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Mathematics · 2021 · Shift 2

19 questions from this JEE Advanced paper

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JEE Advanced
Papers
2021 · Shift 2
Mathematics

Mathematics · 2021 · Shift 2

19 questions from this JEE Advanced paper

Permutations and Combinations1 questions Properties of Triangle1 questions Definite Integration6 questions Application of Integration3 questions Vector Algebra1 questions Parabola1 questions Circle4 questions Probability1 questions Ellipse1 questions

Q20.Let $S_{1} = {(i, j, k) : i, j, k \in {1, 2, ...., 10}}$ , $S_{2} = {(i, j) : 1 \leq i < j + 2 \leq 10, i, j \in {1, 2, ..., 10}}$ , $S_{3} = {(i, j, k, l) : 1 \leq i < j < k < l, i, j, k, l \in {1, 2, ..., 10}}$ …
Mathematics · Permutations and Combinations · Multiple correct
Q21.Consider a triangle PQR having sides of lengths p, q and r opposite to the angles P, Q and R, respectively. Then which of the following statements is (are) TRUE?
Mathematics · Properties of Triangle · Multiple correct
Q22.Let $f : [- \frac{π}{2}, \frac{π}{2}] \to R$ be a continuous function such that $f (0) = 1$ and $\int_{0}^{\frac{π}{3}} f (t) d t = 0$ . Then which of the following statements is(are) TRUE?
Mathematics · Definite Integration · Multiple correct
Q23.For any real numbers $α$ and $β$ , let $y_{α, β} (x)$ , x $\in$ R, be the solution of the differential equation $\frac{d y}{d x} + α y = x e^{β x}, y (1) = 1$ . Let $S = {y_{α, β} (x) : α, β \in R}$ …
Mathematics · Application of Integration · Multiple correct
Q24.Let O be the origin and $O A = 2 i + 2 j + k$ and $O B = i - 2 j + 2 k$ and $O C = \frac{1}{2} (O B - λ O A)$ …
Mathematics · Vector Algebra · Multiple correct
Q25.Let E denote the parabola y2 = 8x. Let P = ( $-$ 2, 4), and let Q and Q' be two distinct points on E such that the lines PQ and PQ' are tangents to E. Let F be the focus of E. Then which of the following statements is(are) TRUE?
Mathematics · Parabola · Multiple correct
Q26.Consider the region R = {(x, y) $\in$ R $\times$ R : x $\geq$ 0 and y2 $\leq$ 4 $-$ x}. Let F be the family of all circles that are contained in R and have centers on the x-axis. Let C be the circle that has largest radius among the circles…
Mathematics · Circle · Numerical
Q27.Consider the region R = {(x, y) $\in$ R $\times$ R : x $\geq$ 0 and y2 $\leq$ 4 $-$ x}. Let F be the family of all circles that are contained in R and have centers on the x-axis. Let C be the circle that has largest radius among the circles…
Mathematics · Circle · Numerical
Q28.Let f1 : (0, $\infty$ ) $\to$ R and f2 : (0, $\infty$ ) $\to$ R be defined by $f_{1} (x) = 0 \int x j = 1 \prod 21 (t - j)^{j} d t$ , x > 0 and $f_{2} (x) = 98 (x - 1)^{50} - 600 (x - 1)^{49} + 2450, x > 0$ ,…
Mathematics · Application of Integration · Numerical
Q29.Let f1 : (0, $\infty$ ) $\to$ R and f2 : (0, $\infty$ ) $\to$ R be defined by $f_{1} (x) = 0 \int x j = 1 \prod 21 (t - j)^{j} d t$ , x > 0 and $f_{2} (x) = 98 (x - 1)^{50} - 600 (x - 1)^{49} + 2450, x > 0$ ,…
Mathematics · Application of Integration · Numerical
Q30.Let $g_{i} : [\frac{π}{8}, \frac{3 π}{8}] \to R, i = 1, 2$ , and $f : [\frac{π}{8}, \frac{3 π}{8}] \to R$ be functions such that $g_{1} (x) = 1, g_{2} (x) = ∣4 x - π ∣$ and $f (x) = sin^{2} x$ , for all $x \in [\frac{π}{8}, \frac{3 π}{8}]$ …
Mathematics · Definite Integration · Numerical
Q31.Let $g_{i} : [\frac{π}{8}, \frac{3 π}{8}] \to R, i = 1, 2$ , and $f : [\frac{π}{8}, \frac{3 π}{8}] \to R$ be functions such that $g_{1} (x) = 1, g_{2} (x) = ∣4 x - π ∣$ and $f (x) = sin^{2} x$ , for all $x \in [\frac{π}{8}, \frac{3 π}{8}]$ …
Mathematics · Definite Integration · Numerical
Q32.Let $M = {(x, y) \in R \times R : x^{2} + y^{2} \leq r^{2}}$ , where r > 0. Consider the geometric progression $a_{n} = \frac{1}{2 ^{n - 1}}$ , n = 1, 2, 3, ...... . Let S0 = 0 and for n $\geq$ 1, let Sn denote the sum of the first n…
Mathematics · Circle · MCQ
Q33.Let $M = {(x, y) \in R \times R : x^{2} + y^{2} \leq r^{2}}$ , where r > 0. Consider the geometric progression $a_{n} = \frac{1}{2 ^{n - 1}}$ , n = 1, 2, 3, ...... . Let S0 = 0 and for n $\geq$ 1, let Sn denote the sum of the first n…
Mathematics · Circle · MCQ
Q34.Let $ψ_{1} : [0, \infty) \to R$ , $ψ_{2} : [0, \infty) \to R$ , f : (0, $\infty$ ) $\to$ R and g : [0, $\infty$ ) $\to$ R be functions such that f(0) = g(0) = 0, $ψ_{1} (x) = e^{- x} + x, x \geq 0$ , $ψ_{2} (x) = x^{2} - 2 x - 2 e^{- x} + 2, x \geq 0$ …
Mathematics · Definite Integration · MCQ
Q35.Let $ψ_{1} : [0, \infty) \to R$ , $ψ_{2} : [0, \infty) \to R$ , f : (0, $\infty$ ) $\to$ R and g : [0, $\infty$ ) $\to$ R be functions such that f(0) = g(0) = 0, $ψ_{1} (x) = e^{- x} + x, x \geq 0$ , $ψ_{2} (x) = x^{2} - 2 x - 2 e^{- x} + 2, x \geq 0$ …
Mathematics · Definite Integration · MCQ
Q36.A number of chosen at random from the set {1, 2, 3, ....., 2000}. Let p be the probability that the chosen number is a multiple of 3 or a multiple of 7. Then the value of 500p is .
Mathematics · Probability · Numerical
Q37.Let E be the ellipse $\frac{x ^{2}}{16} + \frac{y ^{2}}{9} = 1$ . For any three distinct points P, Q and Q' on E, let M(P, Q) be the mid-point of the line segment joining P and Q, and M(P, Q') be the mid-point of the line segment…
Mathematics · Ellipse · Numerical
Q38.For any real number x, let [ x ] denote the largest integer less than or equal to x. If $I = 0 \int 10 [\frac{10 x}{x + 1}] d x$ , then the value of 9I is .
Mathematics · Definite Integration · Numerical