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1201 |
Let f(x) = (1 – x)2 sin2x + x2Consider the statementsStatement 1: There exists some x ∈ ℝ such that f(x) + 2x = 2(1 + x2)Statement 2: There exists some x ∈ ℝ such that 2f(x) + 1 = 2x(x + 1) a) Both 1 and 2 are true b) 1 is true and 2 is false c) 1 is false and 2 is true d) Both 1 and 2 are false
Let f(x) = (1 – x)2 sin2x + x2Consider the statementsStatement 1: There exists some x ∈ ℝ such that f(x) + 2x = 2(1 + x2)Statement 2: There exists some x ∈ ℝ such that 2f(x) + 1 = 2x(x + 1) a) Both 1 and 2 are true b) 1 is true and 2 is false c) 1 is false and 2 is true d) Both 1 and 2 are false
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IIT 2013 |
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1202 |
Let z and ω be two complex numbers such that |z| ≤ 1, |ω| ≤ 1 and then z equals a) 1 or i b) i or –i c) 1 or –1 d) i or –1
Let z and ω be two complex numbers such that |z| ≤ 1, |ω| ≤ 1 and then z equals a) 1 or i b) i or –i c) 1 or –1 d) i or –1
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IIT 1995 |
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1203 |
Given  Prove that 
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IIT 1984 |
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1204 |
The coordinates of the in centre of the triangle that has the co ordinates of the mid points of its sides as (0, 1), (1, 1) and (1, 0) is a) b) c) d)
The coordinates of the in centre of the triangle that has the co ordinates of the mid points of its sides as (0, 1), (1, 1) and (1, 0) is a) b) c) d)
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IIT 2013 |
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1205 |
Using mathematical induction, prove that for n > 1
Using mathematical induction, prove that for n > 1
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IIT 1986 |
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1206 |
If f(x) = then on the interval [0, π] a) tan and are both continuous b) tan and are both discontinuous c) tan and are both continuous d) tan is continuous but is not
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IIT 1989 |
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1207 |
One or more than one correct option A ray of light along gets reflected upon reaching X- axis, the equation of the reflected ray is a) b) c) d)
One or more than one correct option A ray of light along gets reflected upon reaching X- axis, the equation of the reflected ray is a) b) c) d)
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IIT 2013 |
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1208 |
If and where 0 < x ≤1, then in this interval a) Both f (x) and g (x) are increasing functions b) Both f (x) and g (x) are decreasing functions c) f (x) is an increasing function d) g (x) is an increasing function
If and where 0 < x ≤1, then in this interval a) Both f (x) and g (x) are increasing functions b) Both f (x) and g (x) are decreasing functions c) f (x) is an increasing function d) g (x) is an increasing function
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IIT 1997 |
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1209 |
The number of common tangents to the circles x2 + y2 – 4x − 6y – 12 = 0 and x2 + y2 + 6x + 18y + 26 = 0 is a) 1 b) 2 c) 3 d) 4
The number of common tangents to the circles x2 + y2 – 4x − 6y – 12 = 0 and x2 + y2 + 6x + 18y + 26 = 0 is a) 1 b) 2 c) 3 d) 4
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IIT 2015 |
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1210 |
Let p ≥ 3 be an integer and α, β be the roots of x2 – (p + 1) x + 1 = 0. Using mathematical induction show that αn + βn i) is an integer ii) and is not divisible by p.
Let p ≥ 3 be an integer and α, β be the roots of x2 – (p + 1) x + 1 = 0. Using mathematical induction show that αn + βn i) is an integer ii) and is not divisible by p.
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IIT 1992 |
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1211 |
The function is not differentiable at a) – 1 b) 0 c) 1 d) 2
The function is not differentiable at a) – 1 b) 0 c) 1 d) 2
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IIT 1999 |
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1212 |
One or more than one correct option Let RS be a diameter of the circle x2 + y2 = 1 where S is the point (1, 0). Let P be a variable point (other than R and S) on the circle and the tangents to the circle at S and P meet at the point Q. The normal to the circle at P intersect a line drawn through Q parallel to RS at a point E. Then the locus of E passes through the point(s) a) b) c) d)
One or more than one correct option Let RS be a diameter of the circle x2 + y2 = 1 where S is the point (1, 0). Let P be a variable point (other than R and S) on the circle and the tangents to the circle at S and P meet at the point Q. The normal to the circle at P intersect a line drawn through Q parallel to RS at a point E. Then the locus of E passes through the point(s) a) b) c) d)
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IIT 2016 |
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1213 |
If x is not an integral multiple of 2π use mathematical induction to prove that
If x is not an integral multiple of 2π use mathematical induction to prove that
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IIT 1994 |
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1214 |
A circle passing through (1, −2) and touching the axis of X at (3, 0) also passes through the point a) (−5, 2) b) (2, −5) c) (5, −2) d) (−2, 5)
A circle passing through (1, −2) and touching the axis of X at (3, 0) also passes through the point a) (−5, 2) b) (2, −5) c) (5, −2) d) (−2, 5)
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IIT 2013 |
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1215 |
The circles and intersect each other in distinct points if a) r < 2 b) r > 8 c) 2 < r < 8 d) 2 ≤ r ≤ 8
The circles and intersect each other in distinct points if a) r < 2 b) r > 8 c) 2 < r < 8 d) 2 ≤ r ≤ 8
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IIT 1994 |
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1216 |
Prove by induction that Pn = Aαn + Bβn for all n ≥ 1 Where α and β are roots of the quadratic equation x2 – (1 – P) x – P (1 – P) = 0, P1 = 1, P2 = 1 – P2, . . ., Pn = (1 – P) Pn – 1 + P (1 – P) Pn – 2 n ≥ 3, and , 
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IIT 2000 |
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1217 |
Let O be the vertex and Q be any point on the parabola x2 = 8y. If the point P divides the line segment internally in the ratio 1 : 3 then the locus of P is a) x2 = y b) y2 = x c) y2 = 2x d) x2 = 2y
Let O be the vertex and Q be any point on the parabola x2 = 8y. If the point P divides the line segment internally in the ratio 1 : 3 then the locus of P is a) x2 = y b) y2 = x c) y2 = 2x d) x2 = 2y
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IIT 2015 |
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1218 |
Solve the following equation for x a) −1 b)  c) 0 d) −1 and 
Solve the following equation for x a) −1 b)  c) 0 d) −1 and 
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IIT 1978 |
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1219 |
If f is a differentiable function satisfying for all n ≥ 1, n I then a)  b)  c)  d) is not necessarily zero
If f is a differentiable function satisfying for all n ≥ 1, n I then a)  b)  c)  d) is not necessarily zero
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IIT 2005 |
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1220 |
Evaluate 
Evaluate 
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IIT 2005 |
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1221 |
Let S be the focus of the parabola y2 = 8x and PQ be the common chord of the circle x2 + y2 – 2x – 4y = 0 and the given parabola. The area of △QPS is a) 2 sq. units b) 4 sq. units c) 6 sq. units d) 8 sq. units
Let S be the focus of the parabola y2 = 8x and PQ be the common chord of the circle x2 + y2 – 2x – 4y = 0 and the given parabola. The area of △QPS is a) 2 sq. units b) 4 sq. units c) 6 sq. units d) 8 sq. units
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IIT 2012 |
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1222 |
Multiple choices The function f (x) = 1 + |sinx| is a) continuous nowhere b) continuous everywhere c) differentiable nowhere d) not differentiable at x = 0 e) not differentiable at infinite number of points
Multiple choices The function f (x) = 1 + |sinx| is a) continuous nowhere b) continuous everywhere c) differentiable nowhere d) not differentiable at x = 0 e) not differentiable at infinite number of points
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IIT 1986 |
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1223 |
Let a, r, s, t be non-zero real numbers. Let P(at2, 2at), Q, R(ar2, 2ar and S(as2, 2as) be distinct points on the parabola y2 = 4ax. Suppose PQ is the focal chord and QR and PK are parallel, where K is point (2a, 0)If st = 1 then the tangent at P and normal at S to the parabola meet at a point whose ordinate is a) b) c) d)
Let a, r, s, t be non-zero real numbers. Let P(at2, 2at), Q, R(ar2, 2ar and S(as2, 2as) be distinct points on the parabola y2 = 4ax. Suppose PQ is the focal chord and QR and PK are parallel, where K is point (2a, 0)If st = 1 then the tangent at P and normal at S to the parabola meet at a point whose ordinate is a) b) c) d)
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IIT 2014 |
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1224 |
The tangent PT and the normal PN of the parabola y2 = 4ax at the point P on it meet its axis at the points T and N respectively. The locus of the centroid of the triangle PTM is a parabola whose a) Vertex is b) Directrix is x = 0 c) Latus rectum is d) Focus is (a, 0)
The tangent PT and the normal PN of the parabola y2 = 4ax at the point P on it meet its axis at the points T and N respectively. The locus of the centroid of the triangle PTM is a parabola whose a) Vertex is b) Directrix is x = 0 c) Latus rectum is d) Focus is (a, 0)
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IIT 2009 |
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1225 |
Let f and g be increasing and decreasing functions, respectively from [0, ∞) to [0, ∞). Let h(x) =f(g(x)). If h(0) = 0 then h(x) – h(t) is a) Always zero b) Always negative c) Always positive d) Strictly increasing e) None of these
Let f and g be increasing and decreasing functions, respectively from [0, ∞) to [0, ∞). Let h(x) =f(g(x)). If h(0) = 0 then h(x) – h(t) is a) Always zero b) Always negative c) Always positive d) Strictly increasing e) None of these
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IIT 1988 |
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