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WBJEE Maths Test - 6

75 Questions MCQ Test WBJEE Sample Papers, Section Wise & Full Mock Tests | WBJEE Maths Test - 6

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Attempt WBJEE Maths Test - 6 | 75 questions in 120 minutes | Mock test for JEE preparation | Free important questions MCQ to study WBJEE Sample Papers, Section Wise & Full Mock Tests for JEE Exam | Download free PDF with solutions

QUESTION: 1

The value of (7.995)¹/³ correct to four decimal places is

A.
1.9995
B.
1.9996
C.
1.9990
D.
1.9991

Solution:

QUESTION: 2

The approx value of (7.995)¹∕³ correct to four decimal places is

A.
1.9995
B.
1.9996
C.
1.9990
D.
1.9991

Solution:

QUESTION: 3

If in the expansion of (1 + x)²⁰, the coefficients of rth and (r + 4)th terms are equal, then r is

A.
7
B.
8
C.
9
D.
10

Solution:

QUESTION: 4

The coordinates of the point from which equal tangents are drawn to the circle x²+y²=1, x²+y²+8x+15=0 and x²+y²+10y+24=0 are

A.
[(2),-(5/2)]
B.
[-(2),(5/2)]
C.
[-(2),-(5/2)]
D.
[(2),(5/2)]

Solution:

QUESTION: 5

Circles x²+y²+2gx+2fy=0 and x²+y²+2g'x+2f'y=0 touch externally, if

A.
f'g=g'f
B.
fg=fg'
C.
fg+fg'=0
D.
f'g+fg'=0

Solution:

QUESTION: 6

(1 − ω + ω²)⁷ + (1 + ω − ω²)⁷ =

A.
2⁵
B.
2⁷
C.
2⁹
D.
2¹⁰

Solution:

QUESTION: 7

The area bounded by the curve y=sinx, y=0, x=0 and x=(π/2) is

A.
π
B.
2π
C.
1
D.
2

Solution:

QUESTION: 8

If a ≠ 6, b, c satisfy = 0 then abc =

A.
a + b + c
B.
0
C.
b³
D.
ab + bc

Solution:

QUESTION: 9

The reciprocal of 3+√7i is

A.
(3/16) - (√7/16)i
B.
(3/4)-(√7/4)i
C.
√7 + 3i
D.
3 - √7i

Solution:

QUESTION: 10

A.
π ∕ 4
B.
π ∕ 2
C.
2006 sin x 2007
D.
none of these

Solution:

QUESTION: 11

is equal to

A.
2
B.
1
C.
0
D.
3/2

Solution:

QUESTION: 12

A.
B.
log a - log b
C.
log a + log b
D.

Solution:

QUESTION: 13

The differential equation dy/dx+sin 2y=x³cos²y is reduced to linear from dv/dx+Pv=Q where P and Q are function of x alone, by changing the variable as :

A.
sin y=v
B.
cos y=v
C.
tany=v
D.
sin 2y=v

Solution:

QUESTION: 14

The roots of the equation = 0 are

A.
0, -3
B.
0, 0, - 3
C.
0, 0, 0, - 3
D.
0, 0, 3

Solution:

QUESTION: 15

The eccentricity of the ellipse 9x² + 5y² − 30y = 0 is

A.
1/3
B.
2/3
C.
3/4
D.
1/2

Solution:

QUESTION: 16

The derivative of is

Solution:

QUESTION: 17

If f (x) = (x − x₀) φ (x) and φ (x) is continuous at x = x₀ , then f ′(x) 0 is equal to

A.
φ ' (x₀)
B.
φ (x₀)
C.
x₀ φ (x₀)
D.
2φ x₀) ,

Solution:

QUESTION: 18

The solution of differential equation

A.
2xe^tan−1 y = e^2tan−1 y + k
B.
xe^2tan−1 y = e^tan−1 y + k
C.
xe^tan−1 y = tan⁻¹ y + k
D.
(x − 2) = ke^tan−1 y

Solution:

QUESTION: 19

The sum of distances of any point on the ellipse 3x² + 4y² = 24 from its foci is

A.
8√2
B.
4√2
C.
16√2
D.
none of these

Solution:

QUESTION: 20

The locus of the middle points of chords of the hyperbola 3x² − 2y² + 4x − 6^y = 0 parallel to y = 2x is

A.
3x - 4y = 4
B.
3y - 4x = 4
C.
3x - 4y = 2
D.
3y - 4x = 2

Solution:

QUESTION: 21

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

A.
4/5
B.
5/4
C.
4/3
D.
√7

Solution:

QUESTION: 22

If f(x) = x³ + 4x² + λx + 1 is a monotonically decreasing function of x in the largest possible interval (- 2, -2/3) then

A.
λ = 4
B.
λ = 2
C.
λ = -1
D.
λ has no real value

Solution:

QUESTION: 23

If cos⁻¹x+cos⁻¹y+cos⁻¹z = 3π then xy + yz + zx is equal to

A.
1
B.
0
C.
-3
D.
3

Solution:

QUESTION: 24

The maximum area of the rectangle that can be inscribed in a circle of radius r is

A.
πr²
B.
r²
C.
D.
2r²

Solution:

QUESTION: 25

If |z₁|=|z₂| and amp. z₁+amp.z₂=0, then

A.
z₁=z₂
B.
z₁̅ =z₂
C.
z₁+z₂=0
D.
z₁̅=z₂̅

Solution:

QUESTION: 26

A.
27/16
B.
16/27
C.
9/4
D.
4/9

Solution:

L=lim_(n->∞) n(1³+2³+…….n³)²/1²+2²+…..n²)³=?

Prerequisite r. _rΣⁿ=1³+2³+…..n³=n²(n+1)²/4

_r=1Σⁿr²=1²+2²+…..n²=n[(n+1)(2n+1)]/6

L=lim_(n->∞) n[(n²(n+1)²)]/[(n+1)(2n+1)/6]³

L=lim_(n->∞)[n²(n+1)/2]/[(2n+1)³/27]

=(1+1/∞)/(1+1/2n∞)³x (27/16)

=27/16

lim_(n->∞) n(1³+2³+…….n³)²/1²+2²+…..n²)³=27/16

QUESTION: 27

If A and B are any 2 x 2 matrics, then det (A + B) = 0 implies

A.
det A + det B = 0
B.
det A = 0 or det B = 0
C.
det A = 0 and det B = 0
D.
none of these

Solution:

QUESTION: 28

The parabola with directrix x + 2y − 1 = 0 and focus (1, 0) is

A.
4x²- 4xy + y²− 8x + 4y + 4 = 0
B.
4x²+ 4xy + y²− 8x + 4y + 4 = 0
C.
4x²+ 4xy + y² + 8x − 4y + 4 = 0
D.
4x²− 4xy + y²− 8x − 4y + 4 = 0

Solution:

QUESTION: 29

The line y=mx+c touches the parabola x²=4ay if

A.
c=-am
B.
c=-a/m
C.
c=-am²
D.
c=a/m²

Solution:

QUESTION: 30

How many total words can be formed from the letters of the word INSURANCE in which vowels are always together?

A.
18270
B.
17280
C.
12780
D.
12790

Solution:

QUESTION: 31

Twelve students compete for a race. The number of ways in which first three prizes can be taken is

A.
(12)! - 3
B.
(12)! ÷ 3!
C.
3!
D.
12.11.10

Solution:

QUESTION: 32

P,Q,R and S have to deliver lecture, then in how many ways can the lectures be arranged?

A.
4
B.
12
C.
256
D.
24

Solution: Total no. of ways delivering lectures = ⁴P₄ = 4! = 24

QUESTION: 33

Card is drawn at random from a pocket of 100 cards numbered 1 to 100. The probability of drawing a number which is a square is

A.
1/100
B.
2/100
C.
1/10
D.
3/10

Solution:

QUESTION: 34

Three letters are written to different persons, and addresses on three envelopes are also written without looking at the addresses, the probability that the letters go into right envelope is

A.
1/27
B.
1/6
C.
1/9
D.
None of these

Solution:

QUESTION: 35

For any two independent events E₁ and E₂
P{E₁∪E₂)∩(E̅₁∩E̅₂)} is

A.
≤(1/4)
B.
>(1/4)
C.
≥(1/2)
D.
none of these

Solution:

QUESTION: 36

In ΔABC , if = 0 then B =

A.
π/2
B.
π/4
C.
2π/3
D.
π/3

Solution:

QUESTION: 37

Let α, β be the roots of the quadratic equation x² + px + p³ = 0. If (α, β) is a pt. on the parabola y² = x, then the roots of the qudratic equation are

A.
4, -2
B.
-4, -2
C.
4, 2
D.
-4, 2

Solution:

QUESTION: 38

If a, b, c are distinct +ve real numbers and a² + b² + c² = 1 then ab + bc + ca is

A.
less than 1
B.
equal to 1
C.
greater than 1
D.
any real no

Solution:

QUESTION: 39

If f(x) and g(x) are differentiable functions for 0≤x≤1 such that f(0) = 2, g(0) = 0, f(1) = 6, g(1) = 2, then in the interval (0,1),

A.
f'(x) = 0 for all x
B.
f'(x) = 2g'(x) for at least one x
C.
f'(x) = 2g'(x) for at most one x
D.
none of these

Solution:

Since f(x) and g(x) are differentiable in [0,1], h(x) is also differentiable in [0,1]. Hence, h(x) is also continuous in [0,1].So, all the conditions of Rolle's theorem are satisfied. Hence, there exists a point c, 0 < c < 1 for which h'(c) = 0.
∴ f ′(c) − 2 g ′ c = 0, i . e . , f ′(c) = 2 g ′(c)

QUESTION: 40

In ΔABC , if x = tan , then x + y + z in terms of x , y , z only is

A.
xyz
B.
2xyz
C.
-xyz
D.

Solution:

QUESTION: 41

The sum of the integers from 1 to 100 which are not divisible by 3 or 5 is

A.
2489
B.
4735
C.
2632
D.
2317

Solution:

QUESTION: 42

The harmonic Mean of a/1 - ab and a/1 + ab is

A.
a/√1 − a²b²
B.
a/1 − a²b²
C.
a
D.
1/1 - a²b²

Solution:

QUESTION: 43

For a series whose nth term is (n/x) + (y), the sum of r terms is

A.
(r(r + 1)/2x) + (ry)
B.
(r(r - 1)/2x)
C.
(r(r - 1)/2x) - (ry)
D.
(r(r + 1)/2y) - (rx)

Solution:

QUESTION: 44

The two geometric means between 1 and 64 are

A.
1 and 64
B.
8 and 16
C.
4 and 16
D.
3 and 16

Solution:

QUESTION: 45

In Rule Method the null set is represented by

A.
{}
B.
φ
C.
{x:x = x}
D.
{x:x ≠ x}

Solution:

QUESTION: 46

The relation "less than" in the set of natural numbers is

A.
only symmetric
B.
only transistive
C.
only reflexive
D.
equivalence relation

Solution:

QUESTION: 47

The equation of line passing through lines 5x-6y-1=0 and 3x+2y+5=0 and perpendicular to 3x-5y+11=0 is

A.
5x+3y+8=0
B.
5x+3y+11=0
C.
5x+3y-8=0
D.
5x-3y-8=0

Solution:

QUESTION: 48

For what value of a line x = 3, y = 4 and 4x - 3y + a = 0 are con-current.

A.
0
B.
-1
C.
2
D.
3

Solution:

QUESTION: 49

If sec4θ-sec2θ=2, the general value of θ is

A.
(2n+1)(π/4)
B.
(2n+1)(π/10)
C.
nπ+(π/2) or (2nπ/5)+(π/10)
D.
nπ+(π/5)

Solution:

QUESTION: 50

A quadratic equation whose one root is 3 + 2i is_____

A.
x² + 6x + 13 = 0
B.
x² - 6x - 13 = 0
C.
x² - 6x + 13 = 0
D.
2x² - 6x + 13 = 0

Solution: Every quadratic equation has only two roots one root is 3 + 2i, second root will be its conjugate 3 - 2i.

QUESTION: 51

If the pair of lines ax² + 2a + bxy + by² = 0 lie along diameters of a circle and divide the circle into four sectors such that the area of one sector is thrice that of another then

A.
3a²− 2ab + 3b²= 0
B.
3a²− 10ab + 3b² = 0
C.
3a² + 2ab + 3b²= 0
D.
3a²+ 10ab + 3b² = 0

Solution:

QUESTION: 52

If A is the set of the divisors of the number 15, B is the set of prime number smaller than 10 and C is the set of even number smaller than 9, then A ∪ C ∩ B is the following set

A.
{1,3,5}
B.
{1,2,3}
C.
{2,3,5}
D.
{2,5}

Solution:

QUESTION: 53

In a shop there are five types of ice-creams available. A child buys six ice-creams.
Statement-1: The number of different ways the child can buy the six ice-creams is ¹⁰C₅ .
Statement-2: The number of different ways the child can buy the six ice-creams is equal to the number of different ways of arranging 6A's and 4B's in a row.

A.
Statement-1 is false, Statement-2 is true
B.
Statement-1 is true, Statement-2 is true, Statement-2 is a correct explanation for Statement-1
C.
Statement-1 is true, Statement-2 is true, Statement-2 is not a correct explanation for Statement-1
D.
Statement-1 is true, Statement-2 is false

Solution:

QUESTION: 54

If then the value of K is

A.
1/2
B.
1
C.
2
D.
None of these

Solution:

QUESTION: 55

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

A.
a ∗ b = a-b ∀ a, b ∈ Q
B.
C.
a ∗ b = ab² ∀ a, b ∈ Q
D.
None of the above

Solution:

QUESTION: 56

If the line x cosθ + y sinθ = 2 is the equation of a transverse common tangent to the circles x² + y² = 16 and x² + y² − − 6y + 20 = 0, then θ equals

A.
5π/6
B.
2π/3
C.
π/3
D.
π/6

Solution:

QUESTION: 57

A circle with centre at the origin and radius equal to 'a' meets the axis of x at A and B.P( α ) and Q(β) are two points on this circle so that α − β = 2γ , where γ is a constant. The locus of the point of intersection of AP and BQ is

A.
x²− y²− 2ay tan γ = a²
B.
x²+ y²+ 2ay tan γ = a²
C.
x² + y²− 2ay tan γ = a²
D.
x²− y²+ 2ay tan γ = a²

Solution:

QUESTION: 58

The medians PS of a triangle PQR is perpendicular to PQ then the value of tan P + 2tan Q is

A.
1
B.
-1
C.
2
D.
0

Solution:

QUESTION: 59

If then the maximum value of n is

A.
1
B.
5
C.
9
D.
None of these

Solution:

QUESTION: 60

Solution:

QUESTION: 61

Let two non-collinear unit vector â and b̂ and form an acute angle. A point P moves so that at any time t the position vector (where O is the origin) is given by â cos t + b̂ sin t. When P is farthest from origin O, let M be the length of and û be the unit vector along . Then,

Solution:

QUESTION: 62

A.
9/10
B.
1/10
C.
1/2
D.
1/3

Solution:

QUESTION: 63

The values of θ and λ for which the following equations (sinθ)x − (cosθ) y + (λ + 1)z = 0;(cosθ)x − (sinθ) y−λz=0;λx + (λ+1)y − cosθz=0 have non-trivial solution, is