WBJEE Maths Test - 14 - JEE MCQ

# WBJEE Maths Test - 14 - JEE MCQ

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## 75 Questions MCQ Test WBJEE Sample Papers, Section Wise & Full Mock Tests 2025 - WBJEE Maths Test - 14

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WBJEE Maths Test - 14 - Question 1

### The coefficients xp and xq in the expansion of (1+x)(p+q) (where p and q are positive integers) are

WBJEE Maths Test - 14 - Question 2

### The first three terms in the expansion of (1 + ax)n (n ≠ 0) are 1, 6x and 16x 2. Then the values of a and n are respectively

WBJEE Maths Test - 14 - Question 3

### In the expansion of (1+x)(2n+2) the maximum coefficient is :

WBJEE Maths Test - 14 - Question 4
The polar of a point P w.r.t a circle of radius a touching both x and y axis and lying in the first quadrant is x + 2y = 4a. The coordinates of P are
WBJEE Maths Test - 14 - Question 5

The limiting point of the system of co-axial circles x2+y2-6x-6y+4=0, x2+y2-2x-4y+3=0 is

WBJEE Maths Test - 14 - Question 6

Detailed Solution for WBJEE Maths Test - 14 - Question 6

WBJEE Maths Test - 14 - Question 7

Find the value of

WBJEE Maths Test - 14 - Question 8

WBJEE Maths Test - 14 - Question 9

WBJEE Maths Test - 14 - Question 10

WBJEE Maths Test - 14 - Question 11

Let f(x) be a function satisfying f ′ x = f x with f(0) = 1 and g(x) be a function that satisfies f(x) + g(x) = x2, then value of integral
is equal to

WBJEE Maths Test - 14 - Question 12

WBJEE Maths Test - 14 - Question 13

WBJEE Maths Test - 14 - Question 14
If x dy = y(dx + y dy), y > 0 and y (1) = 1, then y (-3) is equal to
WBJEE Maths Test - 14 - Question 15
Solution of cos x dy/dx+y sin x =1 is
WBJEE Maths Test - 14 - Question 16
If y = 2ax and $\frac{dy}{dx}$ = log 256 at x = 1, then a =
WBJEE Maths Test - 14 - Question 17
The number of points at which the function f(x)=|x-0.5|+|x-1|+tanx is not differentiable in (0,2) is
WBJEE Maths Test - 14 - Question 18

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

WBJEE Maths Test - 14 - Question 19
S and T are the foci of an ellipse and B is an end of the minor axis. If STB is an equilateral triangle then the eccentricity of the ellipse is
WBJEE Maths Test - 14 - Question 20

The st. line lx + my + n = 0 touches the hyperbola x2/a2 - y22/b2 = 1 if

WBJEE Maths Test - 14 - Question 21

The line y = 4x + c touches the hyperbola x2 - y2 = 1 if

Detailed Solution for WBJEE Maths Test - 14 - Question 21

We know that the line y = mx + c touches the hyperbola x 2 a 2 - y 2 b 2 = 1, then
c2 = a2m2 - b2
Here the hyperbola is x2 - y2 = 1
ie here a2 = b2 = 1
and comparing y = 4x + c with y = mx + c,
we get
m = 4
∴ c2 = 16 - 1 = 15

WBJEE Maths Test - 14 - Question 22

The equation x + ex = 0 has

Detailed Solution for WBJEE Maths Test - 14 - Question 22

Let f x = x + e x = 0.

Since f − ∞ = − ∞ and f + ∞ = ∞ ,
∴ f x = 0 has a real root.
Let the real root be α . Then f( α ) = 0.
Now , f ′ x = 1 + e x > 0, ∀ x ∈ R
∴ f x is an increasing function ∀ x ∈ R .
∴ for any other real number β ,
f β > f α or f β < f α .
But f a = 0 ; so , f β ≠ 0.
∴ f x = 0 has no other real root.
Hence, the equation has only one real root.

WBJEE Maths Test - 14 - Question 23

tan⁻1(1/7)+2tan⁻1(1/3)=

WBJEE Maths Test - 14 - Question 24

WBJEE Maths Test - 14 - Question 25
If A is a singular matrix, then Adj A is
WBJEE Maths Test - 14 - Question 26
The maximum value of x3 - 3x in [0,2] is
WBJEE Maths Test - 14 - Question 27

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

WBJEE Maths Test - 14 - Question 28

The normals to the parabola y2=4ax from the point (5a,2a) are

WBJEE Maths Test - 14 - Question 29

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

WBJEE Maths Test - 14 - Question 30
The number of five digits telephone numbers having atleast one of their digits repeated is
WBJEE Maths Test - 14 - Question 31

Five digit number divisible by 3 is formed using the digits 0, 1 , 2, 3, 4 and 5 without repetition. Total number of such numbers is

Detailed Solution for WBJEE Maths Test - 14 - Question 31

A number is divisible by 3 if and only if the sum of its digits are divisible by 3
Notice that 1 + 2 + 3 + 4 + 5 = 15, which is divisible by 3
The only other way we can have a sum of 5 digits divisible by 3 is to replace the 3 by the 0 making the sum 3 less:
1 + 2 + 0 + 4 + 5 = 12, which is divisible by 3
No other choice of 5 digits can have a sum divisible by 3, because there is no other way to make the sum 12 or 15, and we certainly can't have a sum of 9 or 18
So the number of 5-digit numbers that can be formed from the digits {1,2,3,4,5} is

Number of ways = 1 x 2 x 3 x 4 x 5 = 120
And the number of 5-digit numbers that can be formed from the digits {1, 2, 0, 4, 5} is figured this way

Number of ways = 1 x 2 x 3 x 4 x 4 = 96
Total Number of ways = 120 + 96 = 216

WBJEE Maths Test - 14 - Question 32
The number of 7 digit numbers which can be formed using the digits 1, 2, 3, 2, 3, 3, 4 is
Detailed Solution for WBJEE Maths Test - 14 - Question 32
There are 7 digits 1, 2, 3, 2, 3, 3, 4 in which 2 occurs 2 times and 3 occurs 3 times
Number of 7 digit numbers
= $\frac{7!}{2! 3!}$ = 420
WBJEE Maths Test - 14 - Question 33
A and B are events such that P(A ∪ B) = 3/4, P(A ∩ B) = 1/4, P(A̅) = 2/3, then P(A̅ ∩ B) is
WBJEE Maths Test - 14 - Question 34
There are 5 volumes of mathematics among 25 books. They are arranged on a shelf in random order. The probability that the volumes of mathematics stand in increasing order from left to right is
WBJEE Maths Test - 14 - Question 35
A card is drawn at random from a pack of cards. The probability of this being red or queen is
WBJEE Maths Test - 14 - Question 36

If two angles of a Δ A B C are 45 º and 60 º then the ratio of the smallest and the greatest sides are

WBJEE Maths Test - 14 - Question 37
If the sides of a triangle are 3,5,7, it has
WBJEE Maths Test - 14 - Question 38

If the equation x3 - ax2 + bx - a = 0 has real roots, then it must be the case that :

WBJEE Maths Test - 14 - Question 39

If α,β are the roots of the equation x2+x+1 = 0 and α/β, β/α are roots of the equation x2+px+q = 0, then p equals

WBJEE Maths Test - 14 - Question 40
A G.P. consist of an even number of terms. If the sum of all the terms is 5 times the sum of the terms occupying odd places, then the common ratio will be
WBJEE Maths Test - 14 - Question 41
If α and β (α < β) be two different real roots of the equation ax2 + bx + c = 0, then
Detailed Solution for WBJEE Maths Test - 14 - Question 41
$\text{Let}\phantom{\rule{0.5em}{0ex}}f\left(x\right)=a{x}^{2}+bx+c.$
$\text{Then},\phantom{\rule{0.5em}{0ex}}f\left(\alpha \right)=0=f\left(\beta \right).$
Also, f(x) is continuous and differentiable in [$\alpha ,\beta$] as it is a polynomial function or x.
Hence, by Rolle's theorem, there exists a k in ($\alpha ,\beta$), such that
$f\prime \left(k\right)=0⇒2ak+b=0⇒k=-\frac{b}{2a}.$
$\therefore \alpha <-\frac{b}{2a}<\beta .$
WBJEE Maths Test - 14 - Question 42
If (a, b), (c, d), (e, f) are the verices of a triangle such that a, c, e are in G.P. with common ratio r and b, d, f are in G.P. with common ratio's then the area of the triangle is
WBJEE Maths Test - 14 - Question 43
The 6th term of a G.P. is 32 and its 8th term is 128 ; then the common ratio of the G.P. is
Detailed Solution for WBJEE Maths Test - 14 - Question 43
The nth tan of a G.P is (tn) = arn - 1
t6 = a.r5 = 32
t8 = a.r7 = 128
$\frac{{r}^{5}}{{r}^{7}}$ = $\frac{32}{128}$
or $\frac{{1}^{}}{{r}^{2}}$ = $\frac{1}{4}$
or r2 = (2)2
or r = 2
WBJEE Maths Test - 14 - Question 44
The sum of 10 terms of the series √2 + √6 + √18 +.... is
WBJEE Maths Test - 14 - Question 45
In a city 20 percent of the population travels by car, 50 percent travels by bus and 10 percent travels by both car and bus. Then persons travelling by car or bus is
WBJEE Maths Test - 14 - Question 46
Set A has 3 elements and set B has 4 elements. The number of injections that can be defined from A to B is
Detailed Solution for WBJEE Maths Test - 14 - Question 46
No. of injections = 4P3 = 24
WBJEE Maths Test - 14 - 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
WBJEE Maths Test - 14 - Question 48
The equation of line passing through (a,b) and perpendicular to ax+by+c=0 is
WBJEE Maths Test - 14 - Question 49
If $\alpha ,\beta ,\gamma$ are the roots of ${x}^{3}+2{x}^{2}-3x-1=0,\phantom{\rule{0.5em}{0ex}}\text{then}\phantom{\rule{0.5em}{0ex}}{\alpha }^{-2}+{\beta }^{-2}+{\gamma }^{-2}=$
WBJEE Maths Test - 14 - Question 50

If cos2θ=(√2+1)(cosθ-(1/√2)), the value of θ is

WBJEE Maths Test - 14 - Question 51

Let f be real valued function satisfying and φ x = ∫ x x + 8 f t d t , then φ ′ x is

WBJEE Maths Test - 14 - Question 52

WBJEE Maths Test - 14 - Question 53

If α , β and γ are the altitudes of the Δ A B C from the vertices A, B and C respectively, then the value of is

WBJEE Maths Test - 14 - Question 54

If f(x) is an even and differentiable function, then the value of

WBJEE Maths Test - 14 - Question 55

If the line x cos θ + y sin θ = 2 is the equation of a transverse common tangent to the circles x 2 + y 2 = 16 and , then θ equals

WBJEE Maths Test - 14 - Question 56

If two events A and B are such that P(Ac) = 0.3, P(B) = 0.4 and , then is equal to

WBJEE Maths Test - 14 - Question 57

The distance between the chords of contact of the tangents to x 2 + y 2 + 2 g x + 2 f y + c = 0 from (0, 0) and (g, f) is

WBJEE Maths Test - 14 - Question 58

Area bounded by , x-axis and ordinates x = 0 and x = 3 2 is

WBJEE Maths Test - 14 - Question 59

For any two complex numbers z1, z2 and a, b ∈ R,

WBJEE Maths Test - 14 - Question 60

Let F denote the set of all onto functions from A = { a 1 , a 2 , a 3 , a 4 } to B = { x , y , z } . A function f is chosen at random from F. The probability that f − 1 consists of exactly one element is

WBJEE Maths Test - 14 - Question 61

Let a and b be non-zero real numbers. Then, the equation (ax2 + by2 + c) (x2 - 5xy + 6y2) = 0 represents

WBJEE Maths Test - 14 - Question 62

The values of a, b, c for which the equation has infinitely many solutions are

WBJEE Maths Test - 14 - Question 63

The function defined by , Where [ ] denotes greatest integer function satisfies

WBJEE Maths Test - 14 - Question 64

Tangents drawn from the point P(1,8) to the circle
x 2 + y 2 − 6 x − 4 y − 11 = 0
touch the circle at the points A and B. The equation of the circumcircle of the triangle PAB is

WBJEE Maths Test - 14 - Question 65

The number of real roots of the equation

*Multiple options can be correct
WBJEE Maths Test - 14 - Question 66

If n is the number of positive integral solutions of X1 X2 X3 X4 = 210 then

*Multiple options can be correct
WBJEE Maths Test - 14 - Question 67

The equation sin x = [ 1 + sin x ] + [ 1 − cos x ] has
{where [x] is the greatest integer less then or equal to x }

*Multiple options can be correct
WBJEE Maths Test - 14 - Question 68

Let a, b, c, d real numbers. Suppose that all the roots of the equation z4 + az3 + bz2 + cz + d = 0 are complex numbers lying on the circle |z| = 1 in the complex plane. The sum of the reciprocals of the roots is necessarily

*Multiple options can be correct
WBJEE Maths Test - 14 - Question 69

In a certain culture of bacteria, the rate of increase is proportional to the number present. It if be known that the number doubles in 4 hours, then

*Multiple options can be correct
WBJEE Maths Test - 14 - Question 70

Two lines 0 where a , b ∈ C − { 0 } and c , d ∈ R , are:

*Multiple options can be correct
WBJEE Maths Test - 14 - Question 71

All points lying inside the triangle formed by the points (1, 3), (5, 0) and (-1, 2) satisfy

*Multiple options can be correct
WBJEE Maths Test - 14 - Question 72

Let A(z1), B(z2), C(z3) be the vertices of a triangle in the Argand plane. then which of the following statements is correct?

*Multiple options can be correct
WBJEE Maths Test - 14 - Question 73

Let f x = max { 1 + sin x , 1 , 1 − cos x } , x ∈ [ 0 , 2 π ] and g x = max { 1 , | x − 1 | } ∀ x ∈ R then,

*Multiple options can be correct
WBJEE Maths Test - 14 - Question 74

*Multiple options can be correct
WBJEE Maths Test - 14 - Question 75

If P(1, 2), Q(4, 6), R(5, 7) and S(a, b) are the vertices of a parallelogram PQRS, then

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