
Answer» Find a point on the $x$ -axis, which is equidistant from the points (5, 4) and (2, 3).

Discussion

7256.	The value of a for which the equation 2sin2θ−√acos2θ=√2+√2−a has solution in θ is
Answer» The value of $a$ for which the equation $2 sin 2 θ - \sqrt{a} cos 2 θ = \sqrt{2} + \sqrt{2 - a}$ has solution in $θ$ is

Discussion

7257.	Ify=sin−1x1−x2, then the value of (1−x2)d2ydx2−3xdydx−y=
Answer» $If y = \frac{{sin}^{- 1} x}{1 - x^{2}}, then the value of (1 - x^{2}) \frac{d^{2} y}{{dx}^{2}} - 3 x \frac{dy}{dx} - y =$

Discussion

7258.	If f(x)=eex. Let g(x) be inverse of f(x). Then g′(x) at x=2 is
Answer» If $f (x) = e^{e^{x}}$ . Let $g (x)$ be inverse of $f (x) .$ Then $g^{'} (x)$ at $x = 2$ is

Discussion

7259.	If y=e^(msin-¹x) (where -1
Answer» If y=e^(msin-¹x) (where -1<=x<=1) then prove that (1-x²)d²y/dx²-xdy/dx=m²y

Discussion

7260.	A line x−y√3=c is drawn through the focus (F) of the parabola y2−8x−16=0. If the two intersection points of the given line and the parabola are P and Q, such that the perpendicular bisector of PQ intersects the x-axis at A, then the length of AF is
Answer» A line $x - \frac{y}{\sqrt{3}} = c$ is drawn through the focus $(F)$ of the parabola $y^{2} - 8 x - 16 = 0.$ If the two intersection points of the given line and the parabola are $P$ and $Q$ , such that the perpendicular bisector of $P Q$ intersects the $x$ -axis at $A$ , then the length of $A F$ is

Discussion

7261.	If y=tan (esinx), then dydx=
Answer» If $y = tan (e^{sin x}),$ then $\frac{d y}{d x} =$

Discussion

7262.	If (2,0) is vertex and y−axis is the directrix of a parabola. Then its focus will be
Answer» If $(2, 0)$ is vertex and $y -$ axis is the directrix of a parabola. Then its focus will be

Discussion

7263.	If A={(x,y):x2+y2≤1;x,y∈R} and B={(x,y):x2+y2≥4;x,y∈R}, then
Answer» If $A = {(x, y) : x^{2} + y^{2} \leq 1; x, y \in R}$ and $B = {(x, y) : x^{2} + y^{2} \geq 4; x, y \in R}$ , then

Discussion

7264.	Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are .
Answer» Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are .

Discussion

7265.	A fruit grower can use two types of fertilizer in his garden, brand P and brand Q. The amounts (in kg) of nitrogen, phosphoric acid, potash, and chlorine in a bag of each brand are given in the table. Tests indicate that the garden needs at least 240 kg of phosphoric acid at least 270 kg of potash and at most 310 kg of chlorine. If the grower wants to minimize the amount of nitrogen added to the garden, how many bags of each brand should be used? What is the minimum amount of nitrogen added in the garden? kg per bag Brand P Brand Q Nitrogen Phosphoric acid Potash Chlorine 3 1 3 1.5 3.5 2 1.5 2
Answer» A fruit grower can use two types of fertilizer in his garden, brand P and brand Q. The amounts (in kg) of nitrogen, phosphoric acid, potash, and chlorine in a bag of each brand are given in the table. Tests indicate that the garden needs at least 240 kg of phosphoric acid at least 270 kg of potash and at most 310 kg of chlorine. If the grower wants to minimize the amount of nitrogen added to the garden, how many bags of each brand should be used? What is the minimum amount of nitrogen added in the garden? kg per bag Brand P Brand Q Nitrogen Phosphoric acid Potash Chlorine 3 1 3 1.5 3.5 2 1.5 2

Discussion

7266.	The value of ‘c’ such that the line joining (0, 3), (5, –2) is a tangent to y=cx+1is ------
Answer» The value of ‘c’ such that the line joining (0, 3), (5, –2) is a tangent to y= $\frac{c}{x + 1}$ is ------

7266.

The value of ‘c’ such that the line joining (0, 3), (5, –2) is a tangent to y=cx+1is ------

Answer»

The value of ‘c’ such that the line joining (0, 3), (5, –2) is a tangent to

y= $\frac{c}{x + 1}$ is ------

Discussion

7267.	The angle 37π4 is equivalent to
Answer» The angle $\frac{37 π}{4}$ is equivalent to

Discussion

7268.	If three distinct numbers a,b,c are in AP and b-a,c-b,a are in G them a:b:c is equal to
Answer» If three distinct numbers a,b,c are in AP and b-a,c-b,a are in G them a:b:c is equal to

Discussion

7269.	If the roots α and β of the equation ax^2+bx+c=0 are real and of opposite sign then the roots of the equationα(x-β)^2 +β(x-α)^2 is/are(1) positive(2) Negative(3) Real and opposite (4) imaginary
Answer» If the roots α and β of the equation ax^2+bx+c=0 are real and of opposite sign then the roots of the equation α(x-β)^2 +β(x-α)^2 is/are (1) positive (2) Negative (3) Real and opposite (4) imaginary

Discussion

7270.	Show that the tangents at the end points of a diameter of a circle are parallel.
Answer» Show that the tangents at the end points of a diameter of a circle are parallel.

7270.

Show that the tangents at the end points of a diameter of a circle are parallel.

Answer»

Show that the tangents at the end points of a diameter of a circle are parallel.

Discussion

7271.	The value of α for which the plane x + αy + z = 5 cuts equal intercepts on the axes, is ____________.
Answer» The value of $α$ for which the plane x + $α y$ + z = 5 cuts equal intercepts on the axes, is ____________.

Discussion

7272.	2 If y=ax+bx+c represents a parabola with vertex A and b=2(b+2ac),Where a is not equal to 0 and a perpendicular from the vertex(A) is draw and named as AP=3 then the distance of point p from the origin is=?
Answer» 2 If y=ax+bx+c represents a parabola with vertex A and b=2(b+2ac),Where a is not equal to 0 and a perpendicular from the vertex(A) is draw and named as AP=3 then the distance of point p from the origin is=?

Discussion

7273.	For what value of 'M' will the equation X square +mx-(m square +m-32)=0 have equal roots
Answer» For what value of 'M' will the equation X square +mx-(m square +m-32)=0 have equal roots

Discussion

7274.	By the method of matrix inversion, solve the system. ⎛⎜⎝11125721−1⎤⎥⎦⎛⎜⎝x1y1x2y2x3y3⎤⎥⎦=⎛⎜⎝9252150−1⎤⎥⎦
Answer» By the method of matrix inversion, solve the system. $⎛ ⎜ ⎝ \begin{matrix} 1 & 1 & 1 2 & 5 & 7 2 & 1 & - 1 \end{matrix} ⎤ ⎥ ⎦ ⎛ ⎜ ⎝ \begin{matrix} x_{1} & y_{1} x_{2} & y_{2} x_{3} & y_{3} \end{matrix} ⎤ ⎥ ⎦ = ⎛ ⎜ ⎝ \begin{matrix} 9 & 2 52 & 15 0 & - 1 \end{matrix} ⎤ ⎥ ⎦$

Discussion

7275.	Let A=⎛⎜⎝02qrpq−rp−qr⎞⎟⎠. If AAT=I3, then \|p\| is:
Answer» Let $A = ⎛ ⎜ ⎝ \begin{matrix} 0 & 2 q & r p & q & - r p & - q & r \end{matrix} ⎞ ⎟ ⎠$ . If $A A^{T} = I_{3}$ , then $\| p \|$ is:

Discussion

7276.	The value of ∣∣∣∣y+zxxyz+xyzzx+y∣∣∣∣ is equal to
Answer» The value of $∣ ∣∣ ∣ \begin{matrix} y + z & x & x y & z + x & y z & z & x + y \end{matrix} ∣ ∣∣ ∣$ is equal to

Discussion

7277.	If PQ is a double ordinate of the hyperbola x2a2−y2b2=1 such that OPQ is an equilateral triangle, O being the centre of the hyperbola, then the eccentricity e of the hyperbola satisfies
Answer» If $P Q$ is a double ordinate of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ such that $O P Q$ is an equilateral triangle, $O$ being the centre of the hyperbola, then the eccentricity $e$ of the hyperbola satisfies

Discussion

7278.	If t1 and t2 are the roots of the equation t2+λt+1=0, where λ is a parameter, then the line joining the points (at21,2at1) and (at22,2at2) always passes through
Answer» If $t_{1}$ and $t_{2}$ are the roots of the equation $t^{2} + λ t + 1 = 0$ , where $λ$ is a parameter, then the line joining the points $(a t_{1}^{2}, 2 a t_{1})$ and $(a t_{2}^{2}, 2 a t_{2})$ always passes through

Discussion

7279.	The equation of line passing through point A(2,−1,1) and parallel to vector 2^i+3^j−^k is
Answer» The equation of line passing through point $A (2, - 1, 1)$ and parallel to vector $2^i + 3^j -^k$ is

Discussion

7280.	Reshma wishes to mix two types of food P and Q in such a way that the vitamin contents of the mixture contains at least 8 units of vitamin A and 11 units of vitamin B. Food P costs ₹60/kg and food Q costs ₹80/kg. Food P contains 3 units/kg of vitamin A and 5 units/kg of vitamin B while food Q contains 4 units/kg of vitamin A and 2 units/kg of vitamin B. Determine the minimum cost of the mixture.
Answer» Reshma wishes to mix two types of food P and Q in such a way that the vitamin contents of the mixture contains at least 8 units of vitamin A and 11 units of vitamin B. Food P costs ₹60/kg and food Q costs ₹80/kg. Food P contains 3 units/kg of vitamin A and 5 units/kg of vitamin B while food Q contains 4 units/kg of vitamin A and 2 units/kg of vitamin B. Determine the minimum cost of the mixture.

Discussion

7281.	If the circles x2+y2=9 and x2+y2−8x−6y+n2=0,n∈Z have exactly two common tangents, then the number of possible values of n is
Answer» If the circles $x^{2} + y^{2} = 9$ and $x^{2} + y^{2} - 8 x - 6 y + n^{2} = 0, n \in Z$ have exactly two common tangents, then the number of possible values of $n$ is

Discussion

7282.	If total 180n,n∈N different matrices can be formed by using all the roots of equation (x−1)2(x−2)3(x−3)4=0. Then the value of n is
Answer» If total $180 n, n \in N$ different matrices can be formed by using all the roots of equation $(x - 1)^{2} (x - 2)^{3} (x - 3)^{4} = 0.$ Then the value of $n$ is

Discussion

7283.	Consider that f: R→ R 1. Let f(x)= x^3+x^2+ax+4 be bijective, then find "a". 2. Let f(x) = ax^3 + bx^2 + cx+ d is bijective, then find the condition.
Answer» Consider that f: R→ R 1. Let f(x)= x^3+x^2+ax+4 be bijective, then find "a". 2. Let f(x) = ax^3 + bx^2 + cx+ d is bijective, then find the condition.

Discussion

7284.	If the each of algebraic expressions (lx^2) + (mx) + n , (mx^2) + (nx) + l and (nx^2) + (lx) + m are perfect squares , then (l+m)/n = _____ (a)-4 (b)6 (c)-8 (d)none of these
Answer» If the each of algebraic expressions (lx^2) + (mx) + n , (mx^2) + (nx) + l and (nx^2) + (lx) + m are perfect squares , then (l+m)/n = _____ (a)-4 (b)6 (c)-8 (d)none of these

Discussion

7285.	4 If p, q and r are real numbers, then rootsof the equation(x-p) (x-q)+ (x - q) (x-r)+ (x-p) (x-r) = 0are equal, if(1) p 1, q 1, r= 0(2) p q=r(3) p 1, q 0, r= 0(4) q=1, r- 1
Answer» 4 If p, q and r are real numbers, then rootsof the equation(x-p) (x-q)+ (x - q) (x-r)+ (x-p) (x-r) = 0are equal, if(1) p 1, q 1, r= 0(2) p q=r(3) p 1, q 0, r= 0(4) q=1, r- 1

Discussion

7286.	If ∫eaxcos(bx)dx=eaxK(acos(bx)+bsin(bx))+C, then the K here would be equal to -
Answer» If $\int e^{a x} cos (b x) d x = \frac{e^{a x}}{K} (a cos (b x) + b sin (b x)) + C$ , then the K here would be equal to -

Discussion

7287.	In the circuit current through source will be ... [Given(cos−1(0.6)=56∘)].The voltage of source is V=10+10√2sin(100πt+45∘)
Answer» In the circuit current through source will be ... $[G i v e n (c o s^{- 1} (0.6) = 56^{\circ})]$ .The voltage of source is $V = 10 + 10 \sqrt{2} s i n (100 π t + 45^{\circ})$

Discussion

7288.	if f(x)=sin(pi*x) then draw graph of the following (i)y=f(\|x\|) (ii) y=f(-\|x\|) (iii) \|y\| =f(x)
Answer» if f(x)=sin(pi*x) then draw graph of the following (i)y=f(\|x\|) (ii) y=f(-\|x\|) (iii) \|y\| =f(x)

Discussion

7289.	find the value of d/dx(x^1/2 + 1/x^1/2)
Answer» find the value of d/dx(x^1/2 + 1/x^1/2)

Discussion

7290.	an = n (n + 2)
Answer» an = n (n + 2)

Discussion

7291.	How Michaelis cons†an t is inversely proportion to turn over number?
Answer» How Michaelis cons†an t is inversely proportion to turn over number?

Discussion

7292.	23. Let S={1,2.100}. The prob of choosing an integer k,1
Answer» 23. Let S={1,2.100}. The prob of choosing an integer k,1

Discussion

7293.	Evaluate I=∫ex(1+sinx)+e−x(1−sinx)1+cosxdx
Answer» Evaluate $I = \int \frac{e^{x} (1 + sin x) + e^{- x} (1 - sin x)}{1 + cos x} d x$

Discussion

7294.	Let O=(0,0),A=(a,11) and B=(b,37) are the vertices of an equilateral triangle OAB, then a and b satisfy the relation :
Answer» Let $O = (0, 0), A = (a, 11)$ and $B = (b, 37)$ are the vertices of an equilateral triangle $O A B$ , then $a$ and $b$ satisfy the relation :

Discussion

7295.	Minimum of the real valued function f(x)=(x−1)2/3 occurs at x equal to
Answer» Minimum of the real valued function $f (x) = (x - 1)^{2 / 3}$ occurs at $x$ equal to

Discussion

7296.	The solution of the differential equation cot y dx = x dy is ________________.
Answer» The solution of the differential equation cot y dx = x dy is ________________.

Discussion

7297.	What are fundamental concept of set ?
Answer» What are fundamental concept of set ?

Discussion

7298.	If b and c are lengths of the segments of any focal chord of the parabola y2=4ax, then write the lengths of its latus-rectum.
Answer» If b and c are lengths of the segments of any focal chord of the parabola $y^{2} = 4 a x,$ then write the lengths of its latus-rectum.

Discussion

7299.	Evaluate the given limit :limx→0cosxπ−x
Answer» Evaluate the given limit : $lim x \to 0 \frac{cos x}{π - x}$

Discussion

7300.	20.Major axis on the x-axis and passes through the points (4,3) and (6,2).
Answer» 20.Major axis on the x-axis and passes through the points (4,3) and (6,2).

Discussion

Explore topic-wise InterviewSolutions in .

∫π3π21-cos2xdx

The minimum value of √ex2−1 is

Area of the region bounded by the curves y=16−x24 and y=sec−1[−sin2x],(where [⋅] denotes greatest integer function ) is (in sq. units)

if 1,α1,α2,α3................α7 are the roots of the equation x8 - 1 = 0.Find of the value of ∑7i=1(αi)2 __

Find a point on the x-axis, which is equidistant from the points (5, 4) and (2, 3).

The value of a for which the equation 2sin2θ−√acos2θ=√2+√2−a has solution in θ is

Ify=sin−1x1−x2, then the value of (1−x2)d2ydx2−3xdydx−y=

If f(x)=eex. Let g(x) be inverse of f(x). Then g′(x) at x=2 is

If y=e^(msin-¹x) (where -1

A line x−y√3=c is drawn through the focus (F) of the parabola y2−8x−16=0. If the two intersection points of the given line and the parabola are P and Q, such that the perpendicular bisector of PQ intersects the x-axis at A, then the length of AF is

If y=tan (esinx), then dydx=

If (2,0) is vertex and y−axis is the directrix of a parabola. Then its focus will be

If A={(x,y):x2+y2≤1;x,y∈R} and B={(x,y):x2+y2≥4;x,y∈R}, then

Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are .

The value of ‘c’ such that the line joining (0, 3), (5, –2) is a tangent to y=cx+1is ------

The angle 37π4 is equivalent to

If three distinct numbers a,b,c are in AP and b-a,c-b,a are in G them a:b:c is equal to

If the roots α and β of the equation ax^2+bx+c=0 are real and of opposite sign then the roots of the equationα(x-β)^2 +β(x-α)^2 is/are(1) positive(2) Negative(3) Real and opposite (4) imaginary

Show that the tangents at the end points of a diameter of a circle are parallel.

The value of α for which the plane x + αy + z = 5 cuts equal intercepts on the axes, is ____________.

2 If y=ax+bx+c represents a parabola with vertex A and b=2(b+2ac),Where a is not equal to 0 and a perpendicular from the vertex(A) is draw and named as AP=3 then the distance of point p from the origin is=?

For what value of 'M' will the equation X square +mx-(m square +m-32)=0 have equal roots

By the method of matrix inversion, solve the system. ⎛⎜⎝11125721−1⎤⎥⎦⎛⎜⎝x1y1x2y2x3y3⎤⎥⎦=⎛⎜⎝9252150−1⎤⎥⎦

Let A=⎛⎜⎝02qrpq−rp−qr⎞⎟⎠. If AAT=I3, then |p| is:

The value of ∣∣∣∣y+zxxyz+xyzzx+y∣∣∣∣ is equal to

If PQ is a double ordinate of the hyperbola x2a2−y2b2=1 such that OPQ is an equilateral triangle, O being the centre of the hyperbola, then the eccentricity e of the hyperbola satisfies

If t1 and t2 are the roots of the equation t2+λt+1=0, where λ is a parameter, then the line joining the points (at21,2at1) and (at22,2at2) always passes through

The equation of line passing through point A(2,−1,1) and parallel to vector 2^i+3^j−^k is

If the circles x2+y2=9 and x2+y2−8x−6y+n2=0,n∈Z have exactly two common tangents, then the number of possible values of n is

If total 180n,n∈N different matrices can be formed by using all the roots of equation (x−1)2(x−2)3(x−3)4=0. Then the value of n is

Consider that f: R→ R 1. Let f(x)= x^3+x^2+ax+4 be bijective, then find "a". 2. Let f(x) = ax^3 + bx^2 + cx+ d is bijective, then find the condition.

If the each of algebraic expressions (lx^2) + (mx) + n , (mx^2) + (nx) + l and (nx^2) + (lx) + m are perfect squares , then (l+m)/n = _____ (a)-4 (b)6 (c)-8 (d)none of these

4 If p, q and r are real numbers, then rootsof the equation(x-p) (x-q)+ (x - q) (x-r)+ (x-p) (x-r) = 0are equal, if(1) p 1, q 1, r= 0(2) p q=r(3) p 1, q 0, r= 0(4) q=1, r- 1

If ∫eaxcos(bx)dx=eaxK(acos(bx)+bsin(bx))+C, then the K here would be equal to -

In the circuit current through source will be ... [Given(cos−1(0.6)=56∘)].The voltage of source is V=10+10√2sin(100πt+45∘)

if f(x)=sin(pi*x) then draw graph of the following (i)y=f(|x|) (ii) y=f(-|x|) (iii) |y| =f(x)

find the value of d/dx(x^1/2 + 1/x^1/2)

an = n (n + 2)

How Michaelis cons†an t is inversely proportion to turn over number?

23. Let S={1,2.100}. The prob of choosing an integer k,1

Evaluate I=∫ex(1+sinx)+e−x(1−sinx)1+cosxdx

Let O=(0,0),A=(a,11) and B=(b,37) are the vertices of an equilateral triangle OAB, then a and b satisfy the relation :

Minimum of the real valued function f(x)=(x−1)2/3 occurs at x equal to

The solution of the differential equation cot y dx = x dy is ________________.

What are fundamental concept of set ?

If b and c are lengths of the segments of any focal chord of the parabola y2=4ax, then write the lengths of its latus-rectum.

Evaluate the given limit :limx→0cosxπ−x

20.Major axis on the x-axis and passes through the points (4,3) and (6,2).