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Mathematics · 2005 · Shift 0

56 questions from this JEE Main paper

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2005 · Shift 0
Mathematics

Mathematics · 2005 · Shift 0

56 questions from this JEE Main paper

Functions2 questions Application of Derivatives4 questions Limits Continuity and Differentiability3 questions Statistics2 questions Quadratic Equation and Inequalities6 questions Complex Numbers3 questions Definite Integration4 questions Matrices and Determinants4 questions Indefinite Integrals1 questions Area Under the Curves3 questions Inverse Trigonometric Functions1 questions Differential Equations2 questions Probability3 questions Vector Algebra3 questions 3D Geometry1 questions Straight Lines and Pair of Straight Lines3 questions Permutations and Combinations1 questions Binomial Theorem3 questions Sequences and Series1 questions Circle3 questions Ellipse1 questions Parabola1 questions Sets and Relations1 questions

Q60.Let $f : (- 1, 1) \to B$ , be a function defined by $f (x) = tan^{- 1} \frac{2 x}{1 - x ^{2}}$ , then $f$ is both one-one and onto when B is the interval
Mathematics · Functions · MCQ
Q61.A real valued function f(x) satisfies the functional equation f(x - y) = f(x)f(y) - f(a - x)f(a + y) where a is given constant and f(0) = 1, f(2a - x) is equal to
Mathematics · Functions · MCQ
Q62.A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched?
Mathematics · Application of Derivatives · MCQ
Q63.Suppose $f (x)$ is differentiable at x = 1 and $h \to 0 lim \frac{1}{h} f (1 + h) = 5$ , then $f^{'} (1)$ equals
Mathematics · Limits Continuity and Differentiability · MCQ
Q64.Let $α$ and $β$ be the distinct roots of $a x^{2} + b x + c = 0$ , then $x \to α lim \frac{1 - c o s ( a x ^{2} + b x + c )}{( x - α ) ^{2}}$ is equal to
Mathematics · Limits Continuity and Differentiability · MCQ
Q65.If $f$ is a real valued differentiable function satisfying $∣ f (x) - f (y) ∣ \leq (x - y)^{2}$ , $x, y \in R$ and $f (0)$ = 0, then $f (1)$ equals
Mathematics · Limits Continuity and Differentiability · MCQ
Q66.If in a frequency distribution, the mean and median are 21 and 22 respectively, then its mode is approximately :
Mathematics · Statistics · MCQ
Q67.Let x1, x2,...........,xn be n observations such that $\sum x_{i}^{2} = 400$ and $\sum x_{i} = 80$ . Then a possible value of n among the following is
Mathematics · Statistics · MCQ
Q68.If the roots of the equation $x^{2} - b x + c = 0$ be two consecutive integers, then $b^{2} - 4 c$ equals
Mathematics · Quadratic Equation and Inequalities · MCQ
Q69.The value of $a$ for which the sum of the squares of the roots of the equation $x^{2} - (a - 2) x - a - 1 = 0$ assume the least value is :
Mathematics · Quadratic Equation and Inequalities · MCQ
Q70.If the cube roots of unity are 1, $ω, ω^{2}$ then the roots of the equation $(x - 1)^{3}$ + 8 = 0, are :
Mathematics · Complex Numbers · MCQ
Q71.The value of $- π \int π \frac{c o s ^{2}}{1 + a ^{x}} d x, a > 0,$ is
Mathematics · Definite Integration · MCQ
Q72.Area of the greatest rectangle that can be inscribed in the ellipse $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$
Mathematics · Application of Derivatives · MCQ
Q73.A spherical iron ball $10$ cm in radius is coated with a layer of ice of uniform thickness that melts at a rate of $50$ cm $^{3}$ /min. When the thickness of ice is $5$ cm, then the rate at which the thickness of ice decreases is
Mathematics · Application of Derivatives · MCQ
Q74.If $A^{2} - A + 1 = 0$ , then the inverse of $A$ is :
Mathematics · Matrices and Determinants · MCQ
Q75.The system of equations $α x + y + z = α - 1 x + α y + z = α - 1 x + y + α z = α - 1$ has no solutions, if $α$ is :
Mathematics · Matrices and Determinants · MCQ
Q76.If $a_{1}, a_{2}, a_{3}, ........, a_{n}, .....$ are in G.P., then the determinant $Δ = lo g a_{n} lo g a_{n + 3} lo g a_{n + 6} lo g a_{n + 1} lo g a_{n + 4} lo g a_{n + 7} lo g a_{n + 2} lo g a_{n + 5} lo g a_{n + 8}$ …
Mathematics · Matrices and Determinants · MCQ
Q77.If $a^{2} + b^{2} + c^{2} = - 2$ and f $(x) = 1 + a^{2} x (1 + a^{2}) x (1 + a^{2}) x (1 + b^{2}) x 1 + b^{2} x (1 + b^{2}) x (1 + c^{2}) x (1 + c^{2}) x 1 + c^{2} x,$ …
Mathematics · Matrices and Determinants · MCQ
Q78. $\int {\frac{( l o g x - 1 )}{1 + ( l o g x ) ^{2}}}^{2} d x$ is equal to
Mathematics · Indefinite Integrals · MCQ
Q79.If $I_{1} = 0 \int 1 2^{x^{2}} d x, I_{2} = 0 \int 1 2^{x^{3}} d x, I_{3} = 1 \int 2 2^{x^{2}} d x$ and $I_{4} = 1 \int 2 2^{x^{3}} d x$ then
Mathematics · Definite Integration · MCQ
Q80.The area enclosed between the curve $y = lo g_{e} (x + e)$ and the coordinate axes is :
Mathematics · Area Under the Curves · MCQ
Q81.The parabolas $y^{2} = 4 x$ and $x^{2} = 4 y$ divide the square region bounded by the lines $x = 4, y = 4$ and the coordinate axes. If $S_{1}, S_{2}, S_{3}$ are respectively the areas of these parts numbered from top to bottom ; then $S_{1}, S_{2}, S_{3}$ …
Mathematics · Area Under the Curves · MCQ
Q82.Let $f (x)$ be a non - negative continuous function such that the area bounded by the curve $y = f (x), x$ -axis and the ordinates $x = \frac{π}{4}$ and $x = β > \frac{π}{4}$ is $(β sin β + \frac{π}{4} cos β + 2 β) .$ …
Mathematics · Area Under the Curves · MCQ
Q83.If $cos^{- 1} x - cos^{- 1} \frac{y}{2} = α,$ then $4 x^{2} - 4 x y cos α + y^{2}$ is equal to :
Mathematics · Inverse Trigonometric Functions · MCQ
Q84.The value of integral, $3 \int 6 \frac{x}{9 - x + x} d x$ is
Mathematics · Definite Integration · MCQ
Q85.The differential equation representing the family of curves $y^{2} = 2 c (x + c),$ where $c > 0,$ is a parameter, is of order and degree as follows:
Mathematics · Differential Equations · MCQ
Q86.If $x \frac{d y}{d x} = y (lo g y - lo g x + 1),$ then the solution of the equation is :
Mathematics · Differential Equations · MCQ
Q87.Three houses are available in a locality. Three persons apply for the houses. Each applies for one house without consulting others. The probability that all the three apply for the same house is :
Mathematics · Probability · MCQ
Q88.A random variable $X$ has Poisson distribution with mean $2$ . Then $P (X > 1.5)$ equals :
Mathematics · Probability · MCQ
Q89.Let $A$ and $B$ two events such that $P (\overline{A \cup B}) = \frac{1}{6}, P (A \cap B) = \frac{1}{4}$ and $P (\overline{A}) = \frac{1}{4},$ where $\overline{A}$ stands for complement of…
Mathematics · Probability · MCQ
Q90.If $C$ is the mid point of $A B$ and $P$ is any point outside $A B,$ then :
Mathematics · Vector Algebra · MCQ
Q91.Let $c$ and $c$ be distinct non-negative numbers. If the vectors $a i + a j + c k, i + k$ and $c i + c j + b k$ lie in a plane, then $c$ is :
Mathematics · Vector Algebra · MCQ
Q92.The angle between the lines $2 x = 3 y = - z$ and $6 x = - y = - 4 z$ is :
Mathematics · 3D Geometry · MCQ
Q93.If a vertex of a triangle is $(1, 1)$ and the mid points of two sides through this vertex are $(- 1, 2)$ and $(3, 2)$ then the centroid of the triangle is :
Mathematics · Straight Lines and Pair of Straight Lines · MCQ
Q94.If $z_{1}$ and $z_{2}$ are two non-zero complex numbers such that $∣ z_{1} + z_{2} ∣ = ∣ z_{1} ∣ + ∣ z_{2} ∣$ , then arg $z_{1}$ - arg $z_{2}$ is equal to :
Mathematics · Complex Numbers · MCQ
Q95.If $ω = \frac{z}{z - \frac{1}{3} i}$ and $∣ ω ∣ = 1$ , then $z$ lies on :
Mathematics · Complex Numbers · MCQ
Q96.In a triangle $P QR, ∠ R = \frac{π}{2} . I f tan (\frac{P}{2})$ and $tan (\frac{Q}{2})$ are the roots of $a x^{2} + b x + c = 0, a e 0$ then
Mathematics · Quadratic Equation and Inequalities · MCQ
Q97.If both the roots of the quadratic equation $x^{2} - 2 k x + k^{2} + k - 5 = 0$ are less than 5, then $k$ lies in the interval
Mathematics · Quadratic Equation and Inequalities · MCQ
Q98.If the letter of the word SACHIN are arranged in all possible ways and these words are written out as in dictionary, then the word SACHIN appears at serial number
Mathematics · Permutations and Combinations · MCQ
Q99.If the coefficient of $x^{7}$ in $[a x^{2} + (\frac{1}{b x})]^{11}$ equals the coefficient of $x^{- 7}$ in $[a x - (\frac{1}{b x ^{2}})]^{11}$ , then $a$ and $b$ satisfy…
Mathematics · Binomial Theorem · MCQ
Q100.If $x$ is so small that $x^{3}$ and higher powers of $x$ may be neglected, then $\frac{( 1 + x ) ^{\frac{3}{2}} - ( 1 + \frac{1}{2} x ) ^{3}}{( 1 - x ) ^{\frac{1}{2}}}$ may be approximated…
Mathematics · Binomial Theorem · MCQ
Q101.If the coefficients of rth, (r+1)th, and (r + 2)th terms in the binomial expansion of $(1 + y)^{m}$ are in A.P., then m and r satisfy the equation
Mathematics · Binomial Theorem · MCQ
Q102.If $x = n = 0 \sum \infty a^{n}, y = n = 0 \sum \infty b^{n}, z = n = 0 \sum \infty c^{n},$ where a, b, c are in A.P and $∣ a ∣ < 1, ∣ b ∣ < 1, ∣ c ∣ < 1$ …
Mathematics · Sequences and Series · MCQ
Q103.The line parallel to the $x$ - axis and passing through the intersection of the lines $a x + 2 b y + 3 b = 0$ and $b x - 2 a y - 3 a = 0,$ where $(a, b) e (0, 0)$ is :
Mathematics · Straight Lines and Pair of Straight Lines · MCQ
Q104.If a circle passes through the point (a, b) and cuts the circle $x^{2} + y^{2} = p^{2}$ orthogonally, then the equation of the locus of its centre is :
Mathematics · Circle · MCQ
Q105.If the circles $x^{2} + y^{2} + 2 a x + cy + a = 0$ and $x^{2} + y^{2} - 3 a x + d y - 1 = 0$ intersect in two ditinct points P and Q then the line 5x + by - a = 0 passes through P and Q for :
Mathematics · Circle · MCQ
Q106.If the pair of lines $a x^{2} + 2 (a + b) x y + b y^{2} = 0$ lie along diameters of a circle and divide the circle into four sectors such that the area of one of the sectors is thrice the area of another sector then :
Mathematics · Circle · MCQ
Q107.An ellipse has $O B$ as semi minor axis, $F$ and $F$ ' its focii and theangle $F B F$ ' is a right angle. Then the eccentricity of the ellipse is :
Mathematics · Ellipse · MCQ
Q108.Let $P$ be the point $(1, 0)$ and $Q$ a point on the parabola $y^{2} = 8 x$ . The locus of mid point of $P Q$ is :
Mathematics · Parabola · MCQ
Q109.The value of $a$ for which the sum of the squares of the roots of the equation $x^{2} - (a - 2) x - a - 1 = 0$ assume the least value is
Mathematics · Quadratic Equation and Inequalities · MCQ
Q110.Let $f : R \to R$ be a differentiable function having $f (2) = 6$ , $f^{'} (2) = (\frac{1}{48})$ . Then $x \to 2 lim 6 \int f (x) \frac{4 t ^{3}}{x - 2} d t$ …
Mathematics · Definite Integration · MCQ
Q111.If the roots of the equation $x^{2} - b x + c = 0$ be two consecutive integers, then $b^{2} - 4 c$ equals
Mathematics · Quadratic Equation and Inequalities · MCQ
Q112.For any vector $a$ , the value of $(a \times i)^{2} + (a \times j)^{2} + (a \times k)^{2}$ is equal…
Mathematics · Vector Algebra · MCQ
Q113.If non zero numbers $a, b, c$ are in $H . P .,$ then the straight line $\frac{x}{a} + \frac{y}{b} + \frac{1}{c} = 0$ always passes through a fixed point. That point is :
Mathematics · Straight Lines and Pair of Straight Lines · MCQ
Q114.Let $R = {(3, 3), (6, 6), (9, 9), (12, 12), (6, 12)$ , $(3, 9), (3, 12), (3, 6)}$ be a relation on the set $A = {3, 6, 9, 12}$ . The relation is :
Mathematics · Sets and Relations · MCQ
Q115.A lizard, at an initial distance of 21 cm behind an insect moves from rest with an acceleration of $2 cm / s^{2}$ and pursues the insect which is crawling uniformly along a straight line at a speed of $20 cm / s$ …
Mathematics · Application of Derivatives · MCQ