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InterviewSolution
This section includes InterviewSolutions, each offering curated multiple-choice questions to sharpen your knowledge and support exam preparation. Choose a topic below to get started.
| 2701. |
Evaluate the definite integral in exercise overset((pi)/(4)) underset(0)int (2 sec^(2)x+x^(3)+2)dx |
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Answer» |
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| 2702. |
Let A be a square matrix of order 3xx3 then |KA| is equal to …… |
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Answer» K|A| |
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| 2703. |
Evaluate the integrals int_(1)^(2)(1/x-1/(2x^(2)))dx |
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| 2704. |
From five consonants and four vowels, how many words consist of three consonants and two vowels ? |
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Answer» SOLUTION :Words of consisting of 3 consonants and 2 vowels are to be formed from FIVE consonants and 4 vowels. `:. "The number of ways" =""^5C_3xx""^4C_2` ` Again , 5 LETTERS can be arranged among themselves in 5! Ways. `:. "The TOTAL number of ways "= ""^5C_3xx ""^4C_2xx5! =10xx6xx120 = 7200.` |
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| 2705. |
Draw the graph of f(x) = x^(2)e^(-|x|) i) Find the point of maxima/minima. ii) Find the asymptote is any. iii) Find the range of the function. iv) Find the number of roots of the equation f(x)=1 |
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Answer» Solution :we have `f(x) = x^(2)e^(-|x|)` Clearly, `f(x)=f(-x)`. So `f(x)` is an even function. Hence the graph is SYMMETRICAL about the y-axis. For `x ge0, f(x) = x^(2)e^(-x)` Now `f^(')(x) = -x^(2)e^(-x) + 2xe^(-x) =XE^(-x)(2-x)` `f^(')(x) = -x^(2)e^(-x)+2xe^(-x) = xe^(-x)(2-x)` `f^(')(x)=0 rArr x=0` or `x=2` `f(0) =0` and `underset(x to infty)"lim"x^(2)/e^(x) = underset(x to infty)"lim"(2x)/e^(x)=underset(x to infty)2/e^(x)=0` So x=2 is the pont of maxima. Hence the graph of `f(x)` is as shown in the following FIGURE.  SINCE, `f(x)` is differentiable at `x=0`,the graph touches the x-axis at x=0 The x-axis is also asymptote to the curve. Also since `f(x)` is an even function, `x=-2` is also the point of maxima. RANGE of the function is `[0,f(2)]` or `[0, 4//e^(2)]` Clearly, `f(x) ne 1`, hence `f(x)=1` has no roots. |
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| 2706. |
A pair of dice is thrown and X denotes the sum of the numbers on uppermost faces. Then the expected value of X is |
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Answer» 12 |
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| 2707. |
If sin^(4)x + cos^(4)y + 2 = 4 sinx cosy and 0 le x, y le pi/2. Then sinx + cos y is equal to: |
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Answer» Solution :`sin^(4)x + cos^(4)y +2 = 4 sinx cosy` `(sin^(2)x-1)^(2) + (cos^(2)y-1)^(2) + 2 sin^(2)x + 2 cos^(2)y - 4 sinx cosy =0` `(sin^(2)x -1)^(2) + (cos^(2)y-1)^(2)+ 2(sinx - cosy)^(2)=0` Which is TRUE if `(sin^(2)x -1)^(2) +(cos^(2)y -1)^(2) + 2(sin x- cosy)^(2)=0` Which is true is `sin^(2)x =1, cos^(2)y, sin^(2)x = cos^(2)y` `RARR sinx = cosy=1 therefore x,y in [0, pi/2]` |
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| 2708. |
Integrate inttan^-1xdx. |
| Answer» SOLUTION :`I=inttan^-1xdx=xtan^-1x-intx1/(1+x^2)dx=xtan^-1x-1/2In(1+x^2)+C` | |
| 2709. |
int (dx)/(sqrt(2n -x))= |
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Answer» `SIN^(-1)((x +a)/(a)) + C` |
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| 2710. |
If the equation x^(2)-cx+d=0has roots equal to thefourth powers of the roots of x^(2)+ax+b=0, where a^(2) gt 4b, then the roots of x^(2)-4bx +2b^(2)-c=0will be |
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Answer» both REAL |
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| 2711. |
Solve the following differential equations.dy/dx=e^y |
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Answer» Solution :`dy/dx=E^yrArre^(-y)dy=dx` Integrating we GET `inte^(-y)dy=intdx` `RARR e^(-y)/-1=x+C rArr -e^(-y)+x+C` `rArrx+e^(-y)+C=0` |
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| 2713. |
If the circle x^(2) + y^(2) + 2kx - 4y+1=0 and x^(2)+y(2)-8x-12 y + 43 = 0 " touch each other theb k =" |
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Answer» 2 |
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| 2714. |
If x,y,z are different andDelta = {:|( x,x^(2) , 1+x^(3)),( y,y^(3) ,1+y^(3)),( z,z^(3) ,1+z^(3)) |:} then value of delta? |
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| 2715. |
If real function f(x) =(x+1)^(2) and g(x) =x^(2) +1 then (fog) (-3) = .......... |
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Answer» 121 |
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| 2716. |
Write down the expansion of (a+b)^8 using Pascal's triangle. |
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Answer» Solution :The ROW N = 8 in the Pascal's TRIANGLE is 1, 8, 28 , 56, 70, 56, 28, 8, 1. therefore `(a+b)^8` = `a^8+8a^(8-1)b^-1+28a^(8-2)b^2+56a^(8-3b)^3+70a^(8-4)b^5+56a^(8-5)b^5+28a^(8-6)b^6+8a^8-7)+b^8` = `a^8+8a^7b+28a^6b^2+56a^5b^3+70a^4b^4+56a^3b^5+28a^2b^6+8ab^7+b^8` |
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| 2717. |
If a reverse mangnus force acting on a blall is 0.620 N and the mass of ball is 155gm speed of projection of the ball is 145 km//h and height of the released of the ball is 1.75m, then how short ball will fall fromthe estimated distance (assuming Bumarah releases the ball horizontally and no air drag acts on ball) (Take :g =10m//s^(2)) |
| Answer» ANSWER :A | |
| 2718. |
Which of the following must be true? (I) The area of triangle P. (II) The area of triangle Q. (III) The area of triangle R. |
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Answer» I = II = III |
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| 2719. |
Compute the magnitude of oversetrarrb=2overset^^i-7overset^^j-3overset^^k,vector |
| Answer» Solution :`|vecb|=sqrt(2^2+(-7)^2+(-3)^2)=sqrt(62)` | |
| 2720. |
The point at which the circle x ^(2) + y ^(2) + 2x + 6y + 4=0 and x ^(2) +y ^(2) + 6x + 2y + 7=0 subtend equal angles lies on the circle |
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Answer» `x ^(2) +y ^(2) + 10X - 2Y + 10=0` |
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| 2721. |
Resolve (1)/(x^(8)(x+1)) into partial fractions. |
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Answer» |
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| 2722. |
If for some c lt 0, the quadratic equation, 2cx^2-2(2x-1)x+3c^2=0 has two distinct real roots , 1/a and 1/b , and Delta=|(1+a,1,1),(1,1+b,1),(1,1,1+c)| . Then Delta is equal to : |
| Answer» ANSWER :D | |
| 2723. |
I : The function f(x) =xe^(-x) has maximum at x=e . II : The function f(x) = sin x(1 + cosx) has maximum at x=pi//3. |
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Answer» only I is TRUE |
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| 2724. |
Evaluate: int(dx)/(a+bcosx)^(2), (a gt b) |
| Answer» SOLUTION :`-(bsinx)/((a^(2)-b^(2))(a+bcosx))+(2A)/((a^(2)-b^(2))^(3//2)tan^(-1)SQRT((a-b)/(a+b))tanx/2+C` | |
| 2725. |
The solution of (dy)/(dx)+(y cos x +sin y+y)/(sin x+x cos y+x)=0 is |
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Answer» `y COS X+x cosy+xy=k` |
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| 2727. |
Differentiate1/2x^(1/2)+1/3x^(1/3) |
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Answer» SOLUTION :`y=1/2x^(1/2)+1/3x^(1/3)` `dy/dx=1/4x^(-1/2)+1/9x^(-2/3)` |
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| 2728. |
What is the condition for the plane ax + by + cz + d = 0 to be perpendicular to xy - plane ? |
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Answer» a = 0 |
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| 2729. |
If a tangent to the ellipse x^2/a^2+y^2/b^2=1(agtb) meets itsmajor axis and minor axis at M and N respectively. Then prove that a^2/(CM)^2+b^2/(CN)^2=1. Where C is the centre of the ellipse. |
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| 2730. |
A block is sliding down a ramp that drops 3 centimeters in elevation for every 5 centimeters along the length of the ramp. The top of the ramp, where the back edge of the block is initally placed, is at 60 centimeters down the ramp. What is the elevation of the ramp, in centimeters, at the point where the back of the block passes t seconds after being released? |
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Answer» `60-(3)/(5)t` |
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| 2731. |
Express with rational denominator((a+sqrt(-1))^3-(a-sqrt(-1))^3)/((a+sqrt(-1))^2-(a-sqrt(-1))^2) |
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Answer» Solution :`((a+sqrt(-1))^3-(a-sqrt(-1))^3)/((a+sqrt(-1))^2-(a-sqrt(-1))^2)` `=((a+i)^3-(a-i)^3)/((a+i)^2-(a-1)^2)` `=(2(3a^2I=i^3))/(4ai)=(2i(3a^2-1))/(4ai)=(3a^2-1)/(2A)` |
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| 2732. |
The solution of (dy)/(dx) = (3x + y + 4)^(2) is |
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Answer» `3X + y - 4 = sqrt3 tan (sqrt3 X + c)` |
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| 2733. |
Using differentials, find the approximate value of each of the following upto 3 place of decimal. (iv) (0.009)^(1//3) |
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| 2734. |
Thevalueof |z^(2)| + |z+3|^(2) + |z - i|^(2) is minimum when z equals. |
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Answer» `2-(2)/(3) i` |
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| 2735. |
Prove that the equality int_(0)^(pi//2) sin ^(m) x dx = int_(0)^(pi//2) cos ^(m)x dx and apply the obtained result in computing the following integrals :int_(0)^(pi//2) cos^(2) "x dx and " int_(0)^(pi//2) sin ^(2) x dx |
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| 2736. |
Find the equation of a plane which is at a distance 3sqrt3 units from origin and the normal to which is equally inclined to coordinate axis. |
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| 2737. |
If z_(1)= 2sqrt(2)(1+i)" and "z_(2)=1+isqrt(3), then z_(1)^(2)z_(2)^(3) is equal to |
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Answer» `128i` |
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| 2740. |
If f(x) is a polynomial function satisfying f(x)f(y)=f(x)+f(y)+f(xy)-2 for all real x and y and f(3) = 10, then (f(4))/(17) is equal to |
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Answer» for `x=1,y=3impliesf(1)=2` `THEREFORE` for `xy=1 " of"y=(1)/(x)` `=f(x)f((1)/(x))=f(x)+f((1)/(x))` `implies f(x)=1pmx^(2)` `because f(3)=10impliesf(x)=1+x^(2)` |
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| 2742. |
If vector hat(i)+hat(j)+hat(k), hat(i)-hat(j)+hat(k) and 2hat(i)+3hat(j)+lambda hat(k) are coplanar, then lambda is equal to |
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Answer» `-2` `|{:(1,1,1),(1,-1,1),(2,3,lamda):}|=0` `rArr""2(-lamda-3)-1(lamda-2)+1(3+2)=0` `rArr""-lamda-3 -lamda+2+5=0` `rArr""2lamda=4lamda=2` |
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| 2743. |
A, B, C try to hit a targetsimultaneouslybut independently. Their respectiveprobabilitiesof hittingthe target are (3)/(4),(1)/(2),(5)/(8) . The probability that target is hit by A or B but not C is |
| Answer» Answer :A | |
| 2744. |
A bag contains 5 red balls and 3 black marbles. Three balls are drawn one by one without replacement.The probability that exactly two of the three balls were red, the first ball being red, is ……… |
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Answer» `(1)/(3)` |
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| 2745. |
Find the slope of the normal to the curve x=1-a sin theta, y= b cos^(2)theta at theta=(pi)/(2). |
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| 2746. |
Find the remainder when 2^(2013) in divided by 17. |
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| 2747. |
If the foot of perpendicular from the origin to a straight line is at the point (3,-4). Then the equation of the line is |
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Answer» 3X – 4Y = 25 |
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| 2749. |
whenfourlettersare insertedin tofourcovers (onein each ) A =eventthat onlyon lettersgoesto thepropercover . B = eventthat exactlythreelettersgo to thepropercovers . C=eventthat lllettersgo topropercoversand then ...... is true |
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Answer» <P>`p TOA , q to c, r to b` |
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| 2750. |
if(3x ^ 2+x + 1 )/((x - 1 ) ^4)= (a)/((x- 1 )) +(b)/((x -1) ^ 2 )+(c )/((x-1)^3)+ (d)/((x-1)^ 4 ),then[{:(a,b),(c,d):}] |
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Answer» `[{:(3,7),(5,0):}] ` `rArr3x ^ 2+ x+1=3 ( x - 1+ 1 ) ^ 2+(x - 1 )+2` `= 3 (x - 1 ) ^ 2+6 ( x - 1 )+3 +(x - 1 )+ 2` `=3 ( x - 1 ) ^ 2 +7 ( x - 1 ) + 5 ` `therefore(3x ^ 2+x + 1 )/( (x - 1 )^4 ) = (3 (x - 1 ) ^2 +7 ( x - 1 )+ 5 )/((x - 1 ) ^ 4 ) ` `= (3 ) /((x - 1 ) ^2 ) +(7 ) /((x - 1 ) ^(3)) +(5 ) /((x -1 ) ^ 4 )+(0)/((x - 1 )) ` `therefore(a ) /((x - 1 )) + (b)/((x - 1 )^ 2 )+(c )/((x -1 ) ^(3))+(d)/((x - 1 ) ^4 ) = (3 ) /((x - 1 ) ^2) ` `+(7 ) /((x -1) ^3 )+(5 ) /((x -1) ^4)+ (0)/((x - 1 )) ` `therefore [{:(a, b ),(c, d ):}] = [{:(0, 3 ) ,(7,5 ) :}] ` |
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