Recent questions in Model Papers

Assume that the chances of a patient having a heart attack is 40%. It is assumed that a meditation and yoga course reduce the risk of heart attack by 30% and prescription of certain drug reduces its chance by 25%. At a time, a patient can choose any one of the two options with equal probabilities. It is given that after going through one of the two options the patient selected at random suffers a heart attack. Find the probability that the patient followed a course of meditation and yoga?

asked Jan 27, 2013 by thanvigandhi_1

1 answer

If $f(x) = \cos \: x \: and \: g(x) = x^3,$ find $fog \:and \:gof.$

asked Jan 27, 2013 by thanvigandhi_1

0 answers

If $sin \bigg( \sin^{-1} \frac{1}{5} + \cos^{-1} x \bigg) = 1,$ find x.

asked Jan 27, 2013 by thanvigandhi_1

0 answers

If $A = \begin{bmatrix} 6 & -5 \\ 2 & 3 \end{bmatrix}$ , then show that A-A' is skew matrix.

asked Jan 27, 2013 by thanvigandhi_1

1 answer

The point (3,-2),(x,2),(8,8) are collinear, find x using determinants.

asked Jan 27, 2013 by thanvigandhi_1

1 answer

Evaluate : $\int\large\frac{x^2+\cos 4x}{4x^3+3\sin 4x}$ $dx$

asked Jan 27, 2013 by thanvigandhi_1

0 answers

If $\int_0^a 3x^2dx=27$ , find the value of a.

asked Jan 27, 2013 by thanvigandhi_1

0 answers

If $| \overrightarrow a | = 10, | \overrightarrow b | = 15$ and the angle between the vector $\overrightarrow a \: and \: \overrightarrow b \: is \:45^{\circ},$ then find $\overrightarrow a.\overrightarrow b$ .

asked Jan 27, 2013 by thanvigandhi_1

0 answers

Find a vector in the direction of $-\hat i + \sqrt3 \hat k,$ whose magnitude is 3.

asked Jan 27, 2013 by thanvigandhi_1

0 answers

Find the angle between the lines $\large \frac{x+2}{1}=\frac{y}{2}=\frac{z-1}{0}$ and $\large\frac{x-1}{3}=\frac{y-4}{2}=\frac{z+5}{1}$

asked Jan 27, 2013 by thanvigandhi_1

0 answers

Find the value of x, such that $A = \begin{bmatrix} x & 5 \\ 4 & 2x \end{bmatrix} = \begin{bmatrix} 2 & 5 \\ 16 & 4 \end{bmatrix}$

asked Jan 27, 2013 by thanvigandhi_1

1 answer

Show that the R defined on the set A of all triangles as $R = \{(T_1, T_2): T_1$ is similar to $T_2 \}$ is an equivalence relation. Consider three right-angled triangles $T_1$ with sides 3, 4, 5, $T_2$ with sides 5, 12, 13 and $T_3$ with sides 6, 8, 10. Which triangles among $T_1, T_2 and T_3$ are related?

asked Jan 27, 2013 by thanvigandhi_1

0 answers

Solve for x : $\tan^{-1}(x+1)+\tan^{-1}(x-1)=\tan^{-1}\large\frac{8}{31}$

asked Jan 27, 2013 by thanvigandhi_1

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If $\cos^{-1}\large\frac{x}{a}$ $+\cos^{-1}\large\frac{y}{b}= \alpha$ , prove that $\large \frac{x^2}{a^2}-\frac{2xy}{ab}$ $\cos \alpha+\large\frac{y^2}{b^2}$ $=\sin^2\alpha.$

asked Jan 27, 2013 by thanvigandhi_1

1 answer

(i)A sum of rupees ten thousand is put into three investments at the rate of 10%, 12% and 15% p.a. The combined income of first and second investment is Rs. 620 more of the income of third investment. If the combined income is Rs. 2420, then find the investment in each using matrices.(ii)"One can buy many things with very little savings".Explain the importance of savings in student's life.

asked Jan 27, 2013 by thanvigandhi_1

0 answers

If $f(x) = \left\{ \begin{array}{l l}x^2, & \quad if\; { x \leq 1}\\ ax+b, & \quad if\; { x > 1} \\ \end{array}. \right.$ is differentiable at x = 1. Find the value of a and b.

asked Jan 27, 2013 by thanvigandhi_1

0 answers

Verify Lagrange's mean value theorem for the following function : $f(x) = \sin \: x - \sin\: 2x \: in \: [0, \pi ]$

asked Jan 27, 2013 by thanvigandhi_1

1 answer

If $x\sqrt{1+y} +y\sqrt{1+x}=0, \: find \:\large \frac{dy}{dx}$

asked Jan 27, 2013 by thanvigandhi_1

0 answers

If $y\sqrt{1-x^2}+x\sqrt{1-y^2}=1,$ prove that $\large\frac{dy}{dx}=-\sqrt{\frac{1-y^2}{1-x^2}}$

asked Jan 27, 2013 by thanvigandhi_1

0 answers

Prove that : $\int_{-a}^a \sqrt{\large\frac{a-x}{a+x}}dx=a\pi$

asked Jan 27, 2013 by thanvigandhi_1

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Show that the curve $4x=y^2\: and \: 4xy=k$ cut at right angles, if $k^2=512$

asked Jan 27, 2013 by thanvigandhi_1

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Solve the following differential equation : $\large\frac{dy}{dx}$ $+2y=xe^{4x}$

asked Jan 27, 2013 by thanvigandhi_1

0 answers

Find the differential equation of all circles which pass through origin and whose centre lies on y - axis.

asked Jan 27, 2013 by thanvigandhi_1

0 answers

Solve the differential equation $(x+y)dy+(x-y)dx=0$

asked Jan 27, 2013 by thanvigandhi_1

1 answer

Find a vector of magnitude 15 which is perpendicular to both the vectors $4\hat i - \hat j + 8\hat k$ and $- \hat j + \hat k$

asked Jan 27, 2013 by thanvigandhi_1

0 answers

Find the equation of the line through A(5,-3,2) and through the intersection $\large \frac{x-2}{1} = \frac{y-3}{5} = \frac{z-4}{4}$ and $\large\frac{x-4}{3} = \frac{y-2}{4} = \frac{z+3}{-3}$

asked Jan 27, 2013 by thanvigandhi_1

0 answers

Arjun throw a die again and again until three fours are obtained. Find the probability of obtaining the third four in the fourth throw of a die.

asked Jan 27, 2013 by thanvigandhi_1

0 answers

If $A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ \end{bmatrix}$ , prove that $A^n = \begin{bmatrix} 3^{n-1} & 3^{n-1} & 3^{n-1} \\ 3^{n-1} & 3^{n-1} & 3^{n-1} \\ 3^{n-1} & 3^{n-1} & 3^{n-1} \\ \end{bmatrix}$ , for $n \in N.$

asked Jan 27, 2013 by thanvigandhi_1

1 answer

The sum of three numbers is -1. If we multiply second number by 2, third number by 3 and add them, we get 5. If we subtract third number from the sum of first and second number, we get -1. Represent the information by the system of equation and find the number using inverse of a matrix.

asked Jan 26, 2013 by thanvigandhi_1

0 answers

A point on the hypotenuse of a right-angled triangle is at distances a and b from the sides. Show that length of the hypotenuse is at least $(a^{\large\frac{2}{3}}+b^{\large\frac{2}{3}})^{\large\frac{3}{2}}$

asked Jan 26, 2013 by thanvigandhi_1

0 answers

Show that semi-vertical angle of the right circular cone of given total surface area and maximum volume is $sin^{-1}\frac{1}{3}$

asked Jan 26, 2013 by thanvigandhi_1

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Find the area enclosed between the parabolas $y^2=4ax\: and \: x^2=4ay$

asked Jan 26, 2013 by thanvigandhi_1

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Evaluate : $\int_0^{\Large\frac{\pi}{4}} \large\frac{\sin x \: \cos x}{\cos^4x+sin^4x}$ $dx$

asked Jan 26, 2013 by thanvigandhi_1

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From the distance of the point (2,3,4) from the line $\large\frac{x+3}{3} = \frac{y-2}{6} = \frac{z}{2}$ measured parallel to the plane $3x+2y+2z+5=0.$

asked Jan 26, 2013 by thanvigandhi_1

0 answers

A company makes two kinds of belts, belt A is of pure leather and belt B is macro based. The respective profits are Rs.40 and Rs.30 per belt. Each belt of type A requires twice as much time as a belt of type B and if all the belts were of type B a company could make 1000 belts per day. The supply of leather and macro is sufficient for only 8000 belts per day (both A and B combined). Belt A requires a fancy buckle and only 4000 buckles are available. There are only 7000 buckles per day available for belt B. What should be the daily production of each belt to have maximum profit?

asked Jan 26, 2013 by thanvigandhi_1

1 answer

If $\tan^{-1}2-\tan^{-1}1=\tan^{-1}x,$ then find x.

asked Jan 25, 2013 by thanvigandhi_1

1 answer

If $A = [ a_{ij}],$ where $a_{ij}= \left\{ \begin{array}{l l}i+j, & \quad if\; { i \geq j}\\ i-j, & \quad if\; { i < j} \\ \end{array}. \right.$ construct a 3 x 2 matrix A.

asked Jan 25, 2013 by thanvigandhi_1

0 answers

If A is a matrix of order 3 x 4 and B is matrix of order 5 x 4, what is the order of matrix (AB)'.

asked Jan 25, 2013 by thanvigandhi_1

1 answer

Evaluate : $\int \large\frac{\sec^2x}{5+tan\: x}$ $dx$

asked Jan 25, 2013 by thanvigandhi_1

0 answers

Find f(x) satisfy the following : $\int e^x (cosec^2x-\cot \: x)dx = e^x\: f(x)+C$

asked Jan 25, 2013 by thanvigandhi_1

0 answers

For what value of k, the vectors $2\hat i - \hat j + \hat k$ and $\hat i+\hat kj - 5\hat k$ are perpendicular to each other.

asked Jan 25, 2013 by thanvigandhi_1

0 answers

For two vectors $\overrightarrow a \: and \: \overrightarrow b,$ show that $\overrightarrow a.\overrightarrow b \leq |\overrightarrow a| \: |\overrightarrow b|.$

asked Jan 25, 2013 by thanvigandhi_1

0 answers

Find the value of p, such that the line $\large\frac{x-3}{12}=\frac{y+1}{p}=\frac{z-3}{17}$ is perpendicular to 4x+y-3z=5.

asked Jan 25, 2013 by thanvigandhi_1

0 answers

If $f : P \rightarrow R$ be function defined by $f(x)=4x^3-7$ , show that function f is a bijective function.

asked Jan 25, 2013 by thanvigandhi_1

0 answers

Let $f(x) = \large\frac{ax}{x+1}$ $, x \neq -1, \: if \: (fof) (x) = x,$ find the value of $a$

asked Jan 25, 2013 by thanvigandhi_1

0 answers

Prove that : $\sin^{-1}\frac{3}{5} + \sin^{-1}\frac{8}{17}=\sin^{-1}\frac{77}{85}$

asked Jan 25, 2013 by thanvigandhi_1

1 answer

Represent the following problem by a system of linear equations. Rs.188000 is the cost of 3 cycles, 1 motorbike and 1 car, the cost of 2 cycles, 2 motorbikes and 1 car is Rs.228000. The cost of 4 cycles 2 cars and 1 motorbike is Rs. 330000. Use matrix to find the cost of each.

asked Jan 25, 2013 by thanvigandhi_1

1 answer

If $x^y=e^{y-x}$ , show that $\large\frac{dy}{dx}=\frac{2-logx}{(1-log\:x)^2.}$

asked Jan 25, 2013 by thanvigandhi_1

0 answers

Show that the function f(x) = |x-1| is continuous but not differentiable at x=1.

asked Jan 25, 2013 by thanvigandhi_1

0 answers

Verify Rolle's theorem for the function $f(x) = e^x\cos \: x\: on \: \bigg[ -\large\frac{\pi}{2}, \frac{\pi}{2} \bigg]$ .

asked Jan 25, 2013 by thanvigandhi_1

0 answers

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