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JEE Mathematics Solved Question Paper 2015

Multiple Choice Questions

71.
Which of the following is not always true?

${|a + b|}^{2} = {|a|}^{2} + {|b|}^{2}$ , if a and b are perpendicular to each other

$|a + λb| \geq |a| for all λ \in R$ , if a and b are perpendicular to each other

${|a + b|}^{2} + {|a - b|}^{2} = 2 ({|a|}^{2} + {|b|}^{2})$

$|a + λb| \geq |a| for all λ \in R$ , if a is parallel to b

$|a + λb| \geq |a| for all λ \in R$ , if a is parallel to b

(a) If a and bare perpendicular to each other, then a . b = 0.

Now consider,

${|a + b|}^{2} = (a + b) . (a + b) = {|a|}^{2} + {|b|}^{2}$

So, option (a) is always true

(b) If a and bare perpendicular to each other, then a · b = 0.

Now consider,

${|a + λb|}^{2} = (a + λb) (a + λb) = {|a|}^{2} + λ {|b|}^{2} \Rightarrow |a + λb| = \sqrt{{|a|}^{2} + λ {|b|}^{2}} \geq |a| for all λ \in R$

So, option (b) is always true.

${|a + b|}^{2} + {|a - b|}^{2} = (a + b) . (a + b) + (a - b) . (a - b) = {|a|}^{2} + a . b + b . a + {|b|}^{2} + {|a|}^{2} - a . b - b . a + {|b|}^{2} = 2 ({|a|}^{2} + {|b|}^{2})$

So, option (c) is always true.

(d) Consider,

$a = - b and b \neq 0 Then, |a + λb| \geq |a| \Rightarrow |- b + λb| \geq |- b| \Rightarrow |b| |λ - 1| \geq |b| \Rightarrow |λ - 1| \geq 1 which is not true for all λ, as we consider λ = \frac{1}{2} then it is not true . Hence, option (d) is not always true .$

72.

If the four points with position vectors $- 2 \hat{i} + \hat{j} + \hat{k}, \hat{i} + \hat{j} + \hat{k}, \hat{j} - \hat{k} and λ \hat{j} + \hat{k}$ are coplanar, then $λ$ is equal to

73.

The area of the region bounded by the curve $y = x^{3}$ , its tangent at (1, 1) and X-axis, is

$\frac{1}{12} sq units$
$\frac{1}{6} sq units$
$\frac{2}{17}$ sq units
$\frac{2}{15}$ sq units

74.

The value of $λ$ for which the straight line $\frac{x - λ}{3} = \frac{y - 1}{2 + λ} = \frac{z - 3}{- 1}$ may lie on the plane x - 2y = 0, is

2
0
$- \frac{1}{2}$
$there is no such λ$

75.

Area of the region bounded by $y = |x| and y = - |x|$ + 2 is

4 sq units
3 sq units
2 sq units
1 sq units

76.

Let f(x) denotes the fractional part of a real number x. Then, the value of $\int_{0}^{\sqrt{3}} f (x^{2}) d x$

$2 \sqrt{3} - \sqrt{2} - 1$
0
$\sqrt{2} - \sqrt{3} + 1$
$\sqrt{3} - \sqrt{2} + 1$

77.

The value of $\int \frac{(x - 2)}{{\{{(x - 2)}^{2} {(x + 3)}^{7}\}}^{1 / 3}} dx$

$\frac{3}{20} {(\frac{x - 2}{x + 3})}^{4 / 3} + C$
$\frac{3}{20} {(\frac{x - 2}{x + 3})}^{3 / 4} + C$
$\frac{5}{12} {(\frac{x - 2}{x + 3})}^{4 / 3} + C$
$\frac{3}{20} {(\frac{x - 2}{x + 3})}^{5 / 3} + C$

78.

If 5 distinct balls are placed at random into 5 cells, then the probability that exactly one cell remains empty, is

48/125
12/125
8/125
1/125

79.

A survey of people in a given region showed that 20% were smokers. The probability of death due to lung cancer, given that a person smoked, "was 10 times the probability of death due to lung cancer, given that a person did not smoke. If the probability of death due to lung cancer in the region is 0.006. What is the probability of death due to lung cancer given that a person is a smoker?

1/140
1/70
3/140
1/10

80.

Suppose a machine produces metal parts that contain some defective parts with probability 0.05. How many parts should be produced in order that the probability of atleast one part being defective is 1/2 or more? (Given that, log₁₀ 95 = 1.977 and log₁₀2 = 0.3)