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Mathematics · 2022 · Shift 2

18 questions from this JEE Advanced paper

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JEE Advanced
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2022 · Shift 2
Mathematics

Mathematics · 2022 · Shift 2

18 questions from this JEE Advanced paper

Trigonometric Functions and Equations2 questions Differential Equations2 questions Definite Integration1 questions Quadratic Equation and Inequalities1 questions Limits Continuity and Differentiability2 questions Matrices and Determinants2 questions Hyperbola1 questions Application of Integration1 questions Application of Derivatives1 questions Complex Numbers1 questions Circle1 questions Vector Algebra1 questions Permutations and Combinations1 questions Probability1 questions

Q19.Let $α$ and $β$ be real numbers such that $- \frac{π}{4} < β < 0 < α < \frac{π}{4}$ . If $sin (α + β) = \frac{1}{3}$ and $cos (α - β) = \frac{2}{3}$ , then the greatest integer less than or equal to $(\frac{s i n α}{c o s β} + \frac{c o s β}{s i n α} + \frac{c o s α}{s i n β} + \frac{s i n β}{c o s α})^{2}$ …
Mathematics · Trigonometric Functions and Equations · Numerical
Q20.If $y (x)$ is the solution of the differential equation $x d y - (y^{2} - 4 y) d x = 0 for x > 0, y (1) = 2,$ and the slope of the curve $y = y (x)$ is never zero, then the value of $10 y (2)$ is
Mathematics · Differential Equations · Numerical
Q21.The greatest integer less than or equal to $\int_{1}^{2} lo g_{2} (x^{3} + 1) d x + \int_{1}^{l o g_{2} 9} (2^{x} - 1)^{\frac{1}{3}} d x$ is .
Mathematics · Definite Integration · Numerical
Q22.The product of all positive real values of $x$ satisfying the equation $x^{(16 (l o g_{5} x)^{3} - 68 l o g_{5} x)} = 5^{- 16}$ is .
Mathematics · Quadratic Equation and Inequalities · Numerical
Q23.If $β = x \to 0 lim \frac{e ^{x^{3}} - ( 1 - x ^{3} ) ^{\frac{1}{3}} + ( ( 1 - x ^{2} ) ^{\frac{1}{2}} - 1 ) s i n x}{x s i n ^{2} x},$ then the value of $6 β$ is .
Mathematics · Limits Continuity and Differentiability · Numerical
Q24.Let $β$ be a real number. Consider the matrix $A = β 23 011 1 - 2 - 2$ If $A^{7} - (β - 1) A^{6} - β A^{5}$ is a singular matrix, then the…
Mathematics · Matrices and Determinants · Numerical
Q25.Consider the hyperbola $\frac{x ^{2}}{100} - \frac{y ^{2}}{64} = 1$ with foci at $S$ and $S_{1}$ , where $S$ lies on the positive $x$ -axis. Let $P$ be a point on the hyperbola, in the first quadrant. Let $∠ S P S_{1} = α$ , with $α < \frac{π}{2}$ …
Mathematics · Hyperbola · Numerical
Q26.Consider the functions $f, g : R \to R$ defined by $f (x) = x^{2} + \frac{5}{12} and g (x) = {2 (1 - \frac{4∣ x ∣}{3}), 0, ∣ x ∣ \leq \frac{3}{4} ∣ x ∣ > \frac{3}{4}$ …
Mathematics · Application of Integration · Numerical
Q27.Let $P QR S$ be a quadrilateral in a plane, where $QR = 1, ∠ P QR = ∠ QR S = 7 0^{\circ}, ∠ P QS = 1 5^{\circ}$ and $∠ P R S = 4 0^{\circ}$ . If $∠ R P S = θ^{\circ}, P Q = α$ and $P S = β$ , then the interval(s)…
Mathematics · Trigonometric Functions and Equations · Multiple correct
Q28.Let $α = k = 1 \sum \infty sin^{2 k} (\frac{π}{6})$ Let $g : [0, 1] \to R$ be the function defined by $g (x) = 2^{α x} + 2^{α (1 - x)} .$ Then, which of the following statements is/are…
Mathematics · Application of Derivatives · Multiple correct
Q29.Let $\overset{z}{ˉ}$ denote the complex conjugate of a complex number $z$ . If $z$ is a non-zero complex number for which both real and imaginary parts of $(\overset{z}{ˉ})^{2} + \frac{1}{z ^{2}}$ are integers, then which of the following is/are possible…
Mathematics · Complex Numbers · Multiple correct
Q30.Let $G$ be a circle of radius $R > 0$ . Let $G_{1}, G_{2}, \dots, G_{n}$ be $n$ circles of equal radius $r > 0$ . Suppose each of the $n$ circles $G_{1}, G_{2}, \dots, G_{n}$ touches the circle $G$ externally. Also, for $i = 1, 2, \dots, n - 1$ …
Mathematics · Circle · Multiple correct
Q31.Let $\overset{}{^}, \overset{}{^}$ and $\hat{k}$ be the unit vectors along the three positive coordinate axes. Let $a = 3 \overset{}{^} + \overset{}{^} - \hat{k}, b = \overset{}{^} + b_{2} \overset{}{^} + b_{3} \hat{k}, b_{2}, b_{3} \in R, c = c_{1} \overset{}{^} + c_{2} \overset{}{^} + c_{3} \hat{k}, c_{1}, c_{2}, c_{3} \in R$ …
Mathematics · Vector Algebra · Multiple correct
Q32.For $x \in R$ , let the function $y (x)$ be the solution of the differential equation $\frac{d y}{d x} + 12 y = cos (\frac{π}{12} x), y (0) = 0$ Then, which of the following statements is/are TRUE ?
Mathematics · Differential Equations · Multiple correct
Q33.Consider 4 boxes, where each box contains 3 red balls and 2 blue balls. Assume that all 20 balls are distinct. In how many different ways can 10 balls be chosen from these 4 boxes so that from each box at least one red ball and one blue…
Mathematics · Permutations and Combinations · MCQ
Q34.If $M = (\frac{5}{2} - \frac{3}{2} \frac{3}{2} - \frac{1}{2})$ , then which of the following matrices is equal to $M^{2022} ?$
Mathematics · Matrices and Determinants · MCQ
Q35.Suppose that Box-I contains 8 red, 3 blue and 5 green balls, Box-II contains 24 red, 9 blue and 15 green balls, Box-III contains 1 blue, 12 green and 3 yellow balls, Box-IV contains 10 green, 16 orange and 6 white balls. A ball is chosen…
Mathematics · Probability · MCQ
Q36.For positive integer $n$ , define $f (n) = n + \frac{16 + 5 n - 3 n ^{2}}{4 n + 3 n ^{2}} + \frac{32 + n - 3 n ^{2}}{8 n + 3 n ^{2}} + \frac{48 - 3 n - 3 n ^{2}}{12 n + 3 n ^{2}} + \dots + \frac{25 n - 7 n ^{2}}{7 n ^{2}} .$ Then, the value of $n \to \infty lim f (n)$ …
Mathematics · Limits Continuity and Differentiability · MCQ