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Mathematics · 2023 · Shift 1

17 questions from this JEE Advanced paper

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JEE Advanced
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2023 · Shift 1
Mathematics

Mathematics · 2023 · Shift 1

17 questions from this JEE Advanced paper

Functions2 questions Ellipse1 questions Definite Integration1 questions Application of Derivatives1 questions Probability1 questions Parabola1 questions Inverse Trigonometric Functions1 questions Application of Integration1 questions Sequences and Series1 questions Complex Numbers2 questions Vector Algebra1 questions Mathematical Induction and Binomial Theorem1 questions Matrices and Determinants1 questions Statistics1 questions 3D Geometry1 questions

Q18.Let $S = (0, 1) \cup (1, 2) \cup (3, 4)$ and $T = {0, 1, 2, 3}$ . Then which of the following statements is(are) true?
Mathematics · Functions · Multiple correct
Q19.Let $T_{1}$ and $T_{2}$ be two distinct common tangents to the ellipse $E : \frac{x ^{2}}{6} + \frac{y ^{2}}{3} = 1$ and the parabola $P : y^{2} = 12 x$ . Suppose that the tangent $T_{1}$ touches $P$ and $E$ at the points $A_{1}$ and $A_{2}$ , respectively and the…
Mathematics · Ellipse · Multiple correct
Q20.Let $f : [0, 1] \to [0, 1]$ be the function defined by $f (x) = \frac{x ^{3}}{3} - x^{2} + \frac{5}{9} x + \frac{17}{36}$ . Consider the square region $S = [0, 1] \times [0, 1]$ . Let $G = {(x, y) \in S : y > f (x)}$ be called the green region and $R = {(x, y) \in S : y < f (x)}$ …
Mathematics · Functions · Multiple correct
Q21.Let $f : (0, 1) \to R$ be the function defined as $f (x) = n$ if $x \in [\frac{1}{n + 1}, \frac{1}{n})$ where $n \in N$ . Let $g : (0, 1) \to R$ be a function such that $x^{2} \int x \frac{1 - t}{t} d t < g (x) < 2 x$ …
Mathematics · Definite Integration · MCQ
Q22.Let $Q$ be the cube with the set of vertices ${(x_{1}, x_{2}, x_{3}) \in R^{3} : x_{1}, x_{2}, x_{3} \in {0, 1}}$ . Let $F$ be the set of all twelve lines containing the diagonals of the six faces of the cube $Q$ . Let $S$ …
Mathematics · Application of Derivatives · MCQ
Q23.Let $X = {(x, y) \in Z \times Z : \frac{x ^{2}}{8} + \frac{y ^{2}}{20} < 1$ and $y^{2} < 5 x}$ . Three distinct points $R$ and $R$ are randomly chosen from $X$ . Then the probability that $R$ and $R$ …
Mathematics · Probability · MCQ
Q24.Let $P$ be a point on the parabola $y^{2} = 4 a x$ , where $a > 0$ . The normal to the parabola at $P$ meets the $x$ -axis at a point $Q$ . The area of the triangle $P F Q$ , where $F$ is the focus of the parabola, is 120 . If the slope $m$ of the…
Mathematics · Parabola · MCQ
Q25.Let $tan^{- 1} (x) \in (- \frac{π}{2}, \frac{π}{2})$ , for $x \in R$ . Then the number of real solutions of the equation $1 + cos (2 x) = 2 tan^{- 1} (tan x)$ in the set $(- \frac{3 π}{2}, - \frac{π}{2}) \cup (- \frac{π}{2}, \frac{π}{2}) \cup (\frac{π}{2}, \frac{3 π}{2})$ …
Mathematics · Inverse Trigonometric Functions · Numerical
Q26.Let $n \geq 2$ be a natural number and $f : [0, 1] \to R$ be the function defined by $f (x) = ⎩ ⎨ ⎧ n (1 - 2 n x) 2 n (2 n x - 1) 4 n (1 - n x) \frac{n}{n - 1} (n x - 1) if 0 \leq x \leq \frac{1}{2 n} if \frac{1}{2 n} \leq x \leq \frac{3}{4 n} if \frac{3}{4 n} \leq x \leq \frac{1}{n} if \frac{1}{n} \leq x \leq 1$ …
Mathematics · Application of Integration · Numerical
Q27.Let $7 5 \dots 5 r 7$ denote the $(r + 2)$ digit number where the first and the last digits are 7 and the remaining $r$ digits are 5 . Consider the sum $S = 77 + 757 + 7557 + \dots + 7 5 \dots 5 98 7$ . If $S = \frac{7 5 \dots 5 99 7 + m}{n}$ …
Mathematics · Sequences and Series · Numerical
Q28.Let $A = {\frac{1967 + 1686 i s i n θ}{7 - 3 i c o s θ} : θ \in R}$ . If $A$ contains exactly one positive integer $n$ , then the value of $n$ is
Mathematics · Complex Numbers · Numerical
Q29.Let $P$ be the plane $3 x + 2 y + 3 z = 16$ and let $S = {α \hat{i} + β \hat{j} + γ \hat{k} : α^{2} + β^{2} + γ^{2} = 1$ and the distance of $(α, β, γ)$ from the plane $P$ is…
Mathematics · Vector Algebra · Numerical
Q30.Let $a$ and $b$ be two nonzero real numbers. If the coefficient of $x^{5}$ in the expansion of $(a x^{2} + \frac{70}{27 b x})^{4}$ is equal to the coefficient of $x^{- 5}$ in the expansion of $(a x - \frac{1}{b x ^{2}})^{7}$ , then…
Mathematics · Mathematical Induction and Binomial Theorem · Numerical
Q31.Let $γ$ and $γ$ be real numbers. Consider the following system of linear equations $x + 2 y + z = 7 x + α z = 11 2 x - 3 y + β z = γ$ Match each entry in List-I to… Includes table
Mathematics · Matrices and Determinants · MCQ
Q32.Consider the given data with frequency distribution $x_{i} f_{i} 35821131025444$ Match each entry in List-I to the… Includes table
Mathematics · Statistics · MCQ
Q33.Let $ℓ_{1}$ and $ℓ_{2}$ be the lines $r_{1} = λ (\hat{i} + \hat{j} + \hat{k})$ and $r_{2} = (\hat{j} - \hat{k}) + μ (\hat{i} + \hat{k})$ , respectively. Let $X$ be the set of all the planes $H$ that contain the line $ℓ_{1}$ . For a plane $H$ … Includes table
Mathematics · 3D Geometry · MCQ
Q34.Let $z$ be a complex number satisfying $∣ z ∣^{3} + 2 z^{2} + 4 \overset{z}{ˉ} - 8 = 0$ , where $\overset{z}{ˉ}$ denotes the complex conjugate of $z$ . Let the imaginary part of $z$ be nonzero. Match each entry in List-I to the correct entries in List-II. The correct… Includes table
Mathematics · Complex Numbers · MCQ