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WBJEE Maths Test - 1

75 Questions MCQ Test WBJEE Sample Papers, Section Wise & Full Mock Tests | WBJEE Maths Test - 1

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Attempt WBJEE Maths Test - 1 | 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 coefficient of x³ in ((√x⁵) + (3/√x³))⁶ is

A.
0
B.
120
C.
420
D.
540

Solution:

QUESTION: 2

If the coefficient of x⁷ in [ax²+(1/bx)]¹¹ equals the coefficient of x^-7 in [ax-(1/bx²)]¹¹, then a and b satisfy the relation

A.
a-b = 1
B.
a+b = 1
C.
a/b = 1
D.
ab = 1

Solution:

QUESTION: 3

The third term in the expansion of ((x²) - (1/x³))ⁿ is independent of x, when n is equal to

A.
2
B.
3
C.
4
D.
5

Solution:

QUESTION: 4

The equation of the circle passing through intersection of circles x²+y²-1=0 and x²+y²-2x-4y+1=0 and touching x+2y=0 is

A.
x²+y²-x-2y=0
B.
x²+y²-x+2y=0
C.
x²+y²+x-2y=0
D.
x²+y²+x+2y=0

Solution:

QUESTION: 5

The square root of the number 5 + 12i is

A.
(3 + 2i)
B.
(3 - 2i)
C.
(3 + 2i)
D.
none

Solution:

QUESTION: 6

The equation of circle which passes through (4,5) and whose centre is (2,2) is

A.
x²+y²+4x+4y-5=0
B.
x²+y²-4x-4y-5=0
C.
x²+y²-4x=13
D.
x²+y²-4x-4y+5=0

Solution:

QUESTION: 7

The volume of a solid which is obtained by revolving area bounded by an ellipse x²+9y²=9 and straight line x+3y=3 about y-axis is

A.
3π
B.
4π
C.
6π
D.
9π

Solution:

QUESTION: 8

A.
B.
C.
π/8
D.
π/16

Solution:

QUESTION: 9

A.
2π
B.
π/2
C.
0
D.
− π

Solution:

QUESTION: 10

A.
0
B.
N
C.
2N + 1
D.
N + 2

Solution:

QUESTION: 11

The degree of the differential equation d²y/dx²+(dy/dx)³+6y=0 is

A.
1
B.
3
C.
2
D.
5

Solution:

QUESTION: 12

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: 13

Differential of x⁶ w.r. to x³ is equal to

A.
6x³
B.
3x²
C.
2x³
D.
x³

Solution:

QUESTION: 14

The solution of the differential equation 2x dy/dx-y=3 represent

A.
st.lines
B.
circles
C.
parabola
D.
ellipse

Solution:

QUESTION: 15

If |z| = 4 and arg z = (5π/6), then the value of z is

A.
2√3 - 2i
B.
2√3 + 2i
C.
-2√3 + 2i
D.
-√3 + i

Solution:

QUESTION: 16

If x = a cos³ θ, y = a sin³ θ, then
√[1 + (dy/dx)²] =

A.
tan² θ
B.
sec² θ
C.
sec θ
D.
∣sec θ∣

Solution:

QUESTION: 17

The eccentricity of the conic 9x² + 25y² = 225 is

A.
2/5
B.
4/5
C.
1/3
D.
1/5

Solution:

QUESTION: 18

The graph represented by the equations x = sin²t,y = 2cost is

A.
A portion of a parabola
B.
A parabola
C.
A part of sine graph
D.
A part of a hyperbola

Solution:

QUESTION: 19

If the major axis of an ellipse is thrice the minor axis, then its eccentricity is equal to

A.
1/3
B.
1/√3
C.
1/√2
D.
(2√2)/3

Solution:

QUESTION: 20

The curve represented by x = a (coshθ + sinhθ) , y = b(coshθ − sinhθ) is

A.
A hyperbola
B.
An ellipse
C.
A parabola
D.
A circle

Solution:

QUESTION: 21

The least possible value of k for which the function f(x) = x² + kx + 1 may be increasing on [1,2] is

A.
2
B.
-2
C.
0
D.
none of these

Solution:

We have , f x = x 2 + k x + 1
⇒ f ′ x = 2 x + k . Also , f ″ x = 2
Now , f ″ x = 2, ∀ x ∈ [ 1,2 ]
⇒ f ″ x > 0, ∀ x ∈ [ 1,2 ]
⇒ f ′ x is an increasing function in the interval [ 1,2 ] .
⇒ f ′ 1 is the least value of f′(x) on [ 1,2 ]
But f ′ x > 0 ∀ x ∈ [ 1,2 ]
[ ∵ f x is increasing on [1,2] ]
∴ f ′ 1 > 0, ∀ x ∈ [ 1,2 ] ⇒ k > − 2
Thus, the least value of k is -2

QUESTION: 22

A.
1
B.
e
C.
e^-1
D.
0

Solution:

QUESTION: 23

The system of equations
αx+y+z = α-1
x+αy+z = α-1
x+y+αz, = α-1 has no solution, if α is

A.
-2
B.
either -2 or 1
C.
not -2
D.
1

Solution:

QUESTION: 24

A.
e
B.
e+1
C.
e-1
D.
2e+1

Solution:

QUESTION: 25

If z=1+i,then the multiplicative inverse of z² is

A.
1-i
B.
i/2
C.
-i/2
D.
2i

Solution:

QUESTION: 26

If a, b, c and d are four positive real numbers such that abcd = 1, what is the minimum value of (1 + a)(1 + b)(1 + c)(1 + d)?

A.
4
B.
1
C.
16
D.
18

Solution:

QUESTION: 27

The equation of the parabola whose vertex is at (2,-1) and focus at (2,-3) is

A.
x²+4x-8y-12=0
B.
x²-4x+8y+12=0
C.
x²+8y=12
D.
x²-4x+12=0

Solution:

QUESTION: 28

sin[(1/2)cos⁻¹(4/5)] is equal to

A.
-(1/√10)
B.
1/√10
C.
-(1/10)
D.
1/10

Solution:

QUESTION: 29

The pole of the line 2x + 3y − 4 = 0 with respect to the parabola y² = 4x is

A.
(4, 6)
B.
− 2, − 3
C.
(1, 1)
D.
2, − 3

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

Number of triangles formed by joining 12 points, no three of which are in the same straight line except 7 of them which are in a straight line, is

A.
185
B.
220
C.
792
D.
none of these

Solution:

QUESTION: 32

The probability of drawing a card which is atleast a spade or a king from a well shuffled pack of cards is

A.
1/13
B.
4/13
C.
1/4
D.
13/51

Solution:

Let A be the event that the card drawn is spade,
B be the event that card drawn is king and S be the sample space,
When a card is drawn from a pack of cards.
Now n(S) = ⁵²C₁ = 52, n(A) = ¹³C₁ = 13, n(B) = ⁴C₁ = 4, n(A ∩ B) = ¹C₁ = 1

QUESTION: 33

A speak the truth in 75% cases and B in 80% cases. In what percent of cases are they likely to contradict each other narring the same incident is

A.
30
B.
35
C.
40
D.
45

Solution:

QUESTION: 34

If ⁵⁶P_r+6 : ⁵⁴P_r+3=30800 : 1, then

A.
r=41
B.
40
C.
39
D.
38

Solution:

∵ 56Pr + 6 : 54Pr + 3 = 30800 : 1

⇒ 56! / (50 - r)! = (30800 * 54!) / (51-r!)

⇒ 56 x 55 = 30800/(51 - r)

⇒ 51 - r = 10

∴ r = 41

QUESTION: 35

In (in terms of x , y , z only) is

A.
xyz
B.
2xyz
C.
-xyz
D.
1/2xyz

Solution:

QUESTION: 36

A.
Δ/s
B.
1/r
C.
2/r
D.
3/r

Solution:

QUESTION: 37

What is the probability of getting a sum 9 from two throws of dice?

A.
1/6
B.
1/8
C.
1/9
D.
1/12

Solution:

QUESTION: 38

If α and β are the roots of x² + px + 1 = 0 and γ and δ are the roots of x² + qx + 1 = 0, the value of (α - γ)(β - γ)(α + δ)(β + δ)

A.
p² - q²
B.
q² - p²
C.
p²
D.
q²

Solution: x² + px + 1 = 0 has roots α, β
x² + qx + 1 = 0 has roots γ, δ
Using the sum and product of roots formulae, we'll get
α + β = -p ; αβ = 1
γ + δ = -q ; γδ = 1
(α - γ)(β - γ)(α + δ)(β + δ)
= (α - γ)(β + δ)(β - γ)(α + δ)
= (αβ + αδ - βγ - γδ)(αβ - βδ - αγ - γδ)
We found out that αβ = γδ = 1
Using these, we'll have
= (1 + αδ - βγ - 1)(1 + βδ - αγ - 1)
= (αδ - βγ)(βδ - αγ)
= αβ.δ² - γδ.α² - γδ.β² + αβγ²
Again, using αβ = γδ = 1
= δ² - α² - β² + γ²
= γ² + δ² - (α² + β²)
Now, 'completing the square' will be used
= [(γ + δ)² - γδ] - [(α + β)² - αβ]
Using α + β = -p, γ + δ = -q, αβ = γδ = 1
= (q² - 1) - (p² - 1)
= q² - p²

QUESTION: 39

If the equation (λ-1)x² + (λ+1)x+(λ-1)=0 has real roots, then λ can have any value in the interval ,

A.
(1/3,2)
B.
(0,3)
C.
(-3,3)
D.
(1/3,3)

Solution:

QUESTION: 40

The sum of the series (1/2) + (1/3) + (1/6) + .... upto 9 terms is

A.
(-5/6)
B.
-(1/2)
C.
1
D.
-(3/2)

Solution:

QUESTION: 41

Let A₁, A₂ be two AMs and G₁, G₂ be two GMS between a and b then (A₁ + A₂)/(G₁ G₂) =

A.
((a + b)/2ab)
B.
(2ab/(a + b))
C.
((a + b)/ab)
D.
((a + b)/√ab)

Solution:

QUESTION: 42

Given that f(x) is continuously differentiable on a ≤ x ≤ b where a < b , f a < 0 and f b > 0, which of the following are always true ?
(i) f(x) is bounded on a ≤ x ≤ b .
(ii) The equation f(x) = 0 has at least one solution in a < x < b .
(iii) The maximum and minimum values of f(x) on a ≤ x ≤ b occur at points where f ′ c = 0
(iv) There is at least one point c with a < c < b where f ′ c > 0.
(v) There is at least one point d with a < d < b where f ′ c < 0.

A.
Only (ii) and (iv) are true
B.
All but (iii) are true
C.
All but (v) are true
D.
Only (i), (ii) and (iv) are true

Solution:

(i) This statement is true, every continuous function is bounded on a closed interval.
(ii) True again, by Intermediate Value Theorem.
(iii) Not ture, because maximum and/or minimum value could also occur at a or b, without the derivatives being O.
(iv) True By the Mean Value Theorem, there exists a point betweem a and b, where the derivative is exactly a clearly positive value.
(v) Not always true, for example, the function might be strictly increasing guaranteeing the derivative to be always positive.
Thus, the true statements are (i), (ii) and (iv).

QUESTION: 43

If the sum of the first 2n terms of the A.P. 2, 5, 8,... is equalto the sum of the first n terms of the A.P. 57, 59, 61,.... then n equals

A.
10
B.
12
C.
11
D.
13

Solution:

QUESTION: 44

If f (x) = {2x - 3, x ≤ 2} {x, x < 2} then f (1) is equal to

A.
2 f (2)
B.
f (2)
C.
-f (2)
D.
(1/2)f (2)

Solution:

QUESTION: 45

If a, b, c, d, e, f are in A.P., then e - c is equal to

A.
2(c - a)
B.
2(f - d)
C.
2(d - c)
D.
d - c

Solution:

Given a, b, c, d, e, f are in A.P.

Let p be the common difference.
b = a+p
c = a+2p
d = a+3p
e = a+4p
f = a+5p
d-c = a+3p – a-2p = p
e-c = a+4p-(a+2p) = 2p
= 2(d-c)

QUESTION: 46

The domain of definition of the function y (x) given by the equation 2^x + 2^y = 2 is

A.
0 < x ≤ 1
B.
0 ≤ x ≤ 1
C.
- ∞ < x ≤ 0
D.
- ∞ < x < 1

Solution:

QUESTION: 47

A line passing through point A(-5,-4) meet other three lines x + 3y + 2 = 0, 2x + y + 4 = 0 and x - y - 5 = 0 at B, C and D respectively. If ( 15 AB )² + ( 10 AC )² = ( 6 AD )², then the equation of line is

A.
2x+3y+22=0
B.
5x-4y+7=0
C.
3x-2y+3=0
D.
3x+2y+3=0

Solution:

Equation of any line through A(-5, -4) is
= r(say)
then the coordinates of any point on this line at a distance r from A are
(r cosθ - 5, r sinθ - 4)
If AB = r₁, AC = r₂, AD = r₃
then (r₁ cosθ - 5, r₁ sinθ - 4) lies on x + 3y + 2 = 0
⇒ r₁ cosθ - 5 + 3 (r₁ sinθ - 4) + 2 = 0
⇒ r₁ (cosθ + 3 sinθ) = 15
⇒ r₁ =

Therefore according to the given condition
(cosθ + 3 sinθ)² + (2 cosθ + sinθ)²
= (cosθ - sinθ)²
⇒ 4 cos²θ + 9 sin²θ + 12 sinθ cosθ = 0
⇒ (2 cosθ + 3 sinθ)² = 0
⇒ 2 + 3 tanθ = 0
⇒ θ = -
Hence the required equation of the line is
y + 4 = - (x + 5)
or 2x + 3y + 22 = 0

QUESTION: 48

The ortho centre of the triangle formed by the lines x = 3, y = 4 and 3x + 4y = 6 is

A.
(0,0)
B.
(3,0)
C.
(0,4)
D.
(3,4)

Solution:

QUESTION: 49

If α₁ , α₂ … α_n be the roots of the equation is equal to_____

Solution:

QUESTION: 50

If a is any real number, the number roots of cot x − tan x = a in the first quadrant is

A.
2
B.
0
C.
1
D.
none of these

Solution:

QUESTION: 51

Evaluate the differentiation of

A.
1
B.
tan^-1⁡x
C.
0
D.
-1

Solution:

Explanation:

QUESTION: 52

If the normals at (x_i, y_i), i = 1,2,3,4 on the rectangular hyperbola xy = c² meet the point α , β then

Solution:

QUESTION: 53

Let = 1 and a x + b y = 1 be two variable lines and a, b be parameters such that a² + b² = ab then the locus of point of intersection of the lines is

A.
x² + y² + xy - 1 = 0
B.
x² + y² - xy + 1 = 0
C.
x² + y² + xy + 1 = 0
D.
x² + y² - xy - 1 = 0

Solution:

QUESTION: 54

The determinant is equal to zero if

A.
a, b, c are in A.P.
B.
a, b, c are in G.P.
C.
a, b, c are in H.P.
D.
None of these

Solution:

QUESTION: 55

The locus of the point of intersection of the lines (t is a parameter) is a hyperbola with eccentricity

A.
√3
B.
2
C.
D.
4/3

Solution:

QUESTION: 56

{x} and [x] represent fractional and integral part of x, then is equal to

A.
[x]
B.
x
C.
2007 {x}
D.

Solution:

QUESTION: 57

If 0 < x < 1, then [{x cos (cot^-1 x) + sin (cot^-1 x)}² - 1]^1/2 =

A.
B.
x
C.
D.

Solution:

QUESTION: 58

If p, q, r are positive integers and ω be an imaginary cube roots of unity and

A.
ω
B.
− ω²
C.
0
D.
None of these

Solution:

QUESTION: 59

The sum of the infinite terms of the series tan⁻¹ + tan⁻¹ + tan⁻¹ + … is equal to

A.
π/6
B.
π/3
C.
π/4
D.
π/2

Solution:

QUESTION: 60

If are non-coplanar non-zero vectors and is any vector in space then,
is equal to

Solution:

QUESTION: 61

(ο) is a binary function such that aοb = a-b+1 where a,b∈z then the given binary function is

A.
commutative and associative
B.
commutative but not associative
C.
associative but not commutative
D.
neither associative nor commutative

Solution:

QUESTION: 62

If a ∗ b = a+b-2 and if x ∗ 3 = 7 then what is the value of x^-1?

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

Solution:

QUESTION: 63

The system of confocal conics
being an arbitary constant

A.
Is self-orthogonal
B.
Has its orthogonal trajectory as
C.
Has its orthogonal trajectory as
D.
Has its orthogonal trajectory as

Solution:

QUESTION: 64

If be any point on a line then the range of values of t for which the point P lies between the parallel lines x + 2y = 1 and 2x + 4y = 15 is:

A.
B.
C.
D.
None of these

Solution:

QUESTION: 65

If , then the value of x is

A.
4
B.
log 8
C.
log 4
D.
log 2

Solution:

*Multiple options can be correct

QUESTION: 66

Which of the following expression are meaningful ?

Solution:

*Multiple options can be correct

QUESTION: 67

The polynomial expression (x³ + ax² + bx + c) is divisible by (x+ 1)², then

A.
a − b + c = 1
B.
2 a − b = 3
C.
a + b + c = 1
D.
None of these

Solution:

*Multiple options can be correct

QUESTION: 68

If y = f(x) be the equation of a parabola (the axis of the parabola being parallel to the y-axis) which is touched by the line y = x at x = 1, then

A.
f ′ (1) = 1
B.
f ′ (0) = f ′ 1
C.
2 f (0) = 1 − f ′ (0)
D.
f (0) + f ′ (0) + f ″ (0) = 1

Solution:

*Multiple options can be correct

QUESTION: 69

If are any two vectors then the vector bisecting the angle between

A.
B.
C.
D.
None of these

Solution:

*Multiple options can be correct

QUESTION: 70

If 8 sinθ + 7 cosθ = 8 , then the value of 7 sin θ − 8 cosθ is equal to

A.
8
B.
7
C.
-7
D.
-8

Solution:

*Multiple options can be correct

QUESTION: 71

Let be vector parallel to line of intersection of planes P₁ and P₂. Plane P₁ is parallel tp the vectors then the angle between vector and a given vector