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WBJEE Maths Test - 4

75 Questions MCQ Test WBJEE Sample Papers, Section Wise & Full Mock Tests | WBJEE Maths Test - 4

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Attempt WBJEE Maths Test - 4 | 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

If the coefficients of 5th, 6th and 7th terms in the expansion of (1 + x)ⁿ, n ∈ N, are in A.P., then n is equal to

A.
5
B.
6
C.
7
D.
7 or 14

Solution:

QUESTION: 2

The angle between tangents drawn from origin to the circle (x-7)² + (y+1)²=25 is

A.
π/3
B.
π/6
C.
π/2
D.
0

Solution:

QUESTION: 3

If the co-efficients of rth and (r+1)th terms in the expansion of (3+7x)²⁹ are equal, the r is equal to

A.
15
B.
21
C.
-21
D.
14

Solution:

QUESTION: 4

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: 5

If = a + ib , then a and b are

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

Solution:

QUESTION: 6

Circles x²+y²-6x-2y+1=0 and x²+y²+2x-8y+13=0

A.
do not touch each other
B.
do not intersect
C.
intersect at two points
D.
touch each other

Solution:

QUESTION: 7

For what values of x and y ,the complex numbers 9y² - 4 - 10xi and 8y² + 20i⁷ are conjugate to each other ?

A.
x = -2,y = 2
B.
x = -2,y = -1
C.
x = 2,y = 2
D.
x = 2,y = -2

Solution:

QUESTION: 8

The area bounded by two curves y²=4ax and x²=4ay is

A.
(16/3)a² sq.unit
B.
(16/3)sq.unit
C.
(32/3)sq.unit
D.
(32/3)a² sq.unit

Solution:

The curves y² = 4ax and x² = 4ay intersects at points where (x²/4a)²=4ax

QUESTION: 9

If a < 0 < b then

A.
b-a
B.
a-b
C.
a+b
D.
0

Solution:

QUESTION: 10

If a ≠ 6, b, c satisfy = 0 then abc =

A.
a + b + c
B.
0
C.
b³
D.
ab + bc

Solution:

QUESTION: 11

A.
50π
B.
25π
C.
75π
D.
150π

Solution:

QUESTION: 12

A.
0.25
B.
2.5
C.
5.2
D.
0.52

Solution:

QUESTION: 13

A.
B.
xyz
C.
D.

Solution:

QUESTION: 14

The solution of the equation ydx-2xdy=0 gives parabolas whose common axis and the tangent at the vertex respectively, are :

A.
x=c,y=0
B.
y=0,x=0
C.
x=0,y=0
D.
x=c,y=0

Solution:

QUESTION: 15

The solution of the equation xdy-ydx=0 is

A.
Rectangular hyperbola
B.
Straight line passing (0,0)
C.
Parabola with vertex at (0,0)
D.
Circle with centre at (0,0)

Solution:

QUESTION: 16

If f(x)=log_x²log(x), then at x=e, f'(x)=

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

Solution:

QUESTION: 17

For real values of x the minimum value of [(1-x+x²)/(1+x+x²)] is

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

Solution:

QUESTION: 18

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

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

Solution:

QUESTION: 19

If the two intersecting lines intersect the hyperbola and neither of them is a tangent to it, then number of intersecting points are

A.
1
B.
2
C.
2, 3 or 4
D.
2 or 3

Solution: The general equation of a straight line is linear: ax + by + c = and the general equation of a hyperbola (but it's the same for all conics) is a second degree equation:
1. Ax² + Bxy + Cy² + Dx + Ey + F = 0
So, finding y ( or x ) from the first equation and substituting in the second, we have a second degree equation that has, at most, two real solutions. The possible cases are:
1) two real distinct solutions: the line is secant at two points
2) two coincident solution: the line is tangent
3) no real solution : the line is e''external''
4) the equation reduce to a first degree equation so we have one real solution and we say that the other solution goes to infinity.
So, for two lines that are not tangent we can have:
no common point with the conic if the two lines are external,
2 points if one is secant at two pints and the other is external or if both have one real common point with the conic an the other at infinity,
3 common points if one is secant at two points and the other at one point,
4 common points if the two lines are both secants at two points.

QUESTION: 20

The eccentricity of the ellipse 4x² + 9y² + 8x + 36y + 4 = 0 is

A.
5/6
B.
3/5
C.
√2/3
D.
√5/3

Solution:

QUESTION: 21

If φ (x) = f (x) + f (1 − x) and f ″ (x) < 0 in − 1,1 , then φ (x) strictly increases in the interval

A.
B.
C.
(-1,0)
D.
(0,1)

Solution:

QUESTION: 22

If sin⁻¹x + sin⁻¹y = 2π/3, then cos⁻¹ x + cos⁻¹ y is equal to

A.
2π/3
B.
π/3
C.
π/6
D.
π

Solution:

QUESTION: 23

if f : R→R be such that f(1) = 4 and f^'(1) = 12 then

A.
e²
B.
C.
D.
e³

Solution:

QUESTION: 24

If each element of a 3 x 3 matrix is multiplie by 3, then the determinant of the newly formed matrix is

A.
3 det A
B.
9 det A
C.
27 det A
D.
(det A)³

Solution:

QUESTION: 25

The minimum value of f(x) = x⁴-x²-2x+6 is

A.
6
B.
8
C.
4
D.
nonexistent

Solution: f'(x) = 4x³ - 2x - 2
∴ f'(x) = 0 ⇒ 2x³ - x - 1 = 0 ⇒ (x - 1)(2x² + 2x + 1) = 0
∴ x = 1.
Now, f" (x) = 12x² - 2 ∴ f" (1) = 12 - 2 > 0.
Hence, global min f(x) = 1 - 1 - 2 + 6 = 4.

QUESTION: 26

The HCF of 155 and 1385 is

A.
5
B.
15
C.
25
D.
none of these

Solution:

QUESTION: 27

The focus of the parabola x²-8x+2y+7=0 is

A.
(0,-1/2)
B.
(4,9/2)
C.
(4,4)
D.
(-4,-9/2)

Solution:

QUESTION: 28

The tangents at the points ( ,2at₁) , ( , 2at₂) on the parabola y² = 4ax are at right angles if

A.
t₁t₂=-1
B.
t₁t₂=1
C.
t₁t₂=2
D.
t₁t₂=-2

Solution:

Tangent at t₁ is yt₁ = x +
Tangent at t₂ is yt₂ = x +
Tangents are at right angles
⇒ t₁t₂ = - 1

QUESTION: 29

In how many different ways can the letters of the word BOOKLET be arranged such that B and T always come together?

A.
360
B.
720
C.
480
D.
5040

Solution:

QUESTION: 30

The equation 16x² + y² + 8xy − 74 x − 78y + 212 = 0 represents

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

Solution:

QUESTION: 31

The numbers of all words formed from the letters of the word CALCUTTA is

A.
2520
B.
5040
C.
10,080
D.
40,320

Solution:

No of letter in the given word is 8
No of repeated word is "C" 2 times, "A" 2 times and "T" 2 times. So the equation is

QUESTION: 32

If A, B, C are three mutually exclusive and exhaustive events of a trial such that P(A) = 2P(B) = 3P(C) then P(A) =

A.
1/3
B.
2/5
C.
3/8
D.
6/11

Solution:

QUESTION: 33

The number of 4-digit even numbers that can be formed using 0, 1, 2, 3, 4, 5, 6 without repetition is

A.
120
B.
300
C.
420
D.
20

Solution:

Each even number must have 0, 2, 4 or 6 in is units place
Here total number of digits = 7

When 0 occurs at units place there is no restriction on other places and when 2 or 4 occurs at units place there is restriction on thousands' place as 0 can not be put at thousands' place
Case I When 0 occurs at units place:

∴ The number of numbers formed in this case

Case II. When 0 does not occur at units place:

The units' place can be filled up by any one of the three digits 2, 4 and 6 in 3 ways
∴ The number of numbers formed in this case

∴ The required number = 120 + 300 = 420

QUESTION: 34

If A and B are two independent events in a sample space, then P(A̅/B̅) is equal to

A.
1-P(A/B)
B.
1-P(A̅/B)
C.
1-P(B)
D.
1-P(A)

Solution:

QUESTION: 35

The probability of a problem being solved by two students are 1/2 and 1/3 . The probability of the problem being solved is

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

Solution:

QUESTION: 36

The diameter of the circumcircle of a triangle with sides 5 cm, 6 cm and 7 cm is

A.
B.
C.
35/48 cm
D.
none of these

Solution:

QUESTION: 37

Let |x| and [x] denote the fractional and integral part of a real number x respectively. Solution of 4 |x|=x+[x] are

A.
2,0
B.
-5/3,0,5/3
C.
4/3,0
D.
2/3,0

Solution:

QUESTION: 38

If the ratio of the roots of the equation x² + px + q = 0 be same as that of the roots of the equation x² + lx + m = 0 then

A.
p²m = l²q
B.
m²p = q²l
C.
pm = ql
D.
p²m + q²l = 0

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 a ΔABC ,

A.
a
B.
b
C.
c
D.
0

Solution:

QUESTION: 41

if the 10th term of a G.P.is 9 and 4th term is 4, then its 7th term is

A.
6
B.
36
C.
4 9
D.
9 4

Solution:

The terms of a G.P. are of form a, ar, ar², ar³ ,.... ar⁹ = 9
So, r⁹ = 9 a .....(1)
ar³ = 4 .....(2)
Cubing both sides of (2), we get,
a³r⁹ = 64
putting value of r⁹ in this, we get,

QUESTION: 42

The sum of the series 1 + 1.3/6 + 1.3.5/6.8 + ....∞ is

A.
1
B.
0
C.
4
D.
∞

Solution:

QUESTION: 43

((1)/(1 x 2)) + ((1/2) x (3)) + ((1/3) x (4)) + ... + ((1)/(n(n + 1))) equals

A.
((n + 1)/n)
B.
(n(n + 1)/6)
C.
(n/(n + 1))
D.
(n²/(n + 1))

Solution:

QUESTION: 44

If the sum of first n positive intergers is 1/5 times the sum of their squares, then n equals :

A.
5
B.
6
C.
7
D.
8

Solution:

QUESTION: 45

The quadratic function f (x) whose graph goes through the points (-1, 4), (1, 0) and (2, 1) is

A.
2x² + 3x + 1
B.
x² - 2x + 1
C.
2x² - 3x + 1
D.
- 2x² - 3x + 1

Solution:

QUESTION: 46

Which of the following is the empty set?

A.
{x : x is a real number and x² - 1 =0}
B.
{x : x is a real number and x² + 1 =0}
C.
{x : x is a real number and x² - 9 =0}
D.
{x : x is a real number and x² = x + 2}

Solution:

QUESTION: 47

A points moves in such a way that the square of its distance from point(3,-2) is equal to numerically its distance from the line 5x-12y=13. The equation of the locus of the point is

A.
x²+y²-11x+16y+26=0
B.
x²+y²-11x+16y=0
C.
13(x²+y²)-83x+64y+182=0
D.
x²+y²-83x+64y+182=0

Solution:

QUESTION: 48

Two vertices of a triangle are (4,-3) and (-2,5) and its ortho centre is (1,2), then its third vertex is

A.
(-33,-26)
B.
(33,26)
C.
(26,33)
D.
(-33,26)

Solution:

QUESTION: 49

If the roots of the equation mx² - 4x +2 (m+1)=0 are real, then

A.
m ≤ 1
B.
m ≥ − 2
C.
m ≥ 1 or m ≤ -2
D.
m ≥ − 2 and m ≤ 1

Solution:

Roots are real
⇒ discriminant ≥ 0

QUESTION: 50

Solution:

QUESTION: 51

Let n be a positive integer with f x = 1 ! + 2 ! + 3 ! + … + x ! and P x , Q x be polynomials in x such that
f x + 2 = P x f x + 1 + Q x f x ∀ x ≥ 1

A.
P x = x + 3
B.
P x = − x − 2
C.
Q x = − x − 2
D.
Q x = x + 3

Solution:

QUESTION: 52

In a triangle ABC with fixed base BC, the vertex A moves such that

\cos B + \cos C = 4 \sin^{2} \frac{A}{2}

.
If a, b and c denote the lengths of the sides of the triangle opposite to the angles A, B and C, respectively, then

A.
b + c = 4a
B.
b + c = 2a
C.
Locus of point A is an ellipse
D.
Locus of point A is a pair of straight lines

Solution:

QUESTION: 53

From a point p on the circle shown with centre O, the chord PA = 8cm is drawn. The radius of the circle is 24 cm let PB be drawn parallel to OA. Suppose BO extended meet PA extended at M. The length of MA equals (in cm).

A.
16
B.
8
C.
10
D.
9

Solution:

QUESTION: 54

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

A.
π/2
B.
C.
π
D.

Solution:

QUESTION: 55

The vertices A and C of a square ABCD are 2 + 3i and 3 − 2i respectively, then the vertices B and D are given by

A.
(0, 0) and (5, 1)
B.
(0, 0) and (4, 0)
C.
(1, 0) and (5, 1)
D.
(1, 0) and (4, 0)

Solution:

QUESTION: 56

Let f be real valued function such that f (2) = 2 and f ′ (2) = 1 , then equals

A.
6
B.
32
C.
12
D.
8

Solution:

QUESTION: 57

The value of f (0) , so that the function f defined as

is continuous at x = 0, is given by

A.
2/3
B.
6
C.
2
D.
4

Solution:

QUESTION: 58

equals

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

Solution:

QUESTION: 59

The ratio of the sum of 10^th and 12^th term of an AP are in the ratio 25 : 36. The ratio of the 31^st term to 29^th term can be

A.
57 : 61
B.
61 : 57
C.
5 : 6
D.
6 : 5

Solution:

QUESTION: 60

For any real is a point on the hyperbola x² - y² = 1. FInd the area bounded by this hyperbola and the line joining its centre to the points corresponding to t₁ and -t₁.