ISC Maths 2017 Class-12 Solved Previous Year Question Paper for practice. Step by step Solutions with section-A, B and C. Visit official website CISCE for detail information about ISC Board Class-12 Maths.

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## ISC Maths 2017 Class-12 Previous Year Question Papers Solved

Section-A

Section-B

Section-C

### ISC Maths 2017 Class-12 Previous Year Question Papers Solved

Time Allowed: 3 Hours
Maximum Marks: 100

(Candidates are allowed additional 15 minutes for only reading the paper. They must NOT start writing during this time.)

• The Question Paper consists of three sections A, B and C.
• Candidates are required to attempt all questions from Section A and all questions either from Section B or Section C.
• Section A: Internal choice has been provided in three questions of four marks each and two questions of six marks each.
• Section B: Internal choice has been provided in two questions of four marks each.
• Section C: Internal choice has been provided in two questions of four marks each.
• All working, including rough work, should be done on the same sheet as, and adjacent to the rest of the answer.
• The intended marks for questions or parts of questions are given in brackets [ ].
• Mathematical tables and graph papers are provided.

### Section – A (80 Marks)

ISC Maths 2017 Class-12 Previous Year Question Papers Solved

Que 1: [10 × 3]
(i) If the matrix $\left(\begin{array}{cc}{6} & {-x^{2}} \\ {2 x-15} & {10}\end{array}\right)$ is symmetric, find the value of x.
(ii) If y – 2x – k = 0 touches the conic 3x2 – 5y2 = 15, find the value of k.
(iii) Prove that $\frac{1}{2} \cos ^{-1}\left(\frac{1-x}{1+x}\right)=\tan ^{-1} \sqrt{x}$
(iv) Using L ‘Hospital’s Rule, evaluate:
$\underset{x \rightarrow \pi / 2}{\mathrm{L}}\left(x \tan x-\frac{\pi}{4} \cdot \sec x\right)$
(v) Evaluate$\int \frac{1}{x^{2}} \sin ^{2}\left(\frac{1}{x}\right) d x$
(vi) Evaluate: $\int_{0}^{\pi / 4} \log (1+\tan \theta) d \theta$
(vii) By using the data $\overline{x}=25, \overline{y}=30 ; b_{y x}=1.6 \text { and } b_{x y}=0.4$, find:
(a) The regression equation y on x.
(b) What is the most likely value of y when x = 60?
(c) What is the coefficient of correlation between x and y?
(viii) A problem is given to three students whose chances of solving it are $\frac { 1 }{ 4 }$$\frac { 1 }{ 5 }$ and $\frac { 1 }{ 3 }$ respectively. Find the probability that the problem is solved.

(x) Solve: $\frac { dy }{ dx }$ = 1 – xy + y – x
Solution 1:

Que 2:
(a) Using properties of determinants, prove that:
$\left|\begin{array}{lll}{a} & {b} & {b+c} \\ {c} & {a} & {c+a} \\ {b} & {c} & {a+b}\end{array}\right|=(a+b+c)(a-c)^{2}$
(b) Given that: $A=\left(\begin{array}{ccc}{1} & {-1} & {0} \\ {2} & {3} & {4} \\ {0} & {1} & {2}\end{array}\right) \text { and } B=\left(\begin{array}{ccc}{2} & {2} & {-4} \\ {-4} & {2} & {-4} \\ {2} & {-1} & {5}\end{array}\right)$, find AB.
Using this result, solve the following system of equation: x – y = 3, 2x + 3y + 4z = 17 and y + 2z = 7
Solution 2:

Que 3:
(a) Solve the equation for x:
$\sin ^{-1} x+\sin ^{-1}(1-x)=\cos ^{-1} x, x \neq 0$
(b) If A, B and C are the elements of Boolean algebra, simplify the expression (A’ + B’) (A + C’) + B’ (B + C). Draw the simplified circuit.
Solution 3:

Que 4:
(a) Verify Langrange’s mean value theorem for the function: [5]
f(x) = x (1 – log x) and find the value of ‘c’ in the interval [1, 2]
(b) Find the coordinates of the centre, foci and equation of directrix of the hyperbola x2 – 3y2 – 4x = 8. [5]
Solution 4:
(a) Given function ‘f’ is continuous in [1, 2] and differentiable in (1, 2)
f(x) = x (1 – log x) = x – x log x
f'(x) = 1 – x × $\frac { 1 }{ x }$ – log x = 1 – 1 – log x
f'(x) = – log x
According to Langrange’s Mean Value Theorem, E a real number c ∈ (1, 2) s.t.,

Que 5:
(a) If y = cos (sin x), show that: [5]
$\frac{d^{2} y}{d x^{2}}+\tan x \frac{d y}{d x}+y \cos ^{2} x=0$
(b) Show that the surface area of a closed cuboid with square base and given volume is minimum when it is a cube. [5]
Solution 5:

(b) Let x be the side of square base of cuboid and other side be y.
Then the volume of a cuboid with square base,
V = x × x × y
⇒ V = x2y
As the volume of the cuboid is given so volume is taken constantly throughout the question, therefore,
$y=\frac{V}{x^{2}}$ …….(i)
In order to show that surface area is minimum when the given cuboid is a cube, we have to show S” > 0 and x = y.
Let S be the surface area of cuboid, then

Que 6:
(a) Evaluate$\int \frac{\sin 2 x}{(1+\sin x)(2+\sin x)} d x$ [5]
(b) Draw a rough sketch of the curve y2 = 4x and find the area of region enclosed by the curve and the line y = x. [5]
Solution 6:

Que 7:
(a) Calculate the Spearman’s rank correlation coefficient for the following data and interpret the result: [5]

(b) Find the line of best fit for the following data, treating x as the dependent variable (Regression equation x on y): [5]

Hence, estimate the value of x when y = 16.
Solution 7:
(a) To determine Spearman’s Rank Correlation:

Question 8:
(a) In a class of 60 students, 30 opted for Mathematics, 32 opted for Biology and 24 opted for both Mathematics and Biology. If one of these students is selected at random, find the probability that: [5]
(i) The student opted for Mathematics or Biology.
(ii) The student has opted neither Mathematics nor Biology.
(iii) The student has opted Mathematics but not Biology.
(b) Bag A contains 1 white, 2 blue and 3 red balls. Bag B contains 3 white, 3 blue and 2 red balls. Bag C contains 2 white, 3 blue and 4 red balls. One bag is selected at random and then two balls are drawn from the selected bag. Find the probability that the balls draw n are white and red. [5]
Solution 8:

U = 60
n(M) = 30
n(B) = 32
n(M ∩ B) = 24
n(M ∪ B) = n(M) + n(B) – n(M ∩ B) = 30 + 32 – 24 = 38
n(M ∪ B)’ = n(∪) – n(M ∪ B) = 60 – 38 = 22
Only Mathematics = n(M) – n(M ∩ B) = 30 – 24 = 6
(i) P(student opted for Mathematics or Biology) = $\frac { 24 }{ 60 }$ = $\frac { 2 }{ 5 }$
(ii) P(student opted neither Mathematics nor Biology) = $\frac { 22 }{ 60 }$ = $\frac { 11 }{ 30 }$
(iii) P(student opted Mathematics but not Biology) = $\frac { 6 }{ 60 }$ = $\frac { 1 }{ 10 }$
(b) Given:
Bag A: 1 white, 2 blue and 3 red balls
Bag B: 3 white, 3 blue and 2 red balls
Bag C: 2 white, 3 blue and 4 red balls
Let B1, B2, B3 and E be the events defined as
B1: Bag A is selected
B2: Bag B is selected
B3: Bag C is selected
And E: 1 white and 1 red ball is drawn

Que 9:
(a) Prove that locus of z is circle and find its centre and radius if $\frac{z-i}{z-1}$ is purely imaginary. [5]
(b) Solve: (x2 – yx2) dy + (y2 + xy2) dx = 0 [5]
Solution 9:

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