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CBSE

Subject

Mathematics

Class

JEE Class 12

JEE Mathematics 2012 Exam Questions

Multiple Choice Questions

1.

In a ∆PQR, if 3 sin P + 4 cos Q = 6 and 4 sin Q + 3 cos P = 1, then the angle R is equal to

  • 5π/6

  • π/6

  • π/4

  • 3π/4


B.

π/6

3 sin P + 4 cos Q = 6 ...... (1)
4 sin Q + 3 cos P = 1 ...... (2)
From (1) and (2) ∠P is obtuse.
(3 sin P + 4 cos Q)2+ (4 sin Q + 3 cos P)2= 37
⇒9 + 16 + 24 (sin P cos Q + cos P sin Q) = 37
⇒ 24 sin (P + Q) = 12
rightwards double arrow space sin space left parenthesis straight P space plus straight Q right parenthesis space equals space 1 half
rightwards double arrow space straight P space plus straight Q space equals space fraction numerator 5 straight pi over denominator 6 end fraction
rightwards double arrow space straight R space equals space straight pi over 6

163 Views

2.

If the line 2x + y = k passes through the point which divides the line segment joining the points (1, 1) and (2, 4) in the ratio 3 : 2, then k equals

  • 29/5

  • 5

  • 6

  • 11/5


C.

6

Line L : 2x +y = k passes through the point (say P) which divides a lie segment (say AB) in ratio 3:2 where A (1,1) and B (2,4).
Using section formula, the coordinates of the point P which divides AB internally in the ratio 3:2 are

straight p space open parentheses fraction numerator 3 space straight x 2 plus 2 space straight x 1 over denominator 3 plus 2 end fraction comma fraction numerator space 3 straight x space 4 plus 2 straight x 1 over denominator 3 plus 2 end fraction close parentheses space equals space straight P open parentheses 8 over 5 comma 14 over 5 close parentheses
Also, since the line L passes through P, hence substituting the coordinates of straight P space open parentheses 8 over 5 comma 14 over 5 close parentheses  in the equation of line L: 2x +y = k, 
we get

2 space open parentheses 8 over 5 close parentheses plus open parentheses 14 over 5 close parentheses space
rightwards double arrow space straight k space equals space 6

224 Views

3.

Let x1, x2, ......, xn be n observations, and let top enclose straight x be their arithmetic mean and σ2 be their variance.
Statement 1: Variance of 2x1, 2x2, ......, 2xn is 4 σ2.
Statement 2: Arithmetic mean of 2x1, 2x2, ......, 2xn is 4top enclose straight x.

  • Statement 1 is false, statement 2 is true

  • Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1

  • Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1

  • Statement 1 is true, statement 2 is false


D.

Statement 1 is true, statement 2 is false

top enclose straight x is the AM and σ2 is the variance of n observations x1, x2, x3......xn
AM of 2x1, 2x2, 2x3, .......2xn

space equals space fraction numerator 2 straight x subscript 1 space plus space 2 straight x subscript 2 space plus 2 straight x subscript 3 space plus.....2 straight x subscript straight n over denominator straight n end fraction

equals space 2 open parentheses fraction numerator straight x subscript 1 plus straight x subscript 2 plus straight x subscript 3 space plus..... plus straight x subscript straight n over denominator straight n end fraction close parentheses space equals space 2 top enclose straight x
Hence, it prove that statement 2 is false.

 
213 Views

4.

Statement 1: The sum of the series 1 + (1 + 2 + 4) + (4 + 6 + 9) + (9 + 12 + 16) + ...... + (361 + 380 +400) is 8000.
Statement 2: sum from straight k space equals 1 to straight n equals space 1 of left square bracket straight k cubed minus left parenthesis straight k minus 1 cubed right parenthesis right square bracket space equals space straight n cubed, for any natural number n.

  • Statement 1 is false, statement 2 is true

  • Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1

  • Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1

  • Statement 1 is true, statement 2 is false


B.

Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1

Statement 1 has 20 terms whose sum is 8000 And statement 2 is true and supporting statement 1.
 kth bracket is (k – 1)2 + k(k – 1) + k2 = 3k2 – 3k + 1.

178 Views

5.

The negation of the statement “If I become a teacher, then I will open a school” is

  • I will become a teacher and I will not open a school

  • Either I will not become a teacher or I will not open a school

  • Neither I will become a teacher nor I will open a school

  • I will not become a teacher or I will open a school


A.

I will become a teacher and I will not open a school

Let us assume that
p: I become a teacher' and 
q: I will open a school

Then, we can easily as certain that
Negation of (p →q)

~(~p ∨ q) = p ∧ ~q
Which means that ' l' will become a teacher and I will not open a school.

208 Views

6.

The equation esinx-e-sinx -4 = 0 has

  • infinite number of real roots

  • No real root

  • exactly one real root

  • exactly four real roots


B.

No real root

straight e to the power of sinx space minus straight e to the power of negative sin space straight x end exponent space equals space 4
rightwards double arrow space straight e to the power of sin space straight x end exponent space equals space straight t
straight t minus 1 over straight t space equals space 4
straight t squared minus 4 straight t minus 1 space equals space 0
rightwards double arrow space straight t space equals space fraction numerator 4 space plus-or-minus square root of 16 plus 4 end root over denominator 2 end fraction
rightwards double arrow straight t space equals space fraction numerator 4 space plus-or-minus 2 square root of 5 over denominator 2 end fraction
rightwards double arrow space straight t space equals space 2 plus-or-minus square root of 5
straight e to the power of sin space straight x end exponent space equals space space 2 plus-or-minus square root of 5 space minus space less or equal than space sin space straight x less or equal than 1
straight e to the power of sin space straight x end exponent space equals space 2 space plus square root of 5 space not space possible
straight e to the power of sin space straight x end exponent space equals space 2 space minus square root of 5 space not space possible
Hence comma space no space solution
312 Views

7.

If 100 times the 100th term of an AP with non zero common difference equals the 50 times its 50th term, then the 150th term of this AP is

  •  –150

  • 150

  • times its 50th term

  • 0


D.

0

The 150 th term of this AP
Let a be the first term and d be the common difference of the given AP, then
T100 =  a+ (100-1)d = a + 99d
T50 = a +(50-1)d = a +49 d
T150 = a + (150-1) d = a +149 d
Now, according to the question,
100 x T100 = 50 x T50
⇒ 100 (a +99d) = 50(a +49d)
2(a +99d) = (a+ 49d)
2a +198 d =a +49d
a +149d = 0

263 Views

8.

If n is a positive integer, then open parentheses square root of 3 plus 1 close parentheses to the power of 2 straight n end exponent minus space left parenthesis square root of 3 minus 1 right parenthesis to the power of 2 straight n end exponent space is

  • an irrational number

  • an odd positive integer

  • an even positive integer

  • a rational number other than positive integers


A.

an irrational number

left parenthesis straight x plus straight a right parenthesis to the power of straight n space equals space to the power of straight n straight C subscript straight o space straight x to the power of straight n space plus to the power of straight n straight C subscript 1 straight x to the power of straight n minus 1 end exponent space straight a space plus to the power of straight n straight C subscript 1 straight x to the power of straight n minus 2 end exponent straight a squared space plus..... plus straight n space to the power of straight n straight C subscript straight n straight a to the power of straight n
and
left parenthesis straight x minus straight a right parenthesis to the power of straight n space equals space to the power of straight n straight C subscript straight o space straight x to the power of straight n space minus to the power of straight n straight C subscript 1 straight x to the power of straight n minus 1 end exponent space straight a space plus to the power of straight n straight C subscript 1 straight x to the power of straight n minus 2 end exponent straight a squared space plus..... plus left parenthesis negative 1 right parenthesis straight n space to the power of straight n straight C subscript straight n straight a to the power of straight n
left parenthesis square root of 3 plus 1 end root right parenthesis to the power of 2 straight n end exponent space equals space to the power of 2 straight n end exponent straight C subscript straight o space left parenthesis square root of 3 right parenthesis to the power of 2 straight n end exponent space plus to the power of 2 straight n end exponent straight C subscript 1 left parenthesis square root of 3 right parenthesis to the power of 2 straight n minus 1 end exponent space plus to the power of 2 straight n end exponent straight C subscript 2 space left parenthesis square root of 3 right parenthesis to the power of 2 straight n minus 2 end exponent space
plus........ plus to the power of 2 straight n end exponent straight C subscript 2 straight n end subscript left parenthesis square root of 3 right parenthesis to the power of 2 straight n minus 2 straight n end exponent

left parenthesis square root of 3 minus 1 end root right parenthesis to the power of 2 straight n end exponent space equals space to the power of 2 straight n end exponent straight C subscript straight o space left parenthesis square root of 3 right parenthesis to the power of 2 straight n end exponent space left parenthesis negative 1 right parenthesis to the power of 0 plus to the power of 2 straight n end exponent straight C subscript 1 left parenthesis square root of 3 right parenthesis to the power of 2 straight n minus 1 end exponent space left parenthesis negative 1 right parenthesis squared space plus
to the power of 2 straight n end exponent straight C subscript 2 space left parenthesis square root of 3 right parenthesis to the power of 2 straight n minus 2 end exponent left parenthesis negative 1 right parenthesis squared space plus........ plus to the power of 2 straight n end exponent straight C subscript 2 straight n end subscript left parenthesis square root of 3 right parenthesis to the power of 2 straight n minus 2 straight n end exponent left parenthesis negative 1 right parenthesis to the power of 2 straight n end exponent
Adding both the binomial expansions above, we get
left parenthesis square root of 3 plus 1 right parenthesis to the power of 2 straight n end exponent space minus space left parenthesis square root of 3 straight n end root minus 1 right parenthesis to the power of 2 straight n end exponent space equals space 2 left square bracket to the power of 2 straight n end exponent straight C subscript 1 left parenthesis square root of 3 right parenthesis to the power of 2 straight n minus 1 end exponent
plus to the power of 2 straight n end exponent straight C subscript 3 space left parenthesis square root of 3 right parenthesis to the power of 2 straight n minus 3 end exponent space plus to the power of 2 straight n end exponent straight C subscript 5 space left parenthesis square root of 3 right parenthesis to the power of 2 straight n minus 5 end exponent space plus....... space plus to the power of 2 straight n end exponent straight C subscript 2 straight n minus 1 end subscript space left parenthesis square root of 3 right parenthesis to the power of 2 straight n minus left parenthesis 2 straight n minus 1 right parenthesis end exponent right square bracket
It is the irrational number because of odd power of square root of 3 appears in each of the terms.


170 Views

9.

Statement I An equation of a common tangent to the parabola straight y squared space equals space 16 space square root of 3 straight x end root and the ellipse 2x2 +y2 =4 is space straight y space equals 2 straight x space plus 2 square root of 3.
Statement II If the line straight Y space equals space mx space plus fraction numerator 4 square root of 3 over denominator straight m end fraction comma space left parenthesis straight m space not equal to 0 right parenthesis is a common tangent to the parabola straight y squared space equals space 16 space square root of 3 straight x and the ellipse 2x2 +y2 =4, then m satisfies m4 +2m2 =24

  • Statement 1 is false, statement 2 is true

  • Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1

  • Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1

  • Statement 1 is true, statement 2 is false


C.

Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1

straight y squared space equals space 16 space square root of 3 straight x
straight x squared over 2 plus straight y squared over 4 space equals 1
straight y space equals space mx space plus fraction numerator 4 square root of 3 over denominator straight m end fraction space is space tangent space to space parabola
which space is space tangent space to space ellipse
rightwards double arrow space straight c squared space equals space straight a squared straight m squared space plus straight b squared
rightwards double arrow space 48 over straight m squared space equals space 2 straight m squared space plus 4
rightwards double arrow straight m to the power of 4 space plus 2 straight m squared space equals space 24
rightwards double arrow space straight m squared space equals space 4
197 Views

10.

An equation of a plane parallel to the plane x – 2y + 2z – 5 = 0 and at a unit distance from the origin is

  • x – 2y + 2z – 3 = 0

  • x – 2y + 2z + 1 = 0

  • x – 2y + 2z – 1 = 0

  • x – 2y + 2z + 5 = 0


A.

x – 2y + 2z – 3 = 0

Perpendicular distance of the plane ax +by + cz +d =0 from the point
(x,y,z) is d =open vertical bar fraction numerator ax subscript 1 space plus by subscript 1 plus cz subscript 1 plus straight d over denominator square root of straight a squared plus straight b squared plus straight c squared end root end fraction close vertical bar
Equation of plane parallel to x – 2y + 2z – 5 = 0 is x – 2y + 2z + k = 0 ...... (1)

perpendicular distance from O(0, 0, 0) to (1) is 1

fraction numerator vertical line straight k vertical line over denominator square root of 1 plus 4 plus 4 end root end fraction space equals space 1
rightwards double arrow space vertical line straight k vertical line space equals space 3
rightwards double arrow space straight k space equals space plus-or-minus 3
therefore space straight x minus 2 straight y plus 2 straight z minus 3 space equals 0

457 Views

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