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WBJEE Mathematics Sample Paper I


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WBJEE Mathematics Sample Paper I - Question 1

The eccentricity of the hyperbola 4x2 – 9y2 = 36 is

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WBJEE Mathematics Sample Paper I - Question 2

The length of the latus rectum of the ellipse 16x2+ 25y2 = 400 is

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Length of latus rectum =

WBJEE Mathematics Sample Paper I - Question 3

The vertex of the parabola y2+ 6x – 2y + 13 = 0 is
(y -1)2 - 6x - 12

Vertex → (2, 1)

WBJEE Mathematics Sample Paper I - Question 4

The coordinates of a moving point p are (2t2+ 4, 4t + 6). Then its locus will be a

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WBJEE Mathematics Sample Paper I - Question 5

The equation 8x2+ 12y2 – 4x + 4y – 1 = 0 represents

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WBJEE Mathematics Sample Paper I - Question 6

If the straight line y = mx lies outside of the circle x2+ y2 – 20y + 90 = 0, then the value of m will satisfy

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WBJEE Mathematics Sample Paper I - Question 7

The locus of the centre of a circle which passes through two variable points (a, 0), (–a, 0) is

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Centre lies on y-axis  locus  x = 0

WBJEE Mathematics Sample Paper I - Question 8

The  coordinates of the two points lying on  x + y = 4 and at a unit distance from the straight line 4x + 3y = 10 are

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WBJEE Mathematics Sample Paper I - Question 9

The intercept on the line y = x by the circle x2+ y2 – 2x = 0 is AB. Equation of the circle with AB as diameter is

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WBJEE Mathematics Sample Paper I - Question 10

If the coordinates of one end of a diameter of the circle x2+y2+4x–8y+5=0, is (2,1), the coordinates of the other end is

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Centre circle (–2, 4)

WBJEE Mathematics Sample Paper I - Question 11

If the three points A(1,6), B(3, –4) and C(x, y) are collinear then the equation satisfying by x and y is

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WBJEE Mathematics Sample Paper I - Question 12

If     and θ lies in the second quadrant, then cosθ is equal to

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θ in 2nd quad Cosθ < 0

WBJEE Mathematics Sample Paper I - Question 13

The solutions set of inequation cos–1x < sin–1x is

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cos–1x < sin–1x is

WBJEE Mathematics Sample Paper I - Question 14

The number of solutions of 2sinx + cos x = 3 is

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√5<3  No solution

WBJEE Mathematics Sample Paper I - Question 15

Let     and    then α + β is

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WBJEE Mathematics Sample Paper I - Question 16

If   

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WBJEE Mathematics Sample Paper I - Question 17

If sinθ and cosθ are the roots of the equation ax2 – bx + c = 0, then a, b and c satisfy the relation

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WBJEE Mathematics Sample Paper I - Question 18

If A and B are two matrices such that A+B and AB are both defined, then

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Addition is defined if order of A is equal to order of B
A B
nxm nxm is defined if m = n
⇒ A, B are square matrices of same order

WBJEE Mathematics Sample Paper I - Question 19

   is a symmetric matrix, then the value of x is

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A = AT

WBJEE Mathematics Sample Paper I - Question 20

If   

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= Real

WBJEE Mathematics Sample Paper I - Question 21

The equation of the locus of the point of intersection of the straight lines x sin θ + (1 – cos θ) y = a sin θ and x sin θ – (1 + cos θ) y + a sin θ = 0 is

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y = a sin θ
x = a cos θ.

WBJEE Mathematics Sample Paper I - Question 22

If sinθ + cosθ = 0 and 0 < θ < π, then θ

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sin θ + cos θ = 0
⇒ tan θ = – 1                   

WBJEE Mathematics Sample Paper I - Question 23

The value of cos 15o – sin 15o is

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WBJEE Mathematics Sample Paper I - Question 24

The period of the function f(x) = cos 4x + tan 3x is

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WBJEE Mathematics Sample Paper I - Question 25

If y = 2x3 – 2x2 + 3x – 5, then for x = 2 and ∆ x = 0.1 value of ∆ y is

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WBJEE Mathematics Sample Paper I - Question 26

The approximate value of     correct to 4 decimal places is

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y = 2 + 1/80

WBJEE Mathematics Sample Paper I - Question 27

The value of    dx is

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WBJEE Mathematics Sample Paper I - Question 28

For the function     Rolle’s theorem is

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WBJEE Mathematics Sample Paper I - Question 29

The general solution of the differential equation  

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auxilary equation m2 + 8m + 16 = 0 ⇒ m = – 4
Solution  

WBJEE Mathematics Sample Paper I - Question 30

If x2+ y2 = 4, then  

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WBJEE Mathematics Sample Paper I - Question 31

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WBJEE Mathematics Sample Paper I - Question 32

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WBJEE Mathematics Sample Paper I - Question 33

The degree and order of the differential equation    are respectively

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WBJEE Mathematics Sample Paper I - Question 34

  The function f (x) is

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WBJEE Mathematics Sample Paper I - Question 35

The function f(x) = ax + b is strictly increasing for all real x if

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f′ (x) = a
f′(x) > 0 ⇒ a > 0

WBJEE Mathematics Sample Paper I - Question 36

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WBJEE Mathematics Sample Paper I - Question 37

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WBJEE Mathematics Sample Paper I - Question 38

The general solution of the differential equation 

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WBJEE Mathematics Sample Paper I - Question 39

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WBJEE Mathematics Sample Paper I - Question 40

If one of the cube roots of 1 be ω, then

Detailed Solution for WBJEE Mathematics Sample Paper I - Question 40

C2 → C2 – C3
C3 → C3 + C2
C3 → C3 + ωC1
C2 → C2 – C1

WBJEE Mathematics Sample Paper I - Question 41

4 boys and 2 girls occupy seats in a row at random. Then the probability that the two girls occupy seats side by side is

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WBJEE Mathematics Sample Paper I - Question 42

A coin is tossed again and again. If tail appears on first three tosses, then the chance that head appears on fourth toss is

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WBJEE Mathematics Sample Paper I - Question 43

The coefficient of xn in the expansion of  

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WBJEE Mathematics Sample Paper I - Question 44

The sum of the series  

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WBJEE Mathematics Sample Paper I - Question 45

The number (101)100 – 1 is divisible by

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WBJEE Mathematics Sample Paper I - Question 46

If A and B are coefficients of xn in the expansions of (1+ x)2n and (1+x)2n – 1 respectively, then A/B is equal to

Detailed Solution for WBJEE Mathematics Sample Paper I - Question 46

A = 2nCn.
B = 2n – 1Cn

WBJEE Mathematics Sample Paper I - Question 47

If n > 1 is an integer and x ≠0, then (1 + x)n – nx – 1is divisible by

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WBJEE Mathematics Sample Paper I - Question 48

If nC4 , nC5 and nC6 are in A.P., then n is

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WBJEE Mathematics Sample Paper I - Question 49

The number of diagonals in a polygon is 20. The number of sides of the polygon is

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nC2 –n = 20
n = 8

WBJEE Mathematics Sample Paper I - Question 50

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WBJEE Mathematics Sample Paper I - Question 51

Let a , b, c be three real numbers such that a + 2b + 4c = 0. Then the equation ax2 + bx + c = 0

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WBJEE Mathematics Sample Paper I - Question 52

If the ratio of the roots of the equation px2 + qx + r = 0 is a : b, then  

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WBJEE Mathematics Sample Paper I - Question 53

If α and β are the roots of the equation x2 + x + 1 = 0, then the equation whose roots are α19 and β7 is

Detailed Solution for WBJEE Mathematics Sample Paper I - Question 53

α and β are the roots of x2 + x + 1 = 0
α = ω                 β = ω2
α19= ω              β7= ω2
x2 – (α19 + β7)x + α19β7= 0
Thou,
x2 – (ω + ω2 ) x + ω . ω2 = 0
x2 + x + 1 = 0

WBJEE Mathematics Sample Paper I - Question 54

For the real parameter t, the locus of the complex number     in the complex plane is

Detailed Solution for WBJEE Mathematics Sample Paper I - Question 54

Given 
Let z = x + iy
x = 1 – t2
y2 = 1 + t2
Thus, x + y2 = 2
y2 = 2 – x
y2 = – (x – 2)
Thus parabola

WBJEE Mathematics Sample Paper I - Question 55

If    , then for any integer n, 

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WBJEE Mathematics Sample Paper I - Question 56

If ω ≠ 1 is a cube root of unity, then the sum of the series S = 1 + 2ω + 3ω² + .......... + 3nω3n – 1 is

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WBJEE Mathematics Sample Paper I - Question 57

If log3x + log3y = 2 + log32 and log3 (x + y) = 2, then

Detailed Solution for WBJEE Mathematics Sample Paper I - Question 57

: log3x + log3y = 2 + log32
⇒ x.y = 18
log (x + y) = 2 ⇒ x + y = 9
we will get x = 3 and y = 6

WBJEE Mathematics Sample Paper I - Question 58

If log7 2 = λ, then the value of log49 (28) is

Detailed Solution for WBJEE Mathematics Sample Paper I - Question 58

log4928 = log724 × 7

WBJEE Mathematics Sample Paper I - Question 59

The sequence log a, 

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log a . (2log a – log b)(3log a – 2 log b)
= T2 – T1 = log a – log b
= T3 – T2 = log a – log b

WBJEE Mathematics Sample Paper I - Question 60

If in a triangle ABC, sin A, sin B, sin C are in A.P., then

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WBJEE Mathematics Sample Paper I - Question 61

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: c1 → c1 + c2 + c3

WBJEE Mathematics Sample Paper I - Question 62

The area enclosed between y2 = x and y = x is

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WBJEE Mathematics Sample Paper I - Question 63

Let f(x) = x3 e–3x, x > 0. Then the maximum value of f(x) is

Detailed Solution for WBJEE Mathematics Sample Paper I - Question 63

f(x) = x3 .e–3x
= f′(x) = 3x2 e–3x + x3 e–3x (–3)
= x23e–3x[1 – x] = 0, x = 1
Maximum at x = 1
f(1) = e–3

WBJEE Mathematics Sample Paper I - Question 64

The area bounded by y2 = 4x and x2 = 4y is

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WBJEE Mathematics Sample Paper I - Question 65

The acceleration of a particle starting from rest moving in a straight line with uniform acceleration is 8m/sec2 . The time taken by the particle to move the second metre is

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WBJEE Mathematics Sample Paper I - Question 66

The solution of

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WBJEE Mathematics Sample Paper I - Question 67

Integrating Factor (I.F.) of the defferential equation

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WBJEE Mathematics Sample Paper I - Question 68

The differential equation of y = aebx (a & b are parameters) is

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y = a.ebx ............ (i)
y1 = abebx
y1 = by ........................... (ii)
y2 = by1 ......................... (iii)

Dividing (ii) & (iii) 

WBJEE Mathematics Sample Paper I - Question 69

The value of 

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WBJEE Mathematics Sample Paper I - Question 70

The value of 

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WBJEE Mathematics Sample Paper I - Question 71

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= 2x f(x)

WBJEE Mathematics Sample Paper I - Question 72

Let f(x) = tan–1x. Then f′(x) + f′′(x) is = 0, when x is equal to

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f(x) = tan–1x

WBJEE Mathematics Sample Paper I - Question 73

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WBJEE Mathematics Sample Paper I - Question 74

The value of 

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WBJEE Mathematics Sample Paper I - Question 75

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= π

WBJEE Mathematics Sample Paper I - Question 76

If the function  ​

is continuous at x = 2, then

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   Put A = 0.

WBJEE Mathematics Sample Paper I - Question 77


​If f(x) is continuous at x = 2, the value of λ will be

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WBJEE Mathematics Sample Paper I - Question 78

The even function of the following is

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WBJEE Mathematics Sample Paper I - Question 79

If f(x + 2y, x – 2y) = xy, then f(x, y) is equal to

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WBJEE Mathematics Sample Paper I - Question 80

The locus of the middle points of all chords of the parabola y2 = 4ax passing through the vertex is

Detailed Solution for WBJEE Mathematics Sample Paper I - Question 80

2h = x, 2k = y
y2 = 4ax
k2 = 2ah
y2 = 2ax

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