ISC Maths 2013 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 2013 Class-12 Previous Year Question Papers Solved

Section-A

Section-B

Section-C

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 2013 Class-12 Previous Year Question Papers Solved

Que 1:
(i) If (A – 2I) (A – 3I) = 0, where $A=\left(\begin{array}{cc}{4} & {2} \\ {-1} & {x}\end{array}\right) \text { and } I=\left(\begin{array}{ll}{1} & {0} \\ {0} & {1}\end{array}\right)$, find the value of x.  (ii) Find the value (s) of k so that the line 2x + y + k = 0 may touch the hyperbola 3x2 – y2 = 3.  (iii) Prove that: $\tan ^{-1} \frac{1}{4}+\tan ^{-1} \frac{2}{9}=\frac{1}{2} \sin ^{-1} \frac{4}{5}$  (iv) Using L’Hospital’s Rule, evaluate: $\lim _{x \rightarrow 0}\left(\frac{e^{x}-e^{-x}-2 x}{x-\sin x}\right)$
(v) Evaluate: $\int \frac{1}{x+\sqrt{x}} d x$  (vi) Evaluate: $\int_{0}^{1} \log \left(\frac{1}{x}-1\right) d x$  (vii) Two regression lines are represented by 4x + 10y = 9 and 6x + 3y = 4. Find the line of regression of y on x.  (viii) If 1, w and w2 are the cube roots of unity, evaluate: (1 – w4 + w8) (1 – w8 + w16)  (ix) Solve the differential equation: $\log \left(\frac{d y}{d x}\right)=2 x-3 y$
(x) If two balls are drawn from a bag containing three red balls and four blue balls, find the probability that:  (a) They are of the same colour
(b) They are of different colours
Solution 1:        Que 2:
(a) Using properties of determinants, prove that: $\left|\begin{array}{ccc}{x} & {y} & {z} \\ {x^{2}} & {y^{2}} & {z^{2}} \\ {y+z} & {z+x} & {x+y}\end{array}\right|=(x-y)(y-z)(z-x)(x+y+z)$
(b) Find A-1, where $A=\left[\begin{array}{ccc}{4} & {2} & {3} \\ {1} & {1} & {1} \\ {3} & {1} & {-2}\end{array}\right]$
Hence, solve the following system of linear equations: 
4x + 2y + 3z = 2
x + y + z = 1
3x + y – 2z = 5
Solution 2:    Que 3:
(a) Solve for x: $\sin ^{-1} x+\sin ^{-1}(1-x)=\cos ^{-1} x$.  (b) Construct a circuit diagram for the following Boolean Function:
(BC + A) (A’B’ + C) + A’B’C’
Using laws of Boolean Algebra, simplify the function and draw the simplifed circuit. 
Solution 3:  Que 4:
(a) Verify Lagrange’s Mean Value Theorem for the function $f(x)=\sqrt{x^{2}-x}$ in the interval [1, 4].  (b) From the following information, find the equation of the Hyperbola and the equation of its Transverse Axis.
Focus: (-2, 1), Directrix: 2x – 3y + 1 = 0, $e=\frac{2}{\sqrt{3}}$  Solution 4:   Which is the required equation of the hyperbola
Transverse axis is ⊥ to directrix
Equation of transverse axis be 3x + 2y + k = 0 ……(ii)
Focus (-2, 1) lies on transverse axis
3 × (-2) + 2 × 1 + k = 0
⇒ -4 + k = 0
⇒ k = 4
Now putting the value of k = 4 in (ii), we have
Equation of transverse axis is 3x + 2y + 4 = 0

Que 5:
(a) If y = (cot-1x)2, show that $\left(1+x^{2}\right)^{2} \frac{d^{2} y}{d x^{2}}+2 x\left(1+x^{2}\right) \frac{d y}{d x}=2$  (b) Find the maximum volume of the blinder which can be inscribed in a sphere of radius 3√3 cm. (Leave the answer in terms of π).
Solution 5:   Que 6:
(a) Evaluate: $\frac{\cos ^{-1} x}{x^{2}} d x$  (b) Find the area bounded by the curve y = 2x – x2, and the line y = x.  Solution 6:
(a) Put cos-1x = t
⇒ x = cos t
⇒ dx = -sin t dt  Que 7:
(a) Find Karl Pearson’s coefficient of correlation between x and y for the following data: (b) The following table shows the mean and standard deviation of the marks of Mathematics and Physics scored by the students in a School: The correlation coefficient between the given marks is 0.86. Estimate the likely marks in Physics if the marks in Mathematics are 92.
Solution 7:
(a) Assume mean A = 20 for the x-variate and B = 30 for the y-variate.  Que 8:
(a) Bag A contains three red and four white Balls; bag B contains two red and three white Balls. If one ball is drawn from bag A and two Balls from bag B, find the probability that: 
(i) One ball is red and two balls are white
(ii) All the three balls are of the same colour.
(b) Three persons, Aman, Bipin and Mohan attempt a Mathematics problem independently. The odds in favour of Aman and Mohan solving the problem are 3 : 2 and 4 : 1 respectively and the odds against Bipin solving the problem are 2 : 1. Find:  (i) The probability that all three will solve the problem.
(ii) The probability that the problem will be solved.
Solution 8:
(a) Here, the possible selection is as follows:
(i) 1 Red from bag A, 2 white from bag B
1 White from bag A, 1 white from bag B 1 Red from B
P (one ball is red and two balls are white) (ii) Possible selection is as follows:
(a) 1 Red from Bag A, 2 Red from Bag B
(b) 1 White from Bag A, 2 white from Bag B
P (All the three balls are of the same colour)    Que 9:
(a) Find the locus of the complex number z = x + iy, satisfying relations arg (z – 1) = $\frac{\pi}{4}$ and |z – 2 – 3i| = 2. Illustrate the locus oh the Argand plane.  (b) Solve the following differential equation: $y e^{y} d x=\left(y^{3}+2 x e^{y}\right) d y$, given that x = 0, y = 1  Solution 9:    Section – B (20 Marks)

### ISC Maths 2013 Class-12 Previous Year Question Papers Solved

Que 10:
(a) If $\vec{a} \text { and } \vec{b}$, are unit vectors and 0 is the angle between them, then show that $|\vec{a}-\vec{b}|=2 \sin \frac{\theta}{2}$.  (b) Find the value of λ for which the four points A, B, C, D with position vectors $-\hat{j}-\hat{k} : 4 \hat{i}+5 \hat{j}+\lambda \hat{k} : 3 \hat{i}+9 \hat{j}+4 \hat{k} \text { and }-4 \hat{i}+4 \hat{j}+4 \hat{k}$ are coplanar.  Solution 10:  Read Next 👇 Click on Page Number Given Below 👇