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Mathematics · 2023 · Shift 2

17 questions from this JEE Advanced paper

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JEE Advanced
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2023 · Shift 2
Mathematics

Mathematics · 2023 · Shift 2

17 questions from this JEE Advanced paper

Differential Equations2 questions Probability4 questions Inverse Trigonometric Functions1 questions Vector Algebra1 questions Matrices and Determinants2 questions Limits Continuity and Differentiability1 questions Differentiation1 questions Definite Integration1 questions Circle2 questions Properties of Triangle1 questions Trigonometric Functions and Equations1 questions

Q18.Let $f : [1, \infty) \to R$ be a differentiable function such that $f (1) = \frac{1}{3}$ and $3 1 \int x f (t) d t = x f (x) - \frac{x ^{3}}{3}, x \in [1, \infty)$ . Let $e$ denote the base of the natural logarithm. Then the value…
Mathematics · Differential Equations · MCQ
Q19.Consider an experiment of tossing a coin repeatedly until the outcomes of two consecutive tosses are same. If the probability of a random toss resulting in head is $\frac{1}{3}$ , then the probability that the experiment stops with head is :
Mathematics · Probability · MCQ
Q20.For any $y \in R$ , let $cot^{- 1} (y) \in (0, π)$ and $tan^{- 1} (y) \in (- \frac{π}{2}, \frac{π}{2})$ . Then the sum of all the solutions of the equation $tan^{- 1} (\frac{6 y}{9 - y ^{2}}) + cot^{- 1} (\frac{9 - y ^{2}}{6 y}) = \frac{2 π}{3}$ …
Mathematics · Inverse Trigonometric Functions · MCQ
Q21.Let the position vectors of the points $P, Q, R$ and $S$ be $a = \hat{i} + 2 \hat{j} - 5 \hat{k}, b = 3 \hat{i} + 6 \hat{j} + 3 \hat{k}$ , $c = \frac{17}{5} \hat{i} + \frac{16}{5} \hat{j} + 7 \hat{k}$ and $\vec{d}=2…
Mathematics · Vector Algebra · MCQ
Q22.Let $M = (a_{ij}), i, j \in {1, 2, 3}$ , be the $3 \times 3$ matrix such that $a_{ij} = 1$ if $j + 1$ is divisible by $i$ , otherwise $a_{ij} = 0$ . Then which of the following statements is(are) true?
Mathematics · Matrices and Determinants · Multiple correct
Q23.Let $f : (0, 1) \to R$ be the function defined as $f (x) = [4 x] (x - \frac{1}{4})^{2} (x - \frac{1}{2})$ , where $[x]$ denotes the greatest integer less than or equal to $x$ . Then which of the following statements…
Mathematics · Limits Continuity and Differentiability · Multiple correct
Q24.Let $S$ be the set of all twice differentiable functions $f$ from $R$ to $R$ such that $\frac{d ^{2} f}{d x ^{2}} (x) > 0$ for all $x \in (- 1, 1)$ . For $f \in S$ , let $X_{f}$ be the number of points $x \in (- 1, 1)$ for which…
Mathematics · Differentiation · Multiple correct
Q25.For $x \in R$ , let $tan^{- 1} (x) \in (- \frac{π}{2}, \frac{π}{2})$ . Then the minimum value of the function $f : R \to R$ defined by $f (x) = 0 \int x t a n^{- 1} x \frac{e ^{(t - c o s t)}}{1 + t ^{2023}} d t$ …
Mathematics · Definite Integration · Numerical
Q26.For $x \in R$ , let $y (x)$ be a solution of the differential equation $(x^{2} - 5) \frac{d y}{d x} - 2 x y = - 2 x (x^{2} - 5)^{2}$ such that $y (2) = 7$ . Then the maximum value of the function $y (x)$ is :
Mathematics · Differential Equations · Numerical
Q27.Let $X$ be the set of all five digit numbers formed using 1,2,2,2,4,4,0. For example, 22240 is in $X$ while 02244 and 44422 are not in $X$ . Suppose that each element of $X$ has an equal chance of being chosen. Let $p$ be the conditional…
Mathematics · Probability · Numerical
Q28.Let $A_{1}, A_{2}, A_{3}, \dots, A_{8}$ be the vertices of a regular octagon that lie on a circle of radius 2 . Let $P$ be a point on the circle and let $P A_{i}$ denote the distance between the points $P$ and $A_{i}$ for $i = 1, 2, \dots, 8$ . If $P$ …
Mathematics · Circle · Numerical
Q29.Let $R = ⎩ ⎨ ⎧ a c 0 325 b d 0 : a, b, c, d \in {0, 3, 5, 7, 11, 13, 17, 19} ⎭ ⎬ ⎫$ . Then the number of invertible matrices in $R$ is :
Mathematics · Matrices and Determinants · Numerical
Q30.Let $C_{1}$ be the circle of radius 1 with center at the origin. Let $C_{2}$ be the circle of radius $r$ with center at the point $A = (4, 1)$ , where $1 < r < 3$ . Two distinct common tangents $P Q$ and $S T$ of $C_{1}$ and $C_{2}$ are drawn. The…
Mathematics · Circle · Numerical
Q31.Consider an obtuse angled triangle $A B C$ in which the difference between the largest and the smallest angle is $\frac{π}{2}$ and whose sides are in arithmetic progression. Suppose that the vertices of this triangle lie on a circle of…
Mathematics · Properties of Triangle · Numerical
Q32.Consider an obtuse angled triangle $A B C$ in which the difference between the largest and the smallest angle is $\frac{π}{2}$ and whose sides are in arithmetic progression. Suppose that the vertices of this triangle lie on a circle of…
Mathematics · Trigonometric Functions and Equations · Numerical
Q33.Consider the $6 \times 6$ square in the figure. Let $A_{1}, A_{2}, \dots, A_{49}$ be the points of intersections (dots in the picture) in some order. We say that $A_{i}$ and $A_{j}$ are friends if they are adjacent along a row or along a column.… Includes diagram
Mathematics · Probability · Numerical
Q34.Consider the $6 \times 6$ square in the figure. Let $A_{1}, A_{2}, \dots, A_{49}$ be the points of intersections (dots in the picture) in some order. We say that $A_{i}$ and $A_{j}$ are friends if they are adjacent along a row or along a column.… Includes diagram
Mathematics · Probability · Numerical