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Mathematics · 2003 · Shift 0

54 questions from this JEE Main paper

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2003 · Shift 0
Mathematics

Mathematics · 2003 · Shift 0

54 questions from this JEE Main paper

Definite Integration5 questions Functions4 questions Limits Continuity and Differentiability4 questions Statistics2 questions Probability2 questions Inverse Trigonometric Functions1 questions Application of Derivatives2 questions Matrices and Determinants3 questions Area Under the Curves1 questions Differential Equations2 questions Vector Algebra7 questions 3D Geometry2 questions Complex Numbers3 questions Quadratic Equation and Inequalities3 questions Binomial Theorem2 questions Permutations and Combinations3 questions Straight Lines and Pair of Straight Lines4 questions Circle1 questions Hyperbola1 questions Differentiation2 questions

Q65.The value of $x \to 0 lim \frac{0 \int x ^{2} s e c ^{2} t d t}{x s in x}$ is
Mathematics · Definite Integration · MCQ
Q66.The function $f (x) = lo g (x + x^{2} + 1)$ , is
Mathematics · Functions · MCQ
Q67.A function $f$ from the set of natural numbers to integers defined by $f (n) = {\frac{n - 1}{2}, w h e n n i s o dd - \frac{n}{2}, w h e n n i s e v e n$ is
Mathematics · Functions · MCQ
Q68.If $f : R \to R$ satisfies $f$ (x + y) = $f$ (x) + $f$ (y), for all x, y $\in$ R and $f$ (1) = 7, then $r = 1 \sum n f (r)$ is
Mathematics · Functions · MCQ
Q69.Domain of definition of the function f(x) = $\frac{3}{4 - x ^{2}}$ + $lo g_{10} (x^{3} - x)$ , is
Mathematics · Functions · MCQ
Q70.If $x \to 0 lim \frac{l o g ( 3 + x ) - l o g ( 3 - x )}{x}$ = k, the value of k is
Mathematics · Limits Continuity and Differentiability · MCQ
Q71.Let $f (a) = g (a) = k$ and their nth derivatives $f^{n} (a)$ , $g^{n} (a)$ exist and are not equal for some n. Further if $x \to a lim \frac{f ( a ) g ( x ) - f ( a ) - g ( a ) f ( x ) + f ( a )}{g ( x ) - f ( x )} = 4$ then the value of k is
Mathematics · Limits Continuity and Differentiability · MCQ
Q72. $x \to \frac{π}{2} lim \frac{[ 1 - t a n ( \frac{x}{2} ) ] [ 1 - s i n x ]}{[ 1 + t a n ( \frac{x}{2} ) ] [ π - 2 x ] ^{3}}$ is
Mathematics · Limits Continuity and Differentiability · MCQ
Q73.If $f (x) = {x e^{- (\frac{1}{∣ x ∣} + \frac{1}{x})} 0, x e 0, x = 0$ then $f (x)$ is
Mathematics · Limits Continuity and Differentiability · MCQ
Q74.In an experiment with 15 observations on $x$ , then following results were available: $\sum x^{2} = 2830$ , $\sum x = 170$ One observation that was 20 was found to be wrong and was replaced by the correct value 30. Then the corrected…
Mathematics · Statistics · MCQ
Q75.The median of a set of 9 distinct observations is 20.5. If each of the largest 4 observations of the set is increased by 2, then the median of the new set :
Mathematics · Statistics · MCQ
Q76.Five horses are in a race. Mr. A selects two of the horses at random and bets on them. The probability that Mr. A selected the winning horse is :
Mathematics · Probability · MCQ
Q77.The trigonometric equation $sin^{- 1} x = 2 sin^{- 1} a$ has a solution for :
Mathematics · Inverse Trigonometric Functions · MCQ
Q78.If the function $f (x) = 2 x^{3} - 9 a x^{2} + 12 a^{2} x + 1,$ where $a > 0,$ attains its maximum and minimum at $p$ and $q$ respectively such that $p^{2} = q$ , then $a$ equals
Mathematics · Application of Derivatives · MCQ
Q79.If $A = [a b b a]$ and $A^{2} = [α β β α]$ , then
Mathematics · Matrices and Determinants · MCQ
Q80.If $1, ω, ω^{2}$ are the cube roots of unity, then $Δ = 1 ω^{n} ω^{2 n} ω^{n} ω^{2 n} 1 ω^{2 n} 1 ω^{n}$ …
Mathematics · Matrices and Determinants · MCQ
Q81.If the system of linear equations $x + 2 a y + a z = 0; x + 3 b y + b z = 0; x + 4 cy + cz = 0;$ has a non - zero solution, then $a, b, c$ .
Mathematics · Matrices and Determinants · MCQ
Q82.The area of the region bounded by the curves $y = ∣ x - 1 ∣$ and $y = 3 - ∣ x ∣$ is :
Mathematics · Area Under the Curves · MCQ
Q83.Let $f (x)$ be a function satisfying $f^{'} (x) = f (x)$ with $f (0) = 1$ and $g (x)$ be a function that satisfies $f (x) + g (x) = x^{2}$ . Then the value of the integral $0 \int 1 f (x) g (x) d x,$ …
Mathematics · Definite Integration · MCQ
Q84.If $f (a + b - x) = f (x)$ then $a \int b x f (x) d x$ is equal to
Mathematics · Definite Integration · MCQ
Q85.The value of the integral $I = 0 \int 1 x (1 - x)^{n} d x$ is
Mathematics · Definite Integration · MCQ
Q86.The solution of the differential equation $(1 + y^{2}) + (x - e^{t a n^{- 1} y}) \frac{d y}{d x} = 0,$ is :
Mathematics · Differential Equations · MCQ
Q87.The degree and order of the differential equation of the family of all parabolas whose axis is $x$ -axis, are respectively.
Mathematics · Differential Equations · MCQ
Q88.Events $A, B, C$ are mutually exclusive events such that $P (A) = \frac{3 x + 1}{3}, P (B) = \frac{1 - x}{4}$ and $P (C) = \frac{1 - 2 x}{2}$ The set of possible values of $x$ are in the interval.
Mathematics · Probability · MCQ
Q89.If $a \times b = b \times c = c \times a$ then $a + b + c =$
Mathematics · Vector Algebra · MCQ
Q90.Let $u = i + j, v = i - j$ and $w = i + 2 j + 3 k .$ If $n$ is a unit vector such that $u . n = 0$ …
Mathematics · Vector Algebra · MCQ
Q91.The vectors $A B = 3 i + 4 k & A C = 5 i - 2 j + 4 k$ are the sides of triangle $A B C .$ The length of the median through $A$ is :
Mathematics · Vector Algebra · MCQ
Q92.The two lines $x = a y + b, z = cy + d$ and $x = a^{'} y + b^{'}, z = c^{'} y + d^{'}$ will be perpendicular, if and only if :
Mathematics · 3D Geometry · MCQ
Q93.The lines $\frac{x - 2}{1} = \frac{y - 3}{1} = \frac{z - 4}{- k}$ and $\frac{x - 1}{k} = \frac{y - 4}{2} = \frac{z - 5}{1}$ are coplanar if :
Mathematics · 3D Geometry · MCQ
Q94. $a, b, c$ are $3$ vectors, such that $a + b + c = 0$ , $a = 1 b = 2, c = 3,$ …
Mathematics · Vector Algebra · MCQ
Q95.A tetrahedron has vertices at $O (0, 0, 0), A (1, 2, 1) B (2, 1, 3)$ and $C (- 1, 1, 2) .$ Then the angle between the faces $O A B$ and $A B C$ will be :
Mathematics · Vector Algebra · MCQ
Q96.If $a b c a^{2} b^{2} c^{2} 1 + a^{3} 1 + b^{3} 1 + c^{3} = 0$ and vectors $(1, a, a^{2}), (1, b, b^{2})$ …
Mathematics · Vector Algebra · MCQ
Q97.Consider points $A, B, C$ and $D$ with position vectors $7 i - 4 j + 7 k, i - 6 j + 10 k, - i - 3 j + 4 k$ and $5 i - j + 5 k$ …
Mathematics · Vector Algebra · MCQ
Q98.Let $Z_{1}$ and $Z_{2}$ be two roots of the equation $Z^{2} + a Z + b = 0$ , Z being complex. Further , assume that the origin, $Z_{1}$ and $Z_{2}$ form an equilateral triangle. Then :
Mathematics · Complex Numbers · MCQ
Q99.If $(\frac{1 + i}{1 - i})^{x} = 1$ then :
Mathematics · Complex Numbers · MCQ
Q100.If the sum of the roots of the quadratic equation $a x^{2} + b x + c = 0$ is equal to the sum of the squares of their reciprocals, then $\frac{a}{c}, \frac{b}{a}$ and $\frac{c}{b}$ are in
Mathematics · Quadratic Equation and Inequalities · MCQ
Q101.The value of ' $a$ ' for which one root of the quadratic equation $(a^{2} - 5 a + 3) x^{2} + (3 a - 1) x + 2 = 0$ is twice as large as the other is
Mathematics · Quadratic Equation and Inequalities · MCQ
Q102.The number of real solutions of the equation $x^{2} - 3 ∣ x ∣ + 2 = 0$ is
Mathematics · Quadratic Equation and Inequalities · MCQ
Q103.The real number $x$ when added to its inverse gives the minimum sum at $x$ equal :
Mathematics · Application of Derivatives · MCQ
Q104.If $x$ is positive, the first negative term in the expansion of $(1 + x)^{27/5}$ is
Mathematics · Binomial Theorem · MCQ
Q105.A student is to answer 10 out of 13 questions in an examination such that he must choose at least 4 from the first five questions. The number of choices available to him is
Mathematics · Permutations and Combinations · MCQ
Q106.The number of integral terms in the expansion of $(3 + 85)^{256}$ is
Mathematics · Binomial Theorem · MCQ
Q107.The number of ways in which 6 men and 5 women can dine at a round table if no two women are to sit together is given by
Mathematics · Permutations and Combinations · MCQ
Q108.If $^{n} C_{r}$ denotes the number of combination of n things taken r at a time, then the expression $^{n} C_{r + 1} +^{n} C_{r - 1} + 2 \times^{n} C_{r}$ equals
Mathematics · Permutations and Combinations · MCQ
Q109.A square of side a lies above the $x$ -axis and has one vertex at the origin. The side passing through the origin makes an angle $α (0 < α < \frac{π}{4})$ with the positive direction of x-axis. The equation…
Mathematics · Straight Lines and Pair of Straight Lines · MCQ
Q110.Locus of centroid of the triangle whose vertices are $(a cos t, a sin t), (b sin t, - b cos t)$ and $(1, 0),$ where $t$ is a parameter, is :
Mathematics · Straight Lines and Pair of Straight Lines · MCQ
Q111.If $x_{1}, x_{2}, x_{3}$ and $y_{1}, y_{2}, y_{3}$ are both in G.P. with the same common ratio, then the points $(x_{1}, y_{1}), (x_{2}, y_{2})$ and $(x_{3}, y_{3})$ :
Mathematics · Straight Lines and Pair of Straight Lines · MCQ
Q112.If the equation of the locus of a point equidistant from the point $(a_{1,} b_{1})$ and $(a_{2,} b_{2})$ is $(a_{1} - a_{2}) x + (b_{1} - b_{2}) y + c = 0$ , then the value…
Mathematics · Straight Lines and Pair of Straight Lines · MCQ
Q113.The lines 2x - 3y = 5 and 3x - 4y = 7 are diameters of a circle having area as 154 sq. units. Then the equation of the circle is :
Mathematics · Circle · MCQ
Q114.The foci of the ellipse $\frac{x ^{2}}{16} + \frac{y ^{2}}{b ^{2}} = 1$ and the hyperbola $\frac{x ^{2}}{144} - \frac{y ^{2}}{81} = \frac{1}{25}$ coincide. Then the value of $b^{2}$ is :
Mathematics · Hyperbola · MCQ
Q115.If $f (y) = e^{y}, g (y) = y; y > 0$ and $F (t) = 0 \int t f (t - y) g (y) d y,$ then :
Mathematics · Definite Integration · MCQ
Q116.If $f (x) = x^{n},$ then the value of $f (1) - \frac{f ^{'} ( 1 )}{1 !} + \frac{f ^{''} ( 1 )}{2 !} - \frac{f ^{'''} ( 1 )}{3 !} + .......... \frac{( - 1 ) ^{n} f ^{n} ( 1 )}{n !}$ …
Mathematics · Differentiation · MCQ
Q117.Let $f (x)$ be a polynomial function of second degree. If $f (1) = f (- 1)$ and $a, b, c$ are in $A . P,$ then $f^{'} (a), f^{'} (b), f^{'} (c)$ are in
Mathematics · Differentiation · MCQ
Q118.If $z$ and $ω$ are two non-zero complex numbers such that $∣ z ω ∣ = 1$ and $A r g (z) - A r g (ω) = \frac{π}{2},$ then $\overline{z} ω$ is equal to
Mathematics · Complex Numbers · MCQ