JEE · Maths
JEE Main 2021 (24 Feb, Shift 1)
2021
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20 questions
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Q61
mcq
Let $f : \mathbf{R} \rightarrow \mathbf{R}$ be defined as $f(x) = 2x - 1$ and $g : \mathbf{R} - \{1\} \rightarrow \mathbf{R}$ be defined as $g(x) = \frac{x - \frac{1}{2}}{x - 1}$.
Then the composition function $f(g(x))$ is :
a.
one-one but not onto
Correct Answer
b.
onto but not one-one
c.
neither one-one nor onto
d.
both one-one and onto
Step-by-Step AI Explanation
Verified Method
The composition is $f(g(x)) = 2g(x) - 1 = 2\left(\frac{x - 1/2}{x - 1}\right) - 1 = \frac{2x - 1 - (x - 1)}{x - 1} = \frac{x}{x - 1} = 1 + \frac{1}{x - 1}$.
For $x_1, x_2 \in \mathbb{R} \setminus \{1\}$, $f(g(x_1)) = f(g(x_2)) \implies x_1 = x_2$, so it is one-one.
Since $\frac{1}{x - 1} \neq 0$, the range of $f(g(x))$ is $\mathbb{R} \setminus \{1\} \neq \mathbb{R}$ (the codomain). Thus, it is not onto.
Q62
mcq
Let p and q be two positive numbers such that $p + q = 2$ and $p^4 + q^4 = 272$. Then p and q are roots of the equation :
a.
$x^2 - 2x + 136 = 0$
b.
$x^2 - 2x + 16 = 0$
Correct Answer
c.
$x^2 - 2x + 8 = 0$
d.
$x^2 - 2x + 2 = 0$
Step-by-Step AI Explanation
Verified Method
We have $p^2 + q^2 = (p+q)^2 - 2pq = 4 - 2pq$.
Then $p^4 + q^4 = (p^2 + q^2)^2 - 2(pq)^2 = (4 - 2pq)^2 - 2(pq)^2 = 2(pq)^2 - 16pq + 16$.
Given $p^4 + q^4 = 272$:
$2(pq)^2 - 16pq + 16 = 272 \implies (pq)^2 - 8pq - 128 = 0 \implies (pq - 16)(pq + 8) = 0$.
Since $p, q > 0$, $pq = 16$.
The quadratic equation having roots $p$ and $q$ is $x^2 - (p+q)x + pq = 0 \implies x^2 - 2x + 16 = 0$.
Q63
mcq
The system of linear equations
$$\begin{aligned} 3x - 2y - kz &= 10 \\ 2x - 4y - 2z &= 6 \\ x + 2y - z &= 5m \end{aligned}$$
is inconsistent if :
a.
$k \neq 3, m \neq \frac{4}{5}$
b.
$k = 3, m = \frac{4}{5}$
c.
$k = 3, m \neq \frac{4}{5}$
Correct Answer
d.
$k \neq 3, m \in \mathbf{R}$
Step-by-Step AI Explanation
Verified Method
The determinant of the coefficient matrix is:
$$D = \begin{vmatrix} 3 & -2 & -k \\ 2 & -4 & -2 \\ 1 & 2 & -1 \end{vmatrix} = 3(4 + 4) + 2(-2 + 2) - k(4 + 4) = 24 - 8k$$
For inconsistency, $D = 0 \implies k = 3$.
For $k = 3$, adding the 2nd and 3rd equations: $(2x - 4y - 2z) + (x + 2y - z) = 3x - 2y - 3z = 6 + 5m$.
For the system to be consistent with the 1st equation $3x - 2y - 3z = 10$, we must have $6 + 5m = 10 \implies m = 4/5$.
Thus, the system is inconsistent if $k = 3$ and $m \neq 4/5$.
Q64
mcq
The value of
$-{}^{15}\text{C}_1 + 2\cdot {}^{15}\text{C}_2 - 3\cdot {}^{15}\text{C}_3 + \dots - 15\cdot {}^{15}\text{C}_{15} + {}^{14}\text{C}_1 + {}^{14}\text{C}_3 + {}^{14}\text{C}_5 + \dots + {}^{14}\text{C}_{11}$ is :
a.
$2^{16} - 1$
b.
$2^{13} - 14$
Correct Answer
c.
$2^{13} - 13$
d.
$2^{14}$
Step-by-Step AI Explanation
Verified Method
The first series is $\sum_{r=1}^{15} (-1)^r r \cdot {}^{15}\text{C}_r = \sum_{r=1}^{15} (-1)^r 15 \cdot {}^{14}\text{C}_{r-1} = -15 \sum_{k=0}^{14} (-1)^k {}^{14}\text{C}_k = -15(1-1)^{14} = 0$.
The second series is ${}^{14}\text{C}_1 + {}^{14}\text{C}_3 + {}^{14}\text{C}_5 + \dots + {}^{14}\text{C}_{11} = \sum_{k=1, 3, \dots, 13} {}^{14}\text{C}_k - {}^{14}\text{C}_{13} = 2^{14-1} - 14 = 2^{13} - 14$.
Thus, the total sum is $0 + 2^{13} - 14 = 2^{13} - 14$.
Q65
mcq
If $e^{\left(\cos^2 x + \cos^4 x + \cos^6 x + \dots \infty\right)\log_e 2}$ satisfies the equation $t^2 - 9t + 8 = 0$, then the value of $\frac{2\sin x}{\sin x + \sqrt{3}\cos x} \left(0 < x < \frac{\pi}{2}\right)$ is :
a.
$\frac{1}{2}$
Correct Answer
b.
$\sqrt{3}$
c.
$\frac{3}{2}$
d.
$2\sqrt{3}$
Step-by-Step AI Explanation
Verified Method
The sum in the exponent is an infinite GP: $S = \frac{\cos^2 x}{1 - \cos^2 x} = \cot^2 x$.
So $t = e^{S \ln 2} = 2^{\cot^2 x}$.
The equation $t^2 - 9t + 8 = 0$ gives $t = 1$ or $t = 8$.
Since $0 < x < \pi/2$, $\cot^2 x > 0$, so $t = 8 \implies 2^{\cot^2 x} = 2^3 \implies \cot^2 x = 3 \implies \cot x = \sqrt{3} \implies x = \pi/6$.
Then $\frac{2\sin x}{\sin x + \sqrt{3}\cos x} = \frac{2(1/2)}{1/2 + \sqrt{3}(\sqrt{3}/2)} = \frac{1}{1/2 + 3/2} = \frac{1}{2}$.
Q66
mcq
$\lim_{x \to 0} \frac{\int_{0}^{x^2} (\sin\sqrt{t})\,\mathrm{d}t}{x^3}$ is equal to :
a.
$\frac{2}{3}$
Correct Answer
b.
$\frac{3}{2}$
c.
$\frac{1}{15}$
d.
$0$
Step-by-Step AI Explanation
Verified Method
Applying L'Hôpital's Rule and the Leibniz integral rule:
$$\lim_{x \to 0} \frac{\frac{\mathrm{d}}{\mathrm{d}x}\int_0^{x^2} \sin\sqrt{t}\,\mathrm{d}t}{\frac{\mathrm{d}}{\mathrm{d}x}(x^3)} = \lim_{x \to 0} \frac{(\sin\sqrt{x^2}) \cdot 2x}{3x^2} = \lim_{x \to 0} \frac{2x \sin x}{3x^2} = \frac{2}{3} \lim_{x \to 0} \frac{\sin x}{x} = \frac{2}{3}$$
Q67
mcq
The function $f(x) = \frac{4x^3 - 3x^2}{6} - 2\sin x + (2x - 1)\cos x$ :
a.
increases in $\left[\frac{1}{2}, \infty\right)$
Correct Answer
b.
decreases in $\left[\frac{1}{2}, \infty\right)$
c.
increases in $\left(-\infty, \frac{1}{2}\right]$
d.
decreases in $\left(-\infty, \frac{1}{2}\right]$
Step-by-Step AI Explanation
Verified Method
$f'(x) = \frac{12x^2 - 6x}{6} - 2\cos x + 2\cos x - (2x - 1)\sin x = 2x^2 - x - (2x - 1)\sin x = (2x - 1)(x - \sin x)$.
For $x > 0$, $x - \sin x > 0$.
For $x \in [1/2, \infty)$, $2x - 1 \ge 0$ and $x - \sin x > 0$, which gives $f'(x) \ge 0$.
Therefore, $f(x)$ increases in $[1/2, \infty)$.
Q68
mcq
A scientific committee is to be formed from 6 Indians and 8 foreigners, which includes at least 2 Indians and double the number of foreigners as Indians. Then the number of ways, the committee can be formed, is :
a.
1050
b.
1625
Correct Answer
c.
560
d.
575
Step-by-Step AI Explanation
Verified Method
Let the number of Indians selected be $k$. Since there are at least 2 Indians, $k \ge 2$.
The number of foreigners is $2k$. Since there are at most 8 foreigners available, $2k \le 8 \implies k \le 4$.
Possible values of $k$:
1. $k = 2$: ${}^6\text{C}_2 \times {}^8\text{C}_4 = 15 \times 70 = 1050$
2. $k = 3$: ${}^6\text{C}_3 \times {}^8\text{C}_6 = 20 \times 28 = 560$
3. $k = 4$: ${}^6\text{C}_4 \times {}^8\text{C}_8 = 15 \times 1 = 15$
Total ways = $1050 + 560 + 15 = 1625$.
Q69
mcq
If $f : \mathbf{R} \rightarrow \mathbf{R}$ is a function defined by $f(x) = [x-1]\cos\left(\frac{2x-1}{2}\right)\pi$, where $[\cdot]$ denotes the greatest integer function, then $f$ is :
a.
discontinuous only at $x=1$
b.
discontinuous at all integral values of $x$ except at $x=1$
c.
continuous only at $x=1$
d.
continuous for every real $x$
Correct Answer
Step-by-Step AI Explanation
Verified Method
For any integer $k$, $\cos\left(\frac{2k-1}{2}\pi\right) = 0$. The left-hand limit at $x=k$ is $(k-2) \cdot 0 = 0$, the right-hand limit is $(k-1) \cdot 0 = 0$, and the function value is $f(k) = (k-1) \cdot 0 = 0$. Since the limits from both sides equal the function value at every integer $k$, and the function is clearly continuous on non-integral intervals, $f$ is continuous for every real $x$.
Q70
mcq
If $\int \frac{\cos x - \sin x}{\sqrt{8-\sin 2x}}\mathrm{d}x = a\sin^{-1}\left(\frac{\sin x + \cos x}{b}\right) + c$, where $c$ is a constant of integration, then the ordered pair $(a, b)$ is equal to :
a.
$(3, 1)$
b.
$(1, 3)$
Correct Answer
c.
$(-1, 3)$
d.
$(1, -3)$
Step-by-Step AI Explanation
Verified Method
Let $t = \sin x + \cos x \implies \mathrm{d}t = (\cos x - \sin x)\mathrm{d}x$. Also, $t^2 = 1 + \sin 2x \implies \sin 2x = t^2 - 1$. Substituting gives $\int \frac{\mathrm{d}t}{\sqrt{8 - (t^2 - 1)}} = \int \frac{\mathrm{d}t}{\sqrt{3^2 - t^2}} = \sin^{-1}\left(\frac{t}{3}\right) + c = \sin^{-1}\left(\frac{\sin x + \cos x}{3}\right) + c$. Comparing gives $a = 1$ and $b = 3$, so $(a, b) = (1, 3)$.
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Q71
mcq
The area (in sq. units) of the part of the circle $x^2 + y^2 = 36$, which is outside the parabola $y^2 = 9x$, is :
a.
$24\pi + 3\sqrt{3}$
b.
$24\pi - 3\sqrt{3}$
c.
$12\pi + 3\sqrt{3}$
d.
$12\pi - 3\sqrt{3}$
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Q72
mcq
The population $\mathrm{P} = \mathrm{P}(t)$ at time '$t$' of a certain species follows the differential equation $\frac{\mathrm{dP}}{\mathrm{d}t} = 0.5\mathrm{P} - 450$. If $\mathrm{P}(0) = 850$, then the time at which population becomes zero is :
a.
$\log_e 9$
b.
$\frac{1}{2}\log_e 18$
c.
$\log_e 18$
d.
$2\log_e 18$
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Q73
mcq
A man is walking on a straight line. The arithmetic mean of the reciprocals of the intercepts of this line on the coordinate axes is $\frac{1}{4}$. Three stones A, B and C are placed at the points $(1, 1)$, $(2, 2)$ and $(4, 4)$ respectively. Then which of these stones is/are on the path of the man ?
a.
A only
b.
B only
c.
C only
d.
All the three
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Q74
mcq
The locus of the mid-point of the line segment joining the focus of the parabola $y^2 = 4ax$ to a moving point of the parabola, is another parabola whose directrix is :
a.
$x = a$
b.
$x = -\frac{a}{2}$
c.
$x = 0$
d.
$x = \frac{a}{2}$
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Q75
mcq
If the tangent to the curve $y = x^3$ at the point $\mathrm{P}(t, t^3)$ meets the curve again at $\mathrm{Q}$, then the ordinate of the point which divides $\mathrm{PQ}$ internally in the ratio $1 : 2$ is :
a.
$0$
b.
$2t^3$
c.
$-t^3$
d.
$-2t^3$
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Q76
mcq
The equation of the plane passing through the point $(1, 2, -3)$ and perpendicular to the planes $3x + y - 2z = 5$ and $2x - 5y - z = 7$, is :
a.
$6x - 5y + 2z + 10 = 0$
b.
$11x + y + 17z + 38 = 0$
c.
$6x - 5y - 2z - 2 = 0$
d.
$3x - 10y - 2z + 11 = 0$
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Q77
mcq
The distance of the point (1, 1, 9) from the point of intersection of the line (x - 3)/1 = (y - 4)/2 = (z - 5)/2 and the plane x + y + z = 17 is :
a.
2\sqrt{19}
b.
19\sqrt{2}
c.
\sqrt{38}
d.
38
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Q78
mcq
An ordinary dice is rolled for a certain number of times. If the probability of getting an odd number 2 times is equal to the probability of getting an even number 3 times, then the probability of getting an odd number for odd number of times is :
a.
1/32
b.
3/16
c.
5/16
d.
1/2
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Q79
mcq
Two vertical poles are 150 m apart and the height of one is three times that of the other. If from the middle point of the line joining their feet, an observer finds the angles of elevation of their tops to be complementary, then the height of the shorter pole (in meters) is :
a.
25
b.
30
c.
20\sqrt{3}
d.
25\sqrt{3}
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Q80
mcq
The statement among the following that is a tautology is :
a.
A \wedge (A \vee B)
b.
A \vee (A \wedge B)
c.
[ A \wedge (A \rightarrow B) ] \rightarrow B
d.
B \rightarrow [ A \wedge (A \rightarrow B) ]
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