UNIVERSITY OF ENGINEERING & MANAGEMENT, JAIPURQUESTION BANK

SUBJECT NAME: MATHEMATICS, SUBJECT CODE: BCA105

BCA, 1ST YEAR, 1ST SEMESTER

GROUP-A

(Objective/Multiple type question)

Q.1 if A ={1,2,3,4,8} ,B = {2,4,6,7} ,then A is

(a) {2,4}

(b) {1,2,3,4,6,7,8}

(c) {1,3,6,7,8}

(d)

Q.2 if be the roots of the equation then is

(a) 2 (b)1 (c) 3 (d)0

Q.3 The value of is equal to

(a) 2 (b)1 (c) 3 (d)0

Q.4 if = -1 is a root of the equation ,then the value of k is

(a) 2 (b)1 (c) (d)0

Q.5 the degree of the polynomial

(a) 2 (b)1 (c) 3 (d)0

Q.6 if then is

(a) (b) (c) (d) 2

Q.7 the value of t for which the matrix is singular is

(a) 2 (b)-2 (c) (d)

Q.8 is equal to

(a) 2 (b)1 (c)e (d)

Q.9 if be the roots of the equation then is

(a) 3 (b)6 (c)2 (d)

Q.10 if A ={1,2,3,} ,B = {2,3,6,} ,then A is

(a) {2,3}

(b) {1,2,3, }

(c) {1,2,3,6}

(d)

Q.11 the value of is equal to

(a) 1 (b)4 (c)2 (d)0

Q.12 The polar equation r = 4 sin θ represents a

a) circle b) ellipse

c) straight line d)none of these.

Q.13 If a, b and c are roots of x 3 − 3x + 9 = 0 then a 2 + b 2 + c 2 isa) 6 b) − 6c) 0 d) 1.

Q.14 Solution of the equation x 3 + 2x + 3 = 0 will give

a) no real positive roots but one real negative root

b) two real positive roots and one real negative root

c) one real positive root and two imaginary roots

d) two real negative roots and only one imaginary root.

Q.15 Which of the following does not satisfy Rolles theorem in [−2, 2] ?

a) x b) 1/x

c) 1/(x − 5) d) x 2 – 5.

Q.16 Area of the region bounded by the curve y = cosx ,x = 0 and x = is

(a) 2 sq. units (b) 3 sq. units (c) 4 sq. units (d) 1 sq. units

Q.17 if sinx = then is equal to

(a) 1 (b) -1 (c) 2 (d) 0

Q.18 if f(x) = 8 then fog(x) is

(a) 8x (b) 8

Q.19 is equal to

(a) (b) (c) 0 (d) doesn’t exists.

Q.20 is equal to

(a) +c (b) - (c) (d)

Q.21 tan( ) +tan( ) is equal to

(a) sec (b) 2sec (c) sec (d) sin

Q.22 is equal to

(a) (b) (c) (d) 0

Q.23 is equal to

(a) 0 (b) – (c) (d)

Q.24 if (x,y) is equidistant from (a+b,b-a) and (a-b,a+b),then

(a) x+y = 0 (b)bx-ay = 0 (c) ax-by = 0. (d) bx+ay = 0

Q.25 if the points (1,0) ,(0,1) and (x,8) are collinear ,then the value of x is equal to

(a) 5 (b) -6 (c) 6 (d) 7

Q.26 The minimum value of the function max{x, } is equal to

(a) 0 (b) 1 (c) 2 (d)

Q.27 if f(9) = f’(9) = 0 then

(a) 0 (b) f(0) (c) f(9) (d) f’(9

Q.28 The value of cos( ) +cos ( ) is

(a) .

Q.29 Area of the triangle with vertices (-2,2) , (1,5) and (6,-1) is

(a) 15 (b) 7 (c) 3 (d) 11

Q.30 The equation of the line passing through (-3,5) and perpendicular to the line through the points (1,0)

and (-4,1) is

(a)5x+y+10 =0 (b) 5x-y+20 = 0 (c) 5x-y-10 = 0 (d) 5y-x-10 =0

Q.31 The area bounded by the curves y = and y = 0 is

(a) (b) (c) 4 (d) 5

Q.32 The distance between (2,1,0) and 2x+y+2z+5 = 0 is

(a) 10 (b) 5 (c) 1 (d)0

Q.33 is equal to

(a) 1 (b) 0 (c)4 (d) 2

Q.34 is equal to

(a) (b) (c) (d)

Q.35 if a and b are the non zero distinct roots of then the minimum value of

is

(a) 1 (b) (c) (d)

Q.36 The angle between the pair of lines and is

(a) (b) (c) (d)

Q.37 The area of triangular region whose sides are y = 2x+1 ,y = 3x+1 and x =4 is

(a) 5 (b)6 (c)7 (d)8

Q.38 Let S = {1,2,3…….10} the number of subsets of S containing only odd number is

(a) 15 (b) 31 (c) 63 (d) 7

Q.39 The area of parallelogram with vertices (0,0), (7,2), (5,9) and (12,11) is

(a) 50 (b) 54 (c) 51 (d) 53

Q.40 The equation of plane passing through (1,2,-3) and (2,-2,1) and parallel to X-axis is

(a) y-z+1 = 0 (b) y-z-1 = 0 (c) y+z-1 = 0 (d) y+z+1 = 0

Q.41 Three lines are drawn from the origin O with direction cosines propotional to (1,-1,1),(2,-3,0) and

(1,0,3) the three lines are

(a) not coplanar (b) coplanar (c) perpendicular to each other (d) coincident

Q.42 if f(x) = , then f’(e) is equal to

(a) (b) (c) (d)

Q.43 Differential coefficients of with respect to log is

(a) (b) x (c) (d)

Q.44 + = if x equals to

(a) 1 (b) (c) (d)

Q.45 if f(x) = then

(a) 1 (b) 4 (c) 3 (d) -4

Q.46 Let = if = 25 ,then =

(a)1/5 (b) 5 (c) 1 (d)

Q.47 if f(x) = 2 +

(a) f’(2) = f’(3) (b) f’(2) = 0 (c) f’( ) = (d) d’( ) = 0.

Q.48 B = is a

(a) rectangular matrix (b) singular matrix (c) square matrix (d) non singular matrix

Let are the roots of the equation 6 where pis a real number.

Q.49 if both are greater than 2 ,then

(a) p < -4 (b) p < -2 (c) p< -6-2 (d) none of these

Q.50 The equation whose roots are and is

(a) 3

(b) 3

(c) 3

(d) none of these.

GROUP-B

(Short answer type questions)

Q.1 Evaluate the integral

Q.2 if prove that ] = 1.

Q.3 In a survey of 320 persons, number of persons taking tea is 210, taking

milk is 100 and coffee is 70. Number of persons who take tea and milk is

50, milk and coffee is 30, tea and coffee is 50. The number of persons

taking all three together is 20. Find the number of people who take

neither tea nor coffee nor milk.

Q.4 express as a sum of a symmetric and a skew symmetric matrix.

Q.5 if be the roots of the equation then find the

equation whose root is 1+ , 1+ , 1+ .

Q.6 In a survey concerning the smoking habits of consumers it was found

that 55% smoke cigarette-A, 50% smoke cigarette-B, 42% smoke

cigarette-C, 28% smoke cigarette-A & B, 20% smoke cigarette-A & C,

12% smoke cigarette-B & C and 10% smoke all the three cigarette.

What percentage do not smoke ?

Q.7 Reduce the following equation to its canonical form and determine the

nature of the conic represented by it :

Q.8

If u = tan-1 x 2 + y2 , show that x ∂u + y ∂u = 1 sin 2u.

x + y ∂x ∂y 2

Q.9 Apply Descerte’s rule of sign to find the nature of the roots of the given equation : x 4 + qx 2 + rx – s = 0 (where q, r, s being positive).

Q.10 Obtain a relation between p, q and r so that x3 + px 2 + qx + r = 0 has 3 roots that are in A.P.Q.11 Evalute

Q.12

Q.13 if A = and , then find α+β+γ .

Q.14 if A = and B = then show that 4A+5B-C =0.

Q.15 if = then find the value of x,y,z,

Q.16 let A = {1,2} and B = {3,4,5} find (i) B (ii) A A AQ.17 let P ={x,y,z} and Q = {3,4} find number of relations from P to Q.

Q.18 let f be a exponential function and g be the logarithmic function (fg)(1).

Q.19 let = and =(3x+2) be two real functions then ,find (f+g)(x).

Q.20 let g = {(1,2),(2,5),(3,8), (4,10),(5,12),(6,12)} is a function? If yes ,find its domain

and range .if no ,give reason .

Q.21 let ,g: be function defined by and

g ={(n, ): n} show that f = g.

Q.22 if f(x) = ,prove that f( ).

Q.23 let R be a relation on the set of natural numbers N defined by

for every .

Q.24 if P = {a,b} and Q = (x,y,z) then show that

Q.25 given f(x) = , g(x) f(f(x)) and h(x) = f(f(f(x).then find the value of

f(x).g(x)h(x).

Q.26 if A and C ,prove that A

Q.27find the distance of the point (-2,4,-5) from the line .

Q.28 if A is a square matrix of order 3 ,then find .

Q.29 equation of line passing throught the point (1,2) and perpendicular to the

line y = 3x-1.

Q.30 find the value of C is mean value theorem for the function f(x) = in

[2,4].

Q.31 evaluate the value of .

Q.32 find , if y = .

Q.33 evaluate : .

Q.34 evaluate :

Q.35 the area of the region bounded by the curve y = and the line y = 16.

Q.36 if then find the value of x.

Q.37 find the perpendicular distance of the point P(6,7,8) fom XY –plane .

Q.38 find 3+5+7+…………….to n terms.

Q.39 the derivative of with respect to

Q.40 if y = log(logx) , then find

Q.41 evaluate : .

Q.42 .

Q.43 the point on the curve where tangent makes an angle with X-axis.

Q.44 the plane 2x-3y+6x-11 = 0 makes an angle with X-axis.find the value

of

Q.45 evaluate .

Q.46 if = then find .

Q.47 find the eccentricity of the ellipse

Q.48 evaluate: .

Q.49 evaluate:

Q.50 find the reflection of the point ( ) in XY plane.

GROUP-C

(Long type questions)

Q.1 verify whether the matrix is orthogonal

Q.2 solve the following system of linear equations:

Q.3 if A = and B = find AB.

Q.4 show that * =

Q.5 evaluate

Q.6 evaluate

Q.7 evaluate

Q.8 evaluate .

Q.9 differentiate with respect to

Q.10 If ,then show that

)

Q.11 if A = then find and show that A = .

Q.12 find the maxima and minima of the equation

Q.13 If PSQ be a focal chord of a conic with focus S and semi latus rectum L, then

prove that 1/SP + 2/SQ = 2/L.

Q.14 Find the point on the conic 6/r = 1 + 4 cos θ whose vertical angle is π/3.

Q.15 If u = cos – 1 { } then show that

Q.16 find out the rank of the matrix

Q.17 solve the following system of linear equation by matrix inversion method

2

Q.18 find the value of a and b such that the following system of linear equation has (a) unique solution

(b) many solution

Q.19 State Cauchy's mean value theorem.

Q.20 Apply Descartes' rule of signs to find the nature of roots of the equation

x 4 + 2x 2 + x – 12 = 0

Q.21 Determine whether the function is continuous at the origin.

) = if (x,y)

0 if (x,y) (0,0)

Q.22 If α, β, γ are the 3 roots of x3 px2 qx r 0 obtain the value of .

Q.23 Reduce the following equation to its canonical form and determine the nature the conic represented by it :3x2 - 8xy - 3y2 10x - 13y 18 0.

Q.24 If by a rotation of rectangular co-ordinate axes without change of origin expressions ax + by and cx + dy are transformed into al xl bl yl and cl xl dl yl Show that al dl ∀ bl cl ad ∀ bc.

Q.25 if then show that

) -(2n+1) + ) =0

Q.26 evaluate

Q.27 if then prove that

Q.28 Solve the equation by Cardan’s method : 2x3 3x2 3x 1

Q.29 State Descart's rule of sign. Using this rule find the nature of the roots of the

Equation x 4 -7x3 + 21x 2 -9 x +21 = 0

Q.30 Solve the following system of linear equations by Cramer's rule

2

Q.31 If by a transformation of one rectangular axis to another with same origin the expression

ax + by changes to a’x + b’y. prove that = .

Q.32 Show that the equation 20x 2 +15xy- 9 x +3y+1 = 0 represents a pair of intersecting straight lines

which are equidistant from the origin.

Q.33 If A = { a, b, c, d, e } , B = { c, a, e, g } and C = { b, e, f, g },

then show that ( A U B ) C = ( A C ) U ( B C ).

Q.34 Reduce the following equation to the canonical form and determine the nature of the

conic represented by it x 2 -4xy+4 y 2 - 12 x -6y-39= 0

Q.35 If α, β, γ are the 3 roots of x3 px2 qx r 0 obtain the value of .

Q.36 Show that if 0< < .

Q.37 Evaluate: ).

Q.38 evaluate:

Q.39 If PSQ be a focal chord of a conic with focus S and semi-latus rectum l, then prove

that

1

+

1

=

2

.SP SQ l

Q.40 if A- 2B = and 2A+B = ,find A and B?

Q.41 show that

Q.42 if then show that

Q.43 If y = sin-1 x, then prove that (1 – x 2) yn+2 – (2n + 1) x yn+1 – n 2 yn = 0

Q.44 In the mean value theorem f (x + h) = f (x ) + h f / (x + θh), if f (x ) = px 2 + qx + r (p ≠

0), then show that θ = ½.

Q.45 Reduce the equation 3x 2 + 2xy + 3y 2 – 16x + 20 = 0 into canonical form and hence determine the nature of the conic.Q.46 Find the nature of the conic 8r = 4 – 5 cos θ

Q.47 Expand e x in ascending powers of x by Taylor’s series.

Q.48 Solve using Carden’s method : x 3 – 9x + 28 = 0

Q.49 If by a transformation of motion of co-ordinate axes,the expression ax2 + 2hxy + by2 changes into

a′x 2 + 2h′x′y′ + b′y′ 2, then show that ab –h 2 = a′ b′ −h′ 2.

Q.50

Solve the equations by matrix inversion method :

x + y + z = 4

2x – y + 3z = 1

3x + 2y – z = 1