Mathematical Reasoning & Induction · JEE Main previous year questions

2024 4 questions

1. Q112024 · 8 Apr 2024 · Shift 1

   For the function $f(x)=(\cos x)-x+1,x\in \mathbb{R}$, between the following two statements (S1) $f(x)=0$ for only one value of $x$ in $[0,\pi ]$. (S2) $f(x)$ is decreasing in $[0,\frac{\pi }{2}]$ and increasing in $[\frac{\pi }{2},\pi ]$.

   1. A.Both (S1) and (S2) are incorrect.
   2. B.Only (S1) is correct.
   3. C.Only (S2) is correct.
   4. D.Both (S1) and (S2) are correct.
   Show answer

   Answer: B

2. Q52024 · 31 Jan 2024 · Shift 1

   For $0<c<b<a$, let $(a+b-2c)x^2+(b+c-2a)x+(c+a-2b)=0$ and $\alpha \ne 1$ be one of its roots. Then, among the two statements (I) If $\alpha \in (-1,0)$, then $b$ cannot be the geometric mean of $a$ and $c$ (II) If $\alpha \in (0,1)$, then $b$ may be the geometric mean of $a$ and $c$

   1. A.only (II) is true
   2. B.Both (I) and (II) are true
   3. C.only (I) is true
   4. D.Neither (I) nor (II) is true
   Show answer

   Answer: B

3. Q92024 · 30 Jan 2024 · Shift 1

   Consider the system of linear equations $x+y+z=4\mu ,x+2y+2\lambda z=10\mu ,x+3y+4{\lambda }^{2}z={\mu }^2+15$ where $\lambda ,\mu \in \mathbf{R}$. Which one of the following statements is NOT correct ?

   1. A.The system has unique solution if $\lambda \ne \frac{1}{2}$ and $\mu \ne 1,15$
   2. B.The system has infinite number of solutions if $\lambda =\frac{1}{2}$ and $\mu =15$
   3. C.The system is consistent if $\lambda \ne \frac{1}{2}$
   4. D.The system is inconsistent if $\lambda =\frac{1}{2}$ and $\mu \ne 1$
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   Answer: D

4. Q162024 · 29 Jan 2024 · Shift 1

   Consider the function $f:\left[\frac{1}{2},1\right]\to \mathbb{R}$ defined by $f(x)=4\sqrt{2}{x}^3-3\sqrt{2}x-1$. Consider the statements (I) The curve $y=f(x)$ intersects the $x$-axis exactly at one point. (II) The curve $y=f(x)$ intersects the $x$-axis at $x=\cos \frac{\pi }{12}$. Then

   1. A.Both (I) and (II) are correct.
   2. B.Only (I) is correct.
   3. C.Both (I) and (II) are incorrect.
   4. D.Only (II) is correct.
   Show answer

   Answer: A