WBJEE Maths Test - 13 - JEE MCQ

# WBJEE Maths Test - 13 - JEE MCQ

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## 75 Questions MCQ Test WBJEE Sample Papers, Section Wise & Full Mock Tests 2025 - WBJEE Maths Test - 13

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WBJEE Maths Test - 13 - Question 1

### The positive integer just greater than (1 + .0001)10000 is

WBJEE Maths Test - 13 - Question 2

### The inverse point of (1, 2) with respect to the circle x2 + y2 - 4x - 6y + 9 = 0 is

WBJEE Maths Test - 13 - Question 3

### If the line lx+my=1 is a tangent line to the circle x2+y2=a2, the locus of (l,m) is

WBJEE Maths Test - 13 - Question 4

[(-1+√-3)/2]3n + [(-1-√-3)/2]3n =

WBJEE Maths Test - 13 - Question 5

The coefficients of three successive terms in the expansion of (1 + x)n are 165, 330 and 462 respectively, then the value of n will be

Detailed Solution for WBJEE Maths Test - 13 - Question 5

Let coefficients of three consecutive terms i.e., (r + 1)th, (r + 2)th and (r + 3)th in expansion of (1 + x)n are 165.330 and 462 respectively then
coefficient of (r + 1)th term = nCr = 165
coefficient of (r + 2)th term = nCr + 1 = 330
coefficient of (r + 3)th term = nCr + 2 = 462
∴ n C r + 1 n C r = n - r r + 1 = 2
or, n - r = 2(r + 1)
or, r = 1 3 (n - 2)
and n C r + 2 n C r + 1 = n - r - 1 r + 2 = 231 165
or, 165(n - r - 1) = 231 (r + 2)
or, 165n - 627 = 396r
or, 165n - 627 = 396 x 1 3 (n - 2)
or, 165n - 627 = 132 (n - 2)
or, 33n = 363
∴ n = 11

WBJEE Maths Test - 13 - Question 6

If x is positive, then the first negative terms in the expansion of 1 + x 27 5 is

Detailed Solution for WBJEE Maths Test - 13 - Question 6

WBJEE Maths Test - 13 - Question 7

If w is a cube root of unity, then the value of (1 + ω - ω)2 (1 - ω + ω2) is

WBJEE Maths Test - 13 - Question 8

WBJEE Maths Test - 13 - Question 9

WBJEE Maths Test - 13 - Question 10

WBJEE Maths Test - 13 - Question 11

If m and n are integers, then what is the value of

Detailed Solution for WBJEE Maths Test - 13 - Question 11

Since sin mx ,sin nx is an odd function if m ≠ n, then ∫ 0 π sin mx . sin nx dx = 0

WBJEE Maths Test - 13 - Question 12

The area bounded by the parabola y2=4ax and the straight line y=2ax is

Detailed Solution for WBJEE Maths Test - 13 - Question 12

On solving y 2 = 4ax and y = 2ax, we get

x = 0 or 1 a
and y = 0 or 2

WBJEE Maths Test - 13 - Question 13

WBJEE Maths Test - 13 - Question 14

The order and degree of the equation are

WBJEE Maths Test - 13 - Question 15

The particular integral of (D2 + 1)y = xe2x is equal to

WBJEE Maths Test - 13 - Question 16

If y=log logx, ey(dy/dx)=

WBJEE Maths Test - 13 - Question 17
(d/dx)(log tan x)=
WBJEE Maths Test - 13 - Question 18

If the normal at the point P(θ) to the ellipse ((x2/14)+(y2/5) = 1) intersects it again at the point Q(2θ), then cosθ is equal to

WBJEE Maths Test - 13 - Question 19
If the latus rectum of an ellipse is one half of its minor axis, then its eccentricity is
WBJEE Maths Test - 13 - Question 20

The parametric equations of the hyperbola x2/a2 - y2/b2 = 1 are

WBJEE Maths Test - 13 - Question 21

The eccentricity of the conic 9x2 - 16y2 = 144 is

WBJEE Maths Test - 13 - Question 22
Let f(x) = tan-1 {φ(x)}, where φ(x)is monotonically increasing for 0 < x < π/2. Then f(x) is
Detailed Solution for WBJEE Maths Test - 13 - Question 22
$f\prime \left(x\right)=\frac{\phi \prime \left(x\right)}{1+\left\{\phi \left(x\right){\right\}}^{2}}>\phantom{\rule{0.5em}{0ex}}\text{0 for 0}\phantom{\rule{0.5em}{0ex}}0,\phi \left(x\right)\phantom{\rule{0.5em}{0ex}}\text{being monotonically increasing}$
WBJEE Maths Test - 13 - Question 23

Sin (sin⁻11/2 + cos⁻11/2) equals

WBJEE Maths Test - 13 - Question 24

If is finite, then the values of a,b are respectively

WBJEE Maths Test - 13 - Question 25
If the value of a third order determinant is11, then the value of the square of the determinant formed by the co-factors will be
WBJEE Maths Test - 13 - Question 26

Largest value of min (2 + x2, 6 - 3x) when x > 0 is :

WBJEE Maths Test - 13 - Question 27
If a complex number lies in the IIIrd quadrant then its conjugate lies in quadrant number
WBJEE Maths Test - 13 - Question 28

The vertex of the parabola x 2 + 8 x + 12 y + 4 = 0 is

WBJEE Maths Test - 13 - Question 29

Axis of the parabola x 2 − 3 y − 6 x + 6 = 0 is

WBJEE Maths Test - 13 - Question 30
Out of 15 points in a plane, no three are in a straight line except 8 points which are collinear. How many triangles can be formed by joining them?
WBJEE Maths Test - 13 - Question 31
How many words can be formed from the letters of the word 'SIGNATURE' be arranged so that the vowels always come together?
WBJEE Maths Test - 13 - Question 32
In how many different ways can the letters of the word 'OPTICAL' be arranged so that the vowels always come together?
WBJEE Maths Test - 13 - Question 33

A box contains n enumerated articles. All the articles are taken out one by one at random. The probability that the numbers of the selected articles are in the sequence 1, 2, .........n is

WBJEE Maths Test - 13 - Question 34

An unbiased coin is tossed to get 2 points for turning up a head and one point for the tail. If three unbiased coins are tossed simultaneously, then the probability of getting a total of odd number of points is

WBJEE Maths Test - 13 - Question 35

Two dice are thrown simultaneously. The probability of getting a pair of 1 is

WBJEE Maths Test - 13 - Question 36

In Δ A B C , i f b = 20 , c = 21 and sin A = 3 5 , then a =

WBJEE Maths Test - 13 - Question 37

In a Δ A B C , 2s = perimeter and R = circumradius. Then s/R is equal to

Detailed Solution for WBJEE Maths Test - 13 - Question 37

WBJEE Maths Test - 13 - Question 38

the greatest value of a non-negative real number λ for which both the equations 2x2 + (λ - 1)x + 8 = 0 and x2 - 8x + λ + 4 = 0 have real roots is :

WBJEE Maths Test - 13 - Question 39

The set of values of p for which the roots of the equation 3x2 + 2x + (p - 1)p = 0 are of opposite sign, is

WBJEE Maths Test - 13 - Question 40

The second drivative f"(x) of the function f(x) exists for all x in [0,1] and satisfies | f ″ x | ≤ 1. If f 0 = f 1 , then for all x in [0,1]

Detailed Solution for WBJEE Maths Test - 13 - Question 40

The first derivative f'(x) exists for all x in [0,1] which implies that f(x) is continuous for all x in [0,1]. Also, it is given that f(0) = f(1)
Thus, applying Rolle's theorem on f(x) in the interval [0,1], we have
f'(c) = 0 for some c in [0,1]
The second derivative f''(x) exists for all x in [0,1] which implies that f'(x) is continuous for all x in [0,1]
Thus, applying Lagrange's theorem on f'(x) in the interval [ c , x ] c < x ≥ 1 , we have

Similarly, applying Lagrange's theorem on f’(x) in the interval [ x , c ] 0 ≤ x < c , we have

WBJEE Maths Test - 13 - Question 41

The fourth, seventh and tenth terms of a G.P. are p, q and r respectively, then

WBJEE Maths Test - 13 - Question 42

If A.M. between two numbers is 5 and their G.M. is 4, then their H.M. is

Detailed Solution for WBJEE Maths Test - 13 - Question 42

If x, y and z respectively represent AM, GM and HM between two numbers a and b, then
y2 = xz
Here x = 5, y = 4
then 16 = 5x z
z = 16/5

WBJEE Maths Test - 13 - Question 43
If 7th and 13th terms of an A.P. be 34 and 64 respectively, then its 18th term is
WBJEE Maths Test - 13 - Question 44
Every term of a G.P. is positive and also every term is the sum of two preceding terms. Then the common ratio of the G.P. is
WBJEE Maths Test - 13 - Question 45

If sin (x + 3α) = 3 sin (α - x), then

WBJEE Maths Test - 13 - Question 46

cot⁻1(-1/2) + cot⁻1(-1/3) is equal to

WBJEE Maths Test - 13 - Question 47
The coordinates of mid-point of portion of line cut by coordinate axis are (3,2), the equation of the line is
WBJEE Maths Test - 13 - Question 48

The angle between lines 3x+y-7=0 and x+2y+9=0 is

WBJEE Maths Test - 13 - Question 49
The condition that the cubic equation x3 - px2 + qx - r = 0 has all of its three roots equal is given by___
Detailed Solution for WBJEE Maths Test - 13 - Question 49
Let $\alpha$ is a root of the equation
${x}^{3}-p{x}^{2}+qx-r=0$
Since, all the roots of the given equation are equal
$\therefore \alpha +\alpha +\alpha =p$
$⇒\alpha =p∕3\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}\dots \left(1\right)$
$\alpha .\alpha +\alpha .\alpha +\alpha .\alpha =q$
$⇒{\alpha }^{2}=q∕3\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}\dots \left(2\right)$
$\alpha .\alpha .\alpha =r$
$⇒{\alpha }^{3}=r\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}\dots \left(3\right)$
$\because {\left({\alpha }^{2}\right)}^{2}={\alpha }^{3}.\alpha$
$⇒{\left(q∕3\right)}^{2}=\left(r\right).\left(p∕3\right)$
$⇒{q}^{2}=3pr$
WBJEE Maths Test - 13 - Question 50

I

WBJEE Maths Test - 13 - Question 51

The mean of the numbers a,b,8,5,10 is 6 and the variance is 6.80. Then which one of the following gives possible values of a and b?

WBJEE Maths Test - 13 - Question 52

A square piece of tin of side 18 cm is to be made into a box without top, by cutting a square from each corner and folding up the flaps to form the box. The maximum possible volume of the box is given by (in cm2)

WBJEE Maths Test - 13 - Question 53

The solution for the equation

WBJEE Maths Test - 13 - Question 54

Let a,b,c be any real numbers. Suppose that there are real numbers x,y,z not all zero such that x=cy+bz, y=az+cx and z=bx+ay. Then a2+b2+c2+2abc is equal to

WBJEE Maths Test - 13 - Question 55

The value of

WBJEE Maths Test - 13 - Question 56

On the interval [0, 1] the function
f x = x 1005 (1 − x )1002
assumes maximum value equal to

WBJEE Maths Test - 13 - Question 57

is a binary operation defined on Q. Find which of the following operation is Associative.

WBJEE Maths Test - 13 - Question 58

In a triangle X Y Z , ∠ Z = π 2 . If tan X 2 and tan Y 2 are the roots of the equation ax2 + bx + c = 0, a ≠ 0 then

WBJEE Maths Test - 13 - Question 59

If
then ∑ r = 1 n Δ r is equal to

WBJEE Maths Test - 13 - Question 60

Statement-1 :
Statement-2 :

WBJEE Maths Test - 13 - Question 61

Given the family of lines, a (3x + 4y +6) + b (x + y +2) = 0. The line of the family situated at the greatest distance from the point P(2, 3) has equation

WBJEE Maths Test - 13 - Question 62

Let P(3, 2, 6) be a point in space and Q be a point on the line

Then the value of μ for which the vector P Q → is parallel to the plane x - 4y+3z = 1 is

WBJEE Maths Test - 13 - Question 63

The differential equation (x4 − 2 x y2 + y4 ) d x − 2x2 y − 4xy3 + sin y) dy = 0 has its solution as

WBJEE Maths Test - 13 - Question 64

If the function f(x) and g(x) are defined on RR such that

WBJEE Maths Test - 13 - Question 65

The area bounded by the curves f (x)   = sin − 1 (sin x)    and (gx)   = [ sin − 1 sin x ] in the interval [ 0 , π ] , where [ . ] is a greatest integer function, is

*Multiple options can be correct
WBJEE Maths Test - 13 - Question 66

The cubes of natural number are grouped as 1 3 , 2 3 , 3 3 , 4 3 , 5 3 , 6 3 ,...Let Sn denotes the sum of cubes in the nth group, then 8Sn is divisible by

*Multiple options can be correct
WBJEE Maths Test - 13 - Question 67

If are any four vectors then is a vector

*Multiple options can be correct
WBJEE Maths Test - 13 - Question 68

The solution of the equation cos 103 x − sin 103 x = 1 are

*Multiple options can be correct
WBJEE Maths Test - 13 - Question 69

In Δ A B C , c , then

*Multiple options can be correct
WBJEE Maths Test - 13 - Question 70

*Multiple options can be correct
WBJEE Maths Test - 13 - Question 71

The point (0,0), (a, 11) and (b, 37) are the vertices of an equilateral triangle. Then

*Multiple options can be correct
WBJEE Maths Test - 13 - Question 72

If then:

*Multiple options can be correct
WBJEE Maths Test - 13 - Question 73

For three vectors u → , v → , w → which of the following expression is not equal to any of the remaining three?

*Multiple options can be correct
WBJEE Maths Test - 13 - Question 74

If a → = i + j ‸ + k ‸ , b → = 4 i ‸ + 3 j ‸ + 4 k ‸ and c → = i ‸ + α j ‸ + β k ‸ are linearly dependent vectors and | c | = 3 , then

*Multiple options can be correct
WBJEE Maths Test - 13 - Question 75

. Then

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