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Mathematics · 2010 · Shift 1

28 questions from this JEE Advanced paper

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JEE Advanced
Papers
2010 · Shift 1
Mathematics

Mathematics · 2010 · Shift 1

28 questions from this JEE Advanced paper

Trigonometric Functions and Equations3 questions Complex Numbers2 questions Quadratic Equation and Inequalities1 questions Sequences and Series1 questions Parabola1 questions Hyperbola3 questions Properties of Triangle2 questions Application of Derivatives1 questions 3D Geometry2 questions Vector Algebra2 questions Probability1 questions Definite Integration3 questions Application of Integration1 questions Matrices and Determinants4 questions Functions1 questions

Q29.The number of values of $θ$ in the interval, $(- \frac{π}{2}, \frac{π}{2})$ such that $θ e \frac{nπ}{5}$ for $n = 0, \pm 1, \pm 2$ and $tan θ = cot 5 θ$ as well as $sin 2 θ = cos 4 θ$ …
Mathematics · Trigonometric Functions and Equations · Numerical
Q30.The number of all possible values of $θ$ where $0 < θ < π,$ for which the system of equations $(y + z) cos 3 θ = (x y z) sin 3 θ x sin 3 θ = \frac{2 c o s 3 θ}{y} + \frac{2 s i n 3 θ}{z} (x y z) sin 3 θ = (y + 2 z) cos 3 θ + y s in 3 θ$ …
Mathematics · Trigonometric Functions and Equations · Numerical
Q31.The maximum value of the expression $\frac{1}{s i n ^{2} θ + 3 s i n θ c o s θ + 5 c o s ^{2} θ}$ is
Mathematics · Trigonometric Functions and Equations · Numerical
Q32.Let $z_{1}$ and $z_{2}$ be two distinct complex number and let z =( 1 - t) $z_{1}$ + t $z_{2}$ for some real number t with 0 < t < 1. IfArg (w) denote the principal argument of a non-zero complex number w, then
Mathematics · Complex Numbers · Multiple correct
Q33.Let $p$ and $q$ be real numbers such that $p e 0, p^{3} e q$ and $p^{3} e - q .$ If $p^{3} e - q .$ and $β$ are nonzero complex numbers satisfying $α + β = - p$ and $α^{3} + β^{3} = q,$ then a quadratic…
Mathematics · Quadratic Equation and Inequalities · MCQ
Q34.Let $S_{k}$ = 1, 2,....., 100, denote the sum of the infinite geometric series whose first term is $\frac{k - 1}{k !}$ and the common ratio is $\frac{1}{k}$ . Then the value of $\frac{100 ^{2}}{100 !} + k = 1 \sum 100 (k^{2} - 3 k + 1) S_{k}$ …
Mathematics · Sequences and Series · Numerical
Q35.Let $A$ and $B$ be two distinct points on the parabola $y^{2} = 4 x$ . If the axis of the parabola touches a circle of radius $r$ having $A B$ as its diameter, then the slope of the line joining $A$ and $B$ can be
Mathematics · Parabola · Multiple correct
Q36.The circle $x^{2} + y^{2} - 8 x = 0$ and hyperbola $\frac{x ^{2}}{9} - \frac{y ^{2}}{4} = 1$ intersect at the points $A$ and $B$ . Equation of a common tangent with positive slope to the circle as well as to the hyperbola is
Mathematics · Hyperbola · MCQ
Q37.The circle $x^{2} + y^{2} - 8 x = 0$ and hyperbola $\frac{x ^{2}}{9} - \frac{y ^{2}}{4} = 1$ intersect at the points $A$ and $B$ . Equation of the circle with $A B$ as its diameter is
Mathematics · Hyperbola · MCQ
Q38.The line $2 x + y = 1$ is tangent to the hyperbola $\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$ . If this line passes through the point of intersection of the nearest directrix and the $x$ -axis, then the eccentricity of the…
Mathematics · Hyperbola · Numerical
Q39.If the angles $C$ and $C$ of a triangle are in an arithmetic progression and if $c$ and $c$ denote the lengths of the sides opposite to $C$ and $C$ respectively, then the value of the expression $\frac{a}{c} sin 2 C + \frac{c}{a} sin 2 A$ …
Mathematics · Properties of Triangle · MCQ
Q40.Let $f$ be a real-valued differentiable function on $R$ (the set of all real numbers) such that $f (1) = 1$ . If the $y$ -intercept of the tangent at any point $P (x, y)$ on the curve $y = f (x)$ is equal to the cube of the abscissa of $P$ , then find…
Mathematics · Application of Derivatives · Numerical
Q41.If the distance between the plane $A x - 2 y + z = d$ and the plane containing the lines $\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}$ and $\frac{x - 2}{3} = \frac{y - 3}{4} = \frac{z - 4}{5}$ is $6,$ then $∣ d ∣$ …
Mathematics · 3D Geometry · Numerical
Q42.If $a$ and $b$ are vectors in space given by $a = \frac{i - 2 j}{5}$ and $b = \frac{2 i + j + 3 k}{14},$ …
Mathematics · Vector Algebra · Numerical
Q43.Equation of the plane containing the straight line $\frac{x}{2} = \frac{y}{3} = \frac{z}{4}$ and perpendicular to the plane containing the straight lines $\frac{x}{3} = \frac{y}{4} = \frac{z}{2}$ and $\frac{x}{4} = \frac{y}{2} = \frac{z}{3}$ …
Mathematics · 3D Geometry · MCQ
Q44.Let $P, Q, R$ and $S$ be the points on the plane with position vectors $- 2 i - j, 4 i, 3 i + 3 j$ and $- 3 i + 2 j$ respectively. The quadrilateral $P QR S$ must be a
Mathematics · Vector Algebra · MCQ
Q45.Let $ω$ be a complex cube root of unity with $ω e 1.$ A fair die is thrown three times. If $r_{1}, r_{2}$ and $r_{3}$ are the numbers obtained on the die, then the probability that $ω^{r_{1}} + ω^{r_{2}} + ω^{r_{3}} = 0$ …
Mathematics · Probability · MCQ
Q46.For any real number $x,$ let $x .$ denote the largest integer less than or equal to $x .$ Let $f$ be a real valued function defined on the interval $[- 10, 10]$ by $f (x) = {x - [x] 1 + [x] - x i f [x] i s o dd, i f [x] i s e v e n$ …
Mathematics · Definite Integration · Numerical
Q47.Let $f$ be a real-valued function defined on the interval $(0, \infty)$ by $f (x) = ln x + 0 \int x 1 + sin t d t .$ then which of the following statement(s) is (are) true?
Mathematics · Application of Integration · Multiple correct
Q48.The value of $0 \int 1 \frac{x ^{4} ( 1 - x ) ^{4}}{1 + x ^{2}} d x$ is (are)
Mathematics · Definite Integration · MCQ
Q49.The value of $x \to 0 lim \frac{1}{x ^{3}} 0 \int x \frac{t l n ( 1 + t )}{t ^{4} + 4} d t$ is
Mathematics · Definite Integration · MCQ
Q50.Let $A B C$ be a triangle such that $∠ A C B = \frac{π}{6}$ and let $c$ and $c$ denote the lengths of the sides opposite to $A$ , $B$ and $C$ respectively. The value(s) of $x$ for which $a = x^{2} + x + 1, b = x^{2} - 1$ …
Mathematics · Properties of Triangle · MCQ
Q51.The number of $3 \times 3$ matrices A whose entries are either 0 or 1 and for which the system $A x y z = 100$ has exactly two distinct…
Mathematics · Matrices and Determinants · MCQ
Q52.Let $h$ and $h$ be real valued functions defined on the interval $[0, 1]$ by $f (x) = e^{x^{2}} + e^{- x^{2}}$ , $g (x) = x e^{x^{2}} + e^{- x^{2}}$ and $h (x) = x^{2} e^{x^{2}} + e^{- x^{2}}$ . If $c$ and $c$ denote, respectively, the absolute maximum of $h$ and $h$ …
Mathematics · Functions · MCQ
Q53.Let $z_{1}$ and $z_{2}$ be two distinct complex numbers let $z = (1 - t) z_{1} + t z_{2}$ for some real number t with $0 I f$ \operatorname{Arg}(w) $d e n o t es t h e p r in c i p a l a r g u m e n t o f an o n z er oco m pl e x n u mb er$ w$, then :
Mathematics · Complex Numbers · Multiple correct
Q54.Let $p$ be an odd prime number and $T_{p}$ be the following set of $2 \times 2$ matrices : $T_{p} = {A = [a c b a] : a, b, c \in {0, 1, 2, \dots, p - 1}}$ …
Mathematics · Matrices and Determinants · MCQ
Q55.Let $p$ be an odd prime number and $T_{p}$ be the following set of $2 \times 2$ matrices : $T_{p} = {A = [a c b a] : a, b, c \in {0, 1, 2, \dots, p - 1}}$ …
Mathematics · Matrices and Determinants · MCQ
Q56.Let $p$ be an odd prime number and $T_{p}$ be the following set of $2 \times 2$ matrices : $T_{p} = {A = [a c b a] : a, b, c \in {0, 1, 2, \dots, p - 1}}$ …
Mathematics · Matrices and Determinants · MCQ