JEE Advance - Mathematics (2017 - Paper 1 Offline)

• 1

  Let X and Y be two events such that $$P(X) = {1 \over 3}$$, $$P(X|Y) = {1 \over 2}$$ and $$P(Y|X) = {2 \over 5}$$. Then

  คำตอบ

  A

  B
• 2

  Let f : R $$ \to $$ (0, 1) be a continuous function. Then, which of the following function(s) has (have) the value zero at some point in the interval (0, 1) ?

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  C

  D
• 3

  Let a, b, x and y be real numbers such that a $$-$$ b = 1 and y $$ \ne $$ 0. If the complex number z = x + iy satisfies $${\mathop{\rm Im}\nolimits} \left( {{{az + b} \over {z + 1}}} \right) = y$$, then which of the following is(are) possible value(s) of x?

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  B

  D
• 4

  If $$2x - y + 1 = 0$$ is a tangent to the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {16}} = 1$$ then which of the following CANNOT be sides of a right angled triangle?

  คำตอบ

  A

  C

  D
• 5

  Let [x] be the greatest integer less than or equals to x. Then, at which of the following point(s) the function $$f(x) = x\cos (\pi (x + [x]))$$ is discontinuous?

  คำตอบ

  A

  B

  D
• 6

  Which of the following is(are) NOT the square of a 3 $$ \times $$ 3 matrix with real entries?

  คำตอบ

  A

  C
• 7

  If a chord, which is not a tangent, of the parabola y² = 16x has the equation 2x + y = p, and mid-point (h, k), then which of the following is(are) possible value(s) of p, h and k?

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  (B)

  p = 2, h = 3, k = $$-$$4
• 8

  For a real number $$\alpha $$, if the system
  
  $$\left[ {\matrix{ 1 & \alpha & {{\alpha ^2}} \cr \alpha & 1 & \alpha \cr {{\alpha ^2}} & \alpha & 1 \cr } } \right]\left[ {\matrix{ x \cr y \cr z \cr } } \right] = \left[ {\matrix{ 1 \cr { - 1} \cr 1 \cr } } \right]$$
  
  of linear equations, has infinitely many solutions, then 1 + $$\alpha $$ + $$\alpha $$² =

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  1
• 9

  The sides of a right angled triangle are in arithmetic progression. If the triangle has area 24, then what is the length of its smallest side?

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  6
• 10

  Let f : R $$ \to $$ R be a differentiable function such that f(0) = 0, $$f\left( {{\pi \over 2}} \right) = 3$$ and f'(0) = 1.
  
  If $$g(x) = \int\limits_x^{\pi /2} {[f'(t)\text{cosec}\,t - \cot t\,\text{cosec}\,t\,f(t)]dt} $$
  
  for $$x \in \left( {0,\,{\pi \over 2}} \right]$$, then $$\mathop {\lim }\limits_{x \to 0} g(x)$$ =

  คำตอบ

  2
• 11

  For how many values of p, the circle x² + y² + 2x + 4y $$-$$ p = 0 and the coordinate axes have exactly three common points?

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  2
• 12

  Words of length 10 are formed using the letters A, B, C, D, E, F, G, H, I, J. Let x be the number of such words where no letter is repeated; and let y be the number of such words where exactly one letter is repeated twice and no other letter is repeated. Then, $${y \over {9x}}$$ = ?

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  5
• 13

  For $$a = \sqrt 2 $$, if a tangent is drawn to a suitable conic (Column 1) at the point of contact ($$-$$1, 1), then which of the following options is the only CORRECT combination for obtaining its equation?

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  (A)

  (I) (ii) Q)
• 14

  The tangent to a suitable conic (Column 1) at $$\left( {\sqrt 3 ,\,{1 \over 2}} \right)$$ is found to be $$\sqrt 3 x + 2y = 4$$, then which of the following options is the only CORRECT combination?

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  (B)

  (II) (iv) (R)
• 15

  If a tangent to a suitable conic (Column 1) is found to be y = x + 8 and its point of contact is (8, 16), then which of the following options is the only CORRECT combination?

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  (A)

  (III) (i) (P)
• 16

  Which of the following options is the only INCORRECT combination?

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  (D)

  (III) (i) (R)
• 17

  Which of the following options is the only CORRECT combination?

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  (C)

  (II) (iii) (S)
• 18

  Which of the following options is the only CORRECT combination?

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  (C)

  (II) (ii) (Q)