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Mathematics · 2024 · 31 Jan · Shift 2

30 questions from this JEE Main paper

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2024 · 31 Jan · Shift 2
Mathematics

Mathematics · 2024 · 31 Jan · Shift 2

30 questions from this JEE Main paper

Permutations and Combinations2 questions Probability1 questions Matrices and Determinants2 questions 3D Geometry3 questions Inverse Trigonometric Functions1 questions Ellipse1 questions Trigonometric Ratio and Identites1 questions Area Under the Curves1 questions Definite Integration2 questions Application of Derivatives2 questions Differential Equations2 questions Sequences and Series1 questions Complex Numbers1 questions Limits Continuity and Differentiability2 questions Circle1 questions Straight Lines and Pair of Straight Lines2 questions Statistics1 questions Vector Algebra1 questions Quadratic Equation and Inequalities1 questions Sets and Relations1 questions Binomial Theorem1 questions

Q31.The number of ways in which 21 identical apples can be distributed among three children such that each child gets at least 2 apples, is
Mathematics · Permutations and Combinations · MCQ
Q32.A coin is biased so that a head is twice as likely to occur as a tail. If the coin is tossed 3 times, then the probability of getting two tails and one head is
Mathematics · Probability · MCQ
Q33.Let $A$ be a $3 \times 3$ real matrix such that $A 101 = 2101, A - 1 01 = 4 - 1 01, A 010 = 2010 .$ …
Mathematics · Matrices and Determinants · MCQ
Q34.Let $(α, β, γ)$ be the mirror image of the point $(2, 3, 5)$ in the line $\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}$ . Then, $2 α + 3 β + 4 γ$ is equal to
Mathematics · 3D Geometry · MCQ
Q35.If $a = sin^{- 1} (sin (5))$ and $b = cos^{- 1} (cos (5))$ , then $a^{2} + b^{2}$ is equal to
Mathematics · Inverse Trigonometric Functions · MCQ
Q36.Let $P$ be a parabola with vertex $(2, 3)$ and directrix $2 x + y = 6$ . Let an ellipse $E : \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1, a > b$ , of eccentricity $\frac{1}{2}$ pass through the focus of the parabola $P$ . Then, the square of the…
Mathematics · Ellipse · MCQ
Q37.The number of solutions, of the equation $e^{s i n x} - 2 e^{- s i n x} = 2$ , is :
Mathematics · Trigonometric Ratio and Identites · MCQ
Q38.The shortest distance, between lines $L_{1}$ and $L_{2}$ , where $L_{1} : \frac{x - 1}{2} = \frac{y + 1}{- 3} = \frac{z + 4}{2}$ and $L_{2}$ is the line, passing through the points $A (- 4, 4, 3), B (- 1, 6, 3)$ and perpendicular to the line $\frac{x - 3}{- 2} = \frac{y}{3} = \frac{z - 1}{1}$ …
Mathematics · 3D Geometry · MCQ
Q39.The area of the region enclosed by the parabolas $y = 4 x - x^{2}$ and $3 y = (x - 4)^{2}$ is equal to :
Mathematics · Area Under the Curves · MCQ
Q40.Let $f, g : (0, \infty) \to R$ be two functions defined by $f (x) = - x \int x (∣ t ∣ - t^{2}) e^{- t^{2}} d t$ and $g (x) = 0 \int x^{2} t^{1/2} e^{- t} d t$ . Then, the value of $9 (f (lo g_{e} 9) + g (lo g_{e} 9))$ …
Mathematics · Definite Integration · MCQ
Q41.Let $f :\to R \to (0, \infty)$ be strictly increasing function such that $lim_{x \to \infty} \frac{f ( 7 x )}{f ( x )} = 1$ . Then, the value of $lim_{x \to \infty} [\frac{f ( 5 x )}{f ( x )} - 1]$ …
Mathematics · Application of Derivatives · MCQ
Q42.The temperature $T (t)$ of a body at time $t = 0$ is $16 0^{\circ} F$ and it decreases continuously as per the differential equation $\frac{d T}{d t} = - K (T - 80)$ , where $K$ is a positive constant. If $T (15) = 12 0^{\circ} F$ , then…
Mathematics · Differential Equations · MCQ
Q43.If the function $f : (- \infty, - 1] \to (a, b]$ defined by $f (x) = e^{x^{3} - 3 x + 1}$ is one - one and onto, then the distance of the point $P (2 b + 4, a + 2)$ from the line $x + e^{- 3} y = 4$ is :
Mathematics · Application of Derivatives · MCQ
Q44.Let $2^{nd}, 8^{th}$ and $4 4^{th}$ terms of a non-constant A. P. be respectively the $1^{st}, 2^{nd}$ and $3^{rd}$ terms of a G. P. If the first term of the A. P. is 1, then the sum of…
Mathematics · Sequences and Series · MCQ
Q45.Let $z_{1}$ and $z_{2}$ be two complex numbers such that $z_{1} + z_{2} = 5$ and $z_{1}^{3} + z_{2}^{3} = 20 + 15 i$ Then, $z_{1}^{4} + z_{2}^{4}$ equals -
Mathematics · Complex Numbers · MCQ
Q46.If for some $m, n;^{6} C_{m} + 2 (^{6} C_{m + 1}) +^{6} C_{m + 2} >^{8} C_{3}$ and $^{n - 1} P_{3} :^{n} P_{4} = 1 : 8$ , then $^{n} P_{m + 1} +^{n + 1} C_{m}$ is equal to
Mathematics · Permutations and Combinations · MCQ
Q47.Consider the function $f : (0, \infty) \to R$ defined by $f (x) = e^{- ∣ l o g_{e} x ∣}$ . If $m$ and $n$ be respectively the number of points at which $f$ is not continuous and $f$ is not differentiable, then $m + n$ is
Mathematics · Limits Continuity and Differentiability · MCQ
Q48.Let a variable line passing through the centre of the circle $x^{2} + y^{2} - 16 x - 4 y = 0$ , meet the positive co-ordinate axes at the points $A$ and $B$ . Then the minimum value of $O A + O B$ , where $O$ is the origin, is equal to
Mathematics · Circle · MCQ
Q49.Let $A (a, b), B (3, 4)$ and $C (- 6, - 8)$ respectively denote the centroid, circumcentre and orthocentre of a triangle. Then, the distance of the point $P (2 a + 3, 7 b + 5)$ from the line $2 x + 3 y - 4 = 0$ measured parallel to the line $x - 2 y - 1 = 0$ is
Mathematics · Straight Lines and Pair of Straight Lines · MCQ
Q50.Let the mean and the variance of 6 observations $a, b, 68, 44, 48, 60$ be $55$ and $194$ , respectively. If $a > b$ , then $a + 3 b$ is
Mathematics · Statistics · MCQ
Q51.Let $a = 3 \hat{i} + 2 \hat{j} + \hat{k}, b = 2 \hat{i} - \hat{j} + 3 \hat{k}$ and $c$ be a vector such that $(a + b) \times c = 2 (a \times b) + 24 \hat{j} - 6 \hat{k}$ and $(a - b + \hat{i}) \cdot c = - 3$ …
Mathematics · Vector Algebra · Numerical
Q52.Let $y = y (x)$ be the solution of the differential equation $sec^{2} x d x + (e^{2 y} tan^{2} x + tan x) d y = 0, 0 < x < \frac{π}{2}, y (π /4) = 0$ . If $y (π /6) = α$ , then $e^{8 α}$ is equal to …
Mathematics · Differential Equations · Numerical
Q53. $\frac{120}{π ^{3}} 0 \int π \frac{x ^{2} s i n x c o s x}{s i n ^{4} x + c o s ^{4} x} d x$ is equal to .
Mathematics · Definite Integration · Numerical
Q54.Let $A (- 2, - 1), B (1, 0), C (α, β)$ and $D (γ, δ)$ be the vertices of a parallelogram $A B C D$ . If the point $C$ lies on $2 x - y = 5$ and the point $D$ lies on $3 x - 2 y = 6$ , then the value of $∣ α + β + γ + δ ∣$ is…
Mathematics · Straight Lines and Pair of Straight Lines · Numerical
Q55.Let $a, b, c$ be the lengths of three sides of a triangle satistying the condition $(a^{2} + b^{2}) x^{2} - 2 b (a + c) x + (b^{2} + c^{2}) = 0$ . If the set of all possible values of $x$ is the interval $(α, β)$ , then $12 (α^{2} + β^{2})$ …
Mathematics · Quadratic Equation and Inequalities · Numerical
Q56.If $lim_{x \to 0} \frac{a x ^{2} e ^{x} - b l o g _{e} ( 1 + x ) + c x e ^{- x}}{x ^{2} s i n x} = 1$ , then $16 (a^{2} + b^{2} + c^{2})$ is equal to .
Mathematics · Limits Continuity and Differentiability · Numerical
Q57.Let $A = {1, 2, 3, \dots\dots\dots\dots, 100}$ . Let $R$ be a relation on $A$ defined by $(x, y) \in R$ if and only if $2 x = 3 y$ . Let $R_{1}$ be a symmetric relation on $A$ such that $R \subset R_{1}$ and the number of elements in…
Mathematics · Sets and Relations · Numerical
Q58.Let the coefficient of $x^{r}$ in the expansion of $(x + 3)^{n - 1} + (x + 3)^{n - 2} (x + 2) + (x + 3)^{n - 3} (x + 2)^{2} + \dots\dots\dots . + (x + 2)^{n - 1}$ be $α_{r}$ . If $\sum_{r = 0}^{n} α_{r} = β^{n} - γ^{n}, β, γ \in N$ , then the…
Mathematics · Binomial Theorem · Numerical
Q59.Let A be a $3 \times 3$ matrix and $det (A) = 2$ . If $n = det (2024 - times adj (adj (\dots .. (adj A))))$ , then the remainder when $n$ is divided…
Mathematics · Matrices and Determinants · Numerical
Q60.A line passes through $A (4, - 6, - 2)$ and $B (16, - 2, 4)$ . The point $P (a, b, c)$ , where $a, b, c$ are non-negative integers, on the line $A B$ lies at a distance of 21 units, from the point $A$ . The distance between the points $P (a, b, c)$ and $Q (4, - 12, 3)$ …
Mathematics · 3D Geometry · Numerical