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Mathematics · 2015 · Shift 2

20 questions from this JEE Advanced paper

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JEE Advanced
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2015 · Shift 2
Mathematics

Mathematics · 2015 · Shift 2

20 questions from this JEE Advanced paper

Quadratic Equation and Inequalities1 questions Complex Numbers1 questions Vector Algebra1 questions Probability2 questions Application of Integration2 questions Sequences and Series2 questions Ellipse1 questions Hyperbola1 questions Parabola1 questions Inverse Trigonometric Functions1 questions Application of Derivatives1 questions Definite Integration4 questions Differentiation1 questions Limits Continuity and Differentiability1 questions

Q21.Let $S$ be the set of all non-zero real numbers $α$ such that the quadratic equation $α x^{2} - x + α = 0$ has two distinct real roots $x_{1}$ and $x_{2}$ satisfying the inequality $∣ x_{1} - x_{2} ∣ < 1.$ …
Mathematics · Quadratic Equation and Inequalities · Multiple correct
Q22.For any integer k, let $a_{k} = cos (\frac{k π}{7}) + i sin (\frac{k π}{7})$ , where $i = - 1$ . The value of the expression $\frac{k = 1 \sum 12 ∣ α _{k + 1} - a _{k} ∣}{k = 1 \sum 3 ∣ α _{4 k - 1} - α _{4 k - 2} ∣}$ …
Mathematics · Complex Numbers · Numerical
Q23.Suppose that $p, q$ and $r$ are three non-coplanar vectors in $R^{3}$ . Let the components of a vector $s$ along $p, q$ and $r$ …
Mathematics · Vector Algebra · Numerical
Q24.Let $n_{1}$ and $n_{2}$ be the number of red and black balls, respectively, in box $I$ . Let $n_{3}$ and $n_{4}$ be the number of red and black balls, respectively, in box $I I .$ One of the two boxes, box $I$ and box $I I,$ …
Mathematics · Probability · Multiple correct
Q25.Let $n_{1}$ and $n_{2}$ be the number of red and black balls, respectively, in box $I$ . Let $n_{3}$ and $n_{4}$ be the number of red and black balls, respectively, in box $I I .$ A ball is drawn at random from box $I$ …
Mathematics · Probability · Multiple correct
Q26.Let $f : R \to R$ be a continuous odd function, which vanishes exactly at one point and $f (1) = \frac{1}{2.}$ Suppose that $F (x) = - 1 \int x f (t) d t$ for all $x \in [- 1, 2]$ …
Mathematics · Application of Integration · Numerical
Q27.Suppose that all the terms of an arithmetic progression (A.P) are natural numbers. If the ratio of the sum of the first seven terms to the sum of the first eleven terms is 6 : 11 and the seventh term lies in between 130 and 140, then the…
Mathematics · Sequences and Series · Numerical
Q28.The coefficient of $x^{9}$ in the expansion of (1 + x) (1 + $x^{2})$ (1 + $x^{3}$ ) .... $(1 + x^{100})$ is
Mathematics · Sequences and Series · Numerical
Q29.Let $E_{1}$ and $E_{2}$ be two ellipses whose centres are at the origin. The major axes of $E_{1}$ and $E_{2}$ lie along the $x$ -axis and the $y$ -axis, respectively. Let $S$ be the circle $x^{2} + (y - 1)^{2} = 2$ . The…
Mathematics · Ellipse · Multiple correct
Q30.Consider the hyperbola $H : x^{2} - y^{2} = 1$ and a circle $S$ with center $N (x_{2}, 0)$ . Suppose that $H$ and $S$ touch each other at a point $P (x_{1}, y_{1})$ with $x_{1} > 1$ and $y_{1} > 0$ . The…
Mathematics · Hyperbola · Multiple correct
Q31.Suppose that the foci of the ellipse $\frac{x ^{2}}{9} + \frac{y ^{2}}{5} = 1$ are $(f_{1}, 0)$ and $(f_{2}, 0)$ where $f_{1} > 0$ and $f_{2} < 0$ . Let $P_{1}$ and $P_{2}$ be two parabolas with a…
Mathematics · Parabola · Numerical
Q32.If $α$ $= 3 sin^{- 1} (\frac{6}{11})$ and $β = 3 cos^{- 1} (\frac{4}{9}),$ where the inverse trigonimetric functions take only the principal values, then the correct options(s) is (are)
Mathematics · Inverse Trigonometric Functions · Multiple correct
Q33.Let $f, g :$ $[- 1, 2] \to R$ be continuous functions which are twice differentiable on the interval $(- 1, 2)$ . Let the values of f and g at the points $- 1, 0$ and $2$ be as given in the following table: In each of the… Includes table
Mathematics · Application of Derivatives · Multiple correct
Q34.Let $f (x) = 7 tan^{8} x + 7 tan^{6} x - 3 tan^{4} x - 3 tan^{2} x$ for all $x \in (- \frac{π}{2}, \frac{π}{2}) .$ Then the correct expression(s) is (are)
Mathematics · Definite Integration · Multiple correct
Q35.The option(s) with the values of a and $L$ that satisfy the following equation is (are) $\frac{0 \int 4 π e ^{t} ( s i n ^{6} a t + c o s ^{4} a t ) d t}{0 \int π e ^{t} ( s i n ^{6} a t + c o s ^{4} a t ) d t} = L ?$ …
Mathematics · Definite Integration · Multiple correct
Q36.Let $f^{'} (x) = \frac{192 x ^{3}}{2 + s i n ^{4} π x}$ for all $x \in R$ with $f (\frac{1}{2}) = 0$ . If $m \leq 1/2 \int 1 f (x) d x \leq M,$ then the possible values of $m$ and…
Mathematics · Definite Integration · MCQ
Q37.Let $F : R \to R$ be a thrice differentiable function. Suppose that $F (1) = 0, F (3) = - 4$ and $F (x) < 0$ for all $x \in (\frac{1}{2}, 3) .$ Let $f (x) = x F (x)$ …
Mathematics · Application of Integration · Multiple correct
Q38.Let $F : R \to R$ be a thrice differentiable function. Suppose that $F (1) = 0, F (3) = - 4$ and $F^{'} (x) < 0$ for all $x \in (\frac{1}{2}, 3) .$ Let $f (x) = x F (x)$ …
Mathematics · Differentiation · Multiple correct
Q39.If $α = 0 \int 1 (e^{9 x + 3 t a n^{- 1} x}) (\frac{12 + 9 x ^{2}}{1 + x ^{2}}) d x$ where $tan^{- 1} x$ takes only principal values, then the value of $(lo g_{e} ∣ 1 + α ∣ - \frac{3 π}{4})$ …
Mathematics · Definite Integration · Numerical
Q40.Let m and n be two positive integers greater than 1. If $α \to 0 lim (\frac{e ^{c o s (α^{n})} - e}{α ^{m}}) = - (\frac{e}{2})$ then the value of $\frac{m}{n}$ …
Mathematics · Limits Continuity and Differentiability · Numerical