ISC Maths 2016 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 2016 Class-12 Previous Year Question Papers Solved

Section-A

Section-B

Section-C

### ISC Maths 2016 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 2016 Class-12 Previous Year Question Papers Solved

Que 1. [10 × 3]
(i) Find the matrix X for which: $\left[\begin{array}{ll}{5} & {4} \\ {1} & {1}\end{array}\right] X=\left[\begin{array}{ll}{1} & {-2} \\ {1} & {3}\end{array}\right]$
(ii) Solve for x, if: $\tan \left(\cos ^{-1} x\right)=\frac{2}{\sqrt{5}}$
(iii) Prove that the line 2x – 3y = 9 touches the conics y2 = -8x. Also, find the point of contact.
(iv) Using L’ Hospital’s rule, evaluate: $\lim _{x \rightarrow 0}\left(\frac{1}{x^{2}}-\frac{\cot x}{x}\right)$
(v) Evaluate: $\int \tan ^{3} x d x$
(vi) Using properties of definite integrals, evaluate: $\int_{0}^{\pi / 2} \frac{\sin x-\cos x}{1+\sin x \cos x} d x$
(vii) The two lines of regressions are x + 2y – 5 = 0 and 2x + 3y – 8 = 0 and the variance of x is 12. Find the variance of y and the coefficient of correlation.
(viii) Express $\frac{2+i}{(1+i)(1-2 i)}$ in the form of a + ib. Find its modulus and argument.
(ix) A pair of dice is thrown. What is the probability of getting an even number on the first die or a total of 8?
(x) Solve the differential equation: $x \frac{d y}{d x}+y=3 x^{2}-2$
Solution 1:         Que 2:
(a) Using properties of determinants, prove that: $\left|\begin{array}{ccc}{b+c} & {a} & {a} \\ {b} & {a+c} & {b} \\ {c} & {c} & {a+b}\end{array}\right|=4 a b c$
(b) Solve the following system of linear equations using matrix method: 
3x + y + z = 1
2x + 2z = 0
5x + y + 2z = 2
Solution 2:   Que 3:
(a) If $\sin ^{-1} x+\tan ^{-1} x=\frac{\pi}{2}$, prove that $2 x^{2}+1=\sqrt{5}$  (b) Write the Boolean function corresponding to the switching circuit given below: A, B and C represent switches in ‘on’ position and A’, B’ and C’ represent them in ‘off position. Using Boolean algebra, simplify the function and construct an equivalent switching circuit.
Solution 3:  Que 4:
(a) Verily the conditions of Rolle’s Theorem for the following function:  f(x) = log (x2 + 2) – log 3 on [-1, 1] Find a point in the interval, where the tangent to the curve is parallel to x-axis.
(b) Find the equation of the standard ellipse, taking its axes as the coordinate axes, whose minor axis is equal to the distance between the foci and whose length of latus rectum is 10. Also, find its eccentricity.  Solution 4:
(a) f(x) = log (x2 + 2) – log 3 in [-1, 1] (i) f(x) is continuous in [-1, 1] (ii) f(x) is derivable in (-1, 1)
(iii) f(-1) = log 3 – log 3 = 0
f(1) = log 3 – log 3 = 0
⇒ f(-1) = f(1) = o
Hence, there exists a point c in (-1, 1) such that f'(c) = 0  Que 5:
(a) If logy = tan-1x, prove that: $\left(1+x^{2}\right) \frac{d^{2} y}{d x^{2}}+(2 x-1) \frac{d y}{d x}=0$
(b) A rectangle is inscribed in a semicircle of radius r with one of its sides on the diameter of the semicircle. Find the dimensions of the rectangle to get maximum area. Also, find the maximum area.  Solution 5:  Que 6:
(a) Evaluate: $\int \frac{\sin x+\cos x}{\sqrt{9+16 \sin 2 x}} d x$
(b) Find the area of the region bound by the curves y = 6x – x2 and y = x2 – 2x  Solution 6:   Que 7:
(a) Calculate Karl Pearson’s coefficient of correlation between x and y for the following data and interpret the result: 
(1, 6), (2, 5), (3, 7), (4, 9), (5, 8), (6, 10), (7, 11), (8, 13), (9, 12)
(b) The marks obtained by 10 candidates in English and Mathematics are given below: Estimate the probable score for Mathematics if the marks obtained in English are 24.
Solution 7:  (b) Here n = 10.
Take the marks obtained in English and Mathematics as x and y respectively.
Let for x assumed mean be 17 and for v assumed mean be 19.
We construct the table as follows: y = $\frac { 596 }{ 22 }$ = 27.09 = 27 marks approx
Probable marks of mathematics is 27, when marks obtained in English are 24.

Que 8:
(a) A committee of 4 persons has to be chosen from 8 boys and 6 girls, consisting of at least one girl. Find the probability that the committee consists of more girls than boys.  (b) An urn contains 10 white and 3 black balls while another urn contains 3 white and 5 black balls. Two balls are drawn from the first urn and put into the second urn and then a ball is drawn from the second urn. Find the probability that the ball drawn from the second urn is a white ball.  Solution 8:  Que 9:
(a) Find the locus of a complex number, z = x + iy, satisfying the relation $\left|\frac{z-3 i}{z+3 i}\right| \leq \sqrt{2}$. Illustrate the locus of z in the Argand plane.  (b) Solve the following differential equation:  x2 dy + (xy + y2) dx = 0, when x = 1 and y = 1
Solution 9:   ### Section – B (20 Marks)

ISC Maths 2016 Class-12 Previous Year Question Papers Solved

Que 10:
(a) For any three vectors $\vec{a}, \vec{b}, \vec{c}$, show that $\vec{a}-\vec{b}, \vec{b}-\vec{c}, \vec{c}-\vec{a}$ are coplanar.  (b) Find a unit vector perpendicular to each of the vectors $\vec{a}+\vec{b}$ and $\vec{a}-\vec{b}$ where $\vec{a}=3 \hat{i}+2 \hat{j}+2 \hat{k}$ and $\vec{b}=\hat{i}+2 \hat{j}-2 \hat{k}$  Solution 10:  Que 11:
(a) Find the image of the point (2, -1, 5) in the line $\frac{x-11}{10}=\frac{y+2}{-4}=\frac{z+8}{-11}$. Also, find the length of the perpendicular from the point (2, -1, 5) to the line.  (b) Find the Cartesian equation of the plane, passing through the line of intersection of the planes $\vec{r} \cdot(2 \hat{i}+3 \hat{j}-4 \hat{k})+5=0 \text { and } \vec{r} \cdot(\hat{i}-5 \hat{j}+7 \hat{k})+2=0$ and intersecting y-axis at (0, 3).  Solution 11:
(a) The given line $\frac{x-11}{10}=\frac{y+2}{-4}=\frac{z+8}{-11}$
Let N be the foot of the perpendicular drawn from the point P(2, -1, 5).
Any point on line (1) is N (11 + 10t, -2 – 4t, -8 – 11t).
Now, direction ratio of NP is: < 9 + 10t, -1 – 4t, -13 -11t >
and direction ratio of line (1) is < 10, -4, -11 >
10(9 + 10t) + 4(1 + 4t) + 11(13 + 11t) = 0
⇒ 90 + 100t + 4 + 16t+ 143+ 121t = 0
⇒ 237t + 237 = 0
⇒ t = -1
N = (1, 2, 3)
Now, image of point P (2, -1, 5) is  Read Next 👇 Click on Page Number Given Below 👇